{"id": "5073fa55b1444226", "problem_id": "14378efbde3f5230", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + \\frac{7 x}{3} - 2$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{7}{3}$, c=$-2$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{2}{3}, -3\\}$", "final_answer": "x \\in \\{\\frac{2}{3}, -3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8e4d8f61ec3bce29", "problem_id": "14378efbde3f5230", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + \\frac{7 x}{3} - 2$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{7}{3}$, c=$-2$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{121}{9}$, so √Δ = $\\frac{11}{3}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{2}{3}$.\n7) x2 = $-3$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{2}{3}, -3\\}$", "final_answer": "x \\in \\{\\frac{2}{3}, -3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7f1148038db360c9", "problem_id": "b491c9b52c9a9ab3", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "You want 10 liters of a 35% salt solution. You have a 30% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 10−x liters is 40% solution.\n2) Salt equation: 30x + 40(10−x) = 35·10.\n3) (30-40)x = (35-40)·10 ⇒ x = 5.\n4) Then 10−x = 5.\nFinal answer: $5 L of 30% and 5 L of 40%$", "final_answer": "5 L of 30% and 5 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "45f5e0ac83db0d46", "problem_id": "aaa146a38fe8ea5e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{2} + 2 x - 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(4 x^{2}) = 8 x^{2}$.\n3) So $f'(x) = 8 x^{2} + 2$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(4 x^{2}) = 8 x$.\nTherefore the correct derivative is $f'(x) = 2 \\left(4 x + 1\\right)$.\nFinal answer: $2 \\left(4 x + 1\\right)$", "final_answer": "2 \\left(4 x + 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b9d7dd51b123395b", "problem_id": "48dbd6030f3b25ea", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x \\sin{\\left(x \\right)}) = 2 x \\cos{\\left(x \\right)}$.\n3) So $f'(x) = 3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x \\sin{\\left(x \\right)}) = 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)}$.\nFinal answer: $3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)}$", "final_answer": "3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "089da17c3ce9e100", "problem_id": "8fe2092ac512f2a4", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Two cards are drawn without replacement from a standard 52-card deck. Given that at least one card is an ace, what is the probability both are aces?", "output": "1) Condition on the set of hands with at least one ace.\n2) Count hands with ≥1 ace = C(52,2) − C(48,2).\n3) Count hands with 2 aces = C(4,2). Probability = C(4,2)/(C(52,2)−C(48,2)).\nFinal answer: $\\frac{1}{33}$", "final_answer": "\\frac{1}{33}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "conditional_probability_cards", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9e0d75cfe74df2fe", "problem_id": "da57f54a86d88836", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{5} x + \\frac{5 x}{6} - 1 + \\frac{2 \\sqrt{5}}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{5}{6} - \\sqrt{5}$, c=$-1 + \\frac{2 \\sqrt{5}}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{349}{36} - \\frac{13 \\sqrt{5}}{3}$, so √Δ = $\\sqrt{\\frac{349}{36} - \\frac{13 \\sqrt{5}}{3}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{5}{12} + \\frac{\\sqrt{349 - 156 \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{2}$.\nx2 = $- \\frac{5}{12} - \\frac{\\sqrt{349 - 156 \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{5}{12} + \\frac{\\sqrt{349 - 156\\cdot \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{2}, - \\frac{5}{12} - \\frac{\\sqrt{349 - 156\\cdot \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{5}{12} + \\frac{\\sqrt{349 - 156\\cdot \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{2}, - \\frac{5}{12} - \\frac{\\sqrt{349 - 156\\cdot \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "54f369be46bbd808", "problem_id": "da57f54a86d88836", "style": "answer_only", "instruction": "Final answer only.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{5} x + \\frac{5 x}{6} - 1 + \\frac{2 \\sqrt{5}}{3}$ = 0.", "output": "$x \\in \\{- \\frac{5}{12} + \\frac{\\sqrt{349 - 156\\cdot \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{2}, - \\frac{5}{12} - \\frac{\\sqrt{349 - 156\\cdot \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{5}{12} + \\frac{\\sqrt{349 - 156\\cdot \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{2}, - \\frac{5}{12} - \\frac{\\sqrt{349 - 156\\cdot \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "021c692573e4b802", "problem_id": "9bf02002349e16bb", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 5), x≡5 (mod 7). Give the solution modulo 35.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 35.\nCompute the CRT solution to get x ≡ 12 (mod 35).\nCheck: 12 mod 5=2 and mod 7=5.\nFinal answer: $x\\equiv 12\\pmod{35}$", "final_answer": "x\\equiv 12\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2e240d68717a67b5", "problem_id": "9bf02002349e16bb", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡2 (mod 5), x≡5 (mod 7). Give the solution modulo 35.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 35.\n3) Compute the CRT solution to get x ≡ 12 (mod 35).\nFinal answer: $x\\equiv 12\\pmod{35}$", "final_answer": "x\\equiv 12\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fef97135234c6515", "problem_id": "56037c522cbd4301", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 2 x^{2} - 4 x + 15$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $2 x^{3} + 2 x^{2} + 3 x$ and a = -3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $6 x^{2} + 4 x + 3$.\n5) Evaluate at a: p'(-3) = $45$.\nFinal answer: $45$", "final_answer": "45", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f58e1c6ee9858cf2", "problem_id": "56037c522cbd4301", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 2 x^{2} - 4 x + 15$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{3} + 2 x^{2} + 3 x$ and a = -3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $45$", "final_answer": "45", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d21655f5be63df45", "problem_id": "874a6edcd96a73da", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡5 (mod 11), x≡12 (mod 13). Give the solution modulo 143.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 143.\n3) Compute the CRT solution to get x ≡ 38 (mod 143).\nFinal answer: $x\\equiv 38\\pmod{143}$", "final_answer": "x\\equiv 38\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6f16e0f23a0f2af5", "problem_id": "874a6edcd96a73da", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡5 (mod 11), x≡12 (mod 13). Give the solution modulo 143.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 143.\nCompute the CRT solution to get x ≡ 38 (mod 143).\nCheck: 38 mod 11=5 and mod 13=12.\nFinal answer: $x\\equiv 38\\pmod{143}$", "final_answer": "x\\equiv 38\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "26e5c435061c0219", "problem_id": "5546d09580a746d4", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Worker A finishes a job in 17 hours and Worker B finishes it in 7 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/17 job/hour. B's rate = 1/7 job/hour.\nCombined rate = 1/17 + 1/7 = 24/119 job/hour.\nTime = 1 ÷ (combined rate) = 119/24 hours.\nCheck: (combined rate)×(time) = 24/119×119/24 = 1.\nFinal answer: $\\frac{119}{24}$", "final_answer": "\\frac{119}{24}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dfc85517d28e273f", "problem_id": "5546d09580a746d4", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Worker A finishes a job in 17 hours and Worker B finishes it in 7 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/17 job/hour. B's rate = 1/7 job/hour.\n2) Combined rate = 1/17 + 1/7 = 24/119 job/hour.\n3) Time = 1 ÷ (combined rate) = 119/24 hours.\nFinal answer: $\\frac{119}{24}$", "final_answer": "\\frac{119}{24}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0a82cfc1e8b70e79", "problem_id": "09e3892ab0cb8f2b", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 2 \\sqrt{6} x + \\frac{x}{2} - \\frac{45}{2} + 9 \\sqrt{6}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{1}{2} - 2 \\sqrt{6}$, c=$- \\frac{45}{2} + 9 \\sqrt{6}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{- \\frac{1}{4} + \\frac{\\sqrt{457 - 152\\cdot \\sqrt{6}}}{4} + \\sqrt{6}, - \\frac{\\sqrt{457 - 152\\cdot \\sqrt{6}}}{4} - \\frac{1}{4} + \\sqrt{6}\\}$", "final_answer": "x \\in \\{- \\frac{1}{4} + \\frac{\\sqrt{457 - 152\\cdot \\sqrt{6}}}{4} + \\sqrt{6}, - \\frac{\\sqrt{457 - 152\\cdot \\sqrt{6}}}{4} - \\frac{1}{4} + \\sqrt{6}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "38bd9a2d8fd433d7", "problem_id": "09e3892ab0cb8f2b", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 2 \\sqrt{6} x + \\frac{x}{2} - \\frac{45}{2} + 9 \\sqrt{6}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{1}{2} - 2 \\sqrt{6}$, c=$- \\frac{45}{2} + 9 \\sqrt{6}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{457}{4} - 38 \\sqrt{6}$, so √Δ = $\\sqrt{\\frac{457}{4} - 38 \\sqrt{6}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{1}{4} + \\frac{\\sqrt{457 - 152 \\sqrt{6}}}{4} + \\sqrt{6}$.\n7) x2 = $- \\frac{\\sqrt{457 - 152 \\sqrt{6}}}{4} - \\frac{1}{4} + \\sqrt{6}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{1}{4} + \\frac{\\sqrt{457 - 152\\cdot \\sqrt{6}}}{4} + \\sqrt{6}, - \\frac{\\sqrt{457 - 152\\cdot \\sqrt{6}}}{4} - \\frac{1}{4} + \\sqrt{6}\\}$", "final_answer": "x \\in \\{- \\frac{1}{4} + \\frac{\\sqrt{457 - 152\\cdot \\sqrt{6}}}{4} + \\sqrt{6}, - \\frac{\\sqrt{457 - 152\\cdot \\sqrt{6}}}{4} - \\frac{1}{4} + \\sqrt{6}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1f0c00f2cf8eed17", "problem_id": "b12c3d48a6da4f67", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A store sells a backpack for $42. It is discounted by 40%. You buy 4 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 42 × (1 − 40/100) = 42 × 60/100 = 126/5.\n2) Subtotal = 4 × 126/5 = 504/5.\n3) Total with tax = 504/5 × 109/100 = 13734/125.\nFinal answer: $\\frac{13734}{125}$", "final_answer": "\\frac{13734}{125}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "014ff1560a8210c3", "problem_id": "b12c3d48a6da4f67", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A store sells a backpack for $42. It is discounted by 40%. You buy 4 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 42 × (1 − 40/100) = 42 × 60/100 = 126/5.\nSubtotal = 4 × 126/5 = 504/5.\nTotal with tax = 504/5 × 109/100 = 13734/125.\nCheck: recomputing the formula yields 13734/125.\nFinal answer: $\\frac{13734}{125}$", "final_answer": "\\frac{13734}{125}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c310f0b5f75657a5", "problem_id": "c15028bb2acc83be", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{21 x}{10} - \\frac{\\sqrt{2} x}{2} - 1 - \\frac{\\sqrt{2}}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{21}{10} - \\frac{\\sqrt{2}}{2}$, c=$-1 - \\frac{\\sqrt{2}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{29 \\sqrt{2}}{10} + \\frac{891}{100}$, so √Δ = $\\sqrt{\\frac{29 \\sqrt{2}}{10} + \\frac{891}{100}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{2}}{4} + \\frac{21}{20} + \\frac{\\sqrt{290 \\sqrt{2} + 891}}{20}$.\n7) x2 = $- \\frac{\\sqrt{290 \\sqrt{2} + 891}}{20} + \\frac{\\sqrt{2}}{4} + \\frac{21}{20}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{2}}{4} + \\frac{21}{20} + \\frac{\\sqrt{290\\cdot \\sqrt{2} + 891}}{20}, - \\frac{\\sqrt{290\\cdot \\sqrt{2} + 891}}{20} + \\frac{\\sqrt{2}}{4} + \\frac{21}{20}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{2}}{4} + \\frac{21}{20} + \\frac{\\sqrt{290\\cdot \\sqrt{2} + 891}}{20}, - \\frac{\\sqrt{290\\cdot \\sqrt{2} + 891}}{20} + \\frac{\\sqrt{2}}{4} + \\frac{21}{20}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4da6abd77940f1c5", "problem_id": "c15028bb2acc83be", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{21 x}{10} - \\frac{\\sqrt{2} x}{2} - 1 - \\frac{\\sqrt{2}}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{21}{10} - \\frac{\\sqrt{2}}{2}$, c=$-1 - \\frac{\\sqrt{2}}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{29 \\sqrt{2}}{10} + \\frac{891}{100}$, so √Δ = $\\sqrt{\\frac{29 \\sqrt{2}}{10} + \\frac{891}{100}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{2}}{4} + \\frac{21}{20} + \\frac{\\sqrt{290 \\sqrt{2} + 891}}{20}$.\nx2 = $- \\frac{\\sqrt{290 \\sqrt{2} + 891}}{20} + \\frac{\\sqrt{2}}{4} + \\frac{21}{20}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{2}}{4} + \\frac{21}{20} + \\frac{\\sqrt{290\\cdot \\sqrt{2} + 891}}{20}, - \\frac{\\sqrt{290\\cdot \\sqrt{2} + 891}}{20} + \\frac{\\sqrt{2}}{4} + \\frac{21}{20}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{2}}{4} + \\frac{21}{20} + \\frac{\\sqrt{290\\cdot \\sqrt{2} + 891}}{20}, - \\frac{\\sqrt{290\\cdot \\sqrt{2} + 891}}{20} + \\frac{\\sqrt{2}}{4} + \\frac{21}{20}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1a32d2e16ef5bea0", "problem_id": "a10c9ebddd46e043", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the congruences: x≡7 (mod 8), x≡6 (mod 7). Give the solution modulo 56.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 56.\n3) Compute the CRT solution to get x ≡ 55 (mod 56).\nFinal answer: $x\\equiv 55\\pmod{56}$", "final_answer": "x\\equiv 55\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "533c294e94442e1f", "problem_id": "a10c9ebddd46e043", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡7 (mod 8), x≡6 (mod 7). Give the solution modulo 56.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 56.\nCompute the CRT solution to get x ≡ 55 (mod 56).\nCheck: 55 mod 8=7 and mod 7=6.\nFinal answer: $x\\equiv 55\\pmod{56}$", "final_answer": "x\\equiv 55\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4c275fe42805a034", "problem_id": "27ff93c1fa6d9223", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- 8 x + 10 y$ = $80$\n$10 x - y$ = $-8$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $10$: $- 80 x + 100 y$ = $800$.\nMultiply the second equation by $-8$: $- 80 x + 8 y$ = $64$.\nSubtract to eliminate x: $92 y$ = $736$.\nSolve for y: y = $8$.\nSubstitute back to find x: x = $0$.\nCheck: plug (x,y)=(0,8) into both equations.\nFinal answer: $(x,y)=\\left(0, 8\\right)$", "final_answer": "(x,y)=\\left(0, 8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "494ba24e284d348c", "problem_id": "6116e17f404a4160", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} - x^{2} + 7 x - 29$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} + 6 x - 29.\n3) Then the limit is -69.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: -73.\nFinal answer: $-73$", "final_answer": "-73", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3e6c63e8191de2f1", "problem_id": "13ea56c2c58b5710", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Find the multiplicative inverse of 34 modulo 53.", "output": "1) We want x such that 34x ≡ 1 (mod 53).\n2) Use extended Euclid to find s,t with 34s + 53t = 1.\n3) One Bezout identity is: 34·(-14) + 53·(9) = 1.\n4) Reduce s modulo 53: x ≡ 39 (mod 53).\nFinal answer: $x\\equiv 39\\pmod{53}$", "final_answer": "x\\equiv 39\\pmod{53}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "23bd891b828dda83", "problem_id": "13ea56c2c58b5710", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the multiplicative inverse of 34 modulo 53.", "output": "We want x such that 34x ≡ 1 (mod 53).\nUse extended Euclid to find s,t with 34s + 53t = 1.\nOne Bezout identity is: 34·(-14) + 53·(9) = 1.\nReduce s modulo 53: x ≡ 39 (mod 53).\nCheck: 34·39 mod 53 = 1.\nFinal answer: $x\\equiv 39\\pmod{53}$", "final_answer": "x\\equiv 39\\pmod{53}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "19e36082324900cb", "problem_id": "49f21da7ba95c98b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(x^{2} - 2 x + 3\\right) \\cos{\\left(x^{2} + x - 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x e^{x}) = x e^{x}$.\n3) So $f'(x) = - 2 x^{3} \\sin{\\left(x^{2} + x - 3 \\right)} + 3 x^{2} \\sin{\\left(x^{2} + x - 3 \\right)} + x e^{x} - 4 x \\sin{\\left(x^{2} + x - 3 \\right)} + 2 x \\cos{\\left(\\left(x^{2} + x\\right) - 3 \\right)} + 2 e^{x} - 3 \\sin{\\left(x^{2} + x - 3 \\right)} - 2 \\cos{\\left(\\left(x^{2} + x\\right) - 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x e^{x}) = x e^{x} + e^{x}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x - 1\\right) \\cos{\\left(\\left(x^{2} + x\\right) - 3 \\right)} + \\left(x + 2\\right) e^{x} - \\left(2 x + 1\\right) \\left(x \\left(x - 2\\right) + 3\\right) \\sin{\\left(x^{2} + x - 3 \\right)} + e^{x}$.\nFinal answer: $2 \\left(x - 1\\right) \\cos{\\left(\\left(x^{2} + x\\right) - 3 \\right)} + \\left(x + 2\\right) e^{x} - \\left(2 x + 1\\right) \\left(x \\left(x - 2\\right) + 3\\right) \\sin{\\left(x^{2} + x - 3 \\right)} + e^{x}$", "final_answer": "2 \\left(x - 1\\right) \\cos{\\left(\\left(x^{2} + x\\right) - 3 \\right)} + \\left(x + 2\\right) e^{x} - \\left(2 x + 1\\right) \\left(x \\left(x - 2\\right) + 3\\right) \\sin{\\left(x^{2} + x - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2a933b1d7b29af75", "problem_id": "a36349e25388376a", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 12 red balls and 10 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(22,4).\n3) Favorable samples: choose k reds and n−k blues: C(12,2)·C(10,2).\n4) Compute totals: C(22,4) = 7315, and favorable = 2970.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{54}{133}$", "final_answer": "\\frac{54}{133}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a88e583806f4de5", "problem_id": "0cf69d286051546d", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A disease has prevalence $\\frac{1}{200}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{13}{125}$.\nCompute P(D|+) = $\\frac{9}{208}$.\nFinal answer: $\\frac{9}{208}$", "final_answer": "\\frac{9}{208}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c718ec41ac60f853", "problem_id": "0cf69d286051546d", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A disease has prevalence $\\frac{1}{200}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{13}{125}$.\n3) Compute P(D|+) = $\\frac{9}{208}$.\nFinal answer: $\\frac{9}{208}$", "final_answer": "\\frac{9}{208}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a682fcbb3230904", "problem_id": "f018355b7ee08b26", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 3 x^{2} - 1$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} - x - 1.\n3) Then the limit is -3.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: -4.\nFinal answer: $-4$", "final_answer": "-4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "468f2ed6510fe6b6", "problem_id": "948c9f90cce495a2", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int \\left(x + 3\\right) e^{x} + \\left(2 x + 1\\right) \\cos{\\left(x \\right)}\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $\\left(x + 3\\right) e^{x} + \\left(2 x + 1\\right) \\cos{\\left(x \\right)}$.\n3) Compute an antiderivative: F(x) = $2 x \\sin{\\left(x \\right)} + \\left(x + 2\\right) e^{x} + \\sin{\\left(x \\right)} + 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\left(x + 2\\right)\\cdot e^{x} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\left(x + 2\\right)\\cdot e^{x} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "afe1e32eba4f4a92", "problem_id": "1a63eda3cf2df06c", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} 2 x^{3} - 3 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $2 x^{3} - 3 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{15 \\pi^{4}}{32} - \\frac{9 \\pi^{3}}{8} - \\frac{9 \\pi}{2} - 2 + \\frac{3 \\pi^{2}}{8}$.\nFinal answer: $- \\frac{15\\cdot \\pi^{4}}{32} - \\frac{9\\cdot \\pi^{3}}{8} - \\frac{9\\cdot \\pi}{2} - 2 + \\frac{3\\cdot \\pi^{2}}{8}$", "final_answer": "- \\frac{15\\cdot \\pi^{4}}{32} - \\frac{9\\cdot \\pi^{3}}{8} - \\frac{9\\cdot \\pi}{2} - 2 + \\frac{3\\cdot \\pi^{2}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cc0876e5d45b29a7", "problem_id": "1a63eda3cf2df06c", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} 2 x^{3} - 3 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{3} - 3 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{15 \\pi^{4}}{32} - \\frac{9 \\pi^{3}}{8} - \\frac{9 \\pi}{2} - 2 + \\frac{3 \\pi^{2}}{8}$.\n4) Simplify the result.\nFinal answer: $- \\frac{15\\cdot \\pi^{4}}{32} - \\frac{9\\cdot \\pi^{3}}{8} - \\frac{9\\cdot \\pi}{2} - 2 + \\frac{3\\cdot \\pi^{2}}{8}$", "final_answer": "- \\frac{15\\cdot \\pi^{4}}{32} - \\frac{9\\cdot \\pi^{3}}{8} - \\frac{9\\cdot \\pi}{2} - 2 + \\frac{3\\cdot \\pi^{2}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "044a1dafb13d1857", "problem_id": "1eb1289b8232fd66", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the linear system:\n$- x - 6 y$ = $-31$\n$- 8 x$ = $40$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-8$: $8 x + 48 y$ = $248$.\nMultiply the second equation by $-1$: $8 x$ = $-40$.\nFinal answer: $(x,y)=\\left(-5, 6\\right)$", "final_answer": "(x,y)=\\left(-5, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "28105609ee6aaab3", "problem_id": "1eb1289b8232fd66", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- x - 6 y$ = $-31$\n$- 8 x$ = $40$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-8$: $8 x + 48 y$ = $248$.\nMultiply the second equation by $-1$: $8 x$ = $-40$.\nSubtract to eliminate x: $48 y$ = $288$.\nSolve for y: y = $6$.\nSubstitute back to find x: x = $-5$.\nCheck: plug (x,y)=(-5,6) into both equations.\nFinal answer: $(x,y)=\\left(-5, 6\\right)$", "final_answer": "(x,y)=\\left(-5, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7a29c0011e2ba7d8", "problem_id": "271e4dbb47e11c50", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{4} + 2 x^{3} + x^{2} - 4 x + 2\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $3 x^{4} + 2 x^{3} + x^{2} - 4 x + 2$.\n3) Compute an antiderivative: F(x) = $\\frac{3 x^{5}}{5} + \\frac{x^{4}}{2} + \\frac{x^{3}}{3} - 2 x^{2} + 2 x$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{3\\cdot x^{5}}{5} + \\frac{x^{4}}{2} + \\frac{x^{3}}{3} - 2\\cdot x^{2} + 2\\cdot x$", "final_answer": "C + \\frac{3\\cdot x^{5}}{5} + \\frac{x^{4}}{2} + \\frac{x^{3}}{3} - 2\\cdot x^{2} + 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d9e1a5ed7d69d642", "problem_id": "683f51bf6c7a9c8e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{4} - x^{3} - x^{2} - 2 x - 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x) = - 2 x$.\n3) So $f'(x) = 4 x^{3} - 3 x^{2} - 4 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x) = -2$.\nTherefore the correct derivative is $f'(x) = 4 x^{3} - 3 x^{2} - 2 x - 2$.\nFinal answer: $4 x^{3} - 3 x^{2} - 2 x - 2$", "final_answer": "4 x^{3} - 3 x^{2} - 2 x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0afa5cd49c8da620", "problem_id": "f66393517368d768", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x^{2} - 2 x - 2$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{2} - 3 x - 2.\n3) Then the limit is -16.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -14.\nFinal answer: $-14$", "final_answer": "-14", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "909c48038086aef5", "problem_id": "0e556ce228c5b409", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A cyclist rides at 24 miles per hour for 2 hours, then at 27 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 24 × 2 = 48 miles.\n2) Second leg distance = 27 × 3 = 81 miles.\n3) Total distance = 48 + 81 = 129 miles.\nFinal answer: $129$", "final_answer": "129", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "67fce362fd5fafde", "problem_id": "02dc49021ae9421e", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - x^{2} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $- 9 x^{2} - 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\n2) Slope at x=-2: m = f'(-2) = $-34 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}$.\n3) Point on curve: (-2, f(-2)) = (-2, $\\cos{\\left(4 \\right)} + e^{-1} + 22$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(4 \\right)} + e^{-1} + 22 = -34 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}(x − -2).\n6) Optionally expand: y = $- 34 x + 2 x \\sin{\\left(4 \\right)} + \\frac{x}{2 e} - 46 + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$.\nFinal answer: $y = - 34\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 46 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$", "final_answer": "y = - 34\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 46 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "441279144d6ada4a", "problem_id": "02dc49021ae9421e", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - x^{2} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=-2$.", "output": "Differentiate: f'(x) = $- 9 x^{2} - 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\nSlope at x=-2: m = f'(-2) = $-34 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}$.\nPoint on curve: (-2, f(-2)) = (-2, $\\cos{\\left(4 \\right)} + e^{-1} + 22$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − \\cos{\\left(4 \\right)} + e^{-1} + 22 = -34 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}(x − -2).\nOptionally expand: y = $- 34 x + 2 x \\sin{\\left(4 \\right)} + \\frac{x}{2 e} - 46 + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 34\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 46 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$", "final_answer": "y = - 34\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 46 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8fb96196602ac267", "problem_id": "93c560e9cdb2555e", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{3} + 4 x^{2} + 10 x + 17$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{4} + 2 x^{2} - 3 x + 2$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $8 x^{3} + 4 x - 3$.\nEvaluate at a: p'(2) = $69$.\nCheck: the identity gives $69$.\nFinal answer: $69$", "final_answer": "69", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5f0203c53d0d38a0", "problem_id": "93c560e9cdb2555e", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{3} + 4 x^{2} + 10 x + 17$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $2 x^{4} + 2 x^{2} - 3 x + 2$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $8 x^{3} + 4 x - 3$.\n5) Evaluate at a: p'(2) = $69$.\nFinal answer: $69$", "final_answer": "69", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6635e299c23bbf25", "problem_id": "3ffd222b20b72af7", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\nSubtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{3 \\pi^{2}}{8} + \\frac{\\pi^{3}}{24} + 2 + \\frac{3 \\pi}{2}$.\nFinal answer: $- \\frac{3\\cdot \\pi^{2}}{8} + \\frac{\\pi^{3}}{24} + 2 + \\frac{3\\cdot \\pi}{2}$", "final_answer": "- \\frac{3\\cdot \\pi^{2}}{8} + \\frac{\\pi^{3}}{24} + 2 + \\frac{3\\cdot \\pi}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "96b300c59c28f927", "problem_id": "3ffd222b20b72af7", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\n3) Subtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{3 \\pi^{2}}{8} + \\frac{\\pi^{3}}{24} + 2 + \\frac{3 \\pi}{2}$.\n4) Simplify the result.\nFinal answer: $- \\frac{3\\cdot \\pi^{2}}{8} + \\frac{\\pi^{3}}{24} + 2 + \\frac{3\\cdot \\pi}{2}$", "final_answer": "- \\frac{3\\cdot \\pi^{2}}{8} + \\frac{\\pi^{3}}{24} + 2 + \\frac{3\\cdot \\pi}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "13657d58ca75def9", "problem_id": "9cb8d66629ce6baa", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "An urn has 7 red balls and 8 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(15,5).\n3) Favorable samples: choose k reds and n−k blues: C(7,1)·C(8,4).\n4) Compute totals: C(15,5) = 3003, and favorable = 490.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{70}{429}$", "final_answer": "\\frac{70}{429}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "848d454ad8d79c0e", "problem_id": "9cb8d66629ce6baa", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 7 red balls and 8 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(15,5).\nFavorable samples: choose k reds and n−k blues: C(7,1)·C(8,4).\nCompute totals: C(15,5) = 3003, and favorable = 490.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{70}{429}$.\nFinal answer: $\\frac{70}{429}$", "final_answer": "\\frac{70}{429}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9795cc40fe84c9f0", "problem_id": "9c1dbbda3b656743", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "How many binary strings of length 9 contain exactly 4 ones with no two ones adjacent?", "output": "1) Reserve 3 mandatory zeros between the 4 ones.\n2) This leaves n−k+1 = 6 valid slots for the ones.\n3) Choose the slots: C(6,4) = 15.\nFinal answer: $15$", "final_answer": "15", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6bccac9769b29120", "problem_id": "9c1dbbda3b656743", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many binary strings of length 9 contain exactly 4 ones with no two ones adjacent?", "output": "Reserve 3 mandatory zeros between the 4 ones.\nThis leaves n−k+1 = 6 valid slots for the ones.\nChoose the slots: C(6,4) = 15.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $15$", "final_answer": "15", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0929941a9f5c7ea5", "problem_id": "16bbaf902fbfb4bf", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Worker A finishes a job in 11 hours and Worker B finishes it in 4 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/11 job/hour. B's rate = 1/4 job/hour.\nCombined rate = 1/11 + 1/4 = 15/44 job/hour.\nTime = 1 ÷ (combined rate) = 44/15 hours.\nCheck: (combined rate)×(time) = 15/44×44/15 = 1.\nFinal answer: $\\frac{44}{15}$", "final_answer": "\\frac{44}{15}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "495a558e34221746", "problem_id": "16bbaf902fbfb4bf", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Worker A finishes a job in 11 hours and Worker B finishes it in 4 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/11 job/hour. B's rate = 1/4 job/hour.\n2) Combined rate = 1/11 + 1/4 = 15/44 job/hour.\n3) Time = 1 ÷ (combined rate) = 44/15 hours.\nFinal answer: $\\frac{44}{15}$", "final_answer": "\\frac{44}{15}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a522cf6befd025b0", "problem_id": "835d3645f761103d", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the congruences: x≡0 (mod 7), x≡2 (mod 13). Give the solution modulo 91.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 91.\n3) Compute the CRT solution to get x ≡ 28 (mod 91).\nFinal answer: $x\\equiv 28\\pmod{91}$", "final_answer": "x\\equiv 28\\pmod{91}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "163faec88c9fd6c2", "problem_id": "145328d3f21fe968", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(- 3 x^{2} - 2 x - 2\\right) \\cos{\\left(2 x^{2} - x + 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 2 \\cos{\\left(\\left(2 x^{2} - x\\right) + 1 \\right)}) = 2 \\sin{\\left(2 x^{2} - x + 1 \\right)}$.\n3) So $f'(x) = 12 x^{3} \\sin{\\left(\\left(2 x^{2} - x\\right) + 1 \\right)} + 5 x^{2} \\sin{\\left(\\left(2 x^{2} - x\\right) + 1 \\right)} + x e^{x} - 2 x \\sin{\\left(\\left(2 x^{2} - x\\right) + 1 \\right)} - 6 x \\cos{\\left(\\left(2 x^{2} - x\\right) + 1 \\right)} + 2 e^{x} + 2 \\sin{\\left(2 x^{2} - x + 1 \\right)} - 2 \\cos{\\left(\\left(2 x^{2} - x\\right) + 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 2 \\cos{\\left(\\left(2 x^{2} - x\\right) + 1 \\right)}) = 2 \\left(4 x - 1\\right) \\sin{\\left(\\left(2 x^{2} - x\\right) + 1 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- 3 x - 1\\right) \\cos{\\left(\\left(2 x^{2} - x\\right) + 1 \\right)} + \\left(x + 1\\right) e^{x} - \\left(4 x - 1\\right) \\left(x \\left(- 3 x - 2\\right) - 2\\right) \\sin{\\left(\\left(2 x^{2} - x\\right) + 1 \\right)} + e^{x}$.\nFinal answer: $2 \\left(- 3 x - 1\\right) \\cos{\\left(\\left(2 x^{2} - x\\right) + 1 \\right)} + \\left(x + 1\\right) e^{x} - \\left(4 x - 1\\right) \\left(x \\left(- 3 x - 2\\right) - 2\\right) \\sin{\\left(\\left(2 x^{2} - x\\right) + 1 \\right)} + e^{x}$", "final_answer": "2 \\left(- 3 x - 1\\right) \\cos{\\left(\\left(2 x^{2} - x\\right) + 1 \\right)} + \\left(x + 1\\right) e^{x} - \\left(4 x - 1\\right) \\left(x \\left(- 3 x - 2\\right) - 2\\right) \\sin{\\left(\\left(2 x^{2} - x\\right) + 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "436a9630cc73a02f", "problem_id": "a2ba257433a1a56b", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $1920 in the ratio 9:6. How much does each receive?", "output": "Total ratio parts = 9+6 = 15.\nEach part = 1920 ÷ 15 = 128.\nFirst share = 9 × 128 = 1152.\nSecond share = 6 × 128 = 768.\nCheck: 1152+768=1920.\nFinal answer: $768$", "final_answer": "768", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a51955115bc30c75", "problem_id": "a2ba257433a1a56b", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Two friends split $1920 in the ratio 9:6. How much does each receive?", "output": "1) Total ratio parts = 9+6 = 15.\n2) Each part = 1920 ÷ 15 = 128.\n3) First share = 9 × 128 = 1152.\n4) Second share = 6 × 128 = 768.\nFinal answer: $768$", "final_answer": "768", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0323e1cfaccc557a", "problem_id": "1cad086fbb1211a7", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int \\frac{\\sin{\\left(x \\right)}}{\\cos{\\left(x \\right)} + 1} + \\frac{2}{2 x + 3}\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $\\frac{\\sin{\\left(x \\right)}}{\\cos{\\left(x \\right)} + 1} + \\frac{2}{2 x + 3}$.\nCompute an antiderivative: F(x) = $\\log{\\left(2 x + 3 \\right)} - \\log{\\left(\\cos{\\left(x \\right)} + 1 \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\log{\\left(2\\cdot x + 3 \\right)} - \\log{\\left(\\cos{\\left(x \\right)} + 1 \\right)}$", "final_answer": "C + \\log{\\left(2\\cdot x + 3 \\right)} - \\log{\\left(\\cos{\\left(x \\right)} + 1 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8999c27e70875d2d", "problem_id": "1cad086fbb1211a7", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int \\frac{\\sin{\\left(x \\right)}}{\\cos{\\left(x \\right)} + 1} + \\frac{2}{2 x + 3}\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $\\frac{\\sin{\\left(x \\right)}}{\\cos{\\left(x \\right)} + 1} + \\frac{2}{2 x + 3}$.\n3) Compute an antiderivative: F(x) = $\\log{\\left(2 x + 3 \\right)} - \\log{\\left(\\cos{\\left(x \\right)} + 1 \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + \\log{\\left(2\\cdot x + 3 \\right)} - \\log{\\left(\\cos{\\left(x \\right)} + 1 \\right)}$", "final_answer": "C + \\log{\\left(2\\cdot x + 3 \\right)} - \\log{\\left(\\cos{\\left(x \\right)} + 1 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4f47dfe9390940ea", "problem_id": "1c9aaca61a26f70f", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} - 2 x^{3} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{3} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $- \\frac{15 \\pi^{4}}{2} - 6 \\pi - 4 + 3 \\pi^{2}$.\n4) Simplify the result.\nFinal answer: $- \\frac{15\\cdot \\pi^{4}}{2} - 6\\cdot \\pi - 4 + 3\\cdot \\pi^{2}$", "final_answer": "- \\frac{15\\cdot \\pi^{4}}{2} - 6\\cdot \\pi - 4 + 3\\cdot \\pi^{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2f319366f79f6d62", "problem_id": "1c9aaca61a26f70f", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} - 2 x^{3} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{3} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $- \\frac{15 \\pi^{4}}{2} - 6 \\pi - 4 + 3 \\pi^{2}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{15\\cdot \\pi^{4}}{2} - 6\\cdot \\pi - 4 + 3\\cdot \\pi^{2}$", "final_answer": "- \\frac{15\\cdot \\pi^{4}}{2} - 6\\cdot \\pi - 4 + 3\\cdot \\pi^{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f2f6052b4d637f36", "problem_id": "2309eed682f1a030", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "You want 40 liters of a 35% salt solution. You have a 30% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 40−x liters is 40% solution.\n2) Salt equation: 30x + 40(40−x) = 35·40.\n3) (30-40)x = (35-40)·40 ⇒ x = 20.\n4) Then 40−x = 20.\nFinal answer: $20 L of 30% and 20 L of 40%$", "final_answer": "20 L of 30% and 20 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e35662aed3501c5b", "problem_id": "b82e99176332d679", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} x \\left(- x - 2\\right)$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} - 3 x.\n3) Then the limit is 2.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: 1.\nFinal answer: $1$", "final_answer": "1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "9c6150b007dea59a", "problem_id": "673e550974014bef", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} + x^{2} + x + 2 \\sin{\\left(x \\right)}$ at $x=1$.", "output": "Differentiate: f'(x) = $- 9 x^{2} + 2 x + 2 \\cos{\\left(x \\right)} + 1$.\nSlope at x=1: m = f'(1) = $-6 + 2 \\cos{\\left(1 \\right)}$.\nPoint on curve: (1, f(1)) = (1, $-1 + 2 \\sin{\\left(1 \\right)}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -1 + 2 \\sin{\\left(1 \\right)} = -6 + 2 \\cos{\\left(1 \\right)}(x − 1).\nOptionally expand: y = $- 6 x + 2 x \\cos{\\left(1 \\right)} - 2 \\cos{\\left(1 \\right)} + 2 \\sin{\\left(1 \\right)} + 5$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 6\\cdot x + 2\\cdot x\\cdot \\cos{\\left(1 \\right)} - 2\\cdot \\cos{\\left(1 \\right)} + 2\\cdot \\sin{\\left(1 \\right)} + 5$", "final_answer": "y = - 6\\cdot x + 2\\cdot x\\cdot \\cos{\\left(1 \\right)} - 2\\cdot \\cos{\\left(1 \\right)} + 2\\cdot \\sin{\\left(1 \\right)} + 5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ffa34bff4837b4a", "problem_id": "6f3a478956ce613a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} \\sin{\\left(x \\right)} + x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x^{2} \\sin{\\left(x \\right)}) = x^{2} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x^{2} \\sin{\\left(x \\right)}) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$.\nFinal answer: $x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$", "final_answer": "x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "85482d4de34881f6", "problem_id": "9a9f431d99569750", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the multiplicative inverse of 8 modulo 11.", "output": "1) We want x such that 8x ≡ 1 (mod 11).\n2) Use extended Euclid to find s,t with 8s + 11t = 1.\n3) One Bezout identity is: 8·(-4) + 11·(3) = 1.\n4) Reduce s modulo 11: x ≡ 7 (mod 11).\nFinal answer: $x\\equiv 7\\pmod{11}$", "final_answer": "x\\equiv 7\\pmod{11}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d9bd6ebef6bfde5", "problem_id": "9a9f431d99569750", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 8 modulo 11.", "output": "We want x such that 8x ≡ 1 (mod 11).\nUse extended Euclid to find s,t with 8s + 11t = 1.\nOne Bezout identity is: 8·(-4) + 11·(3) = 1.\nReduce s modulo 11: x ≡ 7 (mod 11).\nCheck: 8·7 mod 11 = 1.\nFinal answer: $x\\equiv 7\\pmod{11}$", "final_answer": "x\\equiv 7\\pmod{11}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a998314ced062289", "problem_id": "e62155e03a5ae2aa", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "You want 36 liters of a 35% salt solution. You have a 15% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 36−x liters is 50% solution.\n2) Salt equation: 15x + 50(36−x) = 35·36.\n3) (15-50)x = (35-50)·36 ⇒ x = 108/7.\n4) Then 36−x = 144/7.\nFinal answer: $108/7 L of 15% and 144/7 L of 50%$", "final_answer": "108/7 L of 15% and 144/7 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d6ae2f00c9ba4da8", "problem_id": "e62155e03a5ae2aa", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 36 liters of a 35% salt solution. You have a 15% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 36−x liters is 50% solution.\nSalt equation: 15x + 50(36−x) = 35·36.\n(15-50)x = (35-50)·36 ⇒ x = 108/7.\nThen 36−x = 144/7.\nCheck: 15·108/7 + 50·144/7 = 35·36.\nFinal answer: $108/7 L of 15% and 144/7 L of 50%$", "final_answer": "108/7 L of 15% and 144/7 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d722790abf016db6", "problem_id": "bad330931ec058df", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 16 red balls and 28 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(44,10).\n3) Favorable samples: choose k reds and n−k blues: C(16,5)·C(28,5).\n4) Compute totals: C(44,10) = 2481256778, and favorable = 429287040.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{2358720}{13633279}$", "final_answer": "\\frac{2358720}{13633279}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3cb71d9d84f90683", "problem_id": "49578745ecf6750f", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - 2 x - 4$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to - 3 x - 4.\n3) Then the limit is -13.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: -10.\nFinal answer: $-10$", "final_answer": "-10", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5cd3c1294a32347b", "problem_id": "54d401eb1276e4cd", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Two friends split $924 in the ratio 6:6. How much does each receive?", "output": "1) Total ratio parts = 6+6 = 12.\n2) Each part = 924 ÷ 12 = 77.\n3) First share = 6 × 77 = 462.\n4) Second share = 6 × 77 = 462.\nFinal answer: $462$", "final_answer": "462", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "522742b50d475216", "problem_id": "54d401eb1276e4cd", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $924 in the ratio 6:6. How much does each receive?", "output": "Total ratio parts = 6+6 = 12.\nEach part = 924 ÷ 12 = 77.\nFirst share = 6 × 77 = 462.\nSecond share = 6 × 77 = 462.\nCheck: 462+462=924.\nFinal answer: $462$", "final_answer": "462", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5049af6e0f61596e", "problem_id": "fa7b526602d4102b", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A store sells a backpack for $167. It is discounted by 10%. You buy 3 backpacks. What is the total cost?", "output": "1) Discounted price = 167 × (1 − 10/100) = 167 × 90/100 = 1503/10.\n2) Subtotal = 3 × 1503/10 = 4509/10.\nFinal answer: $\\frac{4509}{10}$", "final_answer": "\\frac{4509}{10}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a0da5e1c1cc4805", "problem_id": "fa7b526602d4102b", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A store sells a backpack for $167. It is discounted by 10%. You buy 3 backpacks. What is the total cost?", "output": "Discounted price = 167 × (1 − 10/100) = 167 × 90/100 = 1503/10.\nSubtotal = 3 × 1503/10 = 4509/10.\nFinal answer: $\\frac{4509}{10}$", "final_answer": "\\frac{4509}{10}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c02678ffac7c7b2a", "problem_id": "07199a47c3ef5867", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} 2 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\n3) Subtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{\\pi^{2}}{4} + 2 + \\frac{\\pi^{3}}{12} + \\pi$.\n4) Simplify the result.\nFinal answer: $- \\frac{\\pi^{2}}{4} + 2 + \\frac{\\pi^{3}}{12} + \\pi$", "final_answer": "- \\frac{\\pi^{2}}{4} + 2 + \\frac{\\pi^{3}}{12} + \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "307cffbdb5534202", "problem_id": "07199a47c3ef5867", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} 2 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $2 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\nSubtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{\\pi^{2}}{4} + 2 + \\frac{\\pi^{3}}{12} + \\pi$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{\\pi^{2}}{4} + 2 + \\frac{\\pi^{3}}{12} + \\pi$", "final_answer": "- \\frac{\\pi^{2}}{4} + 2 + \\frac{\\pi^{3}}{12} + \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bb56c26fe6796679", "problem_id": "d143c011702f5f25", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Find the multiplicative inverse of 18 modulo 23.", "output": "1) We want x such that 18x ≡ 1 (mod 23).\n2) Use extended Euclid to find s,t with 18s + 23t = 1.\n3) One Bezout identity is: 18·(9) + 23·(-7) = 1.\n4) Reduce s modulo 23: x ≡ 9 (mod 23).\nFinal answer: $x\\equiv 9\\pmod{23}$", "final_answer": "x\\equiv 9\\pmod{23}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "831152b9669a8b83", "problem_id": "6854d03417f22b81", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} + 4 x + 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x^{2}) = 2 x^{2}$.\n3) So $f'(x) = 2 x^{2} + 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x^{2}) = 2 x$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x + 2\\right)$.\nFinal answer: $2 \\left(x + 2\\right)$", "final_answer": "2 \\left(x + 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c9914dd4f7beb05a", "problem_id": "50ae1a8874323499", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A cyclist rides at 75 miles per hour for 1/2 hours, then at 87 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 75 × 1/2 = 75/2 miles.\n2) Second leg distance = 87 × 5/2 = 435/2 miles.\n3) Total distance = 75/2 + 435/2 = 255 miles.\nFinal answer: $255$", "final_answer": "255", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "916ff6a6ad368116", "problem_id": "50ae1a8874323499", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A cyclist rides at 75 miles per hour for 1/2 hours, then at 87 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 75 × 1/2 = 75/2 miles.\nSecond leg distance = 87 × 5/2 = 435/2 miles.\nTotal distance = 75/2 + 435/2 = 255 miles.\nCheck: adding both legs gives 255.\nFinal answer: $255$", "final_answer": "255", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b110a2d8b25fca7e", "problem_id": "2dc0051335f633be", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{11} x + \\frac{23 x}{12} - \\frac{\\sqrt{11}}{4} + \\frac{5}{12}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{23}{12} - \\sqrt{11}$, c=$\\frac{5}{12} - \\frac{\\sqrt{11}}{4}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{1873}{144} - \\frac{17 \\sqrt{11}}{6}$, so √Δ = $\\sqrt{\\frac{1873}{144} - \\frac{17 \\sqrt{11}}{6}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{23}{24} + \\frac{\\sqrt{1873 - 408 \\sqrt{11}}}{24} + \\frac{\\sqrt{11}}{2}$.\nx2 = $- \\frac{23}{24} - \\frac{\\sqrt{1873 - 408 \\sqrt{11}}}{24} + \\frac{\\sqrt{11}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{23}{24} + \\frac{\\sqrt{1873 - 408\\cdot \\sqrt{11}}}{24} + \\frac{\\sqrt{11}}{2}, - \\frac{23}{24} - \\frac{\\sqrt{1873 - 408\\cdot \\sqrt{11}}}{24} + \\frac{\\sqrt{11}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{23}{24} + \\frac{\\sqrt{1873 - 408\\cdot \\sqrt{11}}}{24} + \\frac{\\sqrt{11}}{2}, - \\frac{23}{24} - \\frac{\\sqrt{1873 - 408\\cdot \\sqrt{11}}}{24} + \\frac{\\sqrt{11}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c27711d87bc45627", "problem_id": "2dc0051335f633be", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{11} x + \\frac{23 x}{12} - \\frac{\\sqrt{11}}{4} + \\frac{5}{12}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{23}{12} - \\sqrt{11}$, c=$\\frac{5}{12} - \\frac{\\sqrt{11}}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{1873}{144} - \\frac{17 \\sqrt{11}}{6}$, so √Δ = $\\sqrt{\\frac{1873}{144} - \\frac{17 \\sqrt{11}}{6}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{23}{24} + \\frac{\\sqrt{1873 - 408 \\sqrt{11}}}{24} + \\frac{\\sqrt{11}}{2}$.\n7) x2 = $- \\frac{23}{24} - \\frac{\\sqrt{1873 - 408 \\sqrt{11}}}{24} + \\frac{\\sqrt{11}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{23}{24} + \\frac{\\sqrt{1873 - 408\\cdot \\sqrt{11}}}{24} + \\frac{\\sqrt{11}}{2}, - \\frac{23}{24} - \\frac{\\sqrt{1873 - 408\\cdot \\sqrt{11}}}{24} + \\frac{\\sqrt{11}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{23}{24} + \\frac{\\sqrt{1873 - 408\\cdot \\sqrt{11}}}{24} + \\frac{\\sqrt{11}}{2}, - \\frac{23}{24} - \\frac{\\sqrt{1873 - 408\\cdot \\sqrt{11}}}{24} + \\frac{\\sqrt{11}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f283b64bb476c675", "problem_id": "a9836c54ff5c5377", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{3} + 3 x^{2} + 2 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{2 x - 3}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 3}) = - \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 3}$.\n3) So $f'(x) = \\frac{8 x^{3}}{8 x^{2} - 24 x + 18} + \\frac{4 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{2} - 24 x + 18} + \\frac{8 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{2} - 24 x + 18} - \\frac{6 x^{2}}{8 x^{2} - 24 x + 18} - \\frac{12 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{2} - 24 x + 18} - \\frac{24 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{2} - 24 x + 18} - \\frac{36 x}{8 x^{2} - 24 x + 18} + \\frac{9 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{2} - 24 x + 18} + \\frac{18 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{2} - 24 x + 18} - \\frac{20}{8 x^{2} - 24 x + 18} - \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{4 x^{2} - 12 x + 9} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{4 x - 6} - \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 3} + \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 3}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 3}) = - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(2 x - 3\\right)} - \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 3} + \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x - 3\\right)^{2}}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 4 x^{3} - 12 x^{2} - 8 x + \\left(2 x - 3\\right) \\left(6 x^{2} + 12 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 4 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8}{2 \\left(2 x - 3\\right)^{2}}$.\nFinal answer: $\\frac{- 4 x^{3} - 12 x^{2} - 8 x + \\left(2 x - 3\\right) \\left(6 x^{2} + 12 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 4 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8}{2 \\left(2 x - 3\\right)^{2}}$", "final_answer": "\\frac{- 4 x^{3} - 12 x^{2} - 8 x + \\left(2 x - 3\\right) \\left(6 x^{2} + 12 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 4 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8}{2 \\left(2 x - 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "01dca67b33c1a600", "problem_id": "383f8185b5358276", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$2 x + 4 y$ = $-34$\n$- 7 x + 9 y$ = $-19$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-7$: $- 14 x - 28 y$ = $238$.\n3) Multiply the second equation by $2$: $- 14 x + 18 y$ = $-38$.\n4) Subtract to eliminate x: $- 46 y$ = $276$.\n5) Solve for y: y = $-6$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, -6\\right)$", "final_answer": "(x,y)=\\left(-5, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "23f69048aba81d86", "problem_id": "05feeace1c3d8a0b", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{25 x}{6} - \\sqrt{2} x + \\frac{8 \\sqrt{2}}{3} + 4$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{25}{6} - \\sqrt{2}$, c=$\\frac{8 \\sqrt{2}}{3} + 4$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{121}{36} - \\frac{7 \\sqrt{2}}{3}$, so √Δ = $\\sqrt{\\frac{121}{36} - \\frac{7 \\sqrt{2}}{3}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{121 - 84 \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{2} + \\frac{25}{12}$.\n7) x2 = $- \\frac{\\sqrt{121 - 84 \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{2} + \\frac{25}{12}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{121 - 84\\cdot \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{2} + \\frac{25}{12}, - \\frac{\\sqrt{121 - 84\\cdot \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{2} + \\frac{25}{12}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{121 - 84\\cdot \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{2} + \\frac{25}{12}, - \\frac{\\sqrt{121 - 84\\cdot \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{2} + \\frac{25}{12}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1e9b7a0b4e408a3c", "problem_id": "05feeace1c3d8a0b", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{25 x}{6} - \\sqrt{2} x + \\frac{8 \\sqrt{2}}{3} + 4$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{25}{6} - \\sqrt{2}$, c=$\\frac{8 \\sqrt{2}}{3} + 4$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{121}{36} - \\frac{7 \\sqrt{2}}{3}$, so √Δ = $\\sqrt{\\frac{121}{36} - \\frac{7 \\sqrt{2}}{3}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{121 - 84 \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{2} + \\frac{25}{12}$.\nx2 = $- \\frac{\\sqrt{121 - 84 \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{2} + \\frac{25}{12}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{121 - 84\\cdot \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{2} + \\frac{25}{12}, - \\frac{\\sqrt{121 - 84\\cdot \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{2} + \\frac{25}{12}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{121 - 84\\cdot \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{2} + \\frac{25}{12}, - \\frac{\\sqrt{121 - 84\\cdot \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{2} + \\frac{25}{12}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ee46575000eda9de", "problem_id": "fb66368f2415621d", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(- 3 x^{3} - x + 1\\right) \\cos{\\left(- x^{2} + x + 2 \\right)}$.", "output": "1) Let f(x) = $2 \\left(x + 2\\right) e^{x} + \\left(- 3 x^{3} - x + 1\\right) \\cos{\\left(- x^{2} + x + 2 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- \\left(1 - 2 x\\right) \\left(- 3 x^{3} - x + 1\\right) \\sin{\\left(- x^{2} + x + 2 \\right)} + 2 \\left(x + 2\\right) e^{x} + \\left(- 9 x^{2} - 1\\right) \\cos{\\left(- x^{2} + x + 2 \\right)} + 2 e^{x}$.\nFinal answer: $\\left(2\\cdot x - 1\\right)\\cdot \\left(- 3\\cdot x^{3} - x + 1\\right)\\cdot \\sin{\\left(- x^{2} + x + 2 \\right)} + \\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(- 9\\cdot x^{2} - 1\\right)\\cdot \\cos{\\left(- x^{2} + x + 2 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x - 1\\right)\\cdot \\left(- 3\\cdot x^{3} - x + 1\\right)\\cdot \\sin{\\left(- x^{2} + x + 2 \\right)} + \\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(- 9\\cdot x^{2} - 1\\right)\\cdot \\cos{\\left(- x^{2} + x + 2 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8ad2791bddb39eb8", "problem_id": "fb66368f2415621d", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(- 3 x^{3} - x + 1\\right) \\cos{\\left(- x^{2} + x + 2 \\right)}$.", "output": "Let f(x) = $2 \\left(x + 2\\right) e^{x} + \\left(- 3 x^{3} - x + 1\\right) \\cos{\\left(- x^{2} + x + 2 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- \\left(1 - 2 x\\right) \\left(- 3 x^{3} - x + 1\\right) \\sin{\\left(- x^{2} + x + 2 \\right)} + 2 \\left(x + 2\\right) e^{x} + \\left(- 9 x^{2} - 1\\right) \\cos{\\left(- x^{2} + x + 2 \\right)} + 2 e^{x}$.\nFinal answer: $\\left(2\\cdot x - 1\\right)\\cdot \\left(- 3\\cdot x^{3} - x + 1\\right)\\cdot \\sin{\\left(- x^{2} + x + 2 \\right)} + \\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(- 9\\cdot x^{2} - 1\\right)\\cdot \\cos{\\left(- x^{2} + x + 2 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x - 1\\right)\\cdot \\left(- 3\\cdot x^{3} - x + 1\\right)\\cdot \\sin{\\left(- x^{2} + x + 2 \\right)} + \\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(- 9\\cdot x^{2} - 1\\right)\\cdot \\cos{\\left(- x^{2} + x + 2 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "08b889a2e6625ec2", "problem_id": "c0fbe97de884fdee", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} + 4 x + 1$.", "output": "1) Differentiate term-by-term:\n2) d/dx(1) = 0.\n3) d/dx(x^{3}) = 3 x^{2}.\n4) d/dx(4 x) = 4.\n5) Combine: f'(x) = 3 x^{2} + 4.\nFinal answer: $3\\cdot x^{2} + 4$", "final_answer": "3\\cdot x^{2} + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2e0dd62264260ce1", "problem_id": "c0fbe97de884fdee", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} + 4 x + 1$.", "output": "Differentiate term-by-term:\nd/dx(1) = 0.\nd/dx(x^{3}) = 3 x^{2}.\nd/dx(4 x) = 4.\nCombine: f'(x) = 3 x^{2} + 4.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $3\\cdot x^{2} + 4$", "final_answer": "3\\cdot x^{2} + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cf9a413f903267d8", "problem_id": "1bea1c540a0aa966", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(- 2 x^{3} + x^{2} - x - 3\\right) \\cos{\\left(x + 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 3 \\cos{\\left(x + 2 \\right)}) = 3 \\sin{\\left(x + 2 \\right)}$.\n3) So $f'(x) = 2 x^{3} \\sin{\\left(x + 2 \\right)} - x^{2} \\sin{\\left(x + 2 \\right)} - 6 x^{2} \\cos{\\left(x + 2 \\right)} + x e^{x} + x \\sin{\\left(x + 2 \\right)} + 2 x \\cos{\\left(x + 2 \\right)} + 2 e^{x} + 3 \\sin{\\left(x + 2 \\right)} - \\cos{\\left(x + 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 3 \\cos{\\left(x + 2 \\right)}) = 3 \\sin{\\left(x + 2 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 1\\right) e^{x} - \\left(x \\left(x \\left(1 - 2 x\\right) - 1\\right) - 3\\right) \\sin{\\left(x + 2 \\right)} + \\left(- 6 x^{2} + 2 x - 1\\right) \\cos{\\left(x + 2 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 1\\right) e^{x} - \\left(x \\left(x \\left(1 - 2 x\\right) - 1\\right) - 3\\right) \\sin{\\left(x + 2 \\right)} + \\left(- 6 x^{2} + 2 x - 1\\right) \\cos{\\left(x + 2 \\right)} + e^{x}$", "final_answer": "\\left(x + 1\\right) e^{x} - \\left(x \\left(x \\left(1 - 2 x\\right) - 1\\right) - 3\\right) \\sin{\\left(x + 2 \\right)} + \\left(- 6 x^{2} + 2 x - 1\\right) \\cos{\\left(x + 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "cfd3b47c1dc07d52", "problem_id": "28f9fbe3d798596d", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + \\frac{33 x}{4} + \\frac{21}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$\\frac{33}{4}$, c=$\\frac{21}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{81}{16}$, so √Δ = $\\frac{9}{4}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $-1$.\n7) x2 = $- \\frac{7}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{-1, - \\frac{7}{4}\\}$", "final_answer": "x \\in \\{-1, - \\frac{7}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec9ff43afc123a0b", "problem_id": "3ef96dfabadc66d8", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 2 x^{3} + 5 x^{2} - 6 x + 8$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{3} + 5 x^{2} - 7 x + 8.\n3) Then the limit is 22.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: 21.\nFinal answer: $21$", "final_answer": "21", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2747f0c74b1c6093", "problem_id": "9788f6ca403b6b1e", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 8 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=8, k=5, p=$\\frac{1}{4}$ and simplify.\n4) Here C(8,5) = 56.\nFinal answer: $\\frac{189}{8192}$", "final_answer": "\\frac{189}{8192}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "90813d1b22a2569d", "problem_id": "89aaf14324e98359", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "An urn has 7 red balls and 7 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(14,7).\n3) Favorable samples: choose k reds and n−k blues: C(7,3)·C(7,4).\n4) Compute totals: C(14,7) = 3432, and favorable = 1225.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1225}{3432}$", "final_answer": "\\frac{1225}{3432}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e72f4de8a186432f", "problem_id": "220b1d991a18d9da", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(- 3 x^{2} - 3 x - 2\\right) \\cos{\\left(x - 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 2 \\cos{\\left(x - 3 \\right)}) = 2 \\sin{\\left(x - 3 \\right)}$.\n3) So $f'(x) = 3 x^{2} \\sin{\\left(x - 3 \\right)} + x e^{x} + 3 x \\sin{\\left(x - 3 \\right)} - 6 x \\cos{\\left(x - 3 \\right)} + 2 e^{x} + 2 \\sin{\\left(x - 3 \\right)} - 3 \\cos{\\left(x - 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 2 \\cos{\\left(x - 3 \\right)}) = 2 \\sin{\\left(x - 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = 3 \\left(- 2 x - 1\\right) \\cos{\\left(x - 3 \\right)} + \\left(x + 1\\right) e^{x} - \\left(3 x \\left(- x - 1\\right) - 2\\right) \\sin{\\left(x - 3 \\right)} + e^{x}$.\nFinal answer: $3 \\left(- 2 x - 1\\right) \\cos{\\left(x - 3 \\right)} + \\left(x + 1\\right) e^{x} - \\left(3 x \\left(- x - 1\\right) - 2\\right) \\sin{\\left(x - 3 \\right)} + e^{x}$", "final_answer": "3 \\left(- 2 x - 1\\right) \\cos{\\left(x - 3 \\right)} + \\left(x + 1\\right) e^{x} - \\left(3 x \\left(- x - 1\\right) - 2\\right) \\sin{\\left(x - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f47652d5ae49b0d2", "problem_id": "139e7461e8f5f798", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Two friends split $1920 in the ratio 4:8. How much does each receive?", "output": "1) Total ratio parts = 4+8 = 12.\n2) Each part = 1920 ÷ 12 = 160.\n3) First share = 4 × 160 = 640.\n4) Second share = 8 × 160 = 1280.\nFinal answer: $1280$", "final_answer": "1280", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a5f0a605e25f9b55", "problem_id": "139e7461e8f5f798", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Two friends split $1920 in the ratio 4:8. How much does each receive?", "output": "$1280$", "final_answer": "1280", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bba008c39b89a1a0", "problem_id": "a838613c875aaecb", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "You want 44 liters of a 30% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 44−x liters is 70% solution.\n2) Salt equation: 20x + 70(44−x) = 30·44.\n3) (20-70)x = (30-70)·44 ⇒ x = 176/5.\n4) Then 44−x = 44/5.\nFinal answer: $176/5 L of 20% and 44/5 L of 70%$", "final_answer": "176/5 L of 20% and 44/5 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4523f425e5781f50", "problem_id": "a838613c875aaecb", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "You want 44 liters of a 30% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 44−x liters is 70% solution.\nSalt equation: 20x + 70(44−x) = 30·44.\n(20-70)x = (30-70)·44 ⇒ x = 176/5.\nThen 44−x = 44/5.\nCheck: 20·176/5 + 70·44/5 = 30·44.\nFinal answer: $176/5 L of 20% and 44/5 L of 70%$", "final_answer": "176/5 L of 20% and 44/5 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4412f96247eb2a4f", "problem_id": "5811b11957ac1755", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "An urn has 7 red balls and 8 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(15,6).\nFavorable samples: choose k reds and n−k blues: C(7,3)·C(8,3).\nFinal answer: $\\frac{56}{143}$", "final_answer": "\\frac{56}{143}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d135f1c147af77ff", "problem_id": "5811b11957ac1755", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 7 red balls and 8 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(15,6).\nFavorable samples: choose k reds and n−k blues: C(7,3)·C(8,3).\nCompute totals: C(15,6) = 5005, and favorable = 1960.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{56}{143}$.\nFinal answer: $\\frac{56}{143}$", "final_answer": "\\frac{56}{143}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "61a55f4925cd2c48", "problem_id": "8b2daf8ae4c816c2", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} 3 x^{3} - x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{3} - x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $- \\frac{\\pi^{3}}{12}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{\\pi^{3}}{12}$", "final_answer": "- \\frac{\\pi^{3}}{12}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cda1a8c7f72ae49b", "problem_id": "8b2daf8ae4c816c2", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} 3 x^{3} - x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{3} - x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $- \\frac{\\pi^{3}}{12}$.\n4) Simplify the result.\nFinal answer: $- \\frac{\\pi^{3}}{12}$", "final_answer": "- \\frac{\\pi^{3}}{12}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0c0de7f4ef48736e", "problem_id": "7d2a7558c5fe119e", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "You want 23 liters of a 35% salt solution. You have a 10% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 23−x liters is 50% solution.\nSalt equation: 10x + 50(23−x) = 35·23.\n(10-50)x = (35-50)·23 ⇒ x = 69/8.\nFinal answer: $69/8 L of 10% and 115/8 L of 50%$", "final_answer": "69/8 L of 10% and 115/8 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b8558b650405af58", "problem_id": "7d2a7558c5fe119e", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "You want 23 liters of a 35% salt solution. You have a 10% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 23−x liters is 50% solution.\n2) Salt equation: 10x + 50(23−x) = 35·23.\n3) (10-50)x = (35-50)·23 ⇒ x = 69/8.\n4) Then 23−x = 115/8.\nFinal answer: $69/8 L of 10% and 115/8 L of 50%$", "final_answer": "69/8 L of 10% and 115/8 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "56a98c506f150c11", "problem_id": "3d4c6f220af45356", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 8 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=8, k=3, p=$\\frac{4}{5}$ and simplify.\n4) Here C(8,3) = 56.\nFinal answer: $\\frac{3584}{390625}$", "final_answer": "\\frac{3584}{390625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e43d54525b960e5f", "problem_id": "ec3295382b14f1e2", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{3} + 2 x^{2} + 3 x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{3 x + 4}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 4}) = \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 4}$.\n3) So $f'(x) = - \\frac{36 x^{3}}{18 x^{2} + 48 x + 32} + \\frac{9 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{18 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} + 48 x + 32} - \\frac{60 x^{2}}{18 x^{2} + 48 x + 32} + \\frac{24 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{48 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{32 x}{18 x^{2} + 48 x + 32} + \\frac{9 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{2} + 24 x + 16} - \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x + 8} + \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 4} - \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 4} + \\frac{16 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{32 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{18}{18 x^{2} + 48 x + 32} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 4}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 4}) = \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(3 x + 4\\right)} + \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 4} - \\frac{9 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(3 x + 4\\right)^{2}} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 4}$.\nTherefore the correct derivative is $f'(x) = \\frac{18 x^{3} - 12 x^{2} - 18 x + \\left(3 x + 4\\right) \\left(- 18 x^{2} + 8 x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 6 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6}{2 \\left(3 x + 4\\right)^{2}}$.\nFinal answer: $\\frac{18 x^{3} - 12 x^{2} - 18 x + \\left(3 x + 4\\right) \\left(- 18 x^{2} + 8 x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 6 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6}{2 \\left(3 x + 4\\right)^{2}}$", "final_answer": "\\frac{18 x^{3} - 12 x^{2} - 18 x + \\left(3 x + 4\\right) \\left(- 18 x^{2} + 8 x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 6 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6}{2 \\left(3 x + 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "270bfe414b2f7cf9", "problem_id": "00cbc312ce6a9074", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "You want 30 liters of a 30% salt solution. You have a 15% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 30−x liters is 70% solution.\n2) Salt equation: 15x + 70(30−x) = 30·30.\n3) (15-70)x = (30-70)·30 ⇒ x = 240/11.\n4) Then 30−x = 90/11.\nFinal answer: $240/11 L of 15% and 90/11 L of 70%$", "final_answer": "240/11 L of 15% and 90/11 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "340fd81113d9879a", "problem_id": "00cbc312ce6a9074", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "You want 30 liters of a 30% salt solution. You have a 15% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 30−x liters is 70% solution.\nSalt equation: 15x + 70(30−x) = 30·30.\n(15-70)x = (30-70)·30 ⇒ x = 240/11.\nThen 30−x = 90/11.\nCheck: 15·240/11 + 70·90/11 = 30·30.\nFinal answer: $240/11 L of 15% and 90/11 L of 70%$", "final_answer": "240/11 L of 15% and 90/11 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f799e530f88df42", "problem_id": "73c64bd7f761ad35", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{2} + x + 3$.", "output": "Differentiate term-by-term:\nd/dx(3) = 0.\nd/dx(x) = 1.\nd/dx(4 x^{2}) = 8 x.\nCombine: f'(x) = 8 x + 1.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $8\\cdot x + 1$", "final_answer": "8\\cdot x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ac82a51d2de5b14f", "problem_id": "73c64bd7f761ad35", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{2} + x + 3$.", "output": "1) Differentiate term-by-term:\n2) d/dx(3) = 0.\n3) d/dx(x) = 1.\n4) d/dx(4 x^{2}) = 8 x.\n5) Combine: f'(x) = 8 x + 1.\nFinal answer: $8\\cdot x + 1$", "final_answer": "8\\cdot x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f1ff7fb63bf28dfc", "problem_id": "c1308b44f8c59648", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "You want 41 liters of a 40% salt solution. You have a 10% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 41−x liters is 70% solution.\n2) Salt equation: 10x + 70(41−x) = 40·41.\n3) (10-70)x = (40-70)·41 ⇒ x = 41/2.\n4) Then 41−x = 41/2.\nFinal answer: $41/2 L of 10% and 41/2 L of 70%$", "final_answer": "41/2 L of 10% and 41/2 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1b38ad3c687457e4", "problem_id": "c1308b44f8c59648", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 41 liters of a 40% salt solution. You have a 10% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 41−x liters is 70% solution.\nSalt equation: 10x + 70(41−x) = 40·41.\n(10-70)x = (40-70)·41 ⇒ x = 41/2.\nThen 41−x = 41/2.\nCheck: 10·41/2 + 70·41/2 = 40·41.\nFinal answer: $41/2 L of 10% and 41/2 L of 70%$", "final_answer": "41/2 L of 10% and 41/2 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "21482a3cafd9f6a7", "problem_id": "d560a66ba953cd2b", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{7}$. If it is flipped 19 times, what is the probability of exactly 2 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=19, k=2, p=$\\frac{4}{7}$ and simplify.\nHere C(19,2) = 171.\nCheck: expression simplifies to $\\frac{353327485968}{11398895185373143}$.\nFinal answer: $\\frac{353327485968}{11398895185373143}$", "final_answer": "\\frac{353327485968}{11398895185373143}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "80b9d075bd05ad84", "problem_id": "d560a66ba953cd2b", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{7}$. If it is flipped 19 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=19, k=2, p=$\\frac{4}{7}$ and simplify.\n4) Here C(19,2) = 171.\nFinal answer: $\\frac{353327485968}{11398895185373143}$", "final_answer": "\\frac{353327485968}{11398895185373143}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8c145224848a3669", "problem_id": "9f0a09f4100f3608", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$- 8 x - 4 y$ = $-56$\n$- 7 x + 2 y$ = $-27$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-7$: $56 x + 28 y$ = $392$.\n3) Multiply the second equation by $-8$: $56 x - 16 y$ = $216$.\n4) Subtract to eliminate x: $44 y$ = $176$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, 4\\right)$", "final_answer": "(x,y)=\\left(5, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "80db00ee5320a1d0", "problem_id": "9f0a09f4100f3608", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- 8 x - 4 y$ = $-56$\n$- 7 x + 2 y$ = $-27$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-7$: $56 x + 28 y$ = $392$.\nMultiply the second equation by $-8$: $56 x - 16 y$ = $216$.\nSubtract to eliminate x: $44 y$ = $176$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $5$.\nCheck: plug (x,y)=(5,4) into both equations.\nFinal answer: $(x,y)=\\left(5, 4\\right)$", "final_answer": "(x,y)=\\left(5, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f798c41a7d244a2b", "problem_id": "ce041b15ba49a3ec", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} - 2 x^{3} - x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{3} - x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\n3) Subtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{\\pi^{4}}{32} - \\frac{\\pi}{2} - \\frac{\\pi^{3}}{24} + \\frac{\\pi^{2}}{8} + 5$.\n4) Simplify the result.\nFinal answer: $- \\frac{\\pi^{4}}{32} - \\frac{\\pi}{2} - \\frac{\\pi^{3}}{24} + \\frac{\\pi^{2}}{8} + 5$", "final_answer": "- \\frac{\\pi^{4}}{32} - \\frac{\\pi}{2} - \\frac{\\pi^{3}}{24} + \\frac{\\pi^{2}}{8} + 5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8475222ea8457ef3", "problem_id": "ce041b15ba49a3ec", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} - 2 x^{3} - x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{3} - x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\nSubtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{\\pi^{4}}{32} - \\frac{\\pi}{2} - \\frac{\\pi^{3}}{24} + \\frac{\\pi^{2}}{8} + 5$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{\\pi^{4}}{32} - \\frac{\\pi}{2} - \\frac{\\pi^{3}}{24} + \\frac{\\pi^{2}}{8} + 5$", "final_answer": "- \\frac{\\pi^{4}}{32} - \\frac{\\pi}{2} - \\frac{\\pi^{3}}{24} + \\frac{\\pi^{2}}{8} + 5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "22bf3e30b366eddd", "problem_id": "1777bb60fe5eee80", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A cyclist rides at 45 miles per hour for 2 hours, then at 42 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 45 × 2 = 90 miles.\n2) Second leg distance = 42 × 5/2 = 105 miles.\n3) Total distance = 90 + 105 = 195 miles.\nFinal answer: $195$", "final_answer": "195", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ccabbb3d217bb0b3", "problem_id": "1777bb60fe5eee80", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A cyclist rides at 45 miles per hour for 2 hours, then at 42 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 45 × 2 = 90 miles.\nSecond leg distance = 42 × 5/2 = 105 miles.\nTotal distance = 90 + 105 = 195 miles.\nFinal answer: $195$", "final_answer": "195", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "90ce1218503de336", "problem_id": "ee3d50db9f336670", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the multiplicative inverse of 16 modulo 31.", "output": "1) We want x such that 16x ≡ 1 (mod 31).\n2) Use extended Euclid to find s,t with 16s + 31t = 1.\n3) One Bezout identity is: 16·(2) + 31·(-1) = 1.\n4) Reduce s modulo 31: x ≡ 2 (mod 31).\nFinal answer: $x\\equiv 2\\pmod{31}$", "final_answer": "x\\equiv 2\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bf1e065b77ae46d4", "problem_id": "ee3d50db9f336670", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 16 modulo 31.", "output": "We want x such that 16x ≡ 1 (mod 31).\nUse extended Euclid to find s,t with 16s + 31t = 1.\nOne Bezout identity is: 16·(2) + 31·(-1) = 1.\nReduce s modulo 31: x ≡ 2 (mod 31).\nCheck: 16·2 mod 31 = 1.\nFinal answer: $x\\equiv 2\\pmod{31}$", "final_answer": "x\\equiv 2\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "163c6cb12a8fa3d6", "problem_id": "4d21b1129349c812", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 2 x - 8$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$-2$, c=$-8$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $100$, so √Δ = $10$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $2$.\nx2 = $- \\frac{4}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{2, - \\frac{4}{3}\\}$", "final_answer": "x \\in \\{2, - \\frac{4}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "16e8e1404fcf8139", "problem_id": "4d21b1129349c812", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 2 x - 8$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$-2$, c=$-8$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $100$, so √Δ = $10$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $2$.\n7) x2 = $- \\frac{4}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{2, - \\frac{4}{3}\\}$", "final_answer": "x \\in \\{2, - \\frac{4}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "68c2aa19f7ef58a4", "problem_id": "87f397b29444a87f", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $217 in the ratio 4:3. How much does each receive?", "output": "Total ratio parts = 4+3 = 7.\nEach part = 217 ÷ 7 = 31.\nFirst share = 4 × 31 = 124.\nSecond share = 3 × 31 = 93.\nCheck: 124+93=217.\nFinal answer: $93$", "final_answer": "93", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "02db7309250a9187", "problem_id": "87f397b29444a87f", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Two friends split $217 in the ratio 4:3. How much does each receive?", "output": "1) Total ratio parts = 4+3 = 7.\n2) Each part = 217 ÷ 7 = 31.\n3) First share = 4 × 31 = 124.\n4) Second share = 3 × 31 = 93.\nFinal answer: $93$", "final_answer": "93", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "25d799014fcd8d8e", "problem_id": "e05c653dafc6d458", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$7 x - 2 y$ = $-52$\n$4 x - 5 y$ = $-22$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $28 x - 8 y$ = $-208$.\n3) Multiply the second equation by $7$: $28 x - 35 y$ = $-154$.\n4) Subtract to eliminate x: $27 y$ = $-54$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $-8$.\nFinal answer: $(x,y)=\\left(-8, -2\\right)$", "final_answer": "(x,y)=\\left(-8, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cfcac09520ff6f7e", "problem_id": "e05c653dafc6d458", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$7 x - 2 y$ = $-52$\n$4 x - 5 y$ = $-22$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $28 x - 8 y$ = $-208$.\nMultiply the second equation by $7$: $28 x - 35 y$ = $-154$.\nSubtract to eliminate x: $27 y$ = $-54$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $-8$.\nCheck: plug (x,y)=(-8,-2) into both equations.\nFinal answer: $(x,y)=\\left(-8, -2\\right)$", "final_answer": "(x,y)=\\left(-8, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e8dcbe7db047bc0e", "problem_id": "a3440b5759ca3171", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Find the multiplicative inverse of 22 modulo 29.", "output": "We want x such that 22x ≡ 1 (mod 29).\nUse extended Euclid to find s,t with 22s + 29t = 1.\nOne Bezout identity is: 22·(4) + 29·(-3) = 1.\nFinal answer: $x\\equiv 4\\pmod{29}$", "final_answer": "x\\equiv 4\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0c34f5a5835fc3f9", "problem_id": "a3440b5759ca3171", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the multiplicative inverse of 22 modulo 29.", "output": "We want x such that 22x ≡ 1 (mod 29).\nUse extended Euclid to find s,t with 22s + 29t = 1.\nOne Bezout identity is: 22·(4) + 29·(-3) = 1.\nReduce s modulo 29: x ≡ 4 (mod 29).\nCheck: 22·4 mod 29 = 1.\nFinal answer: $x\\equiv 4\\pmod{29}$", "final_answer": "x\\equiv 4\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cb4481916927e471", "problem_id": "41bcecd3093e9095", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{3} x}{2} + x - 2 + \\frac{\\sqrt{3}}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$1 - \\frac{\\sqrt{3}}{2}$, c=$-2 + \\frac{\\sqrt{3}}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{39}{4} - 3 \\sqrt{3}$, so √Δ = $\\sqrt{\\frac{39}{4} - 3 \\sqrt{3}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{1}{2} + \\frac{\\sqrt{3}}{4} + \\frac{\\sqrt{39 - 12 \\sqrt{3}}}{4}$.\n7) x2 = $- \\frac{\\sqrt{39 - 12 \\sqrt{3}}}{4} - \\frac{1}{2} + \\frac{\\sqrt{3}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{1}{2} + \\frac{\\sqrt{3}}{4} + \\frac{\\sqrt{39 - 12\\cdot \\sqrt{3}}}{4}, - \\frac{\\sqrt{39 - 12\\cdot \\sqrt{3}}}{4} - \\frac{1}{2} + \\frac{\\sqrt{3}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{1}{2} + \\frac{\\sqrt{3}}{4} + \\frac{\\sqrt{39 - 12\\cdot \\sqrt{3}}}{4}, - \\frac{\\sqrt{39 - 12\\cdot \\sqrt{3}}}{4} - \\frac{1}{2} + \\frac{\\sqrt{3}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a6dc968ae8e51b1f", "problem_id": "e3905de450773855", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} - x - 4$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to - 2 x - 4.\n3) Then the limit is -6.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: -5.\nFinal answer: $-5$", "final_answer": "-5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "16145ce071c9afdd", "problem_id": "b0d184af22df1bb8", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Two friends split $1352 in the ratio 4:9. How much does each receive?", "output": "1) Total ratio parts = 4+9 = 13.\n2) Each part = 1352 ÷ 13 = 104.\n3) First share = 4 × 104 = 416.\n4) Second share = 9 × 104 = 936.\nFinal answer: $936$", "final_answer": "936", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "49005b522696edc6", "problem_id": "b0d184af22df1bb8", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Two friends split $1352 in the ratio 4:9. How much does each receive?", "output": "Total ratio parts = 4+9 = 13.\nEach part = 1352 ÷ 13 = 104.\nFirst share = 4 × 104 = 416.\nSecond share = 9 × 104 = 936.\nCheck: 416+936=1352.\nFinal answer: $936$", "final_answer": "936", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e40fe8c3a26a7e75", "problem_id": "0dc39ef70e848b97", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A disease has prevalence $\\frac{3}{200}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{89}{800}$.\n3) Compute P(D|+) = $\\frac{51}{445}$.\nFinal answer: $\\frac{51}{445}$", "final_answer": "\\frac{51}{445}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "947292bcb7208003", "problem_id": "0dc39ef70e848b97", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A disease has prevalence $\\frac{3}{200}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{89}{800}$.\nCompute P(D|+) = $\\frac{51}{445}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{51}{445}$", "final_answer": "\\frac{51}{445}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cca0d54ad6051653", "problem_id": "ab5bccc079756a86", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Two friends split $204 in the ratio 5:1. How much does each receive?", "output": "1) Total ratio parts = 5+1 = 6.\n2) Each part = 204 ÷ 6 = 34.\n3) First share = 5 × 34 = 170.\n4) Second share = 1 × 34 = 34.\nFinal answer: $34$", "final_answer": "34", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ada7a23108ac8d86", "problem_id": "ab5bccc079756a86", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $204 in the ratio 5:1. How much does each receive?", "output": "Total ratio parts = 5+1 = 6.\nEach part = 204 ÷ 6 = 34.\nFirst share = 5 × 34 = 170.\nSecond share = 1 × 34 = 34.\nCheck: 170+34=204.\nFinal answer: $34$", "final_answer": "34", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "76e72f44ae15961d", "problem_id": "5a0e36baad4173ae", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "You want 17 liters of a 37% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 17−x liters is 50% solution.\n2) Salt equation: 25x + 50(17−x) = 37·17.\n3) (25-50)x = (37-50)·17 ⇒ x = 221/25.\n4) Then 17−x = 204/25.\nFinal answer: $221/25 L of 25% and 204/25 L of 50%$", "final_answer": "221/25 L of 25% and 204/25 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "97f3e1dc01e77ae1", "problem_id": "5a0e36baad4173ae", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "You want 17 liters of a 37% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 17−x liters is 50% solution.\nSalt equation: 25x + 50(17−x) = 37·17.\n(25-50)x = (37-50)·17 ⇒ x = 221/25.\nThen 17−x = 204/25.\nCheck: 25·221/25 + 50·204/25 = 37·17.\nFinal answer: $221/25 L of 25% and 204/25 L of 50%$", "final_answer": "221/25 L of 25% and 204/25 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "58fe4e803309f72d", "problem_id": "c951b856084039ac", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $720. It is discounted by 5%. You buy 3 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 720 × (1 − 5/100) = 720 × 95/100 = 684.\nSubtotal = 3 × 684 = 2052.\nTotal with tax = 2052 × 108/100 = 55404/25.\nCheck: recomputing the formula yields 55404/25.\nFinal answer: $\\frac{55404}{25}$", "final_answer": "\\frac{55404}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3a97355b6a5e95eb", "problem_id": "c951b856084039ac", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A store sells a backpack for $720. It is discounted by 5%. You buy 3 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 720 × (1 − 5/100) = 720 × 95/100 = 684.\n2) Subtotal = 3 × 684 = 2052.\n3) Total with tax = 2052 × 108/100 = 55404/25.\nFinal answer: $\\frac{55404}{25}$", "final_answer": "\\frac{55404}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c273ccfbe94d6d96", "problem_id": "6260cec0336408f3", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Two friends split $1749 in the ratio 3:8. How much does each receive?", "output": "1) Total ratio parts = 3+8 = 11.\n2) Each part = 1749 ÷ 11 = 159.\n3) First share = 3 × 159 = 477.\n4) Second share = 8 × 159 = 1272.\nFinal answer: $1272$", "final_answer": "1272", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "966875c7c7e6d99a", "problem_id": "6260cec0336408f3", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $1749 in the ratio 3:8. How much does each receive?", "output": "Total ratio parts = 3+8 = 11.\nEach part = 1749 ÷ 11 = 159.\nFirst share = 3 × 159 = 477.\nSecond share = 8 × 159 = 1272.\nCheck: 477+1272=1749.\nFinal answer: $1272$", "final_answer": "1272", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "035b017f819f37d9", "problem_id": "d03a80a598c72108", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{11} x + 4 x - \\frac{3 \\sqrt{11}}{2} + \\frac{15}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$4 - \\sqrt{11}$, c=$\\frac{15}{4} - \\frac{3 \\sqrt{11}}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $12 - 2 \\sqrt{11}$, so √Δ = $\\sqrt{12 - 2 \\sqrt{11}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $-2 + \\frac{\\sqrt{12 - 2 \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}$.\n7) x2 = $-2 - \\frac{\\sqrt{12 - 2 \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{-2 + \\frac{\\sqrt{12 - 2\\cdot \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}, -2 - \\frac{\\sqrt{12 - 2\\cdot \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}\\}$", "final_answer": "x \\in \\{-2 + \\frac{\\sqrt{12 - 2\\cdot \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}, -2 - \\frac{\\sqrt{12 - 2\\cdot \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "51dade49f0b8ad25", "problem_id": "d03a80a598c72108", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{11} x + 4 x - \\frac{3 \\sqrt{11}}{2} + \\frac{15}{4}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$4 - \\sqrt{11}$, c=$\\frac{15}{4} - \\frac{3 \\sqrt{11}}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $12 - 2 \\sqrt{11}$, so √Δ = $\\sqrt{12 - 2 \\sqrt{11}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $-2 + \\frac{\\sqrt{12 - 2 \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}$.\nx2 = $-2 - \\frac{\\sqrt{12 - 2 \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-2 + \\frac{\\sqrt{12 - 2\\cdot \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}, -2 - \\frac{\\sqrt{12 - 2\\cdot \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}\\}$", "final_answer": "x \\in \\{-2 + \\frac{\\sqrt{12 - 2\\cdot \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}, -2 - \\frac{\\sqrt{12 - 2\\cdot \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "20b8a38626426623", "problem_id": "9845be3668e5ce47", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + \\frac{7 x}{10} - \\frac{9}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{7}{10}$, c=$- \\frac{9}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{1849}{100}$, so √Δ = $\\frac{43}{10}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{9}{5}$.\n7) x2 = $- \\frac{5}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{9}{5}, - \\frac{5}{2}\\}$", "final_answer": "x \\in \\{\\frac{9}{5}, - \\frac{5}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d916e9e83a3fb10", "problem_id": "72a1763b38dfa521", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(- x^{3} - x^{2}\\right) \\cos{\\left(x^{2} + 2 x - 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x^{2} \\cos{\\left(\\left(x^{2} + 2 x\\right) - 2 \\right)}) = x^{2} \\sin{\\left(x^{2} + 2 x - 2 \\right)}$.\n3) So $f'(x) = 2 x^{4} \\sin{\\left(x^{2} + 2 x - 2 \\right)} + 2 x^{3} \\sin{\\left(x^{2} + 2 x - 2 \\right)} + x^{2} \\sin{\\left(x^{2} + 2 x - 2 \\right)} - 3 x^{2} \\cos{\\left(\\left(x^{2} + 2 x\\right) - 2 \\right)} + x e^{x} - 2 e^{x}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x^{2} \\cos{\\left(\\left(x^{2} + 2 x\\right) - 2 \\right)}) = x^{2} \\left(2 x + 2\\right) \\sin{\\left(x^{2} + 2 x - 2 \\right)} - 2 x \\cos{\\left(\\left(x^{2} + 2 x\\right) - 2 \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{2} \\left(- x - 1\\right) \\left(x + 1\\right) \\sin{\\left(x^{2} + 2 x - 2 \\right)} + x \\left(- 3 x - 2\\right) \\cos{\\left(\\left(x^{2} + 2 x\\right) - 2 \\right)} + \\left(x - 3\\right) e^{x} + e^{x}$.\nFinal answer: $- 2 x^{2} \\left(- x - 1\\right) \\left(x + 1\\right) \\sin{\\left(x^{2} + 2 x - 2 \\right)} + x \\left(- 3 x - 2\\right) \\cos{\\left(\\left(x^{2} + 2 x\\right) - 2 \\right)} + \\left(x - 3\\right) e^{x} + e^{x}$", "final_answer": "- 2 x^{2} \\left(- x - 1\\right) \\left(x + 1\\right) \\sin{\\left(x^{2} + 2 x - 2 \\right)} + x \\left(- 3 x - 2\\right) \\cos{\\left(\\left(x^{2} + 2 x\\right) - 2 \\right)} + \\left(x - 3\\right) e^{x} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "98ec20b52e2e7fcc", "problem_id": "741b598d122c3867", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Worker A finishes a job in 7 hours and Worker B finishes it in 21 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/7 job/hour. B's rate = 1/21 job/hour.\nCombined rate = 1/7 + 1/21 = 4/21 job/hour.\nTime = 1 ÷ (combined rate) = 21/4 hours.\nCheck: (combined rate)×(time) = 4/21×21/4 = 1.\nFinal answer: $\\frac{21}{4}$", "final_answer": "\\frac{21}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a2521f2f893b467a", "problem_id": "741b598d122c3867", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Worker A finishes a job in 7 hours and Worker B finishes it in 21 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/7 job/hour. B's rate = 1/21 job/hour.\n2) Combined rate = 1/7 + 1/21 = 4/21 job/hour.\n3) Time = 1 ÷ (combined rate) = 21/4 hours.\nFinal answer: $\\frac{21}{4}$", "final_answer": "\\frac{21}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7e77bbb12ac7b22e", "problem_id": "d9b0a8bfe84361df", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} - x^{2} + 2 x - 2$ at $x=0$.", "output": "1) Differentiate: f'(x) = $6 x^{2} - 2 x + 2$.\n2) Slope at x=0: m = f'(0) = $2$.\n3) Point on curve: (0, f(0)) = (0, $-2$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -2 = 2(x − 0).\n6) Optionally expand: y = $2 x - 2$.\nFinal answer: $y = 2\\cdot x - 2$", "final_answer": "y = 2\\cdot x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "629ebfa3ab4673f7", "problem_id": "d9b0a8bfe84361df", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} - x^{2} + 2 x - 2$ at $x=0$.", "output": "Differentiate: f'(x) = $6 x^{2} - 2 x + 2$.\nSlope at x=0: m = f'(0) = $2$.\nPoint on curve: (0, f(0)) = (0, $-2$).\nFinal answer: $y = 2\\cdot x - 2$", "final_answer": "y = 2\\cdot x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d7cacc964ad4e49d", "problem_id": "37554ed3741d8a1a", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + \\frac{21 x}{5} + \\frac{18}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{21}{5}$, c=$\\frac{18}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{81}{25}$, so √Δ = $\\frac{9}{5}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{6}{5}$.\n7) x2 = $-3$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{6}{5}, -3\\}$", "final_answer": "x \\in \\{- \\frac{6}{5}, -3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4a0ecd2d18016889", "problem_id": "37554ed3741d8a1a", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + \\frac{21 x}{5} + \\frac{18}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{21}{5}$, c=$\\frac{18}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{81}{25}$, so √Δ = $\\frac{9}{5}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{6}{5}$.\nx2 = $-3$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{6}{5}, -3\\}$", "final_answer": "x \\in \\{- \\frac{6}{5}, -3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "07aa973c37570e60", "problem_id": "67d7e252324e85d1", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡6 (mod 7), x≡3 (mod 11). Give the solution modulo 77.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 77.\nCompute the CRT solution to get x ≡ 69 (mod 77).\nCheck: 69 mod 7=6 and mod 11=3.\nFinal answer: $x\\equiv 69\\pmod{77}$", "final_answer": "x\\equiv 69\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e8cf612b56291a16", "problem_id": "67d7e252324e85d1", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡6 (mod 7), x≡3 (mod 11). Give the solution modulo 77.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 77.\n3) Compute the CRT solution to get x ≡ 69 (mod 77).\nFinal answer: $x\\equiv 69\\pmod{77}$", "final_answer": "x\\equiv 69\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7d959a2e7c5c25a8", "problem_id": "0e50f1b750851aff", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$6 x^{2} - 3 x - 2 \\sqrt{2} x + \\sqrt{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$6$, b=$-3 - 2 \\sqrt{2}$, c=$\\sqrt{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $17 - 12 \\sqrt{2}$, so √Δ = $\\sqrt{17 - 12 \\sqrt{2}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{17 - 12 \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{6} + \\frac{1}{4}$.\n7) x2 = $- \\frac{\\sqrt{17 - 12 \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{6} + \\frac{1}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{17 - 12\\cdot \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{6} + \\frac{1}{4}, - \\frac{\\sqrt{17 - 12\\cdot \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{6} + \\frac{1}{4}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{17 - 12\\cdot \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{6} + \\frac{1}{4}, - \\frac{\\sqrt{17 - 12\\cdot \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{6} + \\frac{1}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "64a46a9e5713b227", "problem_id": "0e50f1b750851aff", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$6 x^{2} - 3 x - 2 \\sqrt{2} x + \\sqrt{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$6$, b=$-3 - 2 \\sqrt{2}$, c=$\\sqrt{2}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $17 - 12 \\sqrt{2}$, so √Δ = $\\sqrt{17 - 12 \\sqrt{2}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{17 - 12 \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{6} + \\frac{1}{4}$.\nx2 = $- \\frac{\\sqrt{17 - 12 \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{6} + \\frac{1}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{17 - 12\\cdot \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{6} + \\frac{1}{4}, - \\frac{\\sqrt{17 - 12\\cdot \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{6} + \\frac{1}{4}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{17 - 12\\cdot \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{6} + \\frac{1}{4}, - \\frac{\\sqrt{17 - 12\\cdot \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{6} + \\frac{1}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "682940107d82eeef", "problem_id": "00892c64f4f3362e", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the linear system:\n$3 x$ = $-3$\n$3 x - 3 y$ = $9$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $9 x$ = $-9$.\nMultiply the second equation by $3$: $9 x - 9 y$ = $27$.\nFinal answer: $(x,y)=\\left(-1, -4\\right)$", "final_answer": "(x,y)=\\left(-1, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "79285d80b317573e", "problem_id": "e9ea8694d5d5651d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - x^{3} + 2 x^{2} - 3 x + 4$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to - x^{3} + 2 x^{2} - 4 x + 4.\n3) Then the limit is 28.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 26.\nFinal answer: $26$", "final_answer": "26", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "bcdf01ba38649e61", "problem_id": "d810d322c3be4d1d", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- 2 x - 5 y$ = $-1$\n$8 x - y$ = $67$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $8$: $- 16 x - 40 y$ = $-8$.\nMultiply the second equation by $-2$: $- 16 x + 2 y$ = $-134$.\nSubtract to eliminate x: $- 42 y$ = $126$.\nSolve for y: y = $-3$.\nSubstitute back to find x: x = $8$.\nCheck: plug (x,y)=(8,-3) into both equations.\nFinal answer: $(x,y)=\\left(8, -3\\right)$", "final_answer": "(x,y)=\\left(8, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c477681a25838695", "problem_id": "d810d322c3be4d1d", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$- 2 x - 5 y$ = $-1$\n$8 x - y$ = $67$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $8$: $- 16 x - 40 y$ = $-8$.\n3) Multiply the second equation by $-2$: $- 16 x + 2 y$ = $-134$.\n4) Subtract to eliminate x: $- 42 y$ = $126$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $8$.\nFinal answer: $(x,y)=\\left(8, -3\\right)$", "final_answer": "(x,y)=\\left(8, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "42417c1e30ef1301", "problem_id": "2ce09d513053e1c7", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int \\left(2 x + 1\\right) \\cos{\\left(x \\right)} + \\left(2 x + 4\\right) e^{2 x}\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $\\left(2 x + 1\\right) \\cos{\\left(x \\right)} + \\left(2 x + 4\\right) e^{2 x}$.\n3) Compute an antiderivative: F(x) = $2 x \\sin{\\left(x \\right)} + \\frac{\\left(2 x + 3\\right) e^{2 x}}{2} + \\sin{\\left(x \\right)} + 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\frac{\\left(2\\cdot x + 3\\right)\\cdot e^{2\\cdot x}}{2} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\frac{\\left(2\\cdot x + 3\\right)\\cdot e^{2\\cdot x}}{2} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c54e2599db8dd10f", "problem_id": "2ce09d513053e1c7", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int \\left(2 x + 1\\right) \\cos{\\left(x \\right)} + \\left(2 x + 4\\right) e^{2 x}\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $\\left(2 x + 1\\right) \\cos{\\left(x \\right)} + \\left(2 x + 4\\right) e^{2 x}$.\nCompute an antiderivative: F(x) = $2 x \\sin{\\left(x \\right)} + \\frac{\\left(2 x + 3\\right) e^{2 x}}{2} + \\sin{\\left(x \\right)} + 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\frac{\\left(2\\cdot x + 3\\right)\\cdot e^{2\\cdot x}}{2} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\frac{\\left(2\\cdot x + 3\\right)\\cdot e^{2\\cdot x}}{2} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0dd49f5a03c5ac6b", "problem_id": "a296c28adb8b96c6", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 4 - 3 x$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 3 x^{2} - 2 x$ and a = -2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 6 x - 2$.\n5) Evaluate at a: p'(-2) = $10$.\nFinal answer: $10$", "final_answer": "10", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dd36bc5ad3cef3c0", "problem_id": "a296c28adb8b96c6", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 4 - 3 x$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 3 x^{2} - 2 x$ and a = -2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 6 x - 2$.\nEvaluate at a: p'(-2) = $10$.\nCheck: the identity gives $10$.\nFinal answer: $10$", "final_answer": "10", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "21c7b8b0102777c7", "problem_id": "2ea26b93b6292c82", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A disease has prevalence $\\frac{1}{100}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{387}{10000}$.\n3) Compute P(D|+) = $\\frac{10}{43}$.\nFinal answer: $\\frac{10}{43}$", "final_answer": "\\frac{10}{43}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d70e670b1a596478", "problem_id": "2ea26b93b6292c82", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A disease has prevalence $\\frac{1}{100}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{387}{10000}$.\nCompute P(D|+) = $\\frac{10}{43}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{10}{43}$", "final_answer": "\\frac{10}{43}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "916f01d7e6ca8d2c", "problem_id": "62793a2b48d98b0f", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A store sells a backpack for $64. It is discounted by 35%. You buy 6 backpacks. What is the total cost?", "output": "Discounted price = 64 × (1 − 35/100) = 64 × 65/100 = 208/5.\nSubtotal = 6 × 208/5 = 1248/5.\nCheck: recomputing the formula yields 1248/5.\nFinal answer: $\\frac{1248}{5}$", "final_answer": "\\frac{1248}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fde932728d8e19e1", "problem_id": "62793a2b48d98b0f", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A store sells a backpack for $64. It is discounted by 35%. You buy 6 backpacks. What is the total cost?", "output": "1) Discounted price = 64 × (1 − 35/100) = 64 × 65/100 = 208/5.\n2) Subtotal = 6 × 208/5 = 1248/5.\nFinal answer: $\\frac{1248}{5}$", "final_answer": "\\frac{1248}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "05582209bc1b2479", "problem_id": "0afa354efc24e7f3", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - 3 x^{2} - 11 x - 43$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} - 12 x - 43.\n3) Then the limit is -139.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: -135.\nFinal answer: $-135$", "final_answer": "-135", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "194eb7749303cb39", "problem_id": "dd0a7baeb2fe7bd4", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$11 x - 6 y$ = $-61$\n$- 2 x + 7 y$ = $17$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $- 22 x + 12 y$ = $122$.\nMultiply the second equation by $11$: $- 22 x + 77 y$ = $187$.\nSubtract to eliminate x: $- 65 y$ = $-65$.\nSolve for y: y = $1$.\nSubstitute back to find x: x = $-5$.\nCheck: plug (x,y)=(-5,1) into both equations.\nFinal answer: $(x,y)=\\left(-5, 1\\right)$", "final_answer": "(x,y)=\\left(-5, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b984600a9b84659a", "problem_id": "d0c5e090ad20e2eb", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int \\left(2 x + 1\\right) \\cos{\\left(x \\right)} + \\left(2 x + 2\\right) e^{2 x}\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $\\left(2 x + 1\\right) \\cos{\\left(x \\right)} + \\left(2 x + 2\\right) e^{2 x}$.\n3) Compute an antiderivative: F(x) = $2 x \\sin{\\left(x \\right)} + \\frac{\\left(2 x + 1\\right) e^{2 x}}{2} + \\sin{\\left(x \\right)} + 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\frac{\\left(2\\cdot x + 1\\right)\\cdot e^{2\\cdot x}}{2} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\frac{\\left(2\\cdot x + 1\\right)\\cdot e^{2\\cdot x}}{2} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4ecd8311f34b496f", "problem_id": "d0c5e090ad20e2eb", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int \\left(2 x + 1\\right) \\cos{\\left(x \\right)} + \\left(2 x + 2\\right) e^{2 x}\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $\\left(2 x + 1\\right) \\cos{\\left(x \\right)} + \\left(2 x + 2\\right) e^{2 x}$.\nCompute an antiderivative: F(x) = $2 x \\sin{\\left(x \\right)} + \\frac{\\left(2 x + 1\\right) e^{2 x}}{2} + \\sin{\\left(x \\right)} + 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\frac{\\left(2\\cdot x + 1\\right)\\cdot e^{2\\cdot x}}{2} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\frac{\\left(2\\cdot x + 1\\right)\\cdot e^{2\\cdot x}}{2} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b1bde8830b0a8a78", "problem_id": "a85de886faa84ba0", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 2 x^{3} + x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 2 x^{3} + x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 2$.\n3) Compute an antiderivative: F(x) = $- \\frac{x^{4}}{2} + \\frac{x^{3}}{3} - \\frac{3 x^{2}}{2} - 2 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{x^{4}}{2} + \\frac{x^{3}}{3} - \\frac{3\\cdot x^{2}}{2} - 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{x^{4}}{2} + \\frac{x^{3}}{3} - \\frac{3\\cdot x^{2}}{2} - 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "75eb6c05ba110ebd", "problem_id": "a85de886faa84ba0", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 2 x^{3} + x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 2 x^{3} + x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 2$.\nCompute an antiderivative: F(x) = $- \\frac{x^{4}}{2} + \\frac{x^{3}}{3} - \\frac{3 x^{2}}{2} - 2 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - \\frac{x^{4}}{2} + \\frac{x^{3}}{3} - \\frac{3\\cdot x^{2}}{2} - 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{x^{4}}{2} + \\frac{x^{3}}{3} - \\frac{3\\cdot x^{2}}{2} - 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "94131f7c169ad38b", "problem_id": "192cc84e7fa41147", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} x^{3} + 4 x^{2} + 17 x + 71$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to x^{3} + 4 x^{2} + 16 x + 71.\n3) Then the limit is 263.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: 267.\nFinal answer: $267$", "final_answer": "267", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "97d9d4b36ba33695", "problem_id": "b4efca31a253f59f", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$5 x - y$ = $44$\n$2 x - 2 y$ = $16$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $10 x - 2 y$ = $88$.\n3) Multiply the second equation by $5$: $10 x - 10 y$ = $80$.\n4) Subtract to eliminate x: $8 y$ = $8$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $9$.\nFinal answer: $(x,y)=\\left(9, 1\\right)$", "final_answer": "(x,y)=\\left(9, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "157f4034e76494df", "problem_id": "425c21396b4824a9", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{2} x + \\frac{2 x}{5} - \\frac{3}{5} + \\frac{3 \\sqrt{2}}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{2}{5} - \\sqrt{2}$, c=$- \\frac{3}{5} + \\frac{3 \\sqrt{2}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{114}{25} - \\frac{16 \\sqrt{2}}{5}$, so √Δ = $\\sqrt{\\frac{114}{25} - \\frac{16 \\sqrt{2}}{5}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{1}{5} + \\frac{\\sqrt{114 - 80 \\sqrt{2}}}{10} + \\frac{\\sqrt{2}}{2}$.\n7) x2 = $- \\frac{1}{5} - \\frac{\\sqrt{114 - 80 \\sqrt{2}}}{10} + \\frac{\\sqrt{2}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{1}{5} + \\frac{\\sqrt{114 - 80\\cdot \\sqrt{2}}}{10} + \\frac{\\sqrt{2}}{2}, - \\frac{1}{5} - \\frac{\\sqrt{114 - 80\\cdot \\sqrt{2}}}{10} + \\frac{\\sqrt{2}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{1}{5} + \\frac{\\sqrt{114 - 80\\cdot \\sqrt{2}}}{10} + \\frac{\\sqrt{2}}{2}, - \\frac{1}{5} - \\frac{\\sqrt{114 - 80\\cdot \\sqrt{2}}}{10} + \\frac{\\sqrt{2}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cb35708ee59296a2", "problem_id": "425c21396b4824a9", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{2} x + \\frac{2 x}{5} - \\frac{3}{5} + \\frac{3 \\sqrt{2}}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{2}{5} - \\sqrt{2}$, c=$- \\frac{3}{5} + \\frac{3 \\sqrt{2}}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{114}{25} - \\frac{16 \\sqrt{2}}{5}$, so √Δ = $\\sqrt{\\frac{114}{25} - \\frac{16 \\sqrt{2}}{5}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{1}{5} + \\frac{\\sqrt{114 - 80 \\sqrt{2}}}{10} + \\frac{\\sqrt{2}}{2}$.\nx2 = $- \\frac{1}{5} - \\frac{\\sqrt{114 - 80 \\sqrt{2}}}{10} + \\frac{\\sqrt{2}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{1}{5} + \\frac{\\sqrt{114 - 80\\cdot \\sqrt{2}}}{10} + \\frac{\\sqrt{2}}{2}, - \\frac{1}{5} - \\frac{\\sqrt{114 - 80\\cdot \\sqrt{2}}}{10} + \\frac{\\sqrt{2}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{1}{5} + \\frac{\\sqrt{114 - 80\\cdot \\sqrt{2}}}{10} + \\frac{\\sqrt{2}}{2}, - \\frac{1}{5} - \\frac{\\sqrt{114 - 80\\cdot \\sqrt{2}}}{10} + \\frac{\\sqrt{2}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6bff6c9184b90c9a", "problem_id": "328d656f1f617207", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_6=22$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_6=16.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=16 and k=6.\n4) Compute C(21,5) = 20349.\nFinal answer: $20349$", "final_answer": "20349", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "30018cf1d213a902", "problem_id": "328d656f1f617207", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_6=22$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_6=16.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=16 and k=6.\nCompute C(21,5) = 20349.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $20349$", "final_answer": "20349", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "45444b94d79c84d1", "problem_id": "835ab262ebbc0169", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡5 (mod 9), x≡3 (mod 5). Give the solution modulo 45.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 45.\n3) Compute the CRT solution to get x ≡ 23 (mod 45).\nFinal answer: $x\\equiv 23\\pmod{45}$", "final_answer": "x\\equiv 23\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "00a6ff33a8ab05b5", "problem_id": "835ab262ebbc0169", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡5 (mod 9), x≡3 (mod 5). Give the solution modulo 45.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 45.\nCompute the CRT solution to get x ≡ 23 (mod 45).\nCheck: 23 mod 9=5 and mod 5=3.\nFinal answer: $x\\equiv 23\\pmod{45}$", "final_answer": "x\\equiv 23\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7e056933880ece8a", "problem_id": "1ea3d1b656c77b34", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $- x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $- x^{2}\\cdot \\cos{\\left(x \\right)} - 2\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- x^{2}\\cdot \\cos{\\left(x \\right)} - 2\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d59c2124f929d08", "problem_id": "1ea3d1b656c77b34", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $- x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- x^{2}\\cdot \\cos{\\left(x \\right)} - 2\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- x^{2}\\cdot \\cos{\\left(x \\right)} - 2\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2688e65898f69f6f", "problem_id": "5aa6dc9c3b985efb", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the equation of the tangent line to $y=x^{2} + 3 x + 2 \\sin{\\left(x \\right)}$ at $x=1$.", "output": "Differentiate: f'(x) = $2 x + 2 \\cos{\\left(x \\right)} + 3$.\nSlope at x=1: m = f'(1) = $2 \\cos{\\left(1 \\right)} + 5$.\nPoint on curve: (1, f(1)) = (1, $2 \\sin{\\left(1 \\right)} + 4$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 2 \\sin{\\left(1 \\right)} + 4 = 2 \\cos{\\left(1 \\right)} + 5(x − 1).\nOptionally expand: y = $2 x \\cos{\\left(1 \\right)} + 5 x - 2 \\cos{\\left(1 \\right)} - 1 + 2 \\sin{\\left(1 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 5\\cdot x - 2\\cdot \\cos{\\left(1 \\right)} - 1 + 2\\cdot \\sin{\\left(1 \\right)}$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 5\\cdot x - 2\\cdot \\cos{\\left(1 \\right)} - 1 + 2\\cdot \\sin{\\left(1 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "04cf5232fedf40e1", "problem_id": "5aa6dc9c3b985efb", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the equation of the tangent line to $y=x^{2} + 3 x + 2 \\sin{\\left(x \\right)}$ at $x=1$.", "output": "1) Differentiate: f'(x) = $2 x + 2 \\cos{\\left(x \\right)} + 3$.\n2) Slope at x=1: m = f'(1) = $2 \\cos{\\left(1 \\right)} + 5$.\n3) Point on curve: (1, f(1)) = (1, $2 \\sin{\\left(1 \\right)} + 4$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 2 \\sin{\\left(1 \\right)} + 4 = 2 \\cos{\\left(1 \\right)} + 5(x − 1).\n6) Optionally expand: y = $2 x \\cos{\\left(1 \\right)} + 5 x - 2 \\cos{\\left(1 \\right)} - 1 + 2 \\sin{\\left(1 \\right)}$.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 5\\cdot x - 2\\cdot \\cos{\\left(1 \\right)} - 1 + 2\\cdot \\sin{\\left(1 \\right)}$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 5\\cdot x - 2\\cdot \\cos{\\left(1 \\right)} - 1 + 2\\cdot \\sin{\\left(1 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "34d1c4a1ad8a63b6", "problem_id": "9b8b08a39061e87f", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Worker A finishes a job in 4 hours and Worker B finishes it in 12 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/4 job/hour. B's rate = 1/12 job/hour.\n2) Combined rate = 1/4 + 1/12 = 1/3 job/hour.\n3) Time = 1 ÷ (combined rate) = 3 hours.\nFinal answer: $3$", "final_answer": "3", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9290628de67c28b6", "problem_id": "a3343dda6c927aaf", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} - x + \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{x + 6}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 6}) = \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 6}$.\n3) So $f'(x) = \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 24 x + 72} + \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + 24 x + 72} - \\frac{2 x^{2}}{2 x^{2} + 24 x + 72} + \\frac{12 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 24 x + 72} + \\frac{24 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + 24 x + 72} - \\frac{24 x}{2 x^{2} + 24 x + 72} + \\frac{36 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 24 x + 72} + \\frac{72 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + 24 x + 72} - \\frac{6}{2 x^{2} + 24 x + 72} + \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 12 x + 36} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 6} - \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 6} + \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 6}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 6}) = \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 6} + \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 6} - \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x + 6\\right)^{2}}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 x^{2} + 2 x + \\left(x + 6\\right) \\left(- 4 x + \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 2 \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6}{2 \\left(x + 6\\right)^{2}}$.\nFinal answer: $\\frac{2 x^{2} + 2 x + \\left(x + 6\\right) \\left(- 4 x + \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 2 \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6}{2 \\left(x + 6\\right)^{2}}$", "final_answer": "\\frac{2 x^{2} + 2 x + \\left(x + 6\\right) \\left(- 4 x + \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 2 \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6}{2 \\left(x + 6\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "9972c2d5816c7121", "problem_id": "8971f17a5c0e0a9a", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Two friends split $624 in the ratio 6:2. How much does each receive?", "output": "1) Total ratio parts = 6+2 = 8.\n2) Each part = 624 ÷ 8 = 78.\n3) First share = 6 × 78 = 468.\n4) Second share = 2 × 78 = 156.\nFinal answer: $156$", "final_answer": "156", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f171092806b38142", "problem_id": "8971f17a5c0e0a9a", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $624 in the ratio 6:2. How much does each receive?", "output": "Total ratio parts = 6+2 = 8.\nEach part = 624 ÷ 8 = 78.\nFirst share = 6 × 78 = 468.\nSecond share = 2 × 78 = 156.\nCheck: 468+156=624.\nFinal answer: $156$", "final_answer": "156", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "69847b313d8dcea4", "problem_id": "b704e44cd2b09098", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "How many binary strings of length 17 contain exactly 3 ones with no two ones adjacent?", "output": "1) Reserve 2 mandatory zeros between the 3 ones.\n2) This leaves n−k+1 = 15 valid slots for the ones.\n3) Choose the slots: C(15,3) = 455.\nFinal answer: $455$", "final_answer": "455", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "613cbeec64656f7f", "problem_id": "b704e44cd2b09098", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many binary strings of length 17 contain exactly 3 ones with no two ones adjacent?", "output": "Reserve 2 mandatory zeros between the 3 ones.\nThis leaves n−k+1 = 15 valid slots for the ones.\nChoose the slots: C(15,3) = 455.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $455$", "final_answer": "455", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b279b9e844cd6c1b", "problem_id": "c7bb301094593b82", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many binary strings of length 24 contain exactly 4 ones with no two ones adjacent?", "output": "1) Reserve 3 mandatory zeros between the 4 ones.\n2) This leaves n−k+1 = 21 valid slots for the ones.\n3) Choose the slots: C(21,4) = 5985.\nFinal answer: $5985$", "final_answer": "5985", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eab00250b2d3ffd8", "problem_id": "c7bb301094593b82", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "How many binary strings of length 24 contain exactly 4 ones with no two ones adjacent?", "output": "Reserve 3 mandatory zeros between the 4 ones.\nThis leaves n−k+1 = 21 valid slots for the ones.\nChoose the slots: C(21,4) = 5985.\nFinal answer: $5985$", "final_answer": "5985", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "89825669d81b5525", "problem_id": "9da88beb2217dd7d", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 29 red balls and 11 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(40,11).\nFavorable samples: choose k reds and n−k blues: C(29,1)·C(11,10).\nCompute totals: C(40,11) = 2311801440, and favorable = 319.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{319}{2311801440}$.\nFinal answer: $\\frac{319}{2311801440}$", "final_answer": "\\frac{319}{2311801440}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a07ddd80ac479146", "problem_id": "9da88beb2217dd7d", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "An urn has 29 red balls and 11 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(40,11).\n3) Favorable samples: choose k reds and n−k blues: C(29,1)·C(11,10).\n4) Compute totals: C(40,11) = 2311801440, and favorable = 319.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{319}{2311801440}$", "final_answer": "\\frac{319}{2311801440}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bc487c0644a84dcb", "problem_id": "14cfd12df0d5dfb0", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $792 in the ratio 7:1. How much does each receive?", "output": "1) Total ratio parts = 7+1 = 8.\n2) Each part = 792 ÷ 8 = 99.\n3) First share = 7 × 99 = 693.\n4) Second share = 1 × 99 = 99.\nFinal answer: $99$", "final_answer": "99", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "476a63fc302e105b", "problem_id": "1bea1c540a0aa966", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(- 2 x^{3} + x^{2} - x - 3\\right) \\cos{\\left(x + 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x^{2} \\cos{\\left(x + 2 \\right)}) = - x^{2} \\sin{\\left(x + 2 \\right)}$.\n3) So $f'(x) = 2 x^{3} \\sin{\\left(x + 2 \\right)} - x^{2} \\sin{\\left(x + 2 \\right)} - 6 x^{2} \\cos{\\left(x + 2 \\right)} + x e^{x} + x \\sin{\\left(x + 2 \\right)} + 2 e^{x} + 3 \\sin{\\left(x + 2 \\right)} - \\cos{\\left(x + 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x^{2} \\cos{\\left(x + 2 \\right)}) = - x^{2} \\sin{\\left(x + 2 \\right)} + 2 x \\cos{\\left(x + 2 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 1\\right) e^{x} - \\left(x \\left(x \\left(1 - 2 x\\right) - 1\\right) - 3\\right) \\sin{\\left(x + 2 \\right)} + \\left(- 6 x^{2} + 2 x - 1\\right) \\cos{\\left(x + 2 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 1\\right) e^{x} - \\left(x \\left(x \\left(1 - 2 x\\right) - 1\\right) - 3\\right) \\sin{\\left(x + 2 \\right)} + \\left(- 6 x^{2} + 2 x - 1\\right) \\cos{\\left(x + 2 \\right)} + e^{x}$", "final_answer": "\\left(x + 1\\right) e^{x} - \\left(x \\left(x \\left(1 - 2 x\\right) - 1\\right) - 3\\right) \\sin{\\left(x + 2 \\right)} + \\left(- 6 x^{2} + 2 x - 1\\right) \\cos{\\left(x + 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3f6741b652f56d1b", "problem_id": "731291ed7828f8a8", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x + 4$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $2 x^{2} + 3$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $4 x$.\n5) Evaluate at a: p'(2) = $8$.\nFinal answer: $8$", "final_answer": "8", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7e3fe8d06b911e4a", "problem_id": "731291ed7828f8a8", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x + 4$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{2} + 3$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $4 x$.\nEvaluate at a: p'(2) = $8$.\nCheck: the identity gives $8$.\nFinal answer: $8$", "final_answer": "8", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "79868f6de568c5ee", "problem_id": "d425f2e063466925", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 2 x^{3} - 3 x^{2} + 8 x - 23$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $2 x^{4} + 3 x^{3} - x^{2} + x + 3$ and a = -3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $8 x^{3} + 9 x^{2} - 2 x + 1$.\n5) Evaluate at a: p'(-3) = $-128$.\nFinal answer: $-128$", "final_answer": "-128", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "66ae4a2a92765b50", "problem_id": "d425f2e063466925", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 2 x^{3} - 3 x^{2} + 8 x - 23$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{4} + 3 x^{3} - x^{2} + x + 3$ and a = -3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $8 x^{3} + 9 x^{2} - 2 x + 1$.\nEvaluate at a: p'(-3) = $-128$.\nCheck: the identity gives $-128$.\nFinal answer: $-128$", "final_answer": "-128", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "81bafecdbfca293b", "problem_id": "9ac51afd01182727", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Worker A finishes a job in 22 hours and Worker B finishes it in 19 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "$\\frac{418}{41}$", "final_answer": "\\frac{418}{41}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e5f64eec8ae0193b", "problem_id": "9ac51afd01182727", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Worker A finishes a job in 22 hours and Worker B finishes it in 19 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/22 job/hour. B's rate = 1/19 job/hour.\n2) Combined rate = 1/22 + 1/19 = 41/418 job/hour.\n3) Time = 1 ÷ (combined rate) = 418/41 hours.\nFinal answer: $\\frac{418}{41}$", "final_answer": "\\frac{418}{41}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "66feb0aca1aa1e46", "problem_id": "b36b8c88d7868394", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 6 x - 27$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} - 6 x - 27$ = $\\left(x - 9\\right) \\left(x + 3\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $9$.\n5) From the second factor: x = $-3$.\nFinal answer: $x \\in \\{9, -3\\}$", "final_answer": "x \\in \\{9, -3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d75ed60911494b8d", "problem_id": "b36b8c88d7868394", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 6 x - 27$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} - 6 x - 27$ = $\\left(x - 9\\right) \\left(x + 3\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $9$.\nFrom the second factor: x = $-3$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{9, -3\\}$", "final_answer": "x \\in \\{9, -3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ee7387ab639d7402", "problem_id": "8d82a0fc214b8c8a", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} - x^{4} + 2 x^{3} - 3 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{4} + 2 x^{3} - 3 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\pi).\n3) Subtract: F(\\pi) − F(- \\pi) = $\\frac{2 \\pi^{3} \\left(- \\pi^{2} - 5\\right)}{5}$.\n4) Simplify the result.\nFinal answer: $\\frac{2\\cdot \\pi^{3}\\cdot \\left(- \\pi^{2} - 5\\right)}{5}$", "final_answer": "\\frac{2\\cdot \\pi^{3}\\cdot \\left(- \\pi^{2} - 5\\right)}{5}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "18c39939b704c596", "problem_id": "8d82a0fc214b8c8a", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} - x^{4} + 2 x^{3} - 3 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{4} + 2 x^{3} - 3 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nEvaluate at the bounds: F(\\pi) and F(- \\pi).\nSubtract: F(\\pi) − F(- \\pi) = $\\frac{2 \\pi^{3} \\left(- \\pi^{2} - 5\\right)}{5}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{2\\cdot \\pi^{3}\\cdot \\left(- \\pi^{2} - 5\\right)}{5}$", "final_answer": "\\frac{2\\cdot \\pi^{3}\\cdot \\left(- \\pi^{2} - 5\\right)}{5}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a45d63473c36776c", "problem_id": "46f9e5f88447c37d", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 14 times, what is the probability of exactly 3 heads?", "output": "$\\frac{20312500000}{96889010407}$", "final_answer": "\\frac{20312500000}{96889010407}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "81bbfd93b554d452", "problem_id": "46f9e5f88447c37d", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 14 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=14, k=3, p=$\\frac{2}{7}$ and simplify.\n4) Here C(14,3) = 364.\nFinal answer: $\\frac{20312500000}{96889010407}$", "final_answer": "\\frac{20312500000}{96889010407}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "da6ab5c873a559e9", "problem_id": "53e2d97f7f5d58dd", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{4} + x^{3} - 2 x^{2} + 4 x + 3$.", "output": "1) Differentiate term-by-term:\n2) d/dx(3) = 0.\n3) d/dx(x^{3}) = 3 x^{2}.\n4) d/dx(- 2 x^{2}) = - 4 x.\n5) d/dx(3 x^{4}) = 12 x^{3}.\n6) d/dx(4 x) = 4.\n7) Combine: f'(x) = 12 x^{3} + 3 x^{2} - 4 x + 4.\nFinal answer: $12\\cdot x^{3} + 3\\cdot x^{2} - 4\\cdot x + 4$", "final_answer": "12\\cdot x^{3} + 3\\cdot x^{2} - 4\\cdot x + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e39a5bda9442751c", "problem_id": "53e2d97f7f5d58dd", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{4} + x^{3} - 2 x^{2} + 4 x + 3$.", "output": "Differentiate term-by-term:\nd/dx(3) = 0.\nd/dx(x^{3}) = 3 x^{2}.\nFinal answer: $12\\cdot x^{3} + 3\\cdot x^{2} - 4\\cdot x + 4$", "final_answer": "12\\cdot x^{3} + 3\\cdot x^{2} - 4\\cdot x + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6b03cee1274354f1", "problem_id": "05190806a6b1cd79", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + \\frac{36 x}{5} + \\frac{21}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$\\frac{36}{5}$, c=$\\frac{21}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{36}{25}$, so √Δ = $\\frac{6}{5}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $-1$.\n7) x2 = $- \\frac{7}{5}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{-1, - \\frac{7}{5}\\}$", "final_answer": "x \\in \\{-1, - \\frac{7}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "af5c41fc28413b5d", "problem_id": "05190806a6b1cd79", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + \\frac{36 x}{5} + \\frac{21}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$\\frac{36}{5}$, c=$\\frac{21}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{36}{25}$, so √Δ = $\\frac{6}{5}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $-1$.\nx2 = $- \\frac{7}{5}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-1, - \\frac{7}{5}\\}$", "final_answer": "x \\in \\{-1, - \\frac{7}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2c1a1adaf5e07551", "problem_id": "10871827149b2f65", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int \\left(2 x + 1\\right) \\cos{\\left(x \\right)} + \\left(2 x + 3\\right) e^{2 x}\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $\\left(2 x + 1\\right) \\cos{\\left(x \\right)} + \\left(2 x + 3\\right) e^{2 x}$.\n3) Compute an antiderivative: F(x) = $2 x \\sin{\\left(x \\right)} + \\left(x + 1\\right) e^{2 x} + \\sin{\\left(x \\right)} + 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\left(x + 1\\right)\\cdot e^{2\\cdot x} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\left(x + 1\\right)\\cdot e^{2\\cdot x} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7977d4a7d332a382", "problem_id": "a4898b82e7b4d45a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(- 2 x^{3} - 2 x + 3\\right) \\cos{\\left(2 x + 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 \\cos{\\left(2 x + 1 \\right)}) = - 3 \\sin{\\left(2 x + 1 \\right)}$.\n3) So $f'(x) = 4 x^{3} \\sin{\\left(2 x + 1 \\right)} - 6 x^{2} \\cos{\\left(2 x + 1 \\right)} + 2 x e^{x} + 4 x \\sin{\\left(2 x + 1 \\right)} + 6 e^{x} - 3 \\sin{\\left(2 x + 1 \\right)} - 2 \\cos{\\left(2 x + 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 \\cos{\\left(2 x + 1 \\right)}) = - 6 \\sin{\\left(2 x + 1 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(\\left(x + 2\\right) e^{x} + \\left(- 3 x^{2} - 1\\right) \\cos{\\left(2 x + 1 \\right)} - \\left(2 x \\left(- x^{2} - 1\\right) + 3\\right) \\sin{\\left(2 x + 1 \\right)} + e^{x}\\right)$.\nFinal answer: $2 \\left(\\left(x + 2\\right) e^{x} + \\left(- 3 x^{2} - 1\\right) \\cos{\\left(2 x + 1 \\right)} - \\left(2 x \\left(- x^{2} - 1\\right) + 3\\right) \\sin{\\left(2 x + 1 \\right)} + e^{x}\\right)$", "final_answer": "2 \\left(\\left(x + 2\\right) e^{x} + \\left(- 3 x^{2} - 1\\right) \\cos{\\left(2 x + 1 \\right)} - \\left(2 x \\left(- x^{2} - 1\\right) + 3\\right) \\sin{\\left(2 x + 1 \\right)} + e^{x}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "67221dccbfa3f98a", "problem_id": "4b8ede2c06626f86", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "An urn has 16 red balls and 18 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(34,11).\n3) Favorable samples: choose k reds and n−k blues: C(16,5)·C(18,6).\n4) Compute totals: C(34,11) = 286097760, and favorable = 81087552.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1274}{4495}$", "final_answer": "\\frac{1274}{4495}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4514169482c73294", "problem_id": "4b8ede2c06626f86", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 16 red balls and 18 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(34,11).\nFavorable samples: choose k reds and n−k blues: C(16,5)·C(18,6).\nCompute totals: C(34,11) = 286097760, and favorable = 81087552.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1274}{4495}$.\nFinal answer: $\\frac{1274}{4495}$", "final_answer": "\\frac{1274}{4495}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "91fccd95faebde27", "problem_id": "5f59fcb81066bf31", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - \\frac{x}{2} - 6$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$- \\frac{1}{2}$, c=$-6$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{289}{4}$, so √Δ = $\\frac{17}{2}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{3}{2}$.\n7) x2 = $- \\frac{4}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{3}{2}, - \\frac{4}{3}\\}$", "final_answer": "x \\in \\{\\frac{3}{2}, - \\frac{4}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "65d89cc7fa7324be", "problem_id": "5f59fcb81066bf31", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - \\frac{x}{2} - 6$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$- \\frac{1}{2}$, c=$-6$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{289}{4}$, so √Δ = $\\frac{17}{2}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{3}{2}$.\nx2 = $- \\frac{4}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{3}{2}, - \\frac{4}{3}\\}$", "final_answer": "x \\in \\{\\frac{3}{2}, - \\frac{4}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "475cc78c267d0d72", "problem_id": "9d4cd9c05c99030c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x^{3} + 4 x^{2} + 7 x + 13$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to x^{3} + 4 x^{2} + 6 x + 13.\n3) Then the limit is 49.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 51.\nFinal answer: $51$", "final_answer": "51", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "602afd9fd5425cbd", "problem_id": "56ad6eb37304924e", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $- 2 x^{2} \\sin{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - 3\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - 3\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e7e517f9ab6b0da9", "problem_id": "1a77ef5ac64de3da", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$- 3 x + 8 y$ = $69$\n$- 2 x - 11 y$ = $-52$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $6 x - 16 y$ = $-138$.\n3) Multiply the second equation by $-3$: $6 x + 33 y$ = $156$.\n4) Subtract to eliminate x: $- 49 y$ = $-294$.\n5) Solve for y: y = $6$.\n6) Substitute back to find x: x = $-7$.\nFinal answer: $(x,y)=\\left(-7, 6\\right)$", "final_answer": "(x,y)=\\left(-7, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d9eee6978f635459", "problem_id": "1a77ef5ac64de3da", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Solve the linear system:\n$- 3 x + 8 y$ = $69$\n$- 2 x - 11 y$ = $-52$", "output": "$(x,y)=\\left(-7, 6\\right)$", "final_answer": "(x,y)=\\left(-7, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1a291187768940bf", "problem_id": "42206942bab5f088", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A store sells a backpack for $490. It is discounted by 15%. You buy 5 backpacks. What is the total cost?", "output": "1) Discounted price = 490 × (1 − 15/100) = 490 × 85/100 = 833/2.\n2) Subtotal = 5 × 833/2 = 4165/2.\nFinal answer: $\\frac{4165}{2}$", "final_answer": "\\frac{4165}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c655f627c5cbc29a", "problem_id": "42206942bab5f088", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A store sells a backpack for $490. It is discounted by 15%. You buy 5 backpacks. What is the total cost?", "output": "Discounted price = 490 × (1 − 15/100) = 490 × 85/100 = 833/2.\nSubtotal = 5 × 833/2 = 4165/2.\nFinal answer: $\\frac{4165}{2}$", "final_answer": "\\frac{4165}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4a0227c19dea6e31", "problem_id": "0a37779126447415", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 20 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=20, k=8, p=$\\frac{2}{3}$ and simplify.\nHere C(20,8) = 125970.\nCheck: expression simplifies to $\\frac{10749440}{1162261467}$.\nFinal answer: $\\frac{10749440}{1162261467}$", "final_answer": "\\frac{10749440}{1162261467}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e154b18a07c94c3", "problem_id": "0a37779126447415", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 20 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=20, k=8, p=$\\frac{2}{3}$ and simplify.\nFinal answer: $\\frac{10749440}{1162261467}$", "final_answer": "\\frac{10749440}{1162261467}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ccf4a3f16e1567ea", "problem_id": "83a3b0f783dfd312", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 4 x^{3} + 3 x^{2} - 4 x - 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(3 x^{2}) = 6 x^{2}$.\n3) So $f'(x) = - 6 x^{2} - 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(3 x^{2}) = 6 x$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- 6 x^{2} + 3 x - 2\\right)$.\nFinal answer: $2 \\left(- 6 x^{2} + 3 x - 2\\right)$", "final_answer": "2 \\left(- 6 x^{2} + 3 x - 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d0261690cf8ab883", "problem_id": "f3d7d9357bcb6775", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Two friends split $901 in the ratio 8:9. How much does each receive?", "output": "Total ratio parts = 8+9 = 17.\nEach part = 901 ÷ 17 = 53.\nFirst share = 8 × 53 = 424.\nSecond share = 9 × 53 = 477.\nCheck: 424+477=901.\nFinal answer: $477$", "final_answer": "477", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "131ad5b6d776ea09", "problem_id": "1ac3439e75f7ba4f", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$7 x - 8 y$ = $-32$\n$- 6 x - 5 y$ = $63$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-6$: $- 42 x + 48 y$ = $192$.\nMultiply the second equation by $7$: $- 42 x - 35 y$ = $441$.\nSubtract to eliminate x: $83 y$ = $-249$.\nSolve for y: y = $-3$.\nSubstitute back to find x: x = $-8$.\nCheck: plug (x,y)=(-8,-3) into both equations.\nFinal answer: $(x,y)=\\left(-8, -3\\right)$", "final_answer": "(x,y)=\\left(-8, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d3f9489d51e8d6eb", "problem_id": "1ac3439e75f7ba4f", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$7 x - 8 y$ = $-32$\n$- 6 x - 5 y$ = $63$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-6$: $- 42 x + 48 y$ = $192$.\n3) Multiply the second equation by $7$: $- 42 x - 35 y$ = $441$.\n4) Subtract to eliminate x: $83 y$ = $-249$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $-8$.\nFinal answer: $(x,y)=\\left(-8, -3\\right)$", "final_answer": "(x,y)=\\left(-8, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "44f2a51b1fa556cb", "problem_id": "1e5973de65beda58", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the multiplicative inverse of 4 modulo 13.", "output": "We want x such that 4x ≡ 1 (mod 13).\nUse extended Euclid to find s,t with 4s + 13t = 1.\nOne Bezout identity is: 4·(-3) + 13·(1) = 1.\nReduce s modulo 13: x ≡ 10 (mod 13).\nCheck: 4·10 mod 13 = 1.\nFinal answer: $x\\equiv 10\\pmod{13}$", "final_answer": "x\\equiv 10\\pmod{13}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bd73828b8b308686", "problem_id": "1e5973de65beda58", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the multiplicative inverse of 4 modulo 13.", "output": "1) We want x such that 4x ≡ 1 (mod 13).\n2) Use extended Euclid to find s,t with 4s + 13t = 1.\n3) One Bezout identity is: 4·(-3) + 13·(1) = 1.\n4) Reduce s modulo 13: x ≡ 10 (mod 13).\nFinal answer: $x\\equiv 10\\pmod{13}$", "final_answer": "x\\equiv 10\\pmod{13}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1fbd896220c5eae9", "problem_id": "dcabcacd885a0374", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$- 8 x + 3 y$ = $77$\n$7 x + 4 y$ = $-21$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $7$: $- 56 x + 21 y$ = $539$.\n3) Multiply the second equation by $-8$: $- 56 x - 32 y$ = $168$.\n4) Subtract to eliminate x: $53 y$ = $371$.\n5) Solve for y: y = $7$.\n6) Substitute back to find x: x = $-7$.\nFinal answer: $(x,y)=\\left(-7, 7\\right)$", "final_answer": "(x,y)=\\left(-7, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d7b9350712c4729f", "problem_id": "dcabcacd885a0374", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- 8 x + 3 y$ = $77$\n$7 x + 4 y$ = $-21$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $7$: $- 56 x + 21 y$ = $539$.\nMultiply the second equation by $-8$: $- 56 x - 32 y$ = $168$.\nSubtract to eliminate x: $53 y$ = $371$.\nSolve for y: y = $7$.\nSubstitute back to find x: x = $-7$.\nCheck: plug (x,y)=(-7,7) into both equations.\nFinal answer: $(x,y)=\\left(-7, 7\\right)$", "final_answer": "(x,y)=\\left(-7, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "017854fce77cb7e5", "problem_id": "96464306420da28b", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$10 x^{2} - 5 \\sqrt{5} x + 18 x - 9 \\sqrt{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$10$, b=$18 - 5 \\sqrt{5}$, c=$- 9 \\sqrt{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $180 \\sqrt{5} + 449$, so √Δ = $\\sqrt{180 \\sqrt{5} + 449}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{9}{10} + \\frac{\\sqrt{5}}{4} + \\frac{\\sqrt{180 \\sqrt{5} + 449}}{20}$.\nx2 = $- \\frac{\\sqrt{180 \\sqrt{5} + 449}}{20} - \\frac{9}{10} + \\frac{\\sqrt{5}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{9}{10} + \\frac{\\sqrt{5}}{4} + \\frac{\\sqrt{180\\cdot \\sqrt{5} + 449}}{20}, - \\frac{\\sqrt{180\\cdot \\sqrt{5} + 449}}{20} - \\frac{9}{10} + \\frac{\\sqrt{5}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{9}{10} + \\frac{\\sqrt{5}}{4} + \\frac{\\sqrt{180\\cdot \\sqrt{5} + 449}}{20}, - \\frac{\\sqrt{180\\cdot \\sqrt{5} + 449}}{20} - \\frac{9}{10} + \\frac{\\sqrt{5}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f844a9efd6afaf97", "problem_id": "96464306420da28b", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$10 x^{2} - 5 \\sqrt{5} x + 18 x - 9 \\sqrt{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$10$, b=$18 - 5 \\sqrt{5}$, c=$- 9 \\sqrt{5}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{- \\frac{9}{10} + \\frac{\\sqrt{5}}{4} + \\frac{\\sqrt{180\\cdot \\sqrt{5} + 449}}{20}, - \\frac{\\sqrt{180\\cdot \\sqrt{5} + 449}}{20} - \\frac{9}{10} + \\frac{\\sqrt{5}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{9}{10} + \\frac{\\sqrt{5}}{4} + \\frac{\\sqrt{180\\cdot \\sqrt{5} + 449}}{20}, - \\frac{\\sqrt{180\\cdot \\sqrt{5} + 449}}{20} - \\frac{9}{10} + \\frac{\\sqrt{5}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ac7f90297923c4bf", "problem_id": "3b64fdac755f4e5e", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_6=32$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_6=20.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=20 and k=6.\n4) Compute C(25,5) = 53130.\nFinal answer: $53130$", "final_answer": "53130", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8eaf8935d675f8dd", "problem_id": "3b64fdac755f4e5e", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_6=32$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_6=20.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=20 and k=6.\nCompute C(25,5) = 53130.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $53130$", "final_answer": "53130", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9de457133f059b48", "problem_id": "2593213492596b00", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 13 red balls and 10 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(23,5).\nFavorable samples: choose k reds and n−k blues: C(13,1)·C(10,4).\nCompute totals: C(23,5) = 33649, and favorable = 2730.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{390}{4807}$.\nFinal answer: $\\frac{390}{4807}$", "final_answer": "\\frac{390}{4807}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "124f8c96f3e1ee02", "problem_id": "2593213492596b00", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "An urn has 13 red balls and 10 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(23,5).\n3) Favorable samples: choose k reds and n−k blues: C(13,1)·C(10,4).\n4) Compute totals: C(23,5) = 33649, and favorable = 2730.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{390}{4807}$", "final_answer": "\\frac{390}{4807}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4d3158e49546dd0e", "problem_id": "135d7ab4d032896a", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Worker A finishes a job in 10 hours and Worker B finishes it in 16 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/10 job/hour. B's rate = 1/16 job/hour.\nCombined rate = 1/10 + 1/16 = 13/80 job/hour.\nTime = 1 ÷ (combined rate) = 80/13 hours.\nFinal answer: $\\frac{80}{13}$", "final_answer": "\\frac{80}{13}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "280e6e0907f51954", "problem_id": "135d7ab4d032896a", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Worker A finishes a job in 10 hours and Worker B finishes it in 16 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/10 job/hour. B's rate = 1/16 job/hour.\n2) Combined rate = 1/10 + 1/16 = 13/80 job/hour.\n3) Time = 1 ÷ (combined rate) = 80/13 hours.\nFinal answer: $\\frac{80}{13}$", "final_answer": "\\frac{80}{13}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c249e7a8c3542764", "problem_id": "28221eb396be6a59", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡0 (mod 9), x≡1 (mod 13). Give the solution modulo 117.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 117.\n3) Compute the CRT solution to get x ≡ 27 (mod 117).\nFinal answer: $x\\equiv 27\\pmod{117}$", "final_answer": "x\\equiv 27\\pmod{117}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5624b0a61960a69b", "problem_id": "28221eb396be6a59", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Solve the congruences: x≡0 (mod 9), x≡1 (mod 13). Give the solution modulo 117.", "output": "$x\\equiv 27\\pmod{117}$", "final_answer": "x\\equiv 27\\pmod{117}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eeef86826ca21dde", "problem_id": "f0f142984a0a177e", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$- x$ = $3$\n$- 2 x + 3 y$ = $12$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $2 x$ = $-6$.\n3) Multiply the second equation by $-1$: $2 x - 3 y$ = $-12$.\n4) Subtract to eliminate x: $3 y$ = $6$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, 2\\right)$", "final_answer": "(x,y)=\\left(-3, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "05f7e5681479855b", "problem_id": "f0f142984a0a177e", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the linear system:\n$- x$ = $3$\n$- 2 x + 3 y$ = $12$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $2 x$ = $-6$.\nMultiply the second equation by $-1$: $2 x - 3 y$ = $-12$.\nFinal answer: $(x,y)=\\left(-3, 2\\right)$", "final_answer": "(x,y)=\\left(-3, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ba21ed71685809c8", "problem_id": "cc19d6803382c812", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int \\left(2 x + 1\\right) e^{2 x} + \\left(2 x + 1\\right) \\cos{\\left(x \\right)}\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $\\left(2 x + 1\\right) e^{2 x} + \\left(2 x + 1\\right) \\cos{\\left(x \\right)}$.\nCompute an antiderivative: F(x) = $x e^{2 x} + 2 x \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $C + x\\cdot e^{2\\cdot x} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + x\\cdot e^{2\\cdot x} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0b2929502164c088", "problem_id": "cc19d6803382c812", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int \\left(2 x + 1\\right) e^{2 x} + \\left(2 x + 1\\right) \\cos{\\left(x \\right)}\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $\\left(2 x + 1\\right) e^{2 x} + \\left(2 x + 1\\right) \\cos{\\left(x \\right)}$.\n3) Compute an antiderivative: F(x) = $x e^{2 x} + 2 x \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + x\\cdot e^{2\\cdot x} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + x\\cdot e^{2\\cdot x} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a9f0e738bc63fd7e", "problem_id": "7b9354e4d818564f", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the linear system:\n$8 x + 3 y$ = $-43$\n$7 x$ = $-56$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $7$: $56 x + 21 y$ = $-301$.\nMultiply the second equation by $8$: $56 x$ = $-448$.\nFinal answer: $(x,y)=\\left(-8, 7\\right)$", "final_answer": "(x,y)=\\left(-8, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8ae7e965d13cb22e", "problem_id": "7b9354e4d818564f", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$8 x + 3 y$ = $-43$\n$7 x$ = $-56$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $7$: $56 x + 21 y$ = $-301$.\n3) Multiply the second equation by $8$: $56 x$ = $-448$.\n4) Subtract to eliminate x: $21 y$ = $147$.\n5) Solve for y: y = $7$.\n6) Substitute back to find x: x = $-8$.\nFinal answer: $(x,y)=\\left(-8, 7\\right)$", "final_answer": "(x,y)=\\left(-8, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1febcfcb15f4bc77", "problem_id": "52119ff68cb17c7e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(- 3 x^{3} + x^{2} + 2 x - 2\\right) \\cos{\\left(x - 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 3 x^{3} \\cos{\\left(x - 3 \\right)}) = 3 x^{3} \\sin{\\left(x - 3 \\right)}$.\n3) So $f'(x) = 3 x^{3} \\sin{\\left(x - 3 \\right)} - x^{2} \\sin{\\left(x - 3 \\right)} + x e^{x} - 2 x \\sin{\\left(x - 3 \\right)} + 2 x \\cos{\\left(x - 3 \\right)} + 3 e^{x} + 2 \\sin{\\left(x - 3 \\right)} + 2 \\cos{\\left(x - 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 3 x^{3} \\cos{\\left(x - 3 \\right)}) = 3 x^{3} \\sin{\\left(x - 3 \\right)} - 9 x^{2} \\cos{\\left(x - 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 2\\right) e^{x} - \\left(x \\left(x \\left(1 - 3 x\\right) + 2\\right) - 2\\right) \\sin{\\left(x - 3 \\right)} + \\left(- 9 x^{2} + 2 x + 2\\right) \\cos{\\left(x - 3 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 2\\right) e^{x} - \\left(x \\left(x \\left(1 - 3 x\\right) + 2\\right) - 2\\right) \\sin{\\left(x - 3 \\right)} + \\left(- 9 x^{2} + 2 x + 2\\right) \\cos{\\left(x - 3 \\right)} + e^{x}$", "final_answer": "\\left(x + 2\\right) e^{x} - \\left(x \\left(x \\left(1 - 3 x\\right) + 2\\right) - 2\\right) \\sin{\\left(x - 3 \\right)} + \\left(- 9 x^{2} + 2 x + 2\\right) \\cos{\\left(x - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "89883c08271bc84d", "problem_id": "4377d7c79ac41493", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A cyclist rides at 73 miles per hour for 1/2 hours, then at 64 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 73 × 1/2 = 73/2 miles.\nSecond leg distance = 64 × 2 = 128 miles.\nTotal distance = 73/2 + 128 = 329/2 miles.\nCheck: adding both legs gives 329/2.\nFinal answer: $\\frac{329}{2}$", "final_answer": "\\frac{329}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e1708093f02d31ed", "problem_id": "35e10b4b38ad0d8f", "style": "answer_only", "instruction": "Final answer only.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{3} - 2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "$C - \\frac{3\\cdot x^{4}}{4} - \\frac{2\\cdot x^{3}}{3} + x^{2} + x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{3\\cdot x^{4}}{4} - \\frac{2\\cdot x^{3}}{3} + x^{2} + x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "52a1e963f343586a", "problem_id": "35e10b4b38ad0d8f", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{3} - 2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 3 x^{3} - 2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 1$.\nCompute an antiderivative: F(x) = $- \\frac{3 x^{4}}{4} - \\frac{2 x^{3}}{3} + x^{2} + x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $C - \\frac{3\\cdot x^{4}}{4} - \\frac{2\\cdot x^{3}}{3} + x^{2} + x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{3\\cdot x^{4}}{4} - \\frac{2\\cdot x^{3}}{3} + x^{2} + x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d467dd64d51802a", "problem_id": "77059cff0e0e9a9a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} x^{2} + 2$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to x^{2} - x + 2.\n3) Then the limit is 2.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: 3.\nFinal answer: $3$", "final_answer": "3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "9f9886ad2370aae2", "problem_id": "6d47e8d1d7158db0", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A cyclist rides at 64 miles per hour for 3/2 hours, then at 46 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 64 × 3/2 = 96 miles.\n2) Second leg distance = 46 × 3 = 138 miles.\n3) Total distance = 96 + 138 = 234 miles.\nFinal answer: $234$", "final_answer": "234", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d63715bbff48f8e6", "problem_id": "2f8d8b07695f4e0f", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 14 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=14, k=5, p=$\\frac{1}{5}$ and simplify.\n4) Here C(14,5) = 2002.\nFinal answer: $\\frac{524812288}{6103515625}$", "final_answer": "\\frac{524812288}{6103515625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "447f92ed55db83f6", "problem_id": "2f8d8b07695f4e0f", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 14 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=14, k=5, p=$\\frac{1}{5}$ and simplify.\nHere C(14,5) = 2002.\nCheck: expression simplifies to $\\frac{524812288}{6103515625}$.\nFinal answer: $\\frac{524812288}{6103515625}$", "final_answer": "\\frac{524812288}{6103515625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "321563dc8ef350eb", "problem_id": "68de494a4b261d7d", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 13 times, what is the probability of exactly 7 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=13, k=7, p=$\\frac{3}{8}$ and simplify.\nHere C(13,7) = 1716.\nCheck: expression simplifies to $\\frac{14659734375}{137438953472}$.\nFinal answer: $\\frac{14659734375}{137438953472}$", "final_answer": "\\frac{14659734375}{137438953472}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f32fcbcfac791aac", "problem_id": "68de494a4b261d7d", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 13 times, what is the probability of exactly 7 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=13, k=7, p=$\\frac{3}{8}$ and simplify.\n4) Here C(13,7) = 1716.\nFinal answer: $\\frac{14659734375}{137438953472}$", "final_answer": "\\frac{14659734375}{137438953472}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6e9f95ce01a26231", "problem_id": "0360536a8d694ff9", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "You want 22 liters of a 25% salt solution. You have a 10% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 22−x liters is 70% solution.\n2) Salt equation: 10x + 70(22−x) = 25·22.\n3) (10-70)x = (25-70)·22 ⇒ x = 33/2.\n4) Then 22−x = 11/2.\nFinal answer: $33/2 L of 10% and 11/2 L of 70%$", "final_answer": "33/2 L of 10% and 11/2 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1f588e30d5c9a267", "problem_id": "0360536a8d694ff9", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "You want 22 liters of a 25% salt solution. You have a 10% solution and a 70% solution. How many liters of each should you mix?", "output": "33/2 L of 10% and 11/2 L of 70%", "final_answer": "33/2 L of 10% and 11/2 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e83ade3e0ddbd3f8", "problem_id": "d485cfd828d2b148", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A cyclist rides at 62 miles per hour for 2 hours, then at 39 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 62 × 2 = 124 miles.\n2) Second leg distance = 39 × 2 = 78 miles.\n3) Total distance = 124 + 78 = 202 miles.\nFinal answer: $202$", "final_answer": "202", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d1a67313eaff8cf", "problem_id": "d485cfd828d2b148", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A cyclist rides at 62 miles per hour for 2 hours, then at 39 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 62 × 2 = 124 miles.\nSecond leg distance = 39 × 2 = 78 miles.\nTotal distance = 124 + 78 = 202 miles.\nCheck: adding both legs gives 202.\nFinal answer: $202$", "final_answer": "202", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bef4c6b89519ed74", "problem_id": "accb830cfaf893ae", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the multiplicative inverse of 5 modulo 13.", "output": "1) We want x such that 5x ≡ 1 (mod 13).\n2) Use extended Euclid to find s,t with 5s + 13t = 1.\n3) One Bezout identity is: 5·(-5) + 13·(2) = 1.\n4) Reduce s modulo 13: x ≡ 8 (mod 13).\nFinal answer: $x\\equiv 8\\pmod{13}$", "final_answer": "x\\equiv 8\\pmod{13}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e76034cd9c57fae4", "problem_id": "accb830cfaf893ae", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the multiplicative inverse of 5 modulo 13.", "output": "We want x such that 5x ≡ 1 (mod 13).\nUse extended Euclid to find s,t with 5s + 13t = 1.\nOne Bezout identity is: 5·(-5) + 13·(2) = 1.\nReduce s modulo 13: x ≡ 8 (mod 13).\nCheck: 5·8 mod 13 = 1.\nFinal answer: $x\\equiv 8\\pmod{13}$", "final_answer": "x\\equiv 8\\pmod{13}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ccfeff5f33d988e", "problem_id": "6bbf3d3938c041e5", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} x^{3} + x^{2} + 7 x + 25$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $x^{4} - 3 x^{3} + 3 x^{2} - 3 x - 2$ and a = 4.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $133$", "final_answer": "133", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8a052d668876d558", "problem_id": "6bbf3d3938c041e5", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} x^{3} + x^{2} + 7 x + 25$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{4} - 3 x^{3} + 3 x^{2} - 3 x - 2$ and a = 4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $4 x^{3} - 9 x^{2} + 6 x - 3$.\n5) Evaluate at a: p'(4) = $133$.\nFinal answer: $133$", "final_answer": "133", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1d88f8aab0ce907a", "problem_id": "de0e5b46ef7f5f12", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 8 red balls and 7 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(15,5).\n3) Favorable samples: choose k reds and n−k blues: C(8,2)·C(7,3).\n4) Compute totals: C(15,5) = 3003, and favorable = 980.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{140}{429}$", "final_answer": "\\frac{140}{429}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c00bf1ff706a0058", "problem_id": "de0e5b46ef7f5f12", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 8 red balls and 7 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(15,5).\nFavorable samples: choose k reds and n−k blues: C(8,2)·C(7,3).\nCompute totals: C(15,5) = 3003, and favorable = 980.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{140}{429}$.\nFinal answer: $\\frac{140}{429}$", "final_answer": "\\frac{140}{429}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f2f9afdac1e0d440", "problem_id": "f494e02783d9eb10", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Two friends split $1400 in the ratio 7:3. How much does each receive?", "output": "1) Total ratio parts = 7+3 = 10.\n2) Each part = 1400 ÷ 10 = 140.\n3) First share = 7 × 140 = 980.\n4) Second share = 3 × 140 = 420.\nFinal answer: $420$", "final_answer": "420", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c5bfc094c08d8812", "problem_id": "f494e02783d9eb10", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $1400 in the ratio 7:3. How much does each receive?", "output": "Total ratio parts = 7+3 = 10.\nEach part = 1400 ÷ 10 = 140.\nFirst share = 7 × 140 = 980.\nSecond share = 3 × 140 = 420.\nCheck: 980+420=1400.\nFinal answer: $420$", "final_answer": "420", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e97090a2ea6e31cd", "problem_id": "9e2fc5a48c97023f", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡2 (mod 3), x≡1 (mod 13). Give the solution modulo 39.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 39.\n3) Compute the CRT solution to get x ≡ 14 (mod 39).\nFinal answer: $x\\equiv 14\\pmod{39}$", "final_answer": "x\\equiv 14\\pmod{39}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9bf0f047f7cde07c", "problem_id": "9e2fc5a48c97023f", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 3), x≡1 (mod 13). Give the solution modulo 39.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 39.\nCompute the CRT solution to get x ≡ 14 (mod 39).\nCheck: 14 mod 3=2 and mod 13=1.\nFinal answer: $x\\equiv 14\\pmod{39}$", "final_answer": "x\\equiv 14\\pmod{39}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d0e0f50467251f26", "problem_id": "e9ea8694d5d5651d", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - x^{3} + 2 x^{2} - 3 x + 4$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{4} + x^{2} - 2 x - 3$ and a = -2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $26$", "final_answer": "26", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0de00ca57378acec", "problem_id": "e9ea8694d5d5651d", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - x^{3} + 2 x^{2} - 3 x + 4$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{4} + x^{2} - 2 x - 3$ and a = -2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 4 x^{3} + 2 x - 2$.\n5) Evaluate at a: p'(-2) = $26$.\nFinal answer: $26$", "final_answer": "26", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "132a27476b0cb323", "problem_id": "e8b4cfaf06219beb", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Find the multiplicative inverse of 8 modulo 17.", "output": "$x\\equiv 15\\pmod{17}$", "final_answer": "x\\equiv 15\\pmod{17}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1b43e9067d31050f", "problem_id": "e8b4cfaf06219beb", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the multiplicative inverse of 8 modulo 17.", "output": "1) We want x such that 8x ≡ 1 (mod 17).\n2) Use extended Euclid to find s,t with 8s + 17t = 1.\n3) One Bezout identity is: 8·(-2) + 17·(1) = 1.\n4) Reduce s modulo 17: x ≡ 15 (mod 17).\nFinal answer: $x\\equiv 15\\pmod{17}$", "final_answer": "x\\equiv 15\\pmod{17}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b13e098e5c04dc14", "problem_id": "2818d624c62634e4", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡3 (mod 9), x≡5 (mod 8). Give the solution modulo 72.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 72.\nCompute the CRT solution to get x ≡ 21 (mod 72).\nCheck: 21 mod 9=3 and mod 8=5.\nFinal answer: $x\\equiv 21\\pmod{72}$", "final_answer": "x\\equiv 21\\pmod{72}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9bfbc4fe13e0c42d", "problem_id": "2818d624c62634e4", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡3 (mod 9), x≡5 (mod 8). Give the solution modulo 72.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 72.\n3) Compute the CRT solution to get x ≡ 21 (mod 72).\nFinal answer: $x\\equiv 21\\pmod{72}$", "final_answer": "x\\equiv 21\\pmod{72}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "223379514a753e83", "problem_id": "ff6fb2f4132718dc", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Two friends split $513 in the ratio 4:5. How much does each receive?", "output": "$285$", "final_answer": "285", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8b04fabff8dc2b26", "problem_id": "ff6fb2f4132718dc", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Two friends split $513 in the ratio 4:5. How much does each receive?", "output": "1) Total ratio parts = 4+5 = 9.\n2) Each part = 513 ÷ 9 = 57.\n3) First share = 4 × 57 = 228.\n4) Second share = 5 × 57 = 285.\nFinal answer: $285$", "final_answer": "285", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c64728304b856ecb", "problem_id": "b06da15d51312a7d", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the equation of the tangent line to $y=3 x^{3} + 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 3$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $9 x^{2} + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 2$.\n2) Slope at x=-1: m = f'(-1) = $\\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)} + 11$.\n3) Point on curve: (-1, f(-1)) = (-1, $-2 + \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -2 + \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} = \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)} + 11(x − -1).\n6) Optionally expand: y = $\\frac{x}{2 e^{\\frac{1}{2}}} + 2 x \\sin{\\left(2 \\right)} + 11 x + \\cos{\\left(2 \\right)} + \\frac{3}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)} + 9$.\nFinal answer: $y = \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + 11\\cdot x + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)} + 9$", "final_answer": "y = \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + 11\\cdot x + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)} + 9", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e9edf903b1692f06", "problem_id": "15a878ae69b5201a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(3 x^{3} - 3 x^{2} + 3 x\\right) \\cos{\\left(3 x + 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 x^{3} \\cos{\\left(3 x + 2 \\right)}) = - 3 x^{3} \\sin{\\left(3 x + 2 \\right)}$.\n3) So $f'(x) = - 3 x^{3} \\sin{\\left(3 x + 2 \\right)} + 9 x^{2} \\sin{\\left(3 x + 2 \\right)} + 2 x e^{x} - 9 x \\sin{\\left(3 x + 2 \\right)} - 6 x \\cos{\\left(3 x + 2 \\right)} - 4 e^{x} + 3 \\cos{\\left(3 x + 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 x^{3} \\cos{\\left(3 x + 2 \\right)}) = - 9 x^{3} \\sin{\\left(3 x + 2 \\right)} + 9 x^{2} \\cos{\\left(3 x + 2 \\right)}$.\nTherefore the correct derivative is $f'(x) = - 9 x \\left(x \\left(x - 1\\right) + 1\\right) \\sin{\\left(3 x + 2 \\right)} + 2 \\left(x - 3\\right) e^{x} + 3 \\left(3 x^{2} - 2 x + 1\\right) \\cos{\\left(3 x + 2 \\right)} + 2 e^{x}$.\nFinal answer: $- 9 x \\left(x \\left(x - 1\\right) + 1\\right) \\sin{\\left(3 x + 2 \\right)} + 2 \\left(x - 3\\right) e^{x} + 3 \\left(3 x^{2} - 2 x + 1\\right) \\cos{\\left(3 x + 2 \\right)} + 2 e^{x}$", "final_answer": "- 9 x \\left(x \\left(x - 1\\right) + 1\\right) \\sin{\\left(3 x + 2 \\right)} + 2 \\left(x - 3\\right) e^{x} + 3 \\left(3 x^{2} - 2 x + 1\\right) \\cos{\\left(3 x + 2 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "43ea7caef10b3e7d", "problem_id": "6113ed9bdfbc176e", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} + x^{2} - x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $- 3 x^{2} + 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 1$.\n2) Slope at x=-1: m = f'(-1) = $-6 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\n3) Point on curve: (-1, f(-1)) = (-1, $\\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 1$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 1 = -6 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}(x − -1).\n6) Optionally expand: y = $- 6 x + \\frac{x}{2 e^{\\frac{1}{2}}} + 2 x \\sin{\\left(2 \\right)} - 5 + \\cos{\\left(2 \\right)} + \\frac{3}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = - 6\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 5 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 6\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 5 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3d18af0db70b41f6", "problem_id": "6113ed9bdfbc176e", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} + x^{2} - x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=-1$.", "output": "Differentiate: f'(x) = $- 3 x^{2} + 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 1$.\nSlope at x=-1: m = f'(-1) = $-6 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nPoint on curve: (-1, f(-1)) = (-1, $\\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 1$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 1 = -6 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}(x − -1).\nOptionally expand: y = $- 6 x + \\frac{x}{2 e^{\\frac{1}{2}}} + 2 x \\sin{\\left(2 \\right)} - 5 + \\cos{\\left(2 \\right)} + \\frac{3}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 6\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 5 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 6\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 5 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "33c4c55dfa332021", "problem_id": "f66f8ac891977545", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$- 8 x + 6 y$ = $26$\n$- x - 8 y$ = $-23$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $8 x - 6 y$ = $-26$.\n3) Multiply the second equation by $-8$: $8 x + 64 y$ = $184$.\n4) Subtract to eliminate x: $- 70 y$ = $-210$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, 3\\right)$", "final_answer": "(x,y)=\\left(-1, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5f0e15346cdb2a26", "problem_id": "f66f8ac891977545", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the linear system:\n$- 8 x + 6 y$ = $26$\n$- x - 8 y$ = $-23$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $8 x - 6 y$ = $-26$.\nMultiply the second equation by $-8$: $8 x + 64 y$ = $184$.\nFinal answer: $(x,y)=\\left(-1, 3\\right)$", "final_answer": "(x,y)=\\left(-1, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "06a624e5475c010b", "problem_id": "2f6e612e17a48446", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 32 liters of a 45% salt solution. You have a 30% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 32−x liters is 60% solution.\nSalt equation: 30x + 60(32−x) = 45·32.\n(30-60)x = (45-60)·32 ⇒ x = 16.\nThen 32−x = 16.\nCheck: 30·16 + 60·16 = 45·32.\nFinal answer: $16 L of 30% and 16 L of 60%$", "final_answer": "16 L of 30% and 16 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "577ee5c482f30bf3", "problem_id": "2f6e612e17a48446", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "You want 32 liters of a 45% salt solution. You have a 30% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 32−x liters is 60% solution.\n2) Salt equation: 30x + 60(32−x) = 45·32.\n3) (30-60)x = (45-60)·32 ⇒ x = 16.\n4) Then 32−x = 16.\nFinal answer: $16 L of 30% and 16 L of 60%$", "final_answer": "16 L of 30% and 16 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ac6c79530b412cf8", "problem_id": "5c564341da0fb8b0", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the multiplicative inverse of 7 modulo 47.", "output": "1) We want x such that 7x ≡ 1 (mod 47).\n2) Use extended Euclid to find s,t with 7s + 47t = 1.\n3) One Bezout identity is: 7·(-20) + 47·(3) = 1.\n4) Reduce s modulo 47: x ≡ 27 (mod 47).\nFinal answer: $x\\equiv 27\\pmod{47}$", "final_answer": "x\\equiv 27\\pmod{47}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f41f7d81570f18f6", "problem_id": "5c564341da0fb8b0", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Find the multiplicative inverse of 7 modulo 47.", "output": "We want x such that 7x ≡ 1 (mod 47).\nUse extended Euclid to find s,t with 7s + 47t = 1.\nOne Bezout identity is: 7·(-20) + 47·(3) = 1.\nFinal answer: $x\\equiv 27\\pmod{47}$", "final_answer": "x\\equiv 27\\pmod{47}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c4989494594f7b36", "problem_id": "554de0999f4fed43", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$3 x - 4 y$ = $15$\n$- y$ = $0$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $3$: $- 3 y$ = $0$.\n4) Subtract to eliminate x: $3 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, 0\\right)$", "final_answer": "(x,y)=\\left(5, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "30b93269a3468d2a", "problem_id": "554de0999f4fed43", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$3 x - 4 y$ = $15$\n$- y$ = $0$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $3$: $- 3 y$ = $0$.\nSubtract to eliminate x: $3 y$ = $0$.\nSolve for y: y = $0$.\nSubstitute back to find x: x = $5$.\nCheck: plug (x,y)=(5,0) into both equations.\nFinal answer: $(x,y)=\\left(5, 0\\right)$", "final_answer": "(x,y)=\\left(5, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e61879afb7cec696", "problem_id": "f00306fb205f1470", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Solve the linear system:\n$4 x - 4 y$ = $4$\n$y$ = $-2$", "output": "$(x,y)=\\left(-1, -2\\right)$", "final_answer": "(x,y)=\\left(-1, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "19bbca7c75e64ac2", "problem_id": "f00306fb205f1470", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$4 x - 4 y$ = $4$\n$y$ = $-2$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $4$: $4 y$ = $-8$.\n4) Subtract to eliminate x: $- 4 y$ = $8$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, -2\\right)$", "final_answer": "(x,y)=\\left(-1, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e3cb9cad9f01a3e6", "problem_id": "bfb3bd98b6de7d92", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 20 times, what is the probability of exactly 8 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=20, k=8, p=$\\frac{1}{2}$ and simplify.\n4) Here C(20,8) = 125970.\nFinal answer: $\\frac{62985}{524288}$", "final_answer": "\\frac{62985}{524288}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "36c45492df3fc348", "problem_id": "bfb3bd98b6de7d92", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 20 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=20, k=8, p=$\\frac{1}{2}$ and simplify.\nFinal answer: $\\frac{62985}{524288}$", "final_answer": "\\frac{62985}{524288}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fcab4315d789622c", "problem_id": "d7896f059e73bfb4", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 12 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=3, p=$\\frac{3}{8}$ and simplify.\n4) Here C(12,3) = 220.\nFinal answer: $\\frac{2900390625}{17179869184}$", "final_answer": "\\frac{2900390625}{17179869184}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7a240ccd6dc02c58", "problem_id": "d7896f059e73bfb4", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 12 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=12, k=3, p=$\\frac{3}{8}$ and simplify.\nFinal answer: $\\frac{2900390625}{17179869184}$", "final_answer": "\\frac{2900390625}{17179869184}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a03ff206dc97b4c7", "problem_id": "20d0a93a724d94ec", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "How many binary strings of length 11 contain exactly 3 ones with no two ones adjacent?", "output": "1) Reserve 2 mandatory zeros between the 3 ones.\n2) This leaves n−k+1 = 9 valid slots for the ones.\n3) Choose the slots: C(9,3) = 84.\nFinal answer: $84$", "final_answer": "84", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d6983237ae53b3f6", "problem_id": "20d0a93a724d94ec", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many binary strings of length 11 contain exactly 3 ones with no two ones adjacent?", "output": "Reserve 2 mandatory zeros between the 3 ones.\nThis leaves n−k+1 = 9 valid slots for the ones.\nChoose the slots: C(9,3) = 84.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $84$", "final_answer": "84", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9a6acf34723613ae", "problem_id": "8d20f7512e80d131", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "A cyclist rides at 80 miles per hour for 3/2 hours, then at 35 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 80 × 3/2 = 120 miles.\nSecond leg distance = 35 × 1 = 35 miles.\nTotal distance = 120 + 35 = 155 miles.\nFinal answer: $155$", "final_answer": "155", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "681a5c308f2fd3cb", "problem_id": "8d20f7512e80d131", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A cyclist rides at 80 miles per hour for 3/2 hours, then at 35 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 80 × 3/2 = 120 miles.\n2) Second leg distance = 35 × 1 = 35 miles.\n3) Total distance = 120 + 35 = 155 miles.\nFinal answer: $155$", "final_answer": "155", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a80a5a4fa3c48fe8", "problem_id": "3acdb37b8a86a99a", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{3} + 4 x^{2} + 3 x - 2$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-2) = 0.\n3) d/dx(- 3 x^{3}) = - 9 x^{2}.\n4) d/dx(3 x) = 3.\n5) d/dx(4 x^{2}) = 8 x.\n6) Combine: f'(x) = - 9 x^{2} + 8 x + 3.\nFinal answer: $- 9\\cdot x^{2} + 8\\cdot x + 3$", "final_answer": "- 9\\cdot x^{2} + 8\\cdot x + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "25bfb719076727b8", "problem_id": "3acdb37b8a86a99a", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{3} + 4 x^{2} + 3 x - 2$.", "output": "Differentiate term-by-term:\nd/dx(-2) = 0.\nd/dx(- 3 x^{3}) = - 9 x^{2}.\nFinal answer: $- 9\\cdot x^{2} + 8\\cdot x + 3$", "final_answer": "- 9\\cdot x^{2} + 8\\cdot x + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "090804c53eae702e", "problem_id": "ef00248462a5f9c6", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} x^{2} - x + 2$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{3} + 2 x^{2} - x - 2$ and a = -3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $3 x^{2} + 4 x - 1$.\n5) Evaluate at a: p'(-3) = $14$.\nFinal answer: $14$", "final_answer": "14", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "477e04ea9647f80d", "problem_id": "ef00248462a5f9c6", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} x^{2} - x + 2$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $x^{3} + 2 x^{2} - x - 2$ and a = -3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $3 x^{2} + 4 x - 1$.\nEvaluate at a: p'(-3) = $14$.\nCheck: the identity gives $14$.\nFinal answer: $14$", "final_answer": "14", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b20f6450b8d2336", "problem_id": "4d367f2543b30c51", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "You want 13 liters of a 40% salt solution. You have a 30% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 13−x liters is 50% solution.\n2) Salt equation: 30x + 50(13−x) = 40·13.\n3) (30-50)x = (40-50)·13 ⇒ x = 13/2.\n4) Then 13−x = 13/2.\nFinal answer: $13/2 L of 30% and 13/2 L of 50%$", "final_answer": "13/2 L of 30% and 13/2 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ccd599f9db3a929", "problem_id": "4d367f2543b30c51", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 13 liters of a 40% salt solution. You have a 30% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 13−x liters is 50% solution.\nSalt equation: 30x + 50(13−x) = 40·13.\n(30-50)x = (40-50)·13 ⇒ x = 13/2.\nThen 13−x = 13/2.\nCheck: 30·13/2 + 50·13/2 = 40·13.\nFinal answer: $13/2 L of 30% and 13/2 L of 50%$", "final_answer": "13/2 L of 30% and 13/2 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a333e306f353283", "problem_id": "9161f5dc63449d6a", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Worker A finishes a job in 11 hours and Worker B finishes it in 8 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/11 job/hour. B's rate = 1/8 job/hour.\n2) Combined rate = 1/11 + 1/8 = 19/88 job/hour.\n3) Time = 1 ÷ (combined rate) = 88/19 hours.\nFinal answer: $\\frac{88}{19}$", "final_answer": "\\frac{88}{19}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ccbc27795fc2afc9", "problem_id": "9161f5dc63449d6a", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Worker A finishes a job in 11 hours and Worker B finishes it in 8 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/11 job/hour. B's rate = 1/8 job/hour.\nCombined rate = 1/11 + 1/8 = 19/88 job/hour.\nTime = 1 ÷ (combined rate) = 88/19 hours.\nCheck: (combined rate)×(time) = 19/88×88/19 = 1.\nFinal answer: $\\frac{88}{19}$", "final_answer": "\\frac{88}{19}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "007a7e267a0ff1bf", "problem_id": "e031eb3db3f118ed", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} - 2 x^{3} + 2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{3} + 2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{9 \\pi}{2} - \\frac{3 \\pi^{2}}{4} - 2 + \\frac{3 \\pi^{3}}{4} + \\frac{15 \\pi^{4}}{32}$.\nFinal answer: $- \\frac{9\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{2}}{4} - 2 + \\frac{3\\cdot \\pi^{3}}{4} + \\frac{15\\cdot \\pi^{4}}{32}$", "final_answer": "- \\frac{9\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{2}}{4} - 2 + \\frac{3\\cdot \\pi^{3}}{4} + \\frac{15\\cdot \\pi^{4}}{32}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bd09a280e0319e7e", "problem_id": "e031eb3db3f118ed", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} - 2 x^{3} + 2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{3} + 2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{9 \\pi}{2} - \\frac{3 \\pi^{2}}{4} - 2 + \\frac{3 \\pi^{3}}{4} + \\frac{15 \\pi^{4}}{32}$.\n4) Simplify the result.\nFinal answer: $- \\frac{9\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{2}}{4} - 2 + \\frac{3\\cdot \\pi^{3}}{4} + \\frac{15\\cdot \\pi^{4}}{32}$", "final_answer": "- \\frac{9\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{2}}{4} - 2 + \\frac{3\\cdot \\pi^{3}}{4} + \\frac{15\\cdot \\pi^{4}}{32}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a52b3c9663c7730", "problem_id": "5af21e0c6ff43dfe", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "How many binary strings of length 19 contain exactly 8 ones with no two ones adjacent?", "output": "Reserve 7 mandatory zeros between the 8 ones.\nThis leaves n−k+1 = 12 valid slots for the ones.\nChoose the slots: C(12,8) = 495.\nFinal answer: $495$", "final_answer": "495", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f4b3e58526016eee", "problem_id": "5af21e0c6ff43dfe", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many binary strings of length 19 contain exactly 8 ones with no two ones adjacent?", "output": "1) Reserve 7 mandatory zeros between the 8 ones.\n2) This leaves n−k+1 = 12 valid slots for the ones.\n3) Choose the slots: C(12,8) = 495.\nFinal answer: $495$", "final_answer": "495", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4a5f4dd782b73b70", "problem_id": "dafbebda8b6b941c", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int \\frac{2 x + 2}{x^{2} + 2 x + 5} + \\frac{\\sin{\\left(x \\right)}}{\\cos{\\left(x \\right)} + 1}\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $\\frac{2 x + 2}{x^{2} + 2 x + 5} + \\frac{\\sin{\\left(x \\right)}}{\\cos{\\left(x \\right)} + 1}$.\n3) Compute an antiderivative: F(x) = $- \\log{\\left(\\cos{\\left(x \\right)} + 1 \\right)} + \\log{\\left(x^{2} + 2 x + 5 \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C - \\log{\\left(\\cos{\\left(x \\right)} + 1 \\right)} + \\log{\\left(x^{2} + 2\\cdot x + 5 \\right)}$", "final_answer": "C - \\log{\\left(\\cos{\\left(x \\right)} + 1 \\right)} + \\log{\\left(x^{2} + 2\\cdot x + 5 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4f8ba56ac2bb0420", "problem_id": "dafbebda8b6b941c", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int \\frac{2 x + 2}{x^{2} + 2 x + 5} + \\frac{\\sin{\\left(x \\right)}}{\\cos{\\left(x \\right)} + 1}\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $\\frac{2 x + 2}{x^{2} + 2 x + 5} + \\frac{\\sin{\\left(x \\right)}}{\\cos{\\left(x \\right)} + 1}$.\nCompute an antiderivative: F(x) = $- \\log{\\left(\\cos{\\left(x \\right)} + 1 \\right)} + \\log{\\left(x^{2} + 2 x + 5 \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - \\log{\\left(\\cos{\\left(x \\right)} + 1 \\right)} + \\log{\\left(x^{2} + 2\\cdot x + 5 \\right)}$", "final_answer": "C - \\log{\\left(\\cos{\\left(x \\right)} + 1 \\right)} + \\log{\\left(x^{2} + 2\\cdot x + 5 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cbc6c9bf49614a38", "problem_id": "665bc9ec56862e48", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A store sells a backpack for $292. It is discounted by 40%. You buy 14 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 292 × (1 − 40/100) = 292 × 60/100 = 876/5.\n2) Subtotal = 14 × 876/5 = 12264/5.\n3) Total with tax = 12264/5 × 107/100 = 328062/125.\nFinal answer: $\\frac{328062}{125}$", "final_answer": "\\frac{328062}{125}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3afdec904382a197", "problem_id": "665bc9ec56862e48", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $292. It is discounted by 40%. You buy 14 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 292 × (1 − 40/100) = 292 × 60/100 = 876/5.\nSubtotal = 14 × 876/5 = 12264/5.\nTotal with tax = 12264/5 × 107/100 = 328062/125.\nCheck: recomputing the formula yields 328062/125.\nFinal answer: $\\frac{328062}{125}$", "final_answer": "\\frac{328062}{125}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "053cabd14cbd199f", "problem_id": "0d01b3351d0191c8", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$- x - y$ = $8$\n$2 x + 7 y$ = $-16$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $- 2 x - 2 y$ = $16$.\n3) Multiply the second equation by $-1$: $- 2 x - 7 y$ = $16$.\n4) Subtract to eliminate x: $5 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $-8$.\nFinal answer: $(x,y)=\\left(-8, 0\\right)$", "final_answer": "(x,y)=\\left(-8, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c10ea051fd1b9e27", "problem_id": "0d01b3351d0191c8", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$- x - y$ = $8$\n$2 x + 7 y$ = $-16$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $2$: $- 2 x - 2 y$ = $16$.\nMultiply the second equation by $-1$: $- 2 x - 7 y$ = $16$.\nSubtract to eliminate x: $5 y$ = $0$.\nSolve for y: y = $0$.\nSubstitute back to find x: x = $-8$.\nCheck: plug (x,y)=(-8,0) into both equations.\nFinal answer: $(x,y)=\\left(-8, 0\\right)$", "final_answer": "(x,y)=\\left(-8, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c37ca09bc8a420e7", "problem_id": "67193628b2d1589d", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 3 x + 7$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{2} + x + 3$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $6 x + 1$.\n5) Evaluate at a: p'(2) = $13$.\nFinal answer: $13$", "final_answer": "13", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "009424444a418a49", "problem_id": "67193628b2d1589d", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 3 x + 7$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $3 x^{2} + x + 3$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $6 x + 1$.\nEvaluate at a: p'(2) = $13$.\nCheck: the identity gives $13$.\nFinal answer: $13$", "final_answer": "13", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db655db7eabf6968", "problem_id": "f7d4f6af5c069b4a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{2} + 3 x - 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- x^{2}) = - 2 x^{2}$.\n3) So $f'(x) = 3 - 2 x^{2}$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- x^{2}) = - 2 x$.\nTherefore the correct derivative is $f'(x) = 3 - 2 x$.\nFinal answer: $3 - 2 x$", "final_answer": "3 - 2 x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "76c3c9af25e87440", "problem_id": "375fdcf0a8b91696", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Worker A finishes a job in 16 hours and Worker B finishes it in 12 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/16 job/hour. B's rate = 1/12 job/hour.\n2) Combined rate = 1/16 + 1/12 = 7/48 job/hour.\n3) Time = 1 ÷ (combined rate) = 48/7 hours.\nFinal answer: $\\frac{48}{7}$", "final_answer": "\\frac{48}{7}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a1ec3768eb85f33", "problem_id": "3e7eb28ea8f3b872", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A disease has prevalence $\\frac{3}{200}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{127}{2000}$.\nCompute P(D|+) = $\\frac{57}{254}$.\nFinal answer: $\\frac{57}{254}$", "final_answer": "\\frac{57}{254}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1b30bc2784ab6b63", "problem_id": "3e7eb28ea8f3b872", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A disease has prevalence $\\frac{3}{200}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{127}{2000}$.\nCompute P(D|+) = $\\frac{57}{254}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{57}{254}$", "final_answer": "\\frac{57}{254}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3ad5e6dba0b7cbad", "problem_id": "543795610c48d7b2", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$- 3 x - 4 y$ = $-7$\n$- x + 2 y$ = $11$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $3 x + 4 y$ = $7$.\n3) Multiply the second equation by $-3$: $3 x - 6 y$ = $-33$.\n4) Subtract to eliminate x: $10 y$ = $40$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, 4\\right)$", "final_answer": "(x,y)=\\left(-3, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "86023b57409130a0", "problem_id": "543795610c48d7b2", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- 3 x - 4 y$ = $-7$\n$- x + 2 y$ = $11$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $3 x + 4 y$ = $7$.\nMultiply the second equation by $-3$: $3 x - 6 y$ = $-33$.\nSubtract to eliminate x: $10 y$ = $40$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $-3$.\nCheck: plug (x,y)=(-3,4) into both equations.\nFinal answer: $(x,y)=\\left(-3, 4\\right)$", "final_answer": "(x,y)=\\left(-3, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6191a92d8489db29", "problem_id": "1375bc126a3f36e6", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "You want 36 liters of a 50% salt solution. You have a 30% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 36−x liters is 70% solution.\n2) Salt equation: 30x + 70(36−x) = 50·36.\n3) (30-70)x = (50-70)·36 ⇒ x = 18.\n4) Then 36−x = 18.\nFinal answer: $18 L of 30% and 18 L of 70%$", "final_answer": "18 L of 30% and 18 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9292b74eb032fd55", "problem_id": "1375bc126a3f36e6", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 36 liters of a 50% salt solution. You have a 30% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 36−x liters is 70% solution.\nSalt equation: 30x + 70(36−x) = 50·36.\n(30-70)x = (50-70)·36 ⇒ x = 18.\nThen 36−x = 18.\nCheck: 30·18 + 70·18 = 50·36.\nFinal answer: $18 L of 30% and 18 L of 70%$", "final_answer": "18 L of 30% and 18 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b43b02d53d4f2269", "problem_id": "9673fa0311086037", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x^{3} + 5 x^{2} + 11 x + 20$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{4} + 3 x^{3} + x^{2} - 2 x + 3$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $4 x^{3} + 9 x^{2} + 2 x - 2$.\n5) Evaluate at a: p'(2) = $70$.\nFinal answer: $70$", "final_answer": "70", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "429d8829430fa541", "problem_id": "9673fa0311086037", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x^{3} + 5 x^{2} + 11 x + 20$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $x^{4} + 3 x^{3} + x^{2} - 2 x + 3$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $4 x^{3} + 9 x^{2} + 2 x - 2$.\nEvaluate at a: p'(2) = $70$.\nCheck: the identity gives $70$.\nFinal answer: $70$", "final_answer": "70", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3a80101e731fcbac", "problem_id": "d425f2e063466925", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 2 x^{3} - 3 x^{2} + 8 x - 23$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} - 3 x^{2} + 7 x - 23.\n3) Then the limit is -125.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -128.\nFinal answer: $-128$", "final_answer": "-128", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "05170a2ab95fddf9", "problem_id": "519015e742e8f165", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the multiplicative inverse of 2 modulo 13.", "output": "1) We want x such that 2x ≡ 1 (mod 13).\n2) Use extended Euclid to find s,t with 2s + 13t = 1.\n3) One Bezout identity is: 2·(-6) + 13·(1) = 1.\n4) Reduce s modulo 13: x ≡ 7 (mod 13).\nFinal answer: $x\\equiv 7\\pmod{13}$", "final_answer": "x\\equiv 7\\pmod{13}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cf8b3a66b9298b5f", "problem_id": "f7b52b58f25245c5", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{3}$. If it is flipped 20 times, what is the probability of exactly 13 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=20, k=13, p=$\\frac{1}{3}$ and simplify.\n4) Here C(20,13) = 77520.\nFinal answer: $\\frac{3307520}{1162261467}$", "final_answer": "\\frac{3307520}{1162261467}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "705bbaa5126f8950", "problem_id": "f7b52b58f25245c5", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{3}$. If it is flipped 20 times, what is the probability of exactly 13 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=20, k=13, p=$\\frac{1}{3}$ and simplify.\nHere C(20,13) = 77520.\nCheck: expression simplifies to $\\frac{3307520}{1162261467}$.\nFinal answer: $\\frac{3307520}{1162261467}$", "final_answer": "\\frac{3307520}{1162261467}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "08d18205ca5a0d30", "problem_id": "42023094d7ce61aa", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Worker A finishes a job in 10 hours and Worker B finishes it in 4 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/10 job/hour. B's rate = 1/4 job/hour.\nCombined rate = 1/10 + 1/4 = 7/20 job/hour.\nTime = 1 ÷ (combined rate) = 20/7 hours.\nCheck: (combined rate)×(time) = 7/20×20/7 = 1.\nFinal answer: $\\frac{20}{7}$", "final_answer": "\\frac{20}{7}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d8ec3152457ddfaf", "problem_id": "42023094d7ce61aa", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Worker A finishes a job in 10 hours and Worker B finishes it in 4 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/10 job/hour. B's rate = 1/4 job/hour.\n2) Combined rate = 1/10 + 1/4 = 7/20 job/hour.\n3) Time = 1 ÷ (combined rate) = 20/7 hours.\nFinal answer: $\\frac{20}{7}$", "final_answer": "\\frac{20}{7}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "23a49a218dfa916b", "problem_id": "f0d40ee7212b4c78", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A cyclist rides at 25 miles per hour for 1 hours, then at 83 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 25 × 1 = 25 miles.\n2) Second leg distance = 83 × 1 = 83 miles.\n3) Total distance = 25 + 83 = 108 miles.\nFinal answer: $108$", "final_answer": "108", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "33f985916aadb779", "problem_id": "f0d40ee7212b4c78", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 25 miles per hour for 1 hours, then at 83 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 25 × 1 = 25 miles.\nSecond leg distance = 83 × 1 = 83 miles.\nTotal distance = 25 + 83 = 108 miles.\nCheck: adding both legs gives 108.\nFinal answer: $108$", "final_answer": "108", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "20b93b758a41ce21", "problem_id": "8b1a348fa710b7f2", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} - 2 x + \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{2 x - 5}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 5}) = \\frac{2 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 5}$.\n3) So $f'(x) = \\frac{4 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{2} - 40 x + 50} + \\frac{8 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{2} - 40 x + 50} - \\frac{4 x^{2}}{8 x^{2} - 40 x + 50} - \\frac{20 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{2} - 40 x + 50} - \\frac{40 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{2} - 40 x + 50} + \\frac{20 x}{8 x^{2} - 40 x + 50} + \\frac{4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{4 x^{2} - 20 x + 25} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 5} - \\frac{2 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 5} + \\frac{2 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 5} + \\frac{25 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{2} - 40 x + 50} + \\frac{50 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{2} - 40 x + 50} + \\frac{12}{8 x^{2} - 40 x + 50} - \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 5}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 5}) = \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 5} + \\frac{2 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 5} - \\frac{4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x - 5\\right)^{2}} + \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 5}$.\nTherefore the correct derivative is $f'(x) = \\frac{4 x^{2} + 8 x + \\left(2 x - 5\\right) \\left(- 4 x + \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4\\right) - 4 \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8}{2 \\left(2 x - 5\\right)^{2}}$.\nFinal answer: $\\frac{4 x^{2} + 8 x + \\left(2 x - 5\\right) \\left(- 4 x + \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4\\right) - 4 \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8}{2 \\left(2 x - 5\\right)^{2}}$", "final_answer": "\\frac{4 x^{2} + 8 x + \\left(2 x - 5\\right) \\left(- 4 x + \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4\\right) - 4 \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8}{2 \\left(2 x - 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2c1a07fcbfc73728", "problem_id": "092ed7546829ca5d", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many binary strings of length 16 contain exactly 5 ones with no two ones adjacent?", "output": "Reserve 4 mandatory zeros between the 5 ones.\nThis leaves n−k+1 = 12 valid slots for the ones.\nChoose the slots: C(12,5) = 792.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $792$", "final_answer": "792", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "29b06f13a8e33aaa", "problem_id": "cc5391c11078205d", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the multiplicative inverse of 10 modulo 41.", "output": "We want x such that 10x ≡ 1 (mod 41).\nUse extended Euclid to find s,t with 10s + 41t = 1.\nOne Bezout identity is: 10·(-4) + 41·(1) = 1.\nReduce s modulo 41: x ≡ 37 (mod 41).\nCheck: 10·37 mod 41 = 1.\nFinal answer: $x\\equiv 37\\pmod{41}$", "final_answer": "x\\equiv 37\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "60fa2ea736137b7a", "problem_id": "cc5391c11078205d", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Find the multiplicative inverse of 10 modulo 41.", "output": "1) We want x such that 10x ≡ 1 (mod 41).\n2) Use extended Euclid to find s,t with 10s + 41t = 1.\n3) One Bezout identity is: 10·(-4) + 41·(1) = 1.\n4) Reduce s modulo 41: x ≡ 37 (mod 41).\nFinal answer: $x\\equiv 37\\pmod{41}$", "final_answer": "x\\equiv 37\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e6ba2f4d1b0ebda9", "problem_id": "563cf2ae649fcfeb", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Worker A finishes a job in 19 hours and Worker B finishes it in 22 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/19 job/hour. B's rate = 1/22 job/hour.\nCombined rate = 1/19 + 1/22 = 41/418 job/hour.\nTime = 1 ÷ (combined rate) = 418/41 hours.\nFinal answer: $\\frac{418}{41}$", "final_answer": "\\frac{418}{41}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c1b03ac4b2e4535c", "problem_id": "563cf2ae649fcfeb", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Worker A finishes a job in 19 hours and Worker B finishes it in 22 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/19 job/hour. B's rate = 1/22 job/hour.\nCombined rate = 1/19 + 1/22 = 41/418 job/hour.\nTime = 1 ÷ (combined rate) = 418/41 hours.\nCheck: (combined rate)×(time) = 41/418×418/41 = 1.\nFinal answer: $\\frac{418}{41}$", "final_answer": "\\frac{418}{41}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "307d71fc95bfd52a", "problem_id": "535b5c7b5f1765ff", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A store sells a backpack for $620. It is discounted by 35%. You buy 10 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 620 × (1 − 35/100) = 620 × 65/100 = 403.\n2) Subtotal = 10 × 403 = 4030.\n3) Total with tax = 4030 × 108/100 = 21762/5.\nFinal answer: $\\frac{21762}{5}$", "final_answer": "\\frac{21762}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3d27820f4082f5d5", "problem_id": "535b5c7b5f1765ff", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A store sells a backpack for $620. It is discounted by 35%. You buy 10 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 620 × (1 − 35/100) = 620 × 65/100 = 403.\nSubtotal = 10 × 403 = 4030.\nTotal with tax = 4030 × 108/100 = 21762/5.\nFinal answer: $\\frac{21762}{5}$", "final_answer": "\\frac{21762}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5690dc74ad50c9d9", "problem_id": "b85e4c7b931ea7f5", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡1 (mod 4), x≡2 (mod 9). Give the solution modulo 36.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 36.\nCompute the CRT solution to get x ≡ 29 (mod 36).\nCheck: 29 mod 4=1 and mod 9=2.\nFinal answer: $x\\equiv 29\\pmod{36}$", "final_answer": "x\\equiv 29\\pmod{36}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "52bcf715e56af948", "problem_id": "b85e4c7b931ea7f5", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡1 (mod 4), x≡2 (mod 9). Give the solution modulo 36.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 36.\n3) Compute the CRT solution to get x ≡ 29 (mod 36).\nFinal answer: $x\\equiv 29\\pmod{36}$", "final_answer": "x\\equiv 29\\pmod{36}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7f4d580b67960bbd", "problem_id": "94d4f7ccb8c073ce", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "You want 13 liters of a 30% salt solution. You have a 20% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 13−x liters is 40% solution.\n2) Salt equation: 20x + 40(13−x) = 30·13.\n3) (20-40)x = (30-40)·13 ⇒ x = 13/2.\n4) Then 13−x = 13/2.\nFinal answer: $13/2 L of 20% and 13/2 L of 40%$", "final_answer": "13/2 L of 20% and 13/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7e4ba4c2aa535232", "problem_id": "94d4f7ccb8c073ce", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 13 liters of a 30% salt solution. You have a 20% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 13−x liters is 40% solution.\nSalt equation: 20x + 40(13−x) = 30·13.\n(20-40)x = (30-40)·13 ⇒ x = 13/2.\nThen 13−x = 13/2.\nCheck: 20·13/2 + 40·13/2 = 30·13.\nFinal answer: $13/2 L of 20% and 13/2 L of 40%$", "final_answer": "13/2 L of 20% and 13/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3d4f9d540ba3b11e", "problem_id": "1c2e30d6a980ffaa", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Worker A finishes a job in 17 hours and Worker B finishes it in 13 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/17 job/hour. B's rate = 1/13 job/hour.\n2) Combined rate = 1/17 + 1/13 = 30/221 job/hour.\n3) Time = 1 ÷ (combined rate) = 221/30 hours.\nFinal answer: $\\frac{221}{30}$", "final_answer": "\\frac{221}{30}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5e804fcf5e444713", "problem_id": "1c2e30d6a980ffaa", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Worker A finishes a job in 17 hours and Worker B finishes it in 13 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/17 job/hour. B's rate = 1/13 job/hour.\nCombined rate = 1/17 + 1/13 = 30/221 job/hour.\nTime = 1 ÷ (combined rate) = 221/30 hours.\nFinal answer: $\\frac{221}{30}$", "final_answer": "\\frac{221}{30}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8dd861bb1ed8847b", "problem_id": "60f2ba8d9ffd3cef", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the linear system:\n$- 8 x - 6 y$ = $-28$\n$- 4 x$ = $4$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $32 x + 24 y$ = $112$.\nMultiply the second equation by $-8$: $32 x$ = $-32$.\nFinal answer: $(x,y)=\\left(-1, 6\\right)$", "final_answer": "(x,y)=\\left(-1, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "78d1bdabf17a10fe", "problem_id": "60f2ba8d9ffd3cef", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$- 8 x - 6 y$ = $-28$\n$- 4 x$ = $4$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $32 x + 24 y$ = $112$.\n3) Multiply the second equation by $-8$: $32 x$ = $-32$.\n4) Subtract to eliminate x: $24 y$ = $144$.\n5) Solve for y: y = $6$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, 6\\right)$", "final_answer": "(x,y)=\\left(-1, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "94bd152cb4b404df", "problem_id": "5e32b80e7d1c7104", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many binary strings of length 22 contain exactly 7 ones with no two ones adjacent?", "output": "Reserve 6 mandatory zeros between the 7 ones.\nThis leaves n−k+1 = 16 valid slots for the ones.\nChoose the slots: C(16,7) = 11440.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $11440$", "final_answer": "11440", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a8ca3e60992e1543", "problem_id": "5e32b80e7d1c7104", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many binary strings of length 22 contain exactly 7 ones with no two ones adjacent?", "output": "1) Reserve 6 mandatory zeros between the 7 ones.\n2) This leaves n−k+1 = 16 valid slots for the ones.\n3) Choose the slots: C(16,7) = 11440.\nFinal answer: $11440$", "final_answer": "11440", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1be9f30c93564b7e", "problem_id": "a7f67051d91e4344", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{5} x}{3} + \\frac{11 x}{6} - \\frac{\\sqrt{5}}{6} + \\frac{2}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{11}{6} - \\frac{\\sqrt{5}}{3}$, c=$\\frac{2}{3} - \\frac{\\sqrt{5}}{6}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{5}{4} - \\frac{5 \\sqrt{5}}{9}$, so √Δ = $\\sqrt{\\frac{5}{4} - \\frac{5 \\sqrt{5}}{9}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{11}{12} + \\frac{\\sqrt{45 - 20 \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{6}$.\n7) x2 = $- \\frac{11}{12} - \\frac{\\sqrt{45 - 20 \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{6}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{11}{12} + \\frac{\\sqrt{45 - 20\\cdot \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{6}, - \\frac{11}{12} - \\frac{\\sqrt{45 - 20\\cdot \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{6}\\}$", "final_answer": "x \\in \\{- \\frac{11}{12} + \\frac{\\sqrt{45 - 20\\cdot \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{6}, - \\frac{11}{12} - \\frac{\\sqrt{45 - 20\\cdot \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{6}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fd6765e007f2cffb", "problem_id": "a7f67051d91e4344", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{5} x}{3} + \\frac{11 x}{6} - \\frac{\\sqrt{5}}{6} + \\frac{2}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{11}{6} - \\frac{\\sqrt{5}}{3}$, c=$\\frac{2}{3} - \\frac{\\sqrt{5}}{6}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{5}{4} - \\frac{5 \\sqrt{5}}{9}$, so √Δ = $\\sqrt{\\frac{5}{4} - \\frac{5 \\sqrt{5}}{9}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{11}{12} + \\frac{\\sqrt{45 - 20 \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{6}$.\nx2 = $- \\frac{11}{12} - \\frac{\\sqrt{45 - 20 \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{6}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{11}{12} + \\frac{\\sqrt{45 - 20\\cdot \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{6}, - \\frac{11}{12} - \\frac{\\sqrt{45 - 20\\cdot \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{6}\\}$", "final_answer": "x \\in \\{- \\frac{11}{12} + \\frac{\\sqrt{45 - 20\\cdot \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{6}, - \\frac{11}{12} - \\frac{\\sqrt{45 - 20\\cdot \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{6}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "48efdc8a638c5043", "problem_id": "bbc10e9e49cfe59c", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A disease has prevalence $\\frac{1}{200}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{173}{5000}$.\n3) Compute P(D|+) = $\\frac{95}{692}$.\nFinal answer: $\\frac{95}{692}$", "final_answer": "\\frac{95}{692}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d929c22f7440532", "problem_id": "ac7aa19d28ca3e0c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 2 x^{3} + 8 x^{2} - 22 x + 68$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{3} + 8 x^{2} - 23 x + 68.\n3) Then the limit is 263.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: 260.\nFinal answer: $260$", "final_answer": "260", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "dced8234202a6b83", "problem_id": "9d1a2fc51efce186", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=28$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=28 and k=6.\n4) Compute C(33,5) = 237336.\nFinal answer: $237336$", "final_answer": "237336", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "31a8444b9c8aaf29", "problem_id": "9d1a2fc51efce186", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=28$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=28 and k=6.\nFinal answer: $237336$", "final_answer": "237336", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7816a9acb818a6e3", "problem_id": "dd274cec83dff208", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Two friends split $462 in the ratio 4:3. How much does each receive?", "output": "1) Total ratio parts = 4+3 = 7.\n2) Each part = 462 ÷ 7 = 66.\n3) First share = 4 × 66 = 264.\n4) Second share = 3 × 66 = 198.\nFinal answer: $198$", "final_answer": "198", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c2c3e4783146e2cf", "problem_id": "37ecf5f605acfe9d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{3 x^{3} + 2 x^{2} - x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{3 x + 4}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 4}) = \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 4}$.\n3) So $f'(x) = \\frac{36 x^{3}}{18 x^{2} + 48 x + 32} + \\frac{9 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{18 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{84 x^{2}}{18 x^{2} + 48 x + 32} + \\frac{24 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{48 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{32 x}{18 x^{2} + 48 x + 32} + \\frac{9 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{2} + 24 x + 16} - \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x + 8} + \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 4} - \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 4} + \\frac{16 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{32 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{4}{18 x^{2} + 48 x + 32} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 4}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 4}) = \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(3 x + 4\\right)} + \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 4} - \\frac{9 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(3 x + 4\\right)^{2}} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 4}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 18 x^{3} - 12 x^{2} + 6 x + \\left(3 x + 4\\right) \\left(18 x^{2} + 8 x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 6 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 12}{2 \\left(3 x + 4\\right)^{2}}$.\nFinal answer: $\\frac{- 18 x^{3} - 12 x^{2} + 6 x + \\left(3 x + 4\\right) \\left(18 x^{2} + 8 x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 6 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 12}{2 \\left(3 x + 4\\right)^{2}}$", "final_answer": "\\frac{- 18 x^{3} - 12 x^{2} + 6 x + \\left(3 x + 4\\right) \\left(18 x^{2} + 8 x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 6 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 12}{2 \\left(3 x + 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2b0e26fbb88f08e6", "problem_id": "4e5d6ca6cd8d5eac", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- x - 4 y$ = $36$\n$- 3 x - y$ = $9$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $3 x + 12 y$ = $-108$.\nMultiply the second equation by $-1$: $3 x + y$ = $-9$.\nSubtract to eliminate x: $11 y$ = $-99$.\nSolve for y: y = $-9$.\nSubstitute back to find x: x = $0$.\nCheck: plug (x,y)=(0,-9) into both equations.\nFinal answer: $(x,y)=\\left(0, -9\\right)$", "final_answer": "(x,y)=\\left(0, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9e4cfc6f9f585641", "problem_id": "f94bb98cea397d4b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{3} \\sin{\\left(x \\right)} + x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 x^{3} \\sin{\\left(x \\right)}) = 3 x^{3} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = 3 x^{3} \\cos{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 x^{3} \\sin{\\left(x \\right)}) = 3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$", "final_answer": "3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0a4c586899a5b67f", "problem_id": "c40e940b96d92007", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "How many binary strings of length 16 contain exactly 4 ones with no two ones adjacent?", "output": "1) Reserve 3 mandatory zeros between the 4 ones.\n2) This leaves n−k+1 = 13 valid slots for the ones.\n3) Choose the slots: C(13,4) = 715.\nFinal answer: $715$", "final_answer": "715", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "45b16115d150029c", "problem_id": "c40e940b96d92007", "style": "answer_only", "instruction": "Final answer only.", "input": "How many binary strings of length 16 contain exactly 4 ones with no two ones adjacent?", "output": "$715$", "final_answer": "715", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c38ce7f0da2f289f", "problem_id": "c0cd2e936c95a954", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A cyclist rides at 78 miles per hour for 1/2 hours, then at 25 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 78 × 1/2 = 39 miles.\n2) Second leg distance = 25 × 1 = 25 miles.\n3) Total distance = 39 + 25 = 64 miles.\nFinal answer: $64$", "final_answer": "64", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aa9c504ad3168d6f", "problem_id": "c0cd2e936c95a954", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 78 miles per hour for 1/2 hours, then at 25 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 78 × 1/2 = 39 miles.\nSecond leg distance = 25 × 1 = 25 miles.\nTotal distance = 39 + 25 = 64 miles.\nCheck: adding both legs gives 64.\nFinal answer: $64$", "final_answer": "64", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7c3fcdfe4fbd6907", "problem_id": "5f137f0736c7c320", "style": "answer_only", "instruction": "Final answer only.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 3 x^{2} + 8 x - 24$.", "output": "$-75$", "final_answer": "-75", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ac8b4d676ad16b5a", "problem_id": "5f137f0736c7c320", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 3 x^{2} + 8 x - 24$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 3 x^{3} - x^{2} + 1$ and a = -3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 9 x^{2} - 2 x$.\n5) Evaluate at a: p'(-3) = $-75$.\nFinal answer: $-75$", "final_answer": "-75", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d0c17843cc74813a", "problem_id": "f7d4f6af5c069b4a", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{2} + 3 x - 3$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-3) = 0.\n3) d/dx(- x^{2}) = - 2 x.\n4) d/dx(3 x) = 3.\n5) Combine: f'(x) = 3 - 2 x.\nFinal answer: $3 - 2\\cdot x$", "final_answer": "3 - 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4065000a8242c917", "problem_id": "7ad602b7d6879177", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- 3 x^{2} - 3 x - 3\\right) \\cos{\\left(2 x + 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x e^{x}) = 2 x e^{x}$.\n3) So $f'(x) = 6 x^{2} \\sin{\\left(2 x + 1 \\right)} + 2 x e^{x} + 6 x \\sin{\\left(2 x + 1 \\right)} - 6 x \\cos{\\left(2 x + 1 \\right)} + 2 e^{x} + 6 \\sin{\\left(2 x + 1 \\right)} - 3 \\cos{\\left(2 x + 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x e^{x}) = 2 x e^{x} + 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = 3 \\left(- 2 x - 1\\right) \\cos{\\left(2 x + 1 \\right)} + 2 \\left(x + 1\\right) e^{x} - 6 \\left(x \\left(- x - 1\\right) - 1\\right) \\sin{\\left(2 x + 1 \\right)} + 2 e^{x}$.\nFinal answer: $3 \\left(- 2 x - 1\\right) \\cos{\\left(2 x + 1 \\right)} + 2 \\left(x + 1\\right) e^{x} - 6 \\left(x \\left(- x - 1\\right) - 1\\right) \\sin{\\left(2 x + 1 \\right)} + 2 e^{x}$", "final_answer": "3 \\left(- 2 x - 1\\right) \\cos{\\left(2 x + 1 \\right)} + 2 \\left(x + 1\\right) e^{x} - 6 \\left(x \\left(- x - 1\\right) - 1\\right) \\sin{\\left(2 x + 1 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2597b7cda7d7665c", "problem_id": "c46864b316ce13a6", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - x^{3} - x^{2} + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- x^{3} - x^{2} + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}$.\n3) Compute an antiderivative: F(x) = $- \\frac{x^{4}}{4} - \\frac{x^{3}}{3} - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{x^{4}}{4} - \\frac{x^{3}}{3} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{x^{4}}{4} - \\frac{x^{3}}{3} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2e27128aa410b9fe", "problem_id": "1b3279da9aef3ca3", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{3} + 2 x^{2} + 2 x + 3 \\left(x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 \\left(x - 1\\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 3}) = \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 1}$.\n3) So $f'(x) = \\frac{4 x^{3}}{6 x^{2} - 12 x + 6} + \\frac{3 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{2} - 12 x + 6} + \\frac{6 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{2} - 12 x + 6} - \\frac{2 x^{2}}{6 x^{2} - 12 x + 6} + \\frac{9 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{2} - 18 x + 9} - \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{2} - 12 x + 6} - \\frac{12 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{2} - 12 x + 6} - \\frac{8 x}{6 x^{2} - 12 x + 6} - \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x - 6} - \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 3} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 1} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{2} - 12 x + 6} + \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{2} - 12 x + 6} - \\frac{4}{6 x^{2} - 12 x + 6} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 3}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 3}) = \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(3 x - 3\\right)} + \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 3} - \\frac{9 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(3 x - 3\\right)^{2}} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 3}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 2 x^{3} - 4 x^{2} - 4 x + \\left(x - 1\\right) \\left(6 x^{2} + 8 x + 3 \\left(x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x - 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 6 \\left(x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 \\left(x - 1\\right)^{2}}$.\nFinal answer: $\\frac{- 2 x^{3} - 4 x^{2} - 4 x + \\left(x - 1\\right) \\left(6 x^{2} + 8 x + 3 \\left(x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x - 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 6 \\left(x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 \\left(x - 1\\right)^{2}}$", "final_answer": "\\frac{- 2 x^{3} - 4 x^{2} - 4 x + \\left(x - 1\\right) \\left(6 x^{2} + 8 x + 3 \\left(x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x - 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 6 \\left(x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 \\left(x - 1\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "22656aea02041ca0", "problem_id": "26928f2e6a8baf4a", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 14 times, what is the probability of exactly 7 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=14, k=7, p=$\\frac{1}{10}$ and simplify.\n4) Here C(14,7) = 3432.\nFinal answer: $\\frac{2051893701}{12500000000000}$", "final_answer": "\\frac{2051893701}{12500000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e414929fc152d3e0", "problem_id": "26928f2e6a8baf4a", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 14 times, what is the probability of exactly 7 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=14, k=7, p=$\\frac{1}{10}$ and simplify.\nHere C(14,7) = 3432.\nCheck: expression simplifies to $\\frac{2051893701}{12500000000000}$.\nFinal answer: $\\frac{2051893701}{12500000000000}$", "final_answer": "\\frac{2051893701}{12500000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d8ed2d5dbe42f4ad", "problem_id": "894f1d80f5d925cc", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=12$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=12 and k=5.\n4) Compute C(16,4) = 1820.\nFinal answer: $1820$", "final_answer": "1820", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b0c46eeae82b1d12", "problem_id": "8e0ddc9f3bc1cce5", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 2 x^{3} - 2 x^{2} + 5 x - 12$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} - 2 x^{2} + 4 x - 12.\n3) Then the limit is -44.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -46.\nFinal answer: $-46$", "final_answer": "-46", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "404dc232a370f7e6", "problem_id": "2024877873f6879b", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - 3 x^{3} + 8 x^{2} - 19 x + 35$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 3 x^{4} + 2 x^{3} - 3 x^{2} - 3 x - 1$ and a = -2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $129$", "final_answer": "129", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fb4fc8c1501df9da", "problem_id": "2024877873f6879b", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - 3 x^{3} + 8 x^{2} - 19 x + 35$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 3 x^{4} + 2 x^{3} - 3 x^{2} - 3 x - 1$ and a = -2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 12 x^{3} + 6 x^{2} - 6 x - 3$.\n5) Evaluate at a: p'(-2) = $129$.\nFinal answer: $129$", "final_answer": "129", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c23b0ac3893a3b09", "problem_id": "980101bad1b99e2e", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the congruences: x≡0 (mod 8), x≡4 (mod 11). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 48 (mod 88).\nFinal answer: $x\\equiv 48\\pmod{88}$", "final_answer": "x\\equiv 48\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0439a716a84a5c28", "problem_id": "192e44a45e2bafd4", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} - x + 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x^{3}) = 3 x^{3}$.\n3) So $f'(x) = 3 x^{3} - 1$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x^{3}) = 3 x^{2}$.\nTherefore the correct derivative is $f'(x) = 3 x^{2} - 1$.\nFinal answer: $3 x^{2} - 1$", "final_answer": "3 x^{2} - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "44b0939b4ec4cf42", "problem_id": "adf7ecfc1f245234", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{11 x}{10} - \\frac{3}{10}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$- \\frac{11}{10}$, c=$- \\frac{3}{10}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{361}{100}$, so √Δ = $\\frac{19}{10}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{3}{4}$.\nx2 = $- \\frac{1}{5}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{3}{4}, - \\frac{1}{5}\\}$", "final_answer": "x \\in \\{\\frac{3}{4}, - \\frac{1}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "51ade745f7468553", "problem_id": "adf7ecfc1f245234", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{11 x}{10} - \\frac{3}{10}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$- \\frac{11}{10}$, c=$- \\frac{3}{10}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{361}{100}$, so √Δ = $\\frac{19}{10}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{3}{4}$.\n7) x2 = $- \\frac{1}{5}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{3}{4}, - \\frac{1}{5}\\}$", "final_answer": "x \\in \\{\\frac{3}{4}, - \\frac{1}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8c0a7df4631484b9", "problem_id": "44da1682f1ccbbea", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 3 x^{3} - 4 x^{2} + 2 x + 1$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} - 4 x^{2} + x + 1.\n3) Then the limit is -7.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: -8.\nFinal answer: $-8$", "final_answer": "-8", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ee822538b020715e", "problem_id": "1d27c5c90a4bf345", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{4} + 2 x^{3} - 4 x - 3\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 3 x^{4} + 2 x^{3} - 4 x - 3$.\n3) Compute an antiderivative: F(x) = $- \\frac{3 x^{5}}{5} + \\frac{x^{4}}{2} - 2 x^{2} - 3 x$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{3\\cdot x^{5}}{5} + \\frac{x^{4}}{2} - 2\\cdot x^{2} - 3\\cdot x$", "final_answer": "C - \\frac{3\\cdot x^{5}}{5} + \\frac{x^{4}}{2} - 2\\cdot x^{2} - 3\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1bd7720138a03c76", "problem_id": "f6b6d6c059020932", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 17 red balls and 17 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(34,11).\nFavorable samples: choose k reds and n−k blues: C(17,3)·C(17,8).\nCompute totals: C(34,11) = 286097760, and favorable = 16530800.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{935}{16182}$.\nFinal answer: $\\frac{935}{16182}$", "final_answer": "\\frac{935}{16182}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ce725a041c18eb8", "problem_id": "22a3089edc84880c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x - 3$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 2 x - 3.\n3) Then the limit is -7.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -5.\nFinal answer: $-5$", "final_answer": "-5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "155617e06777d376", "problem_id": "61597971a8e26afc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - x^{2} - 2 x - 3$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} - 3 x - 3.\n3) Then the limit is -21.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: -18.\nFinal answer: $-18$", "final_answer": "-18", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "391a8f24c3b2321a", "problem_id": "935b98dd99668ec4", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡4 (mod 7), x≡2 (mod 10). Give the solution modulo 70.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 70.\n3) Compute the CRT solution to get x ≡ 32 (mod 70).\nFinal answer: $x\\equiv 32\\pmod{70}$", "final_answer": "x\\equiv 32\\pmod{70}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "224f48eaa65e3b7b", "problem_id": "b46d5bc7fb935668", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 2 x^{2} + 5 x - 5$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 2 x^{3} + 3 x^{2}$ and a = -1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 6 x^{2} + 6 x$.\n5) Evaluate at a: p'(-1) = $-12$.\nFinal answer: $-12$", "final_answer": "-12", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b0bdb92fa6e93cb8", "problem_id": "b46d5bc7fb935668", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 2 x^{2} + 5 x - 5$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 2 x^{3} + 3 x^{2}$ and a = -1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 6 x^{2} + 6 x$.\nEvaluate at a: p'(-1) = $-12$.\nCheck: the identity gives $-12$.\nFinal answer: $-12$", "final_answer": "-12", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "075688ff15f375ae", "problem_id": "5d6f611bbe391b5c", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=27$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=27 and k=5.\n4) Compute C(31,4) = 31465.\nFinal answer: $31465$", "final_answer": "31465", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c08f7224aa575a29", "problem_id": "5d6f611bbe391b5c", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=27$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=27 and k=5.\nCompute C(31,4) = 31465.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $31465$", "final_answer": "31465", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "74fe7fd71082e3d1", "problem_id": "25ec14d24ded7e9a", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many binary strings of length 23 contain exactly 9 ones with no two ones adjacent?", "output": "Reserve 8 mandatory zeros between the 9 ones.\nThis leaves n−k+1 = 15 valid slots for the ones.\nChoose the slots: C(15,9) = 5005.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $5005$", "final_answer": "5005", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bfc5385c04366b8a", "problem_id": "25ec14d24ded7e9a", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "How many binary strings of length 23 contain exactly 9 ones with no two ones adjacent?", "output": "1) Reserve 8 mandatory zeros between the 9 ones.\n2) This leaves n−k+1 = 15 valid slots for the ones.\n3) Choose the slots: C(15,9) = 5005.\nFinal answer: $5005$", "final_answer": "5005", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "95e8a5b222e7cb9f", "problem_id": "17676e3b922393cb", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the linear system:\n$- 3 x - 4 y$ = $-5$\n$- 3 y$ = $-6$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $-3$: $9 y$ = $18$.\nFinal answer: $(x,y)=\\left(-1, 2\\right)$", "final_answer": "(x,y)=\\left(-1, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f55d0fea3a6c25e8", "problem_id": "17676e3b922393cb", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$- 3 x - 4 y$ = $-5$\n$- 3 y$ = $-6$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $-3$: $9 y$ = $18$.\n4) Subtract to eliminate x: $- 9 y$ = $-18$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, 2\\right)$", "final_answer": "(x,y)=\\left(-1, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8b337a91e612fea8", "problem_id": "abd6a74f32262289", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- 4 x + 3 y$ = $9$\n$- 3 x$ = $9$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $12 x - 9 y$ = $-27$.\nMultiply the second equation by $-4$: $12 x$ = $-36$.\nSubtract to eliminate x: $- 9 y$ = $9$.\nSolve for y: y = $-1$.\nSubstitute back to find x: x = $-3$.\nCheck: plug (x,y)=(-3,-1) into both equations.\nFinal answer: $(x,y)=\\left(-3, -1\\right)$", "final_answer": "(x,y)=\\left(-3, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ca508d3b84c8eeec", "problem_id": "abd6a74f32262289", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$- 4 x + 3 y$ = $9$\n$- 3 x$ = $9$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $12 x - 9 y$ = $-27$.\n3) Multiply the second equation by $-4$: $12 x$ = $-36$.\n4) Subtract to eliminate x: $- 9 y$ = $9$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, -1\\right)$", "final_answer": "(x,y)=\\left(-3, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c4503a7c5fc990fa", "problem_id": "23c3b00adedb75f4", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} + 2 x^{2} + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 3$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $- 3 x^{2} + 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)}$.\n2) Slope at x=-1: m = f'(-1) = $-7 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\n3) Point on curve: (-1, f(-1)) = (-1, $\\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} = -7 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}(x − -1).\n6) Optionally expand: y = $- 7 x + \\frac{x}{2 e^{\\frac{1}{2}}} + 2 x \\sin{\\left(2 \\right)} - 7 + \\cos{\\left(2 \\right)} + \\frac{3}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = - 7\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 7 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 7\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 7 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dc5a62ab8ce202d7", "problem_id": "23c3b00adedb75f4", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} + 2 x^{2} + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 3$ at $x=-1$.", "output": "Differentiate: f'(x) = $- 3 x^{2} + 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)}$.\nSlope at x=-1: m = f'(-1) = $-7 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nPoint on curve: (-1, f(-1)) = (-1, $\\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} = -7 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}(x − -1).\nOptionally expand: y = $- 7 x + \\frac{x}{2 e^{\\frac{1}{2}}} + 2 x \\sin{\\left(2 \\right)} - 7 + \\cos{\\left(2 \\right)} + \\frac{3}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 7\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 7 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 7\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 7 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a80c58bd4be1327f", "problem_id": "552ce9f80be21247", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{2} + 3 x + 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 3 x^{2}) = - 6 x^{2}$.\n3) So $f'(x) = 3 - 6 x^{2}$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 3 x^{2}) = - 6 x$.\nTherefore the correct derivative is $f'(x) = 3 \\left(1 - 2 x\\right)$.\nFinal answer: $3 \\left(1 - 2 x\\right)$", "final_answer": "3 \\left(1 - 2 x\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8144633acd3276cf", "problem_id": "f03aebff87bb6c61", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the multiplicative inverse of 13 modulo 23.", "output": "1) We want x such that 13x ≡ 1 (mod 23).\n2) Use extended Euclid to find s,t with 13s + 23t = 1.\n3) One Bezout identity is: 13·(-7) + 23·(4) = 1.\n4) Reduce s modulo 23: x ≡ 16 (mod 23).\nFinal answer: $x\\equiv 16\\pmod{23}$", "final_answer": "x\\equiv 16\\pmod{23}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f75043d2a05994ba", "problem_id": "81c151717a14e873", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=24$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=24 and k=7.\n4) Compute C(30,6) = 593775.\nFinal answer: $593775$", "final_answer": "593775", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "766e3190a226d756", "problem_id": "81c151717a14e873", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=24$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=24 and k=7.\nFinal answer: $593775$", "final_answer": "593775", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f787a3de2d2ef95a", "problem_id": "04bc39be437af424", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_5=33$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_5=28.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=28 and k=5.\nCompute C(32,4) = 35960.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $35960$", "final_answer": "35960", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e02d082aa49912af", "problem_id": "04bc39be437af424", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_5=33$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_5=28.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=28 and k=5.\n4) Compute C(32,4) = 35960.\nFinal answer: $35960$", "final_answer": "35960", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "30d42e76ea7a6036", "problem_id": "211ea4b4bfd17d79", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A cyclist rides at 89 miles per hour for 3/2 hours, then at 25 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 89 × 3/2 = 267/2 miles.\n2) Second leg distance = 25 × 2 = 50 miles.\n3) Total distance = 267/2 + 50 = 367/2 miles.\nFinal answer: $\\frac{367}{2}$", "final_answer": "\\frac{367}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fd45bd21ab5b5a01", "problem_id": "211ea4b4bfd17d79", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A cyclist rides at 89 miles per hour for 3/2 hours, then at 25 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 89 × 3/2 = 267/2 miles.\nSecond leg distance = 25 × 2 = 50 miles.\nTotal distance = 267/2 + 50 = 367/2 miles.\nCheck: adding both legs gives 367/2.\nFinal answer: $\\frac{367}{2}$", "final_answer": "\\frac{367}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6228b208f500aa32", "problem_id": "c59e89210b4f83d4", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the multiplicative inverse of 3 modulo 17.", "output": "1) We want x such that 3x ≡ 1 (mod 17).\n2) Use extended Euclid to find s,t with 3s + 17t = 1.\n3) One Bezout identity is: 3·(6) + 17·(-1) = 1.\n4) Reduce s modulo 17: x ≡ 6 (mod 17).\nFinal answer: $x\\equiv 6\\pmod{17}$", "final_answer": "x\\equiv 6\\pmod{17}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "825f6105b8cc2da8", "problem_id": "c59e89210b4f83d4", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Find the multiplicative inverse of 3 modulo 17.", "output": "We want x such that 3x ≡ 1 (mod 17).\nUse extended Euclid to find s,t with 3s + 17t = 1.\nOne Bezout identity is: 3·(6) + 17·(-1) = 1.\nFinal answer: $x\\equiv 6\\pmod{17}$", "final_answer": "x\\equiv 6\\pmod{17}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c2c0fb87968f9d26", "problem_id": "3a366b87b0d01694", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A disease has prevalence $\\frac{2}{25}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{41}{250}$.\n3) Compute P(D|+) = $\\frac{18}{41}$.\nFinal answer: $\\frac{18}{41}$", "final_answer": "\\frac{18}{41}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "05f48d52d8799935", "problem_id": "3a366b87b0d01694", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A disease has prevalence $\\frac{2}{25}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{41}{250}$.\nCompute P(D|+) = $\\frac{18}{41}$.\nFinal answer: $\\frac{18}{41}$", "final_answer": "\\frac{18}{41}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f7429a6ea65d4462", "problem_id": "b7addf236b39dd38", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} - x + 2$.", "output": "1) Differentiate term-by-term:\n2) d/dx(2) = 0.\n3) d/dx(- x) = -1.\n4) d/dx(2 x^{2}) = 4 x.\n5) Combine: f'(x) = 4 x - 1.\nFinal answer: $4\\cdot x - 1$", "final_answer": "4\\cdot x - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ede31683e70aa639", "problem_id": "b7addf236b39dd38", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} - x + 2$.", "output": "Differentiate term-by-term:\nd/dx(2) = 0.\nd/dx(- x) = -1.\nFinal answer: $4\\cdot x - 1$", "final_answer": "4\\cdot x - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aaa9b48044a17118", "problem_id": "8e73a5c19917465b", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the equation of the tangent line to $y=x^{2} - 3 x + 2 \\sin{\\left(x \\right)} + 3$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $2 x + 2 \\cos{\\left(x \\right)} - 3$.\n2) Slope at x=-2: m = f'(-2) = $-7 + 2 \\cos{\\left(2 \\right)}$.\n3) Point on curve: (-2, f(-2)) = (-2, $13 - 2 \\sin{\\left(2 \\right)}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 13 - 2 \\sin{\\left(2 \\right)} = -7 + 2 \\cos{\\left(2 \\right)}(x − -2).\n6) Optionally expand: y = $- 7 x + 2 x \\cos{\\left(2 \\right)} - 2 \\sin{\\left(2 \\right)} + 4 \\cos{\\left(2 \\right)} - 1$.\nFinal answer: $y = - 7\\cdot x + 2\\cdot x\\cdot \\cos{\\left(2 \\right)} - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)} - 1$", "final_answer": "y = - 7\\cdot x + 2\\cdot x\\cdot \\cos{\\left(2 \\right)} - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)} - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e09b520d78e5af0e", "problem_id": "d903d488291fd67c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 2 x^{2} + 2 x + \\left(x^{2} + 3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{x^{2} + 3 x - 5}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 5}) = \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3 x - 5}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{6 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{12 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 6 x^{3} - x^{2} - 30 x + 25} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{16 x^{2}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{3 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 6 x^{3} - x^{2} - 30 x + 25} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 6 x - 10} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3 x - 5} - \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 5} - \\frac{30 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{60 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{48 x}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 5} + \\frac{25 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{50 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{8}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 5}) = \\frac{x^{2} \\left(- 2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(x^{2} + 3 x\\right) - 5\\right)^{2}} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(x^{2} + 3 x\\right) - 5\\right)} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 5} + \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 5}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(- 2 x - 3\\right) \\left(- 2 x^{2} + 2 x + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(x \\left(x + 3\\right) - 5\\right) \\left(- 8 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x + 3\\right) - 5\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(- 2 x - 3\\right) \\left(- 2 x^{2} + 2 x + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(x \\left(x + 3\\right) - 5\\right) \\left(- 8 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x + 3\\right) - 5\\right)^{2}}$", "final_answer": "\\frac{2 \\left(- 2 x - 3\\right) \\left(- 2 x^{2} + 2 x + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(x \\left(x + 3\\right) - 5\\right) \\left(- 8 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x + 3\\right) - 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d731983712c8bbac", "problem_id": "961ca3c143b686d9", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} 4 x^{4} + 3 x^{3} + x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $4 x^{4} + 3 x^{3} + x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\pi).\n3) Subtract: F(\\pi) − F(- \\pi) = $\\frac{2 \\pi \\left(-45 + 5 \\pi^{2} + 12 \\pi^{4}\\right)}{15}$.\n4) Simplify the result.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(-45 + 5\\cdot \\pi^{2} + 12\\cdot \\pi^{4}\\right)}{15}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(-45 + 5\\cdot \\pi^{2} + 12\\cdot \\pi^{4}\\right)}{15}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8bfc634a4f7e566f", "problem_id": "961ca3c143b686d9", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} 4 x^{4} + 3 x^{3} + x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $4 x^{4} + 3 x^{3} + x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\nEvaluate at the bounds: F(\\pi) and F(- \\pi).\nSubtract: F(\\pi) − F(- \\pi) = $\\frac{2 \\pi \\left(-45 + 5 \\pi^{2} + 12 \\pi^{4}\\right)}{15}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(-45 + 5\\cdot \\pi^{2} + 12\\cdot \\pi^{4}\\right)}{15}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(-45 + 5\\cdot \\pi^{2} + 12\\cdot \\pi^{4}\\right)}{15}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "86b6c4ab6cd316f3", "problem_id": "4d2fd72c18a669c1", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the multiplicative inverse of 7 modulo 11.", "output": "1) We want x such that 7x ≡ 1 (mod 11).\n2) Use extended Euclid to find s,t with 7s + 11t = 1.\n3) One Bezout identity is: 7·(-3) + 11·(2) = 1.\n4) Reduce s modulo 11: x ≡ 8 (mod 11).\nFinal answer: $x\\equiv 8\\pmod{11}$", "final_answer": "x\\equiv 8\\pmod{11}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "41a57e3574826d12", "problem_id": "4d2fd72c18a669c1", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the multiplicative inverse of 7 modulo 11.", "output": "We want x such that 7x ≡ 1 (mod 11).\nUse extended Euclid to find s,t with 7s + 11t = 1.\nOne Bezout identity is: 7·(-3) + 11·(2) = 1.\nReduce s modulo 11: x ≡ 8 (mod 11).\nCheck: 7·8 mod 11 = 1.\nFinal answer: $x\\equiv 8\\pmod{11}$", "final_answer": "x\\equiv 8\\pmod{11}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "676e6f7c74132b8d", "problem_id": "8a31666b19b66863", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} x^{4} + 3 x^{3} + 3 x^{2} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $x^{4} + 3 x^{3} + 3 x^{2} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\n3) Subtract: F(\\frac{\\pi}{2}) − F(0) = $\\frac{\\pi^{5}}{160} + \\frac{\\pi^{3}}{8} + \\frac{3 \\pi^{4}}{64} + \\frac{\\pi^{2}}{2} + 5 + 2 \\pi$.\n4) Simplify the result.\nFinal answer: $\\frac{\\pi^{5}}{160} + \\frac{\\pi^{3}}{8} + \\frac{3\\cdot \\pi^{4}}{64} + \\frac{\\pi^{2}}{2} + 5 + 2\\cdot \\pi$", "final_answer": "\\frac{\\pi^{5}}{160} + \\frac{\\pi^{3}}{8} + \\frac{3\\cdot \\pi^{4}}{64} + \\frac{\\pi^{2}}{2} + 5 + 2\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d28217e1f9ad430f", "problem_id": "8a31666b19b66863", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} x^{4} + 3 x^{3} + 3 x^{2} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $x^{4} + 3 x^{3} + 3 x^{2} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\nSubtract: F(\\frac{\\pi}{2}) − F(0) = $\\frac{\\pi^{5}}{160} + \\frac{\\pi^{3}}{8} + \\frac{3 \\pi^{4}}{64} + \\frac{\\pi^{2}}{2} + 5 + 2 \\pi$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{\\pi^{5}}{160} + \\frac{\\pi^{3}}{8} + \\frac{3\\cdot \\pi^{4}}{64} + \\frac{\\pi^{2}}{2} + 5 + 2\\cdot \\pi$", "final_answer": "\\frac{\\pi^{5}}{160} + \\frac{\\pi^{3}}{8} + \\frac{3\\cdot \\pi^{4}}{64} + \\frac{\\pi^{2}}{2} + 5 + 2\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bbbbbf6abf63038b", "problem_id": "1a985e2d9e4dcec9", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the linear system:\n$- x + 7 y$ = $-63$\n$7 x + 6 y$ = $-54$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $7$: $- 7 x + 49 y$ = $-441$.\nMultiply the second equation by $-1$: $- 7 x - 6 y$ = $54$.\nFinal answer: $(x,y)=\\left(0, -9\\right)$", "final_answer": "(x,y)=\\left(0, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "043f2ed5bee39917", "problem_id": "1a985e2d9e4dcec9", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$- x + 7 y$ = $-63$\n$7 x + 6 y$ = $-54$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $7$: $- 7 x + 49 y$ = $-441$.\n3) Multiply the second equation by $-1$: $- 7 x - 6 y$ = $54$.\n4) Subtract to eliminate x: $55 y$ = $-495$.\n5) Solve for y: y = $-9$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, -9\\right)$", "final_answer": "(x,y)=\\left(0, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cbad449b6643c7e3", "problem_id": "d14866f0ac52d8c3", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(3 x^{2} + x - 2\\right) \\cos{\\left(3 x + 3 \\right)}$.", "output": "Let f(x) = $2 \\left(x + 1\\right) e^{x} + \\left(3 x^{2} + x - 2\\right) \\cos{\\left(3 x + 3 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $2 \\left(x + 1\\right) e^{x} + \\left(6 x + 1\\right) \\cos{\\left(3 x + 3 \\right)} - 3 \\left(3 x^{2} + x - 2\\right) \\sin{\\left(3 x + 3 \\right)} + 2 e^{x}$.\nFinal answer: $\\left(2\\cdot x + 2\\right)\\cdot e^{x} + \\left(6\\cdot x + 1\\right)\\cdot \\cos{\\left(3\\cdot x + 3 \\right)} - \\left(9\\cdot x^{2} + 3\\cdot x - 6\\right)\\cdot \\sin{\\left(3\\cdot x + 3 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x + 2\\right)\\cdot e^{x} + \\left(6\\cdot x + 1\\right)\\cdot \\cos{\\left(3\\cdot x + 3 \\right)} - \\left(9\\cdot x^{2} + 3\\cdot x - 6\\right)\\cdot \\sin{\\left(3\\cdot x + 3 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "edda8d4cc5d45954", "problem_id": "d14866f0ac52d8c3", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(3 x^{2} + x - 2\\right) \\cos{\\left(3 x + 3 \\right)}$.", "output": "1) Let f(x) = $2 \\left(x + 1\\right) e^{x} + \\left(3 x^{2} + x - 2\\right) \\cos{\\left(3 x + 3 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $2 \\left(x + 1\\right) e^{x} + \\left(6 x + 1\\right) \\cos{\\left(3 x + 3 \\right)} - 3 \\left(3 x^{2} + x - 2\\right) \\sin{\\left(3 x + 3 \\right)} + 2 e^{x}$.\nFinal answer: $\\left(2\\cdot x + 2\\right)\\cdot e^{x} + \\left(6\\cdot x + 1\\right)\\cdot \\cos{\\left(3\\cdot x + 3 \\right)} - \\left(9\\cdot x^{2} + 3\\cdot x - 6\\right)\\cdot \\sin{\\left(3\\cdot x + 3 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x + 2\\right)\\cdot e^{x} + \\left(6\\cdot x + 1\\right)\\cdot \\cos{\\left(3\\cdot x + 3 \\right)} - \\left(9\\cdot x^{2} + 3\\cdot x - 6\\right)\\cdot \\sin{\\left(3\\cdot x + 3 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e90fc0f1fe55a8bf", "problem_id": "0bfe1547e47bdb0e", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A disease has prevalence $\\frac{1}{50}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{29}{625}$.\nCompute P(D|+) = $\\frac{85}{232}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{85}{232}$", "final_answer": "\\frac{85}{232}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f286d7d74ee54873", "problem_id": "0bfe1547e47bdb0e", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A disease has prevalence $\\frac{1}{50}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{29}{625}$.\n3) Compute P(D|+) = $\\frac{85}{232}$.\nFinal answer: $\\frac{85}{232}$", "final_answer": "\\frac{85}{232}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d3c2546e3f4785d3", "problem_id": "cb77b073bb8e944c", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$2 x$ = $-4$\n$- x + 4 y$ = $18$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $- 2 x$ = $4$.\n3) Multiply the second equation by $2$: $- 2 x + 8 y$ = $36$.\n4) Subtract to eliminate x: $- 8 y$ = $-32$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, 4\\right)$", "final_answer": "(x,y)=\\left(-2, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f2b76908591d26c3", "problem_id": "52119ff68cb17c7e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(- 3 x^{3} + x^{2} + 2 x - 2\\right) \\cos{\\left(x - 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x^{2} \\cos{\\left(x - 3 \\right)}) = - x^{2} \\sin{\\left(x - 3 \\right)}$.\n3) So $f'(x) = 3 x^{3} \\sin{\\left(x - 3 \\right)} - x^{2} \\sin{\\left(x - 3 \\right)} - 9 x^{2} \\cos{\\left(x - 3 \\right)} + x e^{x} - 2 x \\sin{\\left(x - 3 \\right)} + 3 e^{x} + 2 \\sin{\\left(x - 3 \\right)} + 2 \\cos{\\left(x - 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x^{2} \\cos{\\left(x - 3 \\right)}) = - x^{2} \\sin{\\left(x - 3 \\right)} + 2 x \\cos{\\left(x - 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 2\\right) e^{x} - \\left(x \\left(x \\left(1 - 3 x\\right) + 2\\right) - 2\\right) \\sin{\\left(x - 3 \\right)} + \\left(- 9 x^{2} + 2 x + 2\\right) \\cos{\\left(x - 3 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 2\\right) e^{x} - \\left(x \\left(x \\left(1 - 3 x\\right) + 2\\right) - 2\\right) \\sin{\\left(x - 3 \\right)} + \\left(- 9 x^{2} + 2 x + 2\\right) \\cos{\\left(x - 3 \\right)} + e^{x}$", "final_answer": "\\left(x + 2\\right) e^{x} - \\left(x \\left(x \\left(1 - 3 x\\right) + 2\\right) - 2\\right) \\sin{\\left(x - 3 \\right)} + \\left(- 9 x^{2} + 2 x + 2\\right) \\cos{\\left(x - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a459d1a2f7a1624c", "problem_id": "ded5f5ab78bb3fa0", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "An urn has 15 red balls and 28 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(43,9).\nFavorable samples: choose k reds and n−k blues: C(15,1)·C(28,8).\nFinal answer: $\\frac{102465}{1239389}$", "final_answer": "\\frac{102465}{1239389}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8c1ca3d2fa1691b7", "problem_id": "ded5f5ab78bb3fa0", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "An urn has 15 red balls and 28 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(43,9).\n3) Favorable samples: choose k reds and n−k blues: C(15,1)·C(28,8).\n4) Compute totals: C(43,9) = 563921995, and favorable = 46621575.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{102465}{1239389}$", "final_answer": "\\frac{102465}{1239389}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2b3cbadc0725d94f", "problem_id": "cf186e9e958657cb", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡3 (mod 4), x≡6 (mod 11). Give the solution modulo 44.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 44.\n3) Compute the CRT solution to get x ≡ 39 (mod 44).\nFinal answer: $x\\equiv 39\\pmod{44}$", "final_answer": "x\\equiv 39\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "51ee4255fe45d938", "problem_id": "d559cfa8e57a172d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 \\cos{\\left(2 x \\right)}) = - 3 \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 3 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 \\cos{\\left(2 x \\right)}) = - 6 \\sin{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = 3 \\left(x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}\\right)$.\nFinal answer: $3 \\left(x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}\\right)$", "final_answer": "3 \\left(x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7ae06c4d318e2e45", "problem_id": "9b79e1058bb96e39", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "You want 47 liters of a 35% salt solution. You have a 20% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 47−x liters is 40% solution.\n2) Salt equation: 20x + 40(47−x) = 35·47.\n3) (20-40)x = (35-40)·47 ⇒ x = 47/4.\n4) Then 47−x = 141/4.\nFinal answer: $47/4 L of 20% and 141/4 L of 40%$", "final_answer": "47/4 L of 20% and 141/4 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "478e34cb86030e67", "problem_id": "59c32a17a05252ed", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 3 x^{3} + 11 x^{2} - 31 x + 92$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{3} + 11 x^{2} - 32 x + 92.\n3) Then the limit is 368.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: 365.\nFinal answer: $365$", "final_answer": "365", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7a0508fd678299dc", "problem_id": "d5c83e10cdef30ac", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡1 (mod 11), x≡2 (mod 10). Give the solution modulo 110.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 110.\n3) Compute the CRT solution to get x ≡ 12 (mod 110).\nFinal answer: $x\\equiv 12\\pmod{110}$", "final_answer": "x\\equiv 12\\pmod{110}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f8801999618e5d03", "problem_id": "d5c83e10cdef30ac", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the congruences: x≡1 (mod 11), x≡2 (mod 10). Give the solution modulo 110.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 110.\nCompute the CRT solution to get x ≡ 12 (mod 110).\nFinal answer: $x\\equiv 12\\pmod{110}$", "final_answer": "x\\equiv 12\\pmod{110}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ca0dfcb2b7b45d3d", "problem_id": "a636c7de333ca721", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A store sells a backpack for $840. It is discounted by 15%. You buy 9 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 840 × (1 − 15/100) = 840 × 85/100 = 714.\n2) Subtotal = 9 × 714 = 6426.\n3) Total with tax = 6426 × 110/100 = 35343/5.\nFinal answer: $\\frac{35343}{5}$", "final_answer": "\\frac{35343}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "863dab9d78583859", "problem_id": "a636c7de333ca721", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $840. It is discounted by 15%. You buy 9 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 840 × (1 − 15/100) = 840 × 85/100 = 714.\nSubtotal = 9 × 714 = 6426.\nTotal with tax = 6426 × 110/100 = 35343/5.\nCheck: recomputing the formula yields 35343/5.\nFinal answer: $\\frac{35343}{5}$", "final_answer": "\\frac{35343}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d2ccd40d628236d1", "problem_id": "2c8d970f33f1adea", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 2 x^{3} + 3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $2 x^{3} + 3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $- \\frac{9 \\pi^{2}}{2} - 3 \\pi - 4 + 9 \\pi^{3} + \\frac{15 \\pi^{4}}{2}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{9\\cdot \\pi^{2}}{2} - 3\\cdot \\pi - 4 + 9\\cdot \\pi^{3} + \\frac{15\\cdot \\pi^{4}}{2}$", "final_answer": "- \\frac{9\\cdot \\pi^{2}}{2} - 3\\cdot \\pi - 4 + 9\\cdot \\pi^{3} + \\frac{15\\cdot \\pi^{4}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "80bd4226cf40e586", "problem_id": "2c8d970f33f1adea", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 2 x^{3} + 3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{3} + 3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $- \\frac{9 \\pi^{2}}{2} - 3 \\pi - 4 + 9 \\pi^{3} + \\frac{15 \\pi^{4}}{2}$.\n4) Simplify the result.\nFinal answer: $- \\frac{9\\cdot \\pi^{2}}{2} - 3\\cdot \\pi - 4 + 9\\cdot \\pi^{3} + \\frac{15\\cdot \\pi^{4}}{2}$", "final_answer": "- \\frac{9\\cdot \\pi^{2}}{2} - 3\\cdot \\pi - 4 + 9\\cdot \\pi^{3} + \\frac{15\\cdot \\pi^{4}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e8be039a49d8ef40", "problem_id": "e73242e102210611", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- 3 \\pi + 2 + \\frac{9 \\pi^{2}}{8} + \\frac{3 \\pi^{3}}{4}$.\n4) Simplify the result.\nFinal answer: $- 3\\cdot \\pi + 2 + \\frac{9\\cdot \\pi^{2}}{8} + \\frac{3\\cdot \\pi^{3}}{4}$", "final_answer": "- 3\\cdot \\pi + 2 + \\frac{9\\cdot \\pi^{2}}{8} + \\frac{3\\cdot \\pi^{3}}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "df202bb723a04fcc", "problem_id": "3787fad3e50f603c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} + 4$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(3 x^{2}) = 6 x^{2}$.\n3) So $f'(x) = 6 x^{2}$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(3 x^{2}) = 6 x$.\nTherefore the correct derivative is $f'(x) = 6 x$.\nFinal answer: $6 x$", "final_answer": "6 x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "438ea462874d7ad4", "problem_id": "e1373520055de1b4", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the linear system:\n$- 7 x + 8 y$ = $2$\n$- 2 x - 3 y$ = $27$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $14 x - 16 y$ = $-4$.\nMultiply the second equation by $-7$: $14 x + 21 y$ = $-189$.\nFinal answer: $(x,y)=\\left(-6, -5\\right)$", "final_answer": "(x,y)=\\left(-6, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "79ce3afc14068a8e", "problem_id": "e1373520055de1b4", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$- 7 x + 8 y$ = $2$\n$- 2 x - 3 y$ = $27$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $14 x - 16 y$ = $-4$.\n3) Multiply the second equation by $-7$: $14 x + 21 y$ = $-189$.\n4) Subtract to eliminate x: $- 37 y$ = $185$.\n5) Solve for y: y = $-5$.\n6) Substitute back to find x: x = $-6$.\nFinal answer: $(x,y)=\\left(-6, -5\\right)$", "final_answer": "(x,y)=\\left(-6, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a07ca6de1a36e01a", "problem_id": "b18dc57c389a574a", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - \\frac{19 x}{4} + 1$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$- \\frac{19}{4}$, c=$1$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{4}{3}, \\frac{1}{4}\\}$", "final_answer": "x \\in \\{\\frac{4}{3}, \\frac{1}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "09beafd33fc5bf93", "problem_id": "b18dc57c389a574a", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - \\frac{19 x}{4} + 1$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$- \\frac{19}{4}$, c=$1$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{169}{16}$, so √Δ = $\\frac{13}{4}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{4}{3}$.\n7) x2 = $\\frac{1}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{4}{3}, \\frac{1}{4}\\}$", "final_answer": "x \\in \\{\\frac{4}{3}, \\frac{1}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dc783090ffb9f377", "problem_id": "c1bc557687aa8004", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "You want 16 liters of a 30% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 16−x liters is 70% solution.\nSalt equation: 20x + 70(16−x) = 30·16.\n(20-70)x = (30-70)·16 ⇒ x = 64/5.\nThen 16−x = 16/5.\nCheck: 20·64/5 + 70·16/5 = 30·16.\nFinal answer: $64/5 L of 20% and 16/5 L of 70%$", "final_answer": "64/5 L of 20% and 16/5 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fd08b9a64247db5a", "problem_id": "c1bc557687aa8004", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "You want 16 liters of a 30% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 16−x liters is 70% solution.\n2) Salt equation: 20x + 70(16−x) = 30·16.\n3) (20-70)x = (30-70)·16 ⇒ x = 64/5.\n4) Then 16−x = 16/5.\nFinal answer: $64/5 L of 20% and 16/5 L of 70%$", "final_answer": "64/5 L of 20% and 16/5 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c0ecd50ad317fc5c", "problem_id": "64fcb233dd7aee28", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$- y$ = $-1$\n$x + 3 y$ = $-2$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $- y$ = $-1$.\nMultiply the second equation by $0$: $0$ = $0$.\nSubtract to eliminate x: $- y$ = $-1$.\nSolve for y: y = $1$.\nSubstitute back to find x: x = $-5$.\nCheck: plug (x,y)=(-5,1) into both equations.\nFinal answer: $(x,y)=\\left(-5, 1\\right)$", "final_answer": "(x,y)=\\left(-5, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d4983708fafc9b1f", "problem_id": "64fcb233dd7aee28", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$- y$ = $-1$\n$x + 3 y$ = $-2$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $- y$ = $-1$.\n3) Multiply the second equation by $0$: $0$ = $0$.\n4) Subtract to eliminate x: $- y$ = $-1$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, 1\\right)$", "final_answer": "(x,y)=\\left(-5, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1151a94a98c51a42", "problem_id": "8cb2c612c6b331f3", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the multiplicative inverse of 10 modulo 19.", "output": "1) We want x such that 10x ≡ 1 (mod 19).\n2) Use extended Euclid to find s,t with 10s + 19t = 1.\n3) One Bezout identity is: 10·(2) + 19·(-1) = 1.\n4) Reduce s modulo 19: x ≡ 2 (mod 19).\nFinal answer: $x\\equiv 2\\pmod{19}$", "final_answer": "x\\equiv 2\\pmod{19}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6c16cd338cd9efdc", "problem_id": "5c107b6a4bb1aea3", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - 2 x^{2} - 10 x - 42$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{2} - 11 x - 42.\n3) Then the limit is -118.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: -114.\nFinal answer: $-114$", "final_answer": "-114", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "cd0cb0c9e36b566e", "problem_id": "115b862cd75cf87e", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the multiplicative inverse of 10 modulo 37.", "output": "1) We want x such that 10x ≡ 1 (mod 37).\n2) Use extended Euclid to find s,t with 10s + 37t = 1.\n3) One Bezout identity is: 10·(-11) + 37·(3) = 1.\n4) Reduce s modulo 37: x ≡ 26 (mod 37).\nFinal answer: $x\\equiv 26\\pmod{37}$", "final_answer": "x\\equiv 26\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "013d3da8a5949188", "problem_id": "115b862cd75cf87e", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Find the multiplicative inverse of 10 modulo 37.", "output": "We want x such that 10x ≡ 1 (mod 37).\nUse extended Euclid to find s,t with 10s + 37t = 1.\nOne Bezout identity is: 10·(-11) + 37·(3) = 1.\nFinal answer: $x\\equiv 26\\pmod{37}$", "final_answer": "x\\equiv 26\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9590a11e1254f30b", "problem_id": "53dc42542e8e3cbe", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 10 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=3, p=$\\frac{1}{4}$ and simplify.\n4) Here C(10,3) = 120.\nFinal answer: $\\frac{32805}{131072}$", "final_answer": "\\frac{32805}{131072}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4c433fad79b6ee9c", "problem_id": "43199a59849d810a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(3 x^{3} - 3 x^{2} + 2\\right) \\cos{\\left(x - 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 3 x^{2} \\cos{\\left(x - 3 \\right)}) = 3 x^{2} \\sin{\\left(x - 3 \\right)}$.\n3) So $f'(x) = - 3 x^{3} \\sin{\\left(x - 3 \\right)} + 3 x^{2} \\sin{\\left(x - 3 \\right)} + 9 x^{2} \\cos{\\left(x - 3 \\right)} + x e^{x} + 3 e^{x} - 2 \\sin{\\left(x - 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 3 x^{2} \\cos{\\left(x - 3 \\right)}) = 3 x^{2} \\sin{\\left(x - 3 \\right)} - 6 x \\cos{\\left(x - 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = 3 x \\left(3 x - 2\\right) \\cos{\\left(x - 3 \\right)} + \\left(x + 2\\right) e^{x} - \\left(3 x^{2} \\left(x - 1\\right) + 2\\right) \\sin{\\left(x - 3 \\right)} + e^{x}$.\nFinal answer: $3 x \\left(3 x - 2\\right) \\cos{\\left(x - 3 \\right)} + \\left(x + 2\\right) e^{x} - \\left(3 x^{2} \\left(x - 1\\right) + 2\\right) \\sin{\\left(x - 3 \\right)} + e^{x}$", "final_answer": "3 x \\left(3 x - 2\\right) \\cos{\\left(x - 3 \\right)} + \\left(x + 2\\right) e^{x} - \\left(3 x^{2} \\left(x - 1\\right) + 2\\right) \\sin{\\left(x - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "55742f2f50fc41fd", "problem_id": "99b06a6bb87dc2b6", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 0.\n3) Then the limit is 0.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 2.\nFinal answer: $2$", "final_answer": "2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "254bc9debb013864", "problem_id": "7e3a2f6f278e814c", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Worker A finishes a job in 5 hours and Worker B finishes it in 14 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/5 job/hour. B's rate = 1/14 job/hour.\n2) Combined rate = 1/5 + 1/14 = 19/70 job/hour.\n3) Time = 1 ÷ (combined rate) = 70/19 hours.\nFinal answer: $\\frac{70}{19}$", "final_answer": "\\frac{70}{19}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dc5459bef9af3515", "problem_id": "11024044ec7ec250", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{3} + 4 x^{2} + 9 x + 20$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $2 x^{4} + x^{2} + 2 x - 1$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $8 x^{3} + 2 x + 2$.\n5) Evaluate at a: p'(2) = $70$.\nFinal answer: $70$", "final_answer": "70", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a72daf2cd9d18587", "problem_id": "11024044ec7ec250", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{3} + 4 x^{2} + 9 x + 20$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{4} + x^{2} + 2 x - 1$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $8 x^{3} + 2 x + 2$.\nEvaluate at a: p'(2) = $70$.\nCheck: the identity gives $70$.\nFinal answer: $70$", "final_answer": "70", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c8c66cf932b7b4dc", "problem_id": "2826f3434c0f2591", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Worker A finishes a job in 15 hours and Worker B finishes it in 10 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/15 job/hour. B's rate = 1/10 job/hour.\nCombined rate = 1/15 + 1/10 = 1/6 job/hour.\nTime = 1 ÷ (combined rate) = 6 hours.\nCheck: (combined rate)×(time) = 1/6×6 = 1.\nFinal answer: $6$", "final_answer": "6", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "40dc3284a8b6dd7c", "problem_id": "b46d5bc7fb935668", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 2 x^{2} + 5 x - 5$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{2} + 4 x - 5.\n3) Then the limit is -11.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: -12.\nFinal answer: $-12$", "final_answer": "-12", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "742b2410a9777aea", "problem_id": "fcb9bd0865def3fa", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 3 x - 3$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{2} - 3$ and a = -1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $6 x$.\n5) Evaluate at a: p'(-1) = $-6$.\nFinal answer: $-6$", "final_answer": "-6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "52e5b6d4149a1a82", "problem_id": "06e4bc2eebfcb779", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} 2 x^{3} + x^{2} + 4 x + 1$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} + x^{2} + 3 x + 1.\n3) Then the limit is 7.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: 8.\nFinal answer: $8$", "final_answer": "8", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6b9d0a117feb7952", "problem_id": "eec3e433000893c7", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - x^{3} - 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- x^{3} - 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}$.\n3) Compute an antiderivative: F(x) = $- \\frac{x^{4}}{4} - \\frac{2 x^{3}}{3} + \\frac{3 x^{2}}{2} - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{x^{4}}{4} - \\frac{2\\cdot x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{x^{4}}{4} - \\frac{2\\cdot x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "675b4349cc6a01a8", "problem_id": "eec3e433000893c7", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - x^{3} - 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- x^{3} - 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}$.\nCompute an antiderivative: F(x) = $- \\frac{x^{4}}{4} - \\frac{2 x^{3}}{3} + \\frac{3 x^{2}}{2} - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $C - \\frac{x^{4}}{4} - \\frac{2\\cdot x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{x^{4}}{4} - \\frac{2\\cdot x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "841fbd3763458865", "problem_id": "f4bb53485da72686", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{4} - 3 x^{2} + x - 4$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x) = x$.\n3) So $f'(x) = 16 x^{3} - 5 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x) = 1$.\nTherefore the correct derivative is $f'(x) = 16 x^{3} - 6 x + 1$.\nFinal answer: $16 x^{3} - 6 x + 1$", "final_answer": "16 x^{3} - 6 x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "cd02b58e093e21c7", "problem_id": "bc8fc1d01cf40016", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_6=18$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_6=0.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=0 and k=6.\n4) Compute C(5,5) = 1.\nFinal answer: $1$", "final_answer": "1", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e83168f96dea9876", "problem_id": "8fd197769ad52a9f", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Worker A finishes a job in 10 hours and Worker B finishes it in 15 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/10 job/hour. B's rate = 1/15 job/hour.\n2) Combined rate = 1/10 + 1/15 = 1/6 job/hour.\n3) Time = 1 ÷ (combined rate) = 6 hours.\nFinal answer: $6$", "final_answer": "6", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d9d3c660c2d6a904", "problem_id": "8fd197769ad52a9f", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Worker A finishes a job in 10 hours and Worker B finishes it in 15 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/10 job/hour. B's rate = 1/15 job/hour.\nCombined rate = 1/10 + 1/15 = 1/6 job/hour.\nTime = 1 ÷ (combined rate) = 6 hours.\nCheck: (combined rate)×(time) = 1/6×6 = 1.\nFinal answer: $6$", "final_answer": "6", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2cc1e1f3e29067f9", "problem_id": "0477b9f26d5b5b77", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A store sells a backpack for $75. It is discounted by 10%. You buy 9 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 75 × (1 − 10/100) = 75 × 90/100 = 135/2.\n2) Subtotal = 9 × 135/2 = 1215/2.\n3) Total with tax = 1215/2 × 108/100 = 6561/10.\nFinal answer: $\\frac{6561}{10}$", "final_answer": "\\frac{6561}{10}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "18b45799c5ebb24d", "problem_id": "0477b9f26d5b5b77", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A store sells a backpack for $75. It is discounted by 10%. You buy 9 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 75 × (1 − 10/100) = 75 × 90/100 = 135/2.\nSubtotal = 9 × 135/2 = 1215/2.\nTotal with tax = 1215/2 × 108/100 = 6561/10.\nFinal answer: $\\frac{6561}{10}$", "final_answer": "\\frac{6561}{10}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b2f0a823160cc6a3", "problem_id": "a53bad13f9097225", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{3} + 3 x^{2} - 4 x$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 3 x^{3}) = - 9 x^{3}$.\n3) So $f'(x) = - 9 x^{3} + 6 x - 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 3 x^{3}) = - 9 x^{2}$.\nTherefore the correct derivative is $f'(x) = - 9 x^{2} + 6 x - 4$.\nFinal answer: $- 9 x^{2} + 6 x - 4$", "final_answer": "- 9 x^{2} + 6 x - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5821489d3d63d1f8", "problem_id": "6b0e3c2fd06ac29e", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $- 3 x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.\nFinal answer: $- 3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - \\cos{\\left(x \\right)}$", "final_answer": "- 3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b33ba2381b31321b", "problem_id": "ab726daf6318eaad", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{7}$. If it is flipped 22 times, what is the probability of exactly 7 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=22, k=7, p=$\\frac{1}{7}$ and simplify.\n4) Here C(22,7) = 170544.\nFinal answer: $\\frac{80187228009529344}{3909821048582988049}$", "final_answer": "\\frac{80187228009529344}{3909821048582988049}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "acc18d8209764610", "problem_id": "208ef6ba63abd874", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{3} - 2 x^{2} + 3 x - 4$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x^{2}) = - 4 x^{2}$.\n3) So $f'(x) = 2 x^{2} + 3$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x^{2}) = - 4 x$.\nTherefore the correct derivative is $f'(x) = 6 x^{2} - 4 x + 3$.\nFinal answer: $6 x^{2} - 4 x + 3$", "final_answer": "6 x^{2} - 4 x + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f7acb8d86710913d", "problem_id": "dfbce3242d325829", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$5 x + 8 y$ = $-37$\n$- 3 x - 6 y$ = $21$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $- 15 x - 24 y$ = $111$.\nMultiply the second equation by $5$: $- 15 x - 30 y$ = $105$.\nSubtract to eliminate x: $6 y$ = $6$.\nSolve for y: y = $1$.\nSubstitute back to find x: x = $-9$.\nCheck: plug (x,y)=(-9,1) into both equations.\nFinal answer: $(x,y)=\\left(-9, 1\\right)$", "final_answer": "(x,y)=\\left(-9, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c2315d9d949b105c", "problem_id": "dfbce3242d325829", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$5 x + 8 y$ = $-37$\n$- 3 x - 6 y$ = $21$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $- 15 x - 24 y$ = $111$.\n3) Multiply the second equation by $5$: $- 15 x - 30 y$ = $105$.\n4) Subtract to eliminate x: $6 y$ = $6$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $-9$.\nFinal answer: $(x,y)=\\left(-9, 1\\right)$", "final_answer": "(x,y)=\\left(-9, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1e918ffb9c50cbb1", "problem_id": "aed1a0eaae1d9cef", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{3} + x^{2} + 4 x + 6$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} + x^{2} + 3 x + 6.\n3) Then the limit is 32.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 34.\nFinal answer: $34$", "final_answer": "34", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "fcac51c5b05fa947", "problem_id": "bdefc566f0b334af", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "You want 43 liters of a 27% salt solution. You have a 15% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 43−x liters is 40% solution.\n2) Salt equation: 15x + 40(43−x) = 27·43.\n3) (15-40)x = (27-40)·43 ⇒ x = 559/25.\n4) Then 43−x = 516/25.\nFinal answer: $559/25 L of 15% and 516/25 L of 40%$", "final_answer": "559/25 L of 15% and 516/25 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1ea6238a88159506", "problem_id": "f88fcd176d3a4708", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} - x^{2} + 4 x - 4$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $6 x^{2} - 2 x + 4$.\n2) Slope at x=-2: m = f'(-2) = $32$.\n3) Point on curve: (-2, f(-2)) = (-2, $-32$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -32 = 32(x − -2).\n6) Optionally expand: y = $32 x + 32$.\nFinal answer: $y = 32\\cdot x + 32$", "final_answer": "y = 32\\cdot x + 32", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "811e63c57b60fa52", "problem_id": "f88fcd176d3a4708", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} - x^{2} + 4 x - 4$ at $x=-2$.", "output": "Differentiate: f'(x) = $6 x^{2} - 2 x + 4$.\nSlope at x=-2: m = f'(-2) = $32$.\nPoint on curve: (-2, f(-2)) = (-2, $-32$).\nFinal answer: $y = 32\\cdot x + 32$", "final_answer": "y = 32\\cdot x + 32", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "243a1c33c76e6ecc", "problem_id": "66ca8517bba5380c", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{2}$ at $x=0$.", "output": "Differentiate: f'(x) = $6 x$.\nSlope at x=0: m = f'(0) = $0$.\nPoint on curve: (0, f(0)) = (0, $0$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 0 = 0(x − 0).\nOptionally expand: y = $0$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 0$", "final_answer": "y = 0", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "10c583712d9d4edd", "problem_id": "66ca8517bba5380c", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{2}$ at $x=0$.", "output": "1) Differentiate: f'(x) = $6 x$.\n2) Slope at x=0: m = f'(0) = $0$.\n3) Point on curve: (0, f(0)) = (0, $0$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 0 = 0(x − 0).\n6) Optionally expand: y = $0$.\nFinal answer: $y = 0$", "final_answer": "y = 0", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4a15009e69f69ea8", "problem_id": "0654d4bd6fa02d23", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Worker A finishes a job in 19 hours and Worker B finishes it in 12 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/19 job/hour. B's rate = 1/12 job/hour.\n2) Combined rate = 1/19 + 1/12 = 31/228 job/hour.\n3) Time = 1 ÷ (combined rate) = 228/31 hours.\nFinal answer: $\\frac{228}{31}$", "final_answer": "\\frac{228}{31}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "66c02548c8a27e31", "problem_id": "0654d4bd6fa02d23", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Worker A finishes a job in 19 hours and Worker B finishes it in 12 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/19 job/hour. B's rate = 1/12 job/hour.\nCombined rate = 1/19 + 1/12 = 31/228 job/hour.\nTime = 1 ÷ (combined rate) = 228/31 hours.\nCheck: (combined rate)×(time) = 31/228×228/31 = 1.\nFinal answer: $\\frac{228}{31}$", "final_answer": "\\frac{228}{31}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8b66bf0bfc43d5d7", "problem_id": "3555836a498dc79c", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A cyclist rides at 38 miles per hour for 1/2 hours, then at 68 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 38 × 1/2 = 19 miles.\nSecond leg distance = 68 × 3 = 204 miles.\nTotal distance = 19 + 204 = 223 miles.\nCheck: adding both legs gives 223.\nFinal answer: $223$", "final_answer": "223", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fd48d29a5a2c2555", "problem_id": "3555836a498dc79c", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A cyclist rides at 38 miles per hour for 1/2 hours, then at 68 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 38 × 1/2 = 19 miles.\n2) Second leg distance = 68 × 3 = 204 miles.\n3) Total distance = 19 + 204 = 223 miles.\nFinal answer: $223$", "final_answer": "223", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e7db66cf4a9b5074", "problem_id": "a376e825e8735392", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 25 miles per hour for 1 hours, then at 36 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 25 × 1 = 25 miles.\nSecond leg distance = 36 × 2 = 72 miles.\nTotal distance = 25 + 72 = 97 miles.\nCheck: adding both legs gives 97.\nFinal answer: $97$", "final_answer": "97", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "89b749e251c7c432", "problem_id": "a376e825e8735392", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A cyclist rides at 25 miles per hour for 1 hours, then at 36 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 25 × 1 = 25 miles.\nSecond leg distance = 36 × 2 = 72 miles.\nTotal distance = 25 + 72 = 97 miles.\nFinal answer: $97$", "final_answer": "97", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f78e4497d095fed7", "problem_id": "dcf213243a612250", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} - 3 x^{4} + 4 x^{3} + 4 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{4} + 4 x^{3} + 4 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2$.\nEvaluate at the bounds: F(2 \\pi) and F(0).\nSubtract: F(2 \\pi) − F(0) = $\\frac{2 \\pi \\left(- 144 \\pi^{4} + 30 + 45 \\pi + 80 \\pi^{2} + 120 \\pi^{3}\\right)}{15}$.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(- 144\\cdot \\pi^{4} + 30 + 45\\cdot \\pi + 80\\cdot \\pi^{2} + 120\\cdot \\pi^{3}\\right)}{15}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(- 144\\cdot \\pi^{4} + 30 + 45\\cdot \\pi + 80\\cdot \\pi^{2} + 120\\cdot \\pi^{3}\\right)}{15}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0a41c519159565ab", "problem_id": "dcf213243a612250", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} - 3 x^{4} + 4 x^{3} + 4 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{4} + 4 x^{3} + 4 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(0).\n3) Subtract: F(2 \\pi) − F(0) = $\\frac{2 \\pi \\left(- 144 \\pi^{4} + 30 + 45 \\pi + 80 \\pi^{2} + 120 \\pi^{3}\\right)}{15}$.\n4) Simplify the result.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(- 144\\cdot \\pi^{4} + 30 + 45\\cdot \\pi + 80\\cdot \\pi^{2} + 120\\cdot \\pi^{3}\\right)}{15}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(- 144\\cdot \\pi^{4} + 30 + 45\\cdot \\pi + 80\\cdot \\pi^{2} + 120\\cdot \\pi^{3}\\right)}{15}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f21d05800f12c015", "problem_id": "dfdef6efb964e4c9", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "You want 28 liters of a 45% salt solution. You have a 30% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 28−x liters is 60% solution.\n2) Salt equation: 30x + 60(28−x) = 45·28.\n3) (30-60)x = (45-60)·28 ⇒ x = 14.\n4) Then 28−x = 14.\nFinal answer: $14 L of 30% and 14 L of 60%$", "final_answer": "14 L of 30% and 14 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ce5153d324c62c5", "problem_id": "660f83bc358400a6", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{3} - 3 x^{2} - 2$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(4 x^{3}) = 12 x^{3}$.\n3) So $f'(x) = 12 x^{3} - 6 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(4 x^{3}) = 12 x^{2}$.\nTherefore the correct derivative is $f'(x) = 6 x \\left(2 x - 1\\right)$.\nFinal answer: $6 x \\left(2 x - 1\\right)$", "final_answer": "6 x \\left(2 x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6bb4d141b7efaa4c", "problem_id": "8c2b46771a27cbde", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} \\sin{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x \\sin{\\left(x \\right)}) = 2 x \\cos{\\left(x \\right)}$.\n3) So $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x \\sin{\\left(x \\right)}) = 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.\nFinal answer: $x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$", "final_answer": "x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "65b054a57b5c6483", "problem_id": "f8159c307ddf2c84", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{6} x}{2} + \\frac{x}{10} - \\frac{9 \\sqrt{6}}{10} - \\frac{3}{10}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{1}{10} - \\frac{3 \\sqrt{6}}{2}$, c=$- \\frac{9 \\sqrt{6}}{10} - \\frac{3}{10}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{33 \\sqrt{6}}{10} + \\frac{1471}{100}$, so √Δ = $\\sqrt{\\frac{33 \\sqrt{6}}{10} + \\frac{1471}{100}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{1}{20} + \\frac{3 \\sqrt{6}}{4} + \\frac{\\sqrt{330 \\sqrt{6} + 1471}}{20}$.\n7) x2 = $- \\frac{\\sqrt{330 \\sqrt{6} + 1471}}{20} - \\frac{1}{20} + \\frac{3 \\sqrt{6}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{1}{20} + \\frac{3\\cdot \\sqrt{6}}{4} + \\frac{\\sqrt{330\\cdot \\sqrt{6} + 1471}}{20}, - \\frac{\\sqrt{330\\cdot \\sqrt{6} + 1471}}{20} - \\frac{1}{20} + \\frac{3\\cdot \\sqrt{6}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{1}{20} + \\frac{3\\cdot \\sqrt{6}}{4} + \\frac{\\sqrt{330\\cdot \\sqrt{6} + 1471}}{20}, - \\frac{\\sqrt{330\\cdot \\sqrt{6} + 1471}}{20} - \\frac{1}{20} + \\frac{3\\cdot \\sqrt{6}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "802dda6c8cd40fbf", "problem_id": "1b0f04e333a856fe", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "You want 20 liters of a 35% salt solution. You have a 30% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 20−x liters is 70% solution.\n2) Salt equation: 30x + 70(20−x) = 35·20.\n3) (30-70)x = (35-70)·20 ⇒ x = 35/2.\n4) Then 20−x = 5/2.\nFinal answer: $35/2 L of 30% and 5/2 L of 70%$", "final_answer": "35/2 L of 30% and 5/2 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ac9e4294422c777d", "problem_id": "1b0f04e333a856fe", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 20 liters of a 35% salt solution. You have a 30% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 20−x liters is 70% solution.\nSalt equation: 30x + 70(20−x) = 35·20.\n(30-70)x = (35-70)·20 ⇒ x = 35/2.\nThen 20−x = 5/2.\nCheck: 30·35/2 + 70·5/2 = 35·20.\nFinal answer: $35/2 L of 30% and 5/2 L of 70%$", "final_answer": "35/2 L of 30% and 5/2 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fef2f7b54d3ab1eb", "problem_id": "ce58e42d525a772f", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the multiplicative inverse of 36 modulo 53.", "output": "1) We want x such that 36x ≡ 1 (mod 53).\n2) Use extended Euclid to find s,t with 36s + 53t = 1.\n3) One Bezout identity is: 36·(-25) + 53·(17) = 1.\n4) Reduce s modulo 53: x ≡ 28 (mod 53).\nFinal answer: $x\\equiv 28\\pmod{53}$", "final_answer": "x\\equiv 28\\pmod{53}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "914d2c2481abc5fc", "problem_id": "ce58e42d525a772f", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the multiplicative inverse of 36 modulo 53.", "output": "We want x such that 36x ≡ 1 (mod 53).\nUse extended Euclid to find s,t with 36s + 53t = 1.\nOne Bezout identity is: 36·(-25) + 53·(17) = 1.\nReduce s modulo 53: x ≡ 28 (mod 53).\nCheck: 36·28 mod 53 = 1.\nFinal answer: $x\\equiv 28\\pmod{53}$", "final_answer": "x\\equiv 28\\pmod{53}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "03e30447e0eabc71", "problem_id": "db78d67e139f1c64", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} x^{2} + x + 6$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{3} - 2 x^{2} + 3 x + 3$ and a = 3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $3 x^{2} - 4 x + 3$.\n5) Evaluate at a: p'(3) = $18$.\nFinal answer: $18$", "final_answer": "18", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fe493b64f74ff65f", "problem_id": "db78d67e139f1c64", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} x^{2} + x + 6$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $x^{3} - 2 x^{2} + 3 x + 3$ and a = 3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $18$", "final_answer": "18", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9e076f89896db545", "problem_id": "cd0d4ee7a2ece14f", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A disease has prevalence $\\frac{3}{50}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{101}{1000}$.\n3) Compute P(D|+) = $\\frac{54}{101}$.\nFinal answer: $\\frac{54}{101}$", "final_answer": "\\frac{54}{101}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "92409234e3773886", "problem_id": "cd0d4ee7a2ece14f", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A disease has prevalence $\\frac{3}{50}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{101}{1000}$.\nCompute P(D|+) = $\\frac{54}{101}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{54}{101}$", "final_answer": "\\frac{54}{101}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b42511991c6b2da2", "problem_id": "e827483a0a3add3d", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many binary strings of length 13 contain exactly 3 ones with no two ones adjacent?", "output": "1) Reserve 2 mandatory zeros between the 3 ones.\n2) This leaves n−k+1 = 11 valid slots for the ones.\n3) Choose the slots: C(11,3) = 165.\nFinal answer: $165$", "final_answer": "165", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "abf2a3da2a5b1ea3", "problem_id": "e827483a0a3add3d", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many binary strings of length 13 contain exactly 3 ones with no two ones adjacent?", "output": "Reserve 2 mandatory zeros between the 3 ones.\nThis leaves n−k+1 = 11 valid slots for the ones.\nChoose the slots: C(11,3) = 165.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $165$", "final_answer": "165", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f9d275181646278", "problem_id": "f8d8a7b972f25047", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A disease has prevalence $\\frac{1}{100}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{59}{1000}$.\n3) Compute P(D|+) = $\\frac{19}{118}$.\nFinal answer: $\\frac{19}{118}$", "final_answer": "\\frac{19}{118}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "713e06346bc5ccff", "problem_id": "5d1d9aee6cf3e8b1", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 2 x^{2} + 8 x - 21$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{2} + 7 x - 21.\n3) Then the limit is -60.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -63.\nFinal answer: $-63$", "final_answer": "-63", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c5b184b12535de91", "problem_id": "91caf0c67e2112e1", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A cyclist rides at 27 miles per hour for 1 hours, then at 71 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 27 × 1 = 27 miles.\n2) Second leg distance = 71 × 3/2 = 213/2 miles.\n3) Total distance = 27 + 213/2 = 267/2 miles.\nFinal answer: $\\frac{267}{2}$", "final_answer": "\\frac{267}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "66798eeef9b6e97b", "problem_id": "91caf0c67e2112e1", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A cyclist rides at 27 miles per hour for 1 hours, then at 71 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 27 × 1 = 27 miles.\nSecond leg distance = 71 × 3/2 = 213/2 miles.\nTotal distance = 27 + 213/2 = 267/2 miles.\nCheck: adding both legs gives 267/2.\nFinal answer: $\\frac{267}{2}$", "final_answer": "\\frac{267}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "35cee6033e3a7759", "problem_id": "7470aca26d46d416", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "An urn has 16 red balls and 14 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(30,9).\nFavorable samples: choose k reds and n−k blues: C(16,3)·C(14,6).\nFinal answer: $\\frac{392}{3335}$", "final_answer": "\\frac{392}{3335}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bca3e42ecb9dcfa8", "problem_id": "7470aca26d46d416", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "An urn has 16 red balls and 14 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(30,9).\n3) Favorable samples: choose k reds and n−k blues: C(16,3)·C(14,6).\n4) Compute totals: C(30,9) = 14307150, and favorable = 1681680.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{392}{3335}$", "final_answer": "\\frac{392}{3335}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "76a997d2580e8796", "problem_id": "2bccde6ce3e81170", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{11 x}{4} - \\frac{21}{8}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{11}{4}$, c=$- \\frac{21}{8}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{289}{16}$, so √Δ = $\\frac{17}{4}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{7}{2}$.\n7) x2 = $- \\frac{3}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{7}{2}, - \\frac{3}{4}\\}$", "final_answer": "x \\in \\{\\frac{7}{2}, - \\frac{3}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "863e91d054ab9945", "problem_id": "2bccde6ce3e81170", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{11 x}{4} - \\frac{21}{8}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{11}{4}$, c=$- \\frac{21}{8}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{289}{16}$, so √Δ = $\\frac{17}{4}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{7}{2}$.\nx2 = $- \\frac{3}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{7}{2}, - \\frac{3}{4}\\}$", "final_answer": "x \\in \\{\\frac{7}{2}, - \\frac{3}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f2c2f0000d7f5d49", "problem_id": "436776bc9d992791", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡5 (mod 7), x≡0 (mod 11). Give the solution modulo 77.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 77.\n3) Compute the CRT solution to get x ≡ 33 (mod 77).\nFinal answer: $x\\equiv 33\\pmod{77}$", "final_answer": "x\\equiv 33\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9e53cc8aa2c3d3a4", "problem_id": "52119ff68cb17c7e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(- 3 x^{3} + x^{2} + 2 x - 2\\right) \\cos{\\left(x - 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x \\cos{\\left(x - 3 \\right)}) = - 2 x \\sin{\\left(x - 3 \\right)}$.\n3) So $f'(x) = 3 x^{3} \\sin{\\left(x - 3 \\right)} - x^{2} \\sin{\\left(x - 3 \\right)} - 9 x^{2} \\cos{\\left(x - 3 \\right)} + x e^{x} - 2 x \\sin{\\left(x - 3 \\right)} + 2 x \\cos{\\left(x - 3 \\right)} + 3 e^{x} + 2 \\sin{\\left(x - 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x \\cos{\\left(x - 3 \\right)}) = - 2 x \\sin{\\left(x - 3 \\right)} + 2 \\cos{\\left(x - 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 2\\right) e^{x} - \\left(x \\left(x \\left(1 - 3 x\\right) + 2\\right) - 2\\right) \\sin{\\left(x - 3 \\right)} + \\left(- 9 x^{2} + 2 x + 2\\right) \\cos{\\left(x - 3 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 2\\right) e^{x} - \\left(x \\left(x \\left(1 - 3 x\\right) + 2\\right) - 2\\right) \\sin{\\left(x - 3 \\right)} + \\left(- 9 x^{2} + 2 x + 2\\right) \\cos{\\left(x - 3 \\right)} + e^{x}$", "final_answer": "\\left(x + 2\\right) e^{x} - \\left(x \\left(x \\left(1 - 3 x\\right) + 2\\right) - 2\\right) \\sin{\\left(x - 3 \\right)} + \\left(- 9 x^{2} + 2 x + 2\\right) \\cos{\\left(x - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "91359aae97174280", "problem_id": "e2b613732b2a919b", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$- 9 x + 5 y$ = $88$\n$10 x + 2 y$ = $-60$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $10$: $- 90 x + 50 y$ = $880$.\n3) Multiply the second equation by $-9$: $- 90 x - 18 y$ = $540$.\n4) Subtract to eliminate x: $68 y$ = $340$.\n5) Solve for y: y = $5$.\n6) Substitute back to find x: x = $-7$.\nFinal answer: $(x,y)=\\left(-7, 5\\right)$", "final_answer": "(x,y)=\\left(-7, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9b90d64f1302d4a2", "problem_id": "e2b613732b2a919b", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$- 9 x + 5 y$ = $88$\n$10 x + 2 y$ = $-60$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $10$: $- 90 x + 50 y$ = $880$.\nMultiply the second equation by $-9$: $- 90 x - 18 y$ = $540$.\nSubtract to eliminate x: $68 y$ = $340$.\nSolve for y: y = $5$.\nSubstitute back to find x: x = $-7$.\nCheck: plug (x,y)=(-7,5) into both equations.\nFinal answer: $(x,y)=\\left(-7, 5\\right)$", "final_answer": "(x,y)=\\left(-7, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ef94ad58e44e0cad", "problem_id": "0ab403ead1f9fd5c", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} - x^{3} - 4 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{3} - 4 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\nSubtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{\\pi^{3}}{6} - \\frac{3 \\pi}{2} - \\frac{3 \\pi^{2}}{8} - \\frac{\\pi^{4}}{64} + 5$.\nFinal answer: $- \\frac{\\pi^{3}}{6} - \\frac{3\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{2}}{8} - \\frac{\\pi^{4}}{64} + 5$", "final_answer": "- \\frac{\\pi^{3}}{6} - \\frac{3\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{2}}{8} - \\frac{\\pi^{4}}{64} + 5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1b5ba18621966de3", "problem_id": "0ab403ead1f9fd5c", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} - x^{3} - 4 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{3} - 4 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\n3) Subtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{\\pi^{3}}{6} - \\frac{3 \\pi}{2} - \\frac{3 \\pi^{2}}{8} - \\frac{\\pi^{4}}{64} + 5$.\n4) Simplify the result.\nFinal answer: $- \\frac{\\pi^{3}}{6} - \\frac{3\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{2}}{8} - \\frac{\\pi^{4}}{64} + 5$", "final_answer": "- \\frac{\\pi^{3}}{6} - \\frac{3\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{2}}{8} - \\frac{\\pi^{4}}{64} + 5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5384d1ee46ca6c8f", "problem_id": "dc8e468fb7bb5923", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Two friends split $656 in the ratio 3:1. How much does each receive?", "output": "1) Total ratio parts = 3+1 = 4.\n2) Each part = 656 ÷ 4 = 164.\n3) First share = 3 × 164 = 492.\n4) Second share = 1 × 164 = 164.\nFinal answer: $164$", "final_answer": "164", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "97093a20a7eb3090", "problem_id": "dc8e468fb7bb5923", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $656 in the ratio 3:1. How much does each receive?", "output": "Total ratio parts = 3+1 = 4.\nEach part = 656 ÷ 4 = 164.\nFirst share = 3 × 164 = 492.\nSecond share = 1 × 164 = 164.\nCheck: 492+164=656.\nFinal answer: $164$", "final_answer": "164", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5f4f48318f781c2b", "problem_id": "884ffe89a210d9f4", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - x^{2} + 2 x - 7$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{3} - x^{2} - x - 2$ and a = -3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 3 x^{2} - 2 x - 1$.\n5) Evaluate at a: p'(-3) = $-22$.\nFinal answer: $-22$", "final_answer": "-22", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bdebe14e07404f60", "problem_id": "884ffe89a210d9f4", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - x^{2} + 2 x - 7$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{3} - x^{2} - x - 2$ and a = -3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 3 x^{2} - 2 x - 1$.\nEvaluate at a: p'(-3) = $-22$.\nCheck: the identity gives $-22$.\nFinal answer: $-22$", "final_answer": "-22", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cf7f58234199a279", "problem_id": "af3ae32b5ff0539c", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the equation of the tangent line to $y=- 4 x^{3} + 3 x^{2} - 2 x + 2$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $- 12 x^{2} + 6 x - 2$.\n2) Slope at x=-2: m = f'(-2) = $-62$.\n3) Point on curve: (-2, f(-2)) = (-2, $50$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 50 = -62(x − -2).\n6) Optionally expand: y = $- 62 x - 74$.\nFinal answer: $y = - 62\\cdot x - 74$", "final_answer": "y = - 62\\cdot x - 74", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "210812df459a339e", "problem_id": "af3ae32b5ff0539c", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Find the equation of the tangent line to $y=- 4 x^{3} + 3 x^{2} - 2 x + 2$ at $x=-2$.", "output": "Differentiate: f'(x) = $- 12 x^{2} + 6 x - 2$.\nSlope at x=-2: m = f'(-2) = $-62$.\nPoint on curve: (-2, f(-2)) = (-2, $50$).\nFinal answer: $y = - 62\\cdot x - 74$", "final_answer": "y = - 62\\cdot x - 74", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "326b26d99f0eff2c", "problem_id": "46344485094f2e67", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "You want 39 liters of a 40% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 39−x liters is 70% solution.\nSalt equation: 20x + 70(39−x) = 40·39.\n(20-70)x = (40-70)·39 ⇒ x = 117/5.\nThen 39−x = 78/5.\nCheck: 20·117/5 + 70·78/5 = 40·39.\nFinal answer: $117/5 L of 20% and 78/5 L of 70%$", "final_answer": "117/5 L of 20% and 78/5 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fed161a39cab708b", "problem_id": "46344485094f2e67", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "You want 39 liters of a 40% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 39−x liters is 70% solution.\n2) Salt equation: 20x + 70(39−x) = 40·39.\n3) (20-70)x = (40-70)·39 ⇒ x = 117/5.\n4) Then 39−x = 78/5.\nFinal answer: $117/5 L of 20% and 78/5 L of 70%$", "final_answer": "117/5 L of 20% and 78/5 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cbd550f53abafbc9", "problem_id": "00c1f3665c5625ae", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Worker A finishes a job in 6 hours and Worker B finishes it in 21 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/6 job/hour. B's rate = 1/21 job/hour.\n2) Combined rate = 1/6 + 1/21 = 3/14 job/hour.\n3) Time = 1 ÷ (combined rate) = 14/3 hours.\nFinal answer: $\\frac{14}{3}$", "final_answer": "\\frac{14}{3}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6bcb2fd9eac7348c", "problem_id": "00c1f3665c5625ae", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Worker A finishes a job in 6 hours and Worker B finishes it in 21 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/6 job/hour. B's rate = 1/21 job/hour.\nCombined rate = 1/6 + 1/21 = 3/14 job/hour.\nTime = 1 ÷ (combined rate) = 14/3 hours.\nFinal answer: $\\frac{14}{3}$", "final_answer": "\\frac{14}{3}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f8fbdd10a03fabab", "problem_id": "d71de699f38b7741", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Worker A finishes a job in 16 hours and Worker B finishes it in 19 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/16 job/hour. B's rate = 1/19 job/hour.\n2) Combined rate = 1/16 + 1/19 = 35/304 job/hour.\n3) Time = 1 ÷ (combined rate) = 304/35 hours.\nFinal answer: $\\frac{304}{35}$", "final_answer": "\\frac{304}{35}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "92449c38cd5250fb", "problem_id": "d71de699f38b7741", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Worker A finishes a job in 16 hours and Worker B finishes it in 19 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/16 job/hour. B's rate = 1/19 job/hour.\nCombined rate = 1/16 + 1/19 = 35/304 job/hour.\nTime = 1 ÷ (combined rate) = 304/35 hours.\nCheck: (combined rate)×(time) = 35/304×304/35 = 1.\nFinal answer: $\\frac{304}{35}$", "final_answer": "\\frac{304}{35}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e819adc40095abe0", "problem_id": "cd0a8efd3c4bcaac", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the linear system:\n$- 3 x + y$ = $25$\n$- x + 2 y$ = $15$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $3 x - y$ = $-25$.\nMultiply the second equation by $-3$: $3 x - 6 y$ = $-45$.\nFinal answer: $(x,y)=\\left(-7, 4\\right)$", "final_answer": "(x,y)=\\left(-7, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e602202b93d08ecd", "problem_id": "cd0a8efd3c4bcaac", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$- 3 x + y$ = $25$\n$- x + 2 y$ = $15$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $3 x - y$ = $-25$.\n3) Multiply the second equation by $-3$: $3 x - 6 y$ = $-45$.\n4) Subtract to eliminate x: $5 y$ = $20$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $-7$.\nFinal answer: $(x,y)=\\left(-7, 4\\right)$", "final_answer": "(x,y)=\\left(-7, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f5a97bcc0a54c61e", "problem_id": "3639556817eef436", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} - 4 x^{4} - x^{3} - 3 x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 4 x^{4} - x^{3} - 3 x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{33 \\pi^{5}}{40} - \\frac{9 \\pi^{3}}{8} - \\frac{15 \\pi^{4}}{64} - \\frac{3 \\pi^{2}}{2} - \\frac{3 \\pi}{2} - 1$.\n4) Simplify the result.\nFinal answer: $- \\frac{33\\cdot \\pi^{5}}{40} - \\frac{9\\cdot \\pi^{3}}{8} - \\frac{15\\cdot \\pi^{4}}{64} - \\frac{3\\cdot \\pi^{2}}{2} - \\frac{3\\cdot \\pi}{2} - 1$", "final_answer": "- \\frac{33\\cdot \\pi^{5}}{40} - \\frac{9\\cdot \\pi^{3}}{8} - \\frac{15\\cdot \\pi^{4}}{64} - \\frac{3\\cdot \\pi^{2}}{2} - \\frac{3\\cdot \\pi}{2} - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "70726776ab0c7893", "problem_id": "3639556817eef436", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} - 4 x^{4} - x^{3} - 3 x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 4 x^{4} - x^{3} - 3 x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1$.\nEvaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{33 \\pi^{5}}{40} - \\frac{9 \\pi^{3}}{8} - \\frac{15 \\pi^{4}}{64} - \\frac{3 \\pi^{2}}{2} - \\frac{3 \\pi}{2} - 1$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{33\\cdot \\pi^{5}}{40} - \\frac{9\\cdot \\pi^{3}}{8} - \\frac{15\\cdot \\pi^{4}}{64} - \\frac{3\\cdot \\pi^{2}}{2} - \\frac{3\\cdot \\pi}{2} - 1$", "final_answer": "- \\frac{33\\cdot \\pi^{5}}{40} - \\frac{9\\cdot \\pi^{3}}{8} - \\frac{15\\cdot \\pi^{4}}{64} - \\frac{3\\cdot \\pi^{2}}{2} - \\frac{3\\cdot \\pi}{2} - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1e1433511962d77b", "problem_id": "491b7f2e6c86b567", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Worker A finishes a job in 14 hours and Worker B finishes it in 6 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/14 job/hour. B's rate = 1/6 job/hour.\nCombined rate = 1/14 + 1/6 = 5/21 job/hour.\nTime = 1 ÷ (combined rate) = 21/5 hours.\nCheck: (combined rate)×(time) = 5/21×21/5 = 1.\nFinal answer: $\\frac{21}{5}$", "final_answer": "\\frac{21}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "79f3270df4aa7368", "problem_id": "491b7f2e6c86b567", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Worker A finishes a job in 14 hours and Worker B finishes it in 6 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/14 job/hour. B's rate = 1/6 job/hour.\n2) Combined rate = 1/14 + 1/6 = 5/21 job/hour.\n3) Time = 1 ÷ (combined rate) = 21/5 hours.\nFinal answer: $\\frac{21}{5}$", "final_answer": "\\frac{21}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6c522855680c18b3", "problem_id": "f2c176114317ba81", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Two friends split $231 in the ratio 9:2. How much does each receive?", "output": "1) Total ratio parts = 9+2 = 11.\n2) Each part = 231 ÷ 11 = 21.\n3) First share = 9 × 21 = 189.\n4) Second share = 2 × 21 = 42.\nFinal answer: $42$", "final_answer": "42", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e8d343cb424420f1", "problem_id": "f2c176114317ba81", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Two friends split $231 in the ratio 9:2. How much does each receive?", "output": "Total ratio parts = 9+2 = 11.\nEach part = 231 ÷ 11 = 21.\nFirst share = 9 × 21 = 189.\nFinal answer: $42$", "final_answer": "42", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "02a2f0c284036f36", "problem_id": "99b2ac978a71b042", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} - 4 x^{3} - 4 x^{2} + 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 4 x^{3} - 4 x^{2} + 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3$.\n2) Evaluate at the bounds: F(2 \\pi) and F(0).\n3) Subtract: F(2 \\pi) − F(0) = $\\frac{2 \\pi \\left(- 24 \\pi^{3} - 16 \\pi^{2} + 9 + 6 \\pi\\right)}{3}$.\n4) Simplify the result.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(- 24\\cdot \\pi^{3} - 16\\cdot \\pi^{2} + 9 + 6\\cdot \\pi\\right)}{3}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(- 24\\cdot \\pi^{3} - 16\\cdot \\pi^{2} + 9 + 6\\cdot \\pi\\right)}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ee26b3f2bbc75e34", "problem_id": "99b2ac978a71b042", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} - 4 x^{3} - 4 x^{2} + 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 4 x^{3} - 4 x^{2} + 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3$.\nEvaluate at the bounds: F(2 \\pi) and F(0).\nSubtract: F(2 \\pi) − F(0) = $\\frac{2 \\pi \\left(- 24 \\pi^{3} - 16 \\pi^{2} + 9 + 6 \\pi\\right)}{3}$.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(- 24\\cdot \\pi^{3} - 16\\cdot \\pi^{2} + 9 + 6\\cdot \\pi\\right)}{3}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(- 24\\cdot \\pi^{3} - 16\\cdot \\pi^{2} + 9 + 6\\cdot \\pi\\right)}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a8c9d3597c1f311c", "problem_id": "0cff101993da0818", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 55 liters of a 50% salt solution. You have a 30% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 55−x liters is 70% solution.\nSalt equation: 30x + 70(55−x) = 50·55.\n(30-70)x = (50-70)·55 ⇒ x = 55/2.\nThen 55−x = 55/2.\nCheck: 30·55/2 + 70·55/2 = 50·55.\nFinal answer: $55/2 L of 30% and 55/2 L of 70%$", "final_answer": "55/2 L of 30% and 55/2 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5207cab2608f765a", "problem_id": "0cff101993da0818", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "You want 55 liters of a 50% salt solution. You have a 30% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 55−x liters is 70% solution.\n2) Salt equation: 30x + 70(55−x) = 50·55.\n3) (30-70)x = (50-70)·55 ⇒ x = 55/2.\n4) Then 55−x = 55/2.\nFinal answer: $55/2 L of 30% and 55/2 L of 70%$", "final_answer": "55/2 L of 30% and 55/2 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "23623872b38cd6be", "problem_id": "b13c4cfef1efa69b", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} 2 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\n3) Subtract: F(\\frac{\\pi}{2}) − F(0) = $\\frac{\\pi^{2}}{8} + 2 + \\frac{\\pi^{3}}{12}$.\n4) Simplify the result.\nFinal answer: $\\frac{\\pi^{2}}{8} + 2 + \\frac{\\pi^{3}}{12}$", "final_answer": "\\frac{\\pi^{2}}{8} + 2 + \\frac{\\pi^{3}}{12}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7f1c2261c619b589", "problem_id": "b13c4cfef1efa69b", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} 2 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $2 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\nSubtract: F(\\frac{\\pi}{2}) − F(0) = $\\frac{\\pi^{2}}{8} + 2 + \\frac{\\pi^{3}}{12}$.\nFinal answer: $\\frac{\\pi^{2}}{8} + 2 + \\frac{\\pi^{3}}{12}$", "final_answer": "\\frac{\\pi^{2}}{8} + 2 + \\frac{\\pi^{3}}{12}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b5e46092cbb647fe", "problem_id": "4ea5079d8469786a", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Worker A finishes a job in 14 hours and Worker B finishes it in 8 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/14 job/hour. B's rate = 1/8 job/hour.\nCombined rate = 1/14 + 1/8 = 11/56 job/hour.\nTime = 1 ÷ (combined rate) = 56/11 hours.\nFinal answer: $\\frac{56}{11}$", "final_answer": "\\frac{56}{11}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "225fc6bebcf3f84d", "problem_id": "4ea5079d8469786a", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Worker A finishes a job in 14 hours and Worker B finishes it in 8 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/14 job/hour. B's rate = 1/8 job/hour.\n2) Combined rate = 1/14 + 1/8 = 11/56 job/hour.\n3) Time = 1 ÷ (combined rate) = 56/11 hours.\nFinal answer: $\\frac{56}{11}$", "final_answer": "\\frac{56}{11}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6126a16efa73a9ee", "problem_id": "37620aeca56e1ed5", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} - 2 x^{3} - x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{3} - x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\n3) Subtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{3 \\pi}{2} - \\frac{\\pi^{4}}{32} - \\frac{\\pi^{3}}{24} + 2 + \\frac{\\pi^{2}}{4}$.\n4) Simplify the result.\nFinal answer: $- \\frac{3\\cdot \\pi}{2} - \\frac{\\pi^{4}}{32} - \\frac{\\pi^{3}}{24} + 2 + \\frac{\\pi^{2}}{4}$", "final_answer": "- \\frac{3\\cdot \\pi}{2} - \\frac{\\pi^{4}}{32} - \\frac{\\pi^{3}}{24} + 2 + \\frac{\\pi^{2}}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a64951d6f2a6d345", "problem_id": "37620aeca56e1ed5", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} - 2 x^{3} - x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{3} - x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\nSubtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{3 \\pi}{2} - \\frac{\\pi^{4}}{32} - \\frac{\\pi^{3}}{24} + 2 + \\frac{\\pi^{2}}{4}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{3\\cdot \\pi}{2} - \\frac{\\pi^{4}}{32} - \\frac{\\pi^{3}}{24} + 2 + \\frac{\\pi^{2}}{4}$", "final_answer": "- \\frac{3\\cdot \\pi}{2} - \\frac{\\pi^{4}}{32} - \\frac{\\pi^{3}}{24} + 2 + \\frac{\\pi^{2}}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ee701f2f5d7567d4", "problem_id": "0f0288a6315ca943", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{5 x}{2} - \\frac{\\sqrt{2} x}{3} + \\frac{\\sqrt{2}}{2} + \\frac{3}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{5}{2} - \\frac{\\sqrt{2}}{3}$, c=$\\frac{\\sqrt{2}}{2} + \\frac{3}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{17}{36} - \\frac{\\sqrt{2}}{3}$, so √Δ = $\\sqrt{\\frac{17}{36} - \\frac{\\sqrt{2}}{3}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{17 - 12 \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{6} + \\frac{5}{4}$.\nx2 = $- \\frac{\\sqrt{17 - 12 \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{6} + \\frac{5}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{17 - 12\\cdot \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{6} + \\frac{5}{4}, - \\frac{\\sqrt{17 - 12\\cdot \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{6} + \\frac{5}{4}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{17 - 12\\cdot \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{6} + \\frac{5}{4}, - \\frac{\\sqrt{17 - 12\\cdot \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{6} + \\frac{5}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "039368ab73a5e49f", "problem_id": "0f0288a6315ca943", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{5 x}{2} - \\frac{\\sqrt{2} x}{3} + \\frac{\\sqrt{2}}{2} + \\frac{3}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{5}{2} - \\frac{\\sqrt{2}}{3}$, c=$\\frac{\\sqrt{2}}{2} + \\frac{3}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{17}{36} - \\frac{\\sqrt{2}}{3}$, so √Δ = $\\sqrt{\\frac{17}{36} - \\frac{\\sqrt{2}}{3}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{17 - 12 \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{6} + \\frac{5}{4}$.\n7) x2 = $- \\frac{\\sqrt{17 - 12 \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{6} + \\frac{5}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{17 - 12\\cdot \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{6} + \\frac{5}{4}, - \\frac{\\sqrt{17 - 12\\cdot \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{6} + \\frac{5}{4}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{17 - 12\\cdot \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{6} + \\frac{5}{4}, - \\frac{\\sqrt{17 - 12\\cdot \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{6} + \\frac{5}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7a19a214ea778d18", "problem_id": "8892f8e39d81465c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} + 2 x^{2} - 4 x + 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x^{3}) = - 6 x^{3}$.\n3) So $f'(x) = - 6 x^{3} + 4 x - 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x^{3}) = - 6 x^{2}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- 3 x^{2} + 2 x - 2\\right)$.\nFinal answer: $2 \\left(- 3 x^{2} + 2 x - 2\\right)$", "final_answer": "2 \\left(- 3 x^{2} + 2 x - 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "daaba0fa8a424da8", "problem_id": "520ddbf6f2200ffc", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡3 (mod 8), x≡4 (mod 9). Give the solution modulo 72.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 72.\n3) Compute the CRT solution to get x ≡ 67 (mod 72).\nFinal answer: $x\\equiv 67\\pmod{72}$", "final_answer": "x\\equiv 67\\pmod{72}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8c0dee1ae5b299cb", "problem_id": "813f91b2fb3d6981", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 4 x - 7$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$4$, c=$-7$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $100$, so √Δ = $10$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $1$.\nx2 = $- \\frac{7}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{1, - \\frac{7}{3}\\}$", "final_answer": "x \\in \\{1, - \\frac{7}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "378944ac0c632254", "problem_id": "813f91b2fb3d6981", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 4 x - 7$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$4$, c=$-7$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $100$, so √Δ = $10$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $1$.\n7) x2 = $- \\frac{7}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{1, - \\frac{7}{3}\\}$", "final_answer": "x \\in \\{1, - \\frac{7}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4b6214079af98a74", "problem_id": "08dad0ffebbc8c3d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x - 4$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 2 x - 4.\n3) Then the limit is -8.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -6.\nFinal answer: $-6$", "final_answer": "-6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "586031c352532b6c", "problem_id": "122d6fd58f2b29d3", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{3} x}{3} + \\frac{7 x}{3} - \\frac{2 \\sqrt{3}}{9} + \\frac{10}{9}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{7}{3} - \\frac{\\sqrt{3}}{3}$, c=$\\frac{10}{9} - \\frac{2 \\sqrt{3}}{9}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{4}{3} - \\frac{2 \\sqrt{3}}{3}$, so √Δ = $\\sqrt{\\frac{4}{3} - \\frac{2 \\sqrt{3}}{3}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{7}{6} + \\frac{\\sqrt{12 - 6 \\sqrt{3}}}{6} + \\frac{\\sqrt{3}}{6}$.\n7) x2 = $- \\frac{7}{6} - \\frac{\\sqrt{12 - 6 \\sqrt{3}}}{6} + \\frac{\\sqrt{3}}{6}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{7}{6} + \\frac{\\sqrt{12 - 6\\cdot \\sqrt{3}}}{6} + \\frac{\\sqrt{3}}{6}, - \\frac{7}{6} - \\frac{\\sqrt{12 - 6\\cdot \\sqrt{3}}}{6} + \\frac{\\sqrt{3}}{6}\\}$", "final_answer": "x \\in \\{- \\frac{7}{6} + \\frac{\\sqrt{12 - 6\\cdot \\sqrt{3}}}{6} + \\frac{\\sqrt{3}}{6}, - \\frac{7}{6} - \\frac{\\sqrt{12 - 6\\cdot \\sqrt{3}}}{6} + \\frac{\\sqrt{3}}{6}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d56af1e13f4bfe2a", "problem_id": "1d964fb66434edfe", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{3} + x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $2 x^{3} + x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2$.\n3) Compute an antiderivative: F(x) = $\\frac{x^{4}}{2} + \\frac{x^{3}}{3} + \\frac{3 x^{2}}{2} + 2 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{x^{4}}{2} + \\frac{x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{4}}{2} + \\frac{x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dbbcf3f546e767b6", "problem_id": "1d964fb66434edfe", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{3} + x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "$C + \\frac{x^{4}}{2} + \\frac{x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{4}}{2} + \\frac{x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "310d4510cd5bc6f0", "problem_id": "c6c6cff949ce8571", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡6 (mod 9), x≡8 (mod 13). Give the solution modulo 117.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 117.\n3) Compute the CRT solution to get x ≡ 60 (mod 117).\nFinal answer: $x\\equiv 60\\pmod{117}$", "final_answer": "x\\equiv 60\\pmod{117}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "76eb73b9eab193b1", "problem_id": "c6c6cff949ce8571", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡6 (mod 9), x≡8 (mod 13). Give the solution modulo 117.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 117.\nCompute the CRT solution to get x ≡ 60 (mod 117).\nCheck: 60 mod 9=6 and mod 13=8.\nFinal answer: $x\\equiv 60\\pmod{117}$", "final_answer": "x\\equiv 60\\pmod{117}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a3a57f70b80ca45", "problem_id": "d75a4689127d8117", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Two friends split $220 in the ratio 8:3. How much does each receive?", "output": "1) Total ratio parts = 8+3 = 11.\n2) Each part = 220 ÷ 11 = 20.\n3) First share = 8 × 20 = 160.\n4) Second share = 3 × 20 = 60.\nFinal answer: $60$", "final_answer": "60", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ded08e420e6f7b96", "problem_id": "d75a4689127d8117", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Two friends split $220 in the ratio 8:3. How much does each receive?", "output": "Total ratio parts = 8+3 = 11.\nEach part = 220 ÷ 11 = 20.\nFirst share = 8 × 20 = 160.\nFinal answer: $60$", "final_answer": "60", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "036c2c91846167b1", "problem_id": "18f8e0c1dc1e7d63", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$6 x + 8 y$ = $4$\n$6 x + 2 y$ = $10$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $6$: $36 x + 48 y$ = $24$.\nMultiply the second equation by $6$: $36 x + 12 y$ = $60$.\nSubtract to eliminate x: $36 y$ = $-36$.\nSolve for y: y = $-1$.\nSubstitute back to find x: x = $2$.\nCheck: plug (x,y)=(2,-1) into both equations.\nFinal answer: $(x,y)=\\left(2, -1\\right)$", "final_answer": "(x,y)=\\left(2, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a6a7f2745a65a617", "problem_id": "18f8e0c1dc1e7d63", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$6 x + 8 y$ = $4$\n$6 x + 2 y$ = $10$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $6$: $36 x + 48 y$ = $24$.\n3) Multiply the second equation by $6$: $36 x + 12 y$ = $60$.\n4) Subtract to eliminate x: $36 y$ = $-36$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $2$.\nFinal answer: $(x,y)=\\left(2, -1\\right)$", "final_answer": "(x,y)=\\left(2, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a6dc968ae8e51b1f", "problem_id": "e3905de450773855", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} - x - 4$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to - 2 x - 4.\n3) Then the limit is -6.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: -5.\nFinal answer: $-5$", "final_answer": "-5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6ad87423b5d58bb5", "problem_id": "b15b4812339e667d", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} x^{3} + 2 x^{2} + 2 x + 4$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{4} + x^{3} + 2 x - 2$ and a = 1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $4 x^{3} + 3 x^{2} + 2$.\n5) Evaluate at a: p'(1) = $9$.\nFinal answer: $9$", "final_answer": "9", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "faf499919fbc42a7", "problem_id": "b2ea521ab8090dcc", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(2 x^{3} + x^{2} + 3 x - 3\\right) \\cos{\\left(2 x + 2 \\right)}$.", "output": "1) Let f(x) = $2 \\left(x - 3\\right) e^{x} + \\left(2 x^{3} + x^{2} + 3 x - 3\\right) \\cos{\\left(2 x + 2 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $2 \\left(x - 3\\right) e^{x} + \\left(6 x^{2} + 2 x + 3\\right) \\cos{\\left(2 x + 2 \\right)} - 2 \\left(2 x^{3} + x^{2} + 3 x - 3\\right) \\sin{\\left(2 x + 2 \\right)} + 2 e^{x}$.\nFinal answer: $\\left(2\\cdot x - 6\\right)\\cdot e^{x} + \\left(6\\cdot x^{2} + 2\\cdot x + 3\\right)\\cdot \\cos{\\left(2\\cdot x + 2 \\right)} - \\left(4\\cdot x^{3} + 2\\cdot x^{2} + 6\\cdot x - 6\\right)\\cdot \\sin{\\left(2\\cdot x + 2 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x - 6\\right)\\cdot e^{x} + \\left(6\\cdot x^{2} + 2\\cdot x + 3\\right)\\cdot \\cos{\\left(2\\cdot x + 2 \\right)} - \\left(4\\cdot x^{3} + 2\\cdot x^{2} + 6\\cdot x - 6\\right)\\cdot \\sin{\\left(2\\cdot x + 2 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "90dc36125dcec9d6", "problem_id": "b2ea521ab8090dcc", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(2 x^{3} + x^{2} + 3 x - 3\\right) \\cos{\\left(2 x + 2 \\right)}$.", "output": "Let f(x) = $2 \\left(x - 3\\right) e^{x} + \\left(2 x^{3} + x^{2} + 3 x - 3\\right) \\cos{\\left(2 x + 2 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $2 \\left(x - 3\\right) e^{x} + \\left(6 x^{2} + 2 x + 3\\right) \\cos{\\left(2 x + 2 \\right)} - 2 \\left(2 x^{3} + x^{2} + 3 x - 3\\right) \\sin{\\left(2 x + 2 \\right)} + 2 e^{x}$.\nFinal answer: $\\left(2\\cdot x - 6\\right)\\cdot e^{x} + \\left(6\\cdot x^{2} + 2\\cdot x + 3\\right)\\cdot \\cos{\\left(2\\cdot x + 2 \\right)} - \\left(4\\cdot x^{3} + 2\\cdot x^{2} + 6\\cdot x - 6\\right)\\cdot \\sin{\\left(2\\cdot x + 2 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x - 6\\right)\\cdot e^{x} + \\left(6\\cdot x^{2} + 2\\cdot x + 3\\right)\\cdot \\cos{\\left(2\\cdot x + 2 \\right)} - \\left(4\\cdot x^{3} + 2\\cdot x^{2} + 6\\cdot x - 6\\right)\\cdot \\sin{\\left(2\\cdot x + 2 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "923102c84d2a7de6", "problem_id": "c196be9343ddb06a", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A cyclist rides at 68 miles per hour for 3/2 hours, then at 44 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 68 × 3/2 = 102 miles.\n2) Second leg distance = 44 × 1 = 44 miles.\n3) Total distance = 102 + 44 = 146 miles.\nFinal answer: $146$", "final_answer": "146", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3c826897223c346c", "problem_id": "c196be9343ddb06a", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A cyclist rides at 68 miles per hour for 3/2 hours, then at 44 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 68 × 3/2 = 102 miles.\nSecond leg distance = 44 × 1 = 44 miles.\nTotal distance = 102 + 44 = 146 miles.\nCheck: adding both legs gives 146.\nFinal answer: $146$", "final_answer": "146", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "65428d19898cf49d", "problem_id": "651acd3883b89752", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Worker A finishes a job in 20 hours and Worker B finishes it in 15 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/20 job/hour. B's rate = 1/15 job/hour.\n2) Combined rate = 1/20 + 1/15 = 7/60 job/hour.\n3) Time = 1 ÷ (combined rate) = 60/7 hours.\nFinal answer: $\\frac{60}{7}$", "final_answer": "\\frac{60}{7}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b070811d4f787139", "problem_id": "651acd3883b89752", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Worker A finishes a job in 20 hours and Worker B finishes it in 15 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/20 job/hour. B's rate = 1/15 job/hour.\nCombined rate = 1/20 + 1/15 = 7/60 job/hour.\nTime = 1 ÷ (combined rate) = 60/7 hours.\nCheck: (combined rate)×(time) = 7/60×60/7 = 1.\nFinal answer: $\\frac{60}{7}$", "final_answer": "\\frac{60}{7}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d862bfdca1053ab", "problem_id": "8d0b14fdb707f032", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 4 x^{2} - 4 x + 1\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 4 x^{2} - 4 x + 1$.\n3) Compute an antiderivative: F(x) = $- \\frac{4 x^{3}}{3} - 2 x^{2} + x$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{4\\cdot x^{3}}{3} - 2\\cdot x^{2} + x$", "final_answer": "C - \\frac{4\\cdot x^{3}}{3} - 2\\cdot x^{2} + x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "55baed1284b19ba7", "problem_id": "8d0b14fdb707f032", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 4 x^{2} - 4 x + 1\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 4 x^{2} - 4 x + 1$.\nCompute an antiderivative: F(x) = $- \\frac{4 x^{3}}{3} - 2 x^{2} + x$.\nFinal answer: $C - \\frac{4\\cdot x^{3}}{3} - 2\\cdot x^{2} + x$", "final_answer": "C - \\frac{4\\cdot x^{3}}{3} - 2\\cdot x^{2} + x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "99296dea9196900a", "problem_id": "cdee861e0d8a6fcb", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A disease has prevalence $\\frac{3}{50}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{151}{1000}$.\nCompute P(D|+) = $\\frac{57}{151}$.\nFinal answer: $\\frac{57}{151}$", "final_answer": "\\frac{57}{151}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "38b64486c4b9fdc5", "problem_id": "cdee861e0d8a6fcb", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A disease has prevalence $\\frac{3}{50}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{151}{1000}$.\n3) Compute P(D|+) = $\\frac{57}{151}$.\nFinal answer: $\\frac{57}{151}$", "final_answer": "\\frac{57}{151}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e1ba87cdc7983bf6", "problem_id": "89d9a08431d8a66d", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 6 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=6, k=3, p=$\\frac{1}{8}$ and simplify.\n4) Here C(6,3) = 20.\nFinal answer: $\\frac{1715}{65536}$", "final_answer": "\\frac{1715}{65536}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7dfd08ee5d530b75", "problem_id": "89d9a08431d8a66d", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 6 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=6, k=3, p=$\\frac{1}{8}$ and simplify.\nFinal answer: $\\frac{1715}{65536}$", "final_answer": "\\frac{1715}{65536}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cbec811043de8406", "problem_id": "4563cfecfc7b5e89", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "You want 22 liters of a 35% salt solution. You have a 30% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 22−x liters is 40% solution.\n2) Salt equation: 30x + 40(22−x) = 35·22.\n3) (30-40)x = (35-40)·22 ⇒ x = 11.\n4) Then 22−x = 11.\nFinal answer: $11 L of 30% and 11 L of 40%$", "final_answer": "11 L of 30% and 11 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a00ab053479ee03c", "problem_id": "4563cfecfc7b5e89", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 22 liters of a 35% salt solution. You have a 30% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 22−x liters is 40% solution.\nSalt equation: 30x + 40(22−x) = 35·22.\n(30-40)x = (35-40)·22 ⇒ x = 11.\nThen 22−x = 11.\nCheck: 30·11 + 40·11 = 35·22.\nFinal answer: $11 L of 30% and 11 L of 40%$", "final_answer": "11 L of 30% and 11 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a62a89426d11ec10", "problem_id": "6170c06ae7c1aab6", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡5 (mod 7), x≡7 (mod 11). Give the solution modulo 77.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 77.\n3) Compute the CRT solution to get x ≡ 40 (mod 77).\nFinal answer: $x\\equiv 40\\pmod{77}$", "final_answer": "x\\equiv 40\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7200b6e528879cbd", "problem_id": "da68ca0158e4bdcc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{4} - 2 x^{2} - 2 x + 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- x^{4}) = - 4 x^{4}$.\n3) So $f'(x) = - 4 x^{4} - 4 x - 2$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- x^{4}) = - 4 x^{3}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- 2 x^{3} - 2 x - 1\\right)$.\nFinal answer: $2 \\left(- 2 x^{3} - 2 x - 1\\right)$", "final_answer": "2 \\left(- 2 x^{3} - 2 x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b4d9c98d415729b4", "problem_id": "c33172b1ff8bb2f4", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_6=35$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_6=29.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=29 and k=6.\n4) Compute C(34,5) = 278256.\nFinal answer: $278256$", "final_answer": "278256", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dd24d50276e3551b", "problem_id": "c33172b1ff8bb2f4", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_6=35$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_6=29.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=29 and k=6.\nCompute C(34,5) = 278256.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $278256$", "final_answer": "278256", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1924243b3ba4ea21", "problem_id": "f62ebe46a37fde17", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{7} x + \\frac{9 x}{2} - \\frac{9 \\sqrt{7}}{2} + \\frac{9}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{9}{2} - 3 \\sqrt{7}$, c=$\\frac{9}{2} - \\frac{9 \\sqrt{7}}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{261}{4} - 9 \\sqrt{7}$, so √Δ = $\\sqrt{\\frac{261}{4} - 9 \\sqrt{7}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{9}{4} + \\frac{3 \\sqrt{29 - 4 \\sqrt{7}}}{4} + \\frac{3 \\sqrt{7}}{2}$.\nx2 = $- \\frac{3 \\sqrt{29 - 4 \\sqrt{7}}}{4} - \\frac{9}{4} + \\frac{3 \\sqrt{7}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{9}{4} + \\frac{3\\cdot \\sqrt{29 - 4\\cdot \\sqrt{7}}}{4} + \\frac{3\\cdot \\sqrt{7}}{2}, - \\frac{3\\cdot \\sqrt{29 - 4\\cdot \\sqrt{7}}}{4} - \\frac{9}{4} + \\frac{3\\cdot \\sqrt{7}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{9}{4} + \\frac{3\\cdot \\sqrt{29 - 4\\cdot \\sqrt{7}}}{4} + \\frac{3\\cdot \\sqrt{7}}{2}, - \\frac{3\\cdot \\sqrt{29 - 4\\cdot \\sqrt{7}}}{4} - \\frac{9}{4} + \\frac{3\\cdot \\sqrt{7}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "97e495e98e3e7511", "problem_id": "f62ebe46a37fde17", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{7} x + \\frac{9 x}{2} - \\frac{9 \\sqrt{7}}{2} + \\frac{9}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{9}{2} - 3 \\sqrt{7}$, c=$\\frac{9}{2} - \\frac{9 \\sqrt{7}}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{- \\frac{9}{4} + \\frac{3\\cdot \\sqrt{29 - 4\\cdot \\sqrt{7}}}{4} + \\frac{3\\cdot \\sqrt{7}}{2}, - \\frac{3\\cdot \\sqrt{29 - 4\\cdot \\sqrt{7}}}{4} - \\frac{9}{4} + \\frac{3\\cdot \\sqrt{7}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{9}{4} + \\frac{3\\cdot \\sqrt{29 - 4\\cdot \\sqrt{7}}}{4} + \\frac{3\\cdot \\sqrt{7}}{2}, - \\frac{3\\cdot \\sqrt{29 - 4\\cdot \\sqrt{7}}}{4} - \\frac{9}{4} + \\frac{3\\cdot \\sqrt{7}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3fc072f744b75d4f", "problem_id": "74e14a7e0c3c8a84", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} 4 x^{3} + 2 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $4 x^{3} + 2 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nEvaluate at the bounds: F(2 \\pi) and F(0).\nSubtract: F(2 \\pi) − F(0) = $\\pi^{2} \\left(6 + \\frac{16 \\pi}{3} + 16 \\pi^{2}\\right)$.\nFinal answer: $\\pi^{2}\\cdot \\left(6 + \\frac{16\\cdot \\pi}{3} + 16\\cdot \\pi^{2}\\right)$", "final_answer": "\\pi^{2}\\cdot \\left(6 + \\frac{16\\cdot \\pi}{3} + 16\\cdot \\pi^{2}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3266fcc5703e9a4f", "problem_id": "74e14a7e0c3c8a84", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} 4 x^{3} + 2 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $4 x^{3} + 2 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\n2) Evaluate at the bounds: F(2 \\pi) and F(0).\n3) Subtract: F(2 \\pi) − F(0) = $\\pi^{2} \\left(6 + \\frac{16 \\pi}{3} + 16 \\pi^{2}\\right)$.\n4) Simplify the result.\nFinal answer: $\\pi^{2}\\cdot \\left(6 + \\frac{16\\cdot \\pi}{3} + 16\\cdot \\pi^{2}\\right)$", "final_answer": "\\pi^{2}\\cdot \\left(6 + \\frac{16\\cdot \\pi}{3} + 16\\cdot \\pi^{2}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b913f407286b9a40", "problem_id": "b7a82b378f3b92fa", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many binary strings of length 14 contain exactly 6 ones with no two ones adjacent?", "output": "1) Reserve 5 mandatory zeros between the 6 ones.\n2) This leaves n−k+1 = 9 valid slots for the ones.\n3) Choose the slots: C(9,6) = 84.\nFinal answer: $84$", "final_answer": "84", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b69a2d26e637dcee", "problem_id": "b7a82b378f3b92fa", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "How many binary strings of length 14 contain exactly 6 ones with no two ones adjacent?", "output": "Reserve 5 mandatory zeros between the 6 ones.\nThis leaves n−k+1 = 9 valid slots for the ones.\nChoose the slots: C(9,6) = 84.\nFinal answer: $84$", "final_answer": "84", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ad75001c112ae9c", "problem_id": "4d877d7731b44b3a", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{10} x - 2 x + 1 + 3 \\sqrt{10}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- 3 \\sqrt{10} - 2$, c=$1 + 3 \\sqrt{10}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $90$, so √Δ = $3 \\sqrt{10}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $1 + 3 \\sqrt{10}$.\n7) x2 = $1$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{1 + 3\\cdot \\sqrt{10}, 1\\}$", "final_answer": "x \\in \\{1 + 3\\cdot \\sqrt{10}, 1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9b3ce17c9d7e6acc", "problem_id": "4d877d7731b44b3a", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{10} x - 2 x + 1 + 3 \\sqrt{10}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- 3 \\sqrt{10} - 2$, c=$1 + 3 \\sqrt{10}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $90$, so √Δ = $3 \\sqrt{10}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $1 + 3 \\sqrt{10}$.\nx2 = $1$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{1 + 3\\cdot \\sqrt{10}, 1\\}$", "final_answer": "x \\in \\{1 + 3\\cdot \\sqrt{10}, 1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72bc097bc0884d5b", "problem_id": "f82b7d196392660b", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 2 \\sqrt{3} x + \\frac{21 x}{5} - \\frac{2 \\sqrt{3}}{5} + \\frac{4}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{21}{5} - 2 \\sqrt{3}$, c=$\\frac{4}{5} - \\frac{2 \\sqrt{3}}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{- \\frac{21}{10} + \\frac{\\sqrt{661 - 380\\cdot \\sqrt{3}}}{10} + \\sqrt{3}, - \\frac{21}{10} - \\frac{\\sqrt{661 - 380\\cdot \\sqrt{3}}}{10} + \\sqrt{3}\\}$", "final_answer": "x \\in \\{- \\frac{21}{10} + \\frac{\\sqrt{661 - 380\\cdot \\sqrt{3}}}{10} + \\sqrt{3}, - \\frac{21}{10} - \\frac{\\sqrt{661 - 380\\cdot \\sqrt{3}}}{10} + \\sqrt{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6e95ead95020fba9", "problem_id": "f82b7d196392660b", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 2 \\sqrt{3} x + \\frac{21 x}{5} - \\frac{2 \\sqrt{3}}{5} + \\frac{4}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{21}{5} - 2 \\sqrt{3}$, c=$\\frac{4}{5} - \\frac{2 \\sqrt{3}}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{661}{25} - \\frac{76 \\sqrt{3}}{5}$, so √Δ = $\\sqrt{\\frac{661}{25} - \\frac{76 \\sqrt{3}}{5}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{21}{10} + \\frac{\\sqrt{661 - 380 \\sqrt{3}}}{10} + \\sqrt{3}$.\nx2 = $- \\frac{21}{10} - \\frac{\\sqrt{661 - 380 \\sqrt{3}}}{10} + \\sqrt{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{21}{10} + \\frac{\\sqrt{661 - 380\\cdot \\sqrt{3}}}{10} + \\sqrt{3}, - \\frac{21}{10} - \\frac{\\sqrt{661 - 380\\cdot \\sqrt{3}}}{10} + \\sqrt{3}\\}$", "final_answer": "x \\in \\{- \\frac{21}{10} + \\frac{\\sqrt{661 - 380\\cdot \\sqrt{3}}}{10} + \\sqrt{3}, - \\frac{21}{10} - \\frac{\\sqrt{661 - 380\\cdot \\sqrt{3}}}{10} + \\sqrt{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "022a90041d6580b0", "problem_id": "eafb85e476ebe806", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Worker A finishes a job in 18 hours and Worker B finishes it in 17 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/18 job/hour. B's rate = 1/17 job/hour.\n2) Combined rate = 1/18 + 1/17 = 35/306 job/hour.\n3) Time = 1 ÷ (combined rate) = 306/35 hours.\nFinal answer: $\\frac{306}{35}$", "final_answer": "\\frac{306}{35}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "090c9a79ca69b0af", "problem_id": "eafb85e476ebe806", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Worker A finishes a job in 18 hours and Worker B finishes it in 17 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/18 job/hour. B's rate = 1/17 job/hour.\nCombined rate = 1/18 + 1/17 = 35/306 job/hour.\nTime = 1 ÷ (combined rate) = 306/35 hours.\nCheck: (combined rate)×(time) = 35/306×306/35 = 1.\nFinal answer: $\\frac{306}{35}$", "final_answer": "\\frac{306}{35}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4a43530ca363c896", "problem_id": "8706bc0f7ed6ee87", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$x - 2 y$ = $-9$\n$4 x + y$ = $-18$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $4 x - 8 y$ = $-36$.\n3) Multiply the second equation by $1$: $4 x + y$ = $-18$.\n4) Subtract to eliminate x: $- 9 y$ = $-18$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, 2\\right)$", "final_answer": "(x,y)=\\left(-5, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "847e8ca90b5b5240", "problem_id": "8706bc0f7ed6ee87", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$x - 2 y$ = $-9$\n$4 x + y$ = $-18$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $4 x - 8 y$ = $-36$.\nMultiply the second equation by $1$: $4 x + y$ = $-18$.\nSubtract to eliminate x: $- 9 y$ = $-18$.\nSolve for y: y = $2$.\nSubstitute back to find x: x = $-5$.\nCheck: plug (x,y)=(-5,2) into both equations.\nFinal answer: $(x,y)=\\left(-5, 2\\right)$", "final_answer": "(x,y)=\\left(-5, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "366351f1e5c06d71", "problem_id": "6569d2f9d94d6b47", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 47 miles per hour for 3/2 hours, then at 34 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 47 × 3/2 = 141/2 miles.\nSecond leg distance = 34 × 5/2 = 85 miles.\nTotal distance = 141/2 + 85 = 311/2 miles.\nCheck: adding both legs gives 311/2.\nFinal answer: $\\frac{311}{2}$", "final_answer": "\\frac{311}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cef5014335293ade", "problem_id": "6569d2f9d94d6b47", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A cyclist rides at 47 miles per hour for 3/2 hours, then at 34 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 47 × 3/2 = 141/2 miles.\nSecond leg distance = 34 × 5/2 = 85 miles.\nTotal distance = 141/2 + 85 = 311/2 miles.\nFinal answer: $\\frac{311}{2}$", "final_answer": "\\frac{311}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4367aa87ce083993", "problem_id": "0fa2bc7c7db85418", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{32 x}{15} - \\frac{\\sqrt{2} x}{3} + \\frac{3}{5} + \\frac{3 \\sqrt{2}}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{32}{15} - \\frac{\\sqrt{2}}{3}$, c=$\\frac{3}{5} + \\frac{3 \\sqrt{2}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{178}{75} - \\frac{44 \\sqrt{2}}{45}$, so √Δ = $\\sqrt{\\frac{178}{75} - \\frac{44 \\sqrt{2}}{45}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{2}}{6} + \\frac{\\sqrt{534 - 220 \\sqrt{2}}}{30} + \\frac{16}{15}$.\n7) x2 = $- \\frac{\\sqrt{534 - 220 \\sqrt{2}}}{30} + \\frac{\\sqrt{2}}{6} + \\frac{16}{15}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{2}}{6} + \\frac{\\sqrt{534 - 220\\cdot \\sqrt{2}}}{30} + \\frac{16}{15}, - \\frac{\\sqrt{534 - 220\\cdot \\sqrt{2}}}{30} + \\frac{\\sqrt{2}}{6} + \\frac{16}{15}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{2}}{6} + \\frac{\\sqrt{534 - 220\\cdot \\sqrt{2}}}{30} + \\frac{16}{15}, - \\frac{\\sqrt{534 - 220\\cdot \\sqrt{2}}}{30} + \\frac{\\sqrt{2}}{6} + \\frac{16}{15}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ae5f2407293dcd7f", "problem_id": "0fa2bc7c7db85418", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{32 x}{15} - \\frac{\\sqrt{2} x}{3} + \\frac{3}{5} + \\frac{3 \\sqrt{2}}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{32}{15} - \\frac{\\sqrt{2}}{3}$, c=$\\frac{3}{5} + \\frac{3 \\sqrt{2}}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{178}{75} - \\frac{44 \\sqrt{2}}{45}$, so √Δ = $\\sqrt{\\frac{178}{75} - \\frac{44 \\sqrt{2}}{45}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{2}}{6} + \\frac{\\sqrt{534 - 220 \\sqrt{2}}}{30} + \\frac{16}{15}$.\nx2 = $- \\frac{\\sqrt{534 - 220 \\sqrt{2}}}{30} + \\frac{\\sqrt{2}}{6} + \\frac{16}{15}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{2}}{6} + \\frac{\\sqrt{534 - 220\\cdot \\sqrt{2}}}{30} + \\frac{16}{15}, - \\frac{\\sqrt{534 - 220\\cdot \\sqrt{2}}}{30} + \\frac{\\sqrt{2}}{6} + \\frac{16}{15}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{2}}{6} + \\frac{\\sqrt{534 - 220\\cdot \\sqrt{2}}}{30} + \\frac{16}{15}, - \\frac{\\sqrt{534 - 220\\cdot \\sqrt{2}}}{30} + \\frac{\\sqrt{2}}{6} + \\frac{16}{15}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8411ff310231cdd2", "problem_id": "798fa5b4e4c4f411", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} \\frac{- 2 x^{4} + 3 x^{2} + 20}{x + 2}$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to - \\frac{2 x^{4}}{x + 2} + \\frac{3 x^{2}}{x + 2} - x + \\frac{20}{x + 2}.\n3) Then the limit is \\text{NaN}.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 52.\nFinal answer: $52$", "final_answer": "52", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "196e45e620e9f6e5", "problem_id": "434c85258aff593f", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "You want 70 liters of a 40% salt solution. You have a 30% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 70−x liters is 50% solution.\nSalt equation: 30x + 50(70−x) = 40·70.\n(30-50)x = (40-50)·70 ⇒ x = 35.\nThen 70−x = 35.\nCheck: 30·35 + 50·35 = 40·70.\nFinal answer: $35 L of 30% and 35 L of 50%$", "final_answer": "35 L of 30% and 35 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d63c472aa4993c4a", "problem_id": "d509fdf1a3421ace", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$- x + y$ = $2$\n$- x - y$ = $-4$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $x - y$ = $-2$.\n3) Multiply the second equation by $-1$: $x + y$ = $4$.\n4) Subtract to eliminate x: $- 2 y$ = $-6$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $1$.\nFinal answer: $(x,y)=\\left(1, 3\\right)$", "final_answer": "(x,y)=\\left(1, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d3cb82fb3e6f4ab", "problem_id": "d509fdf1a3421ace", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- x + y$ = $2$\n$- x - y$ = $-4$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $x - y$ = $-2$.\nMultiply the second equation by $-1$: $x + y$ = $4$.\nSubtract to eliminate x: $- 2 y$ = $-6$.\nSolve for y: y = $3$.\nSubstitute back to find x: x = $1$.\nCheck: plug (x,y)=(1,3) into both equations.\nFinal answer: $(x,y)=\\left(1, 3\\right)$", "final_answer": "(x,y)=\\left(1, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7471a57d4d5d46c2", "problem_id": "1f3a4a5569bc077d", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int \\frac{2 x}{x^{2} + 1} + \\frac{\\sin{\\left(x \\right)}}{\\cos{\\left(x \\right)} + 1}\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $\\frac{2 x}{x^{2} + 1} + \\frac{\\sin{\\left(x \\right)}}{\\cos{\\left(x \\right)} + 1}$.\n3) Compute an antiderivative: F(x) = $\\log{\\left(x^{2} + 1 \\right)} - \\log{\\left(\\cos{\\left(x \\right)} + 1 \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + \\log{\\left(x^{2} + 1 \\right)} - \\log{\\left(\\cos{\\left(x \\right)} + 1 \\right)}$", "final_answer": "C + \\log{\\left(x^{2} + 1 \\right)} - \\log{\\left(\\cos{\\left(x \\right)} + 1 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "74fa82a11aecfed0", "problem_id": "1f3a4a5569bc077d", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int \\frac{2 x}{x^{2} + 1} + \\frac{\\sin{\\left(x \\right)}}{\\cos{\\left(x \\right)} + 1}\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $\\frac{2 x}{x^{2} + 1} + \\frac{\\sin{\\left(x \\right)}}{\\cos{\\left(x \\right)} + 1}$.\nCompute an antiderivative: F(x) = $\\log{\\left(x^{2} + 1 \\right)} - \\log{\\left(\\cos{\\left(x \\right)} + 1 \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\log{\\left(x^{2} + 1 \\right)} - \\log{\\left(\\cos{\\left(x \\right)} + 1 \\right)}$", "final_answer": "C + \\log{\\left(x^{2} + 1 \\right)} - \\log{\\left(\\cos{\\left(x \\right)} + 1 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b1d9558145543e21", "problem_id": "24d123f4406e305d", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{3} - x^{2} - \\frac{3 x}{2} + \\left(x^{2} - x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\frac{3}{2}}{x^{2} - x + 3}$.", "output": "1) Let f(x) = $\\frac{x^{3} - x^{2} - \\frac{3 x}{2} + \\left(x^{2} - x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\frac{3}{2}}{x^{2} - x + 3}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{\\left(1 - 2 x\\right) \\left(2 x^{3} - 2 x^{2} - 3 x + 2 \\left(x^{2} - x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x^{2} - x + 3\\right) \\left(6 x^{2} - 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} - x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - x + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 3\\right)}{2 \\left(x^{2} - x + 3\\right)^{2}}$.\nFinal answer: $\\frac{\\left(1 - 2\\cdot x\\right)\\cdot \\left(2\\cdot x^{3} - 2\\cdot x^{2} - 3\\cdot x + \\left(2\\cdot x^{2} - 2\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 3\\right) + \\left(x^{2} - x + 3\\right)\\cdot \\left(6\\cdot x^{2} - 4\\cdot x + \\left(4\\cdot x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(x^{2} - x + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} - 2\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 3\\right)}{2\\cdot \\left(x^{2} - x + 3\\right)^{2}}$", "final_answer": "\\frac{\\left(1 - 2\\cdot x\\right)\\cdot \\left(2\\cdot x^{3} - 2\\cdot x^{2} - 3\\cdot x + \\left(2\\cdot x^{2} - 2\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 3\\right) + \\left(x^{2} - x + 3\\right)\\cdot \\left(6\\cdot x^{2} - 4\\cdot x + \\left(4\\cdot x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(x^{2} - x + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} - 2\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 3\\right)}{2\\cdot \\left(x^{2} - x + 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3cb71d9d84f90683", "problem_id": "49578745ecf6750f", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - 2 x - 4$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to - 3 x - 4.\n3) Then the limit is -13.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: -10.\nFinal answer: $-10$", "final_answer": "-10", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "53cecc7a0a93d316", "problem_id": "06754139572efb3d", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $197. It is discounted by 40%. You buy 6 backpacks. What is the total cost?", "output": "Discounted price = 197 × (1 − 40/100) = 197 × 60/100 = 591/5.\nSubtotal = 6 × 591/5 = 3546/5.\nCheck: recomputing the formula yields 3546/5.\nFinal answer: $\\frac{3546}{5}$", "final_answer": "\\frac{3546}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9bb1e82046dd3472", "problem_id": "06754139572efb3d", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A store sells a backpack for $197. It is discounted by 40%. You buy 6 backpacks. What is the total cost?", "output": "1) Discounted price = 197 × (1 − 40/100) = 197 × 60/100 = 591/5.\n2) Subtotal = 6 × 591/5 = 3546/5.\nFinal answer: $\\frac{3546}{5}$", "final_answer": "\\frac{3546}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ebdda232cb7b8af0", "problem_id": "d131dddabc018d34", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A disease has prevalence $\\frac{1}{25}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{41}{500}$.\n3) Compute P(D|+) = $\\frac{17}{41}$.\nFinal answer: $\\frac{17}{41}$", "final_answer": "\\frac{17}{41}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "758e69922d97cbd3", "problem_id": "d131dddabc018d34", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A disease has prevalence $\\frac{1}{25}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{41}{500}$.\nCompute P(D|+) = $\\frac{17}{41}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{17}{41}$", "final_answer": "\\frac{17}{41}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "48f66df6170fcc1d", "problem_id": "5bdf9ca50b152367", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $2125 in the ratio 8:9. How much does each receive?", "output": "1) Total ratio parts = 8+9 = 17.\n2) Each part = 2125 ÷ 17 = 125.\n3) First share = 8 × 125 = 1000.\n4) Second share = 9 × 125 = 1125.\nFinal answer: $1125$", "final_answer": "1125", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "368b12ea9b2e7a38", "problem_id": "5bdf9ca50b152367", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $2125 in the ratio 8:9. How much does each receive?", "output": "Total ratio parts = 8+9 = 17.\nEach part = 2125 ÷ 17 = 125.\nFirst share = 8 × 125 = 1000.\nSecond share = 9 × 125 = 1125.\nCheck: 1000+1125=2125.\nFinal answer: $1125$", "final_answer": "1125", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e4583b6d3f8ffcf4", "problem_id": "022f12113f5afcbc", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the linear system:\n$7 x - 8 y$ = $29$\n$- 5 x$ = $-15$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-5$: $- 35 x + 40 y$ = $-145$.\n3) Multiply the second equation by $7$: $- 35 x$ = $-105$.\n4) Subtract to eliminate x: $40 y$ = $-40$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, -1\\right)$", "final_answer": "(x,y)=\\left(3, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d2439cbbee771c7a", "problem_id": "022f12113f5afcbc", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$7 x - 8 y$ = $29$\n$- 5 x$ = $-15$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-5$: $- 35 x + 40 y$ = $-145$.\nMultiply the second equation by $7$: $- 35 x$ = $-105$.\nSubtract to eliminate x: $40 y$ = $-40$.\nSolve for y: y = $-1$.\nSubstitute back to find x: x = $3$.\nCheck: plug (x,y)=(3,-1) into both equations.\nFinal answer: $(x,y)=\\left(3, -1\\right)$", "final_answer": "(x,y)=\\left(3, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "219ba56d0cfef7f4", "problem_id": "ccd7565c3e0cd78f", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "How many binary strings of length 12 contain exactly 4 ones with no two ones adjacent?", "output": "1) Reserve 3 mandatory zeros between the 4 ones.\n2) This leaves n−k+1 = 9 valid slots for the ones.\n3) Choose the slots: C(9,4) = 126.\nFinal answer: $126$", "final_answer": "126", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1d613d9923823b2c", "problem_id": "ccd7565c3e0cd78f", "style": "answer_only", "instruction": "Final answer only.", "input": "How many binary strings of length 12 contain exactly 4 ones with no two ones adjacent?", "output": "$126$", "final_answer": "126", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d66f79373bd578d", "problem_id": "d8d6f54510aad5d4", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Two friends split $80 in the ratio 1:3. How much does each receive?", "output": "1) Total ratio parts = 1+3 = 4.\n2) Each part = 80 ÷ 4 = 20.\n3) First share = 1 × 20 = 20.\n4) Second share = 3 × 20 = 60.\nFinal answer: $60$", "final_answer": "60", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "354273001bba34fa", "problem_id": "d8d6f54510aad5d4", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Two friends split $80 in the ratio 1:3. How much does each receive?", "output": "Total ratio parts = 1+3 = 4.\nEach part = 80 ÷ 4 = 20.\nFirst share = 1 × 20 = 20.\nSecond share = 3 × 20 = 60.\nCheck: 20+60=80.\nFinal answer: $60$", "final_answer": "60", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7714b9b83a9cf993", "problem_id": "1ab66f27378cde36", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Worker A finishes a job in 12 hours and Worker B finishes it in 13 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/12 job/hour. B's rate = 1/13 job/hour.\nCombined rate = 1/12 + 1/13 = 25/156 job/hour.\nTime = 1 ÷ (combined rate) = 156/25 hours.\nCheck: (combined rate)×(time) = 25/156×156/25 = 1.\nFinal answer: $\\frac{156}{25}$", "final_answer": "\\frac{156}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "96103d271c95e9db", "problem_id": "1ab66f27378cde36", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Worker A finishes a job in 12 hours and Worker B finishes it in 13 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/12 job/hour. B's rate = 1/13 job/hour.\n2) Combined rate = 1/12 + 1/13 = 25/156 job/hour.\n3) Time = 1 ÷ (combined rate) = 156/25 hours.\nFinal answer: $\\frac{156}{25}$", "final_answer": "\\frac{156}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b1181d2b1ae68aba", "problem_id": "c4149ba95f659905", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_9=28$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=28 and k=9.\n4) Compute C(36,8) = 30260340.\nFinal answer: $30260340$", "final_answer": "30260340", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f123f3febe3598c2", "problem_id": "c4149ba95f659905", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_9=28$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=28 and k=9.\nCompute C(36,8) = 30260340.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $30260340$", "final_answer": "30260340", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dfcf052e04b1e1e0", "problem_id": "ff9684d2baa206f0", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$- 8 x + 2 y$ = $44$\n$7 x + 10 y$ = $-62$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $7$: $- 56 x + 14 y$ = $308$.\n3) Multiply the second equation by $-8$: $- 56 x - 80 y$ = $496$.\n4) Subtract to eliminate x: $94 y$ = $-188$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $-6$.\nFinal answer: $(x,y)=\\left(-6, -2\\right)$", "final_answer": "(x,y)=\\left(-6, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "098daf54b22c36b0", "problem_id": "e07c8f3c1d5f8a5b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - 3 x^{3} - 12 x^{2} - 38 x - 113$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{3} - 12 x^{2} - 39 x - 113.\n3) Then the limit is -419.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: -416.\nFinal answer: $-416$", "final_answer": "-416", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5aa2387e0f540392", "problem_id": "28243ad772e62ae5", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int x^{2} - 2 x\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $x^{2} - 2 x$.\n3) Compute an antiderivative: F(x) = $\\frac{x^{3}}{3} - x^{2}$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{x^{3}}{3} - x^{2}$", "final_answer": "C + \\frac{x^{3}}{3} - x^{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c7a50f40a984216e", "problem_id": "67193628b2d1589d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 3 x + 7$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 2 x + 7.\n3) Then the limit is 11.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 13.\nFinal answer: $13$", "final_answer": "13", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "af477bcd8081921a", "problem_id": "daa330c82d21967f", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡9 (mod 11), x≡12 (mod 13). Give the solution modulo 143.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 143.\nCompute the CRT solution to get x ≡ 64 (mod 143).\nCheck: 64 mod 11=9 and mod 13=12.\nFinal answer: $x\\equiv 64\\pmod{143}$", "final_answer": "x\\equiv 64\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a3f5c782c72d1b95", "problem_id": "daa330c82d21967f", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the congruences: x≡9 (mod 11), x≡12 (mod 13). Give the solution modulo 143.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 143.\nCompute the CRT solution to get x ≡ 64 (mod 143).\nFinal answer: $x\\equiv 64\\pmod{143}$", "final_answer": "x\\equiv 64\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f0a246b4b7774cd4", "problem_id": "e1636336dac3aceb", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{4} - 2 x^{3} + 2 x^{2} + 4 x + 2$.", "output": "1) Differentiate term-by-term:\n2) d/dx(2) = 0.\n3) d/dx(- 2 x^{3}) = - 6 x^{2}.\n4) d/dx(2 x^{2}) = 4 x.\n5) d/dx(2 x^{4}) = 8 x^{3}.\n6) d/dx(4 x) = 4.\n7) Combine: f'(x) = 2 \\left(4 x^{3} - 3 x^{2} + 2 x + 2\\right).\nFinal answer: $8\\cdot x^{3} - 6\\cdot x^{2} + 4\\cdot x + 4$", "final_answer": "8\\cdot x^{3} - 6\\cdot x^{2} + 4\\cdot x + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5ed5da545dada59f", "problem_id": "e1636336dac3aceb", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{4} - 2 x^{3} + 2 x^{2} + 4 x + 2$.", "output": "Differentiate term-by-term:\nd/dx(2) = 0.\nd/dx(- 2 x^{3}) = - 6 x^{2}.\nd/dx(2 x^{2}) = 4 x.\nd/dx(2 x^{4}) = 8 x^{3}.\nd/dx(4 x) = 4.\nCombine: f'(x) = 2 \\left(4 x^{3} - 3 x^{2} + 2 x + 2\\right).\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $8\\cdot x^{3} - 6\\cdot x^{2} + 4\\cdot x + 4$", "final_answer": "8\\cdot x^{3} - 6\\cdot x^{2} + 4\\cdot x + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec9e46fc21ef2398", "problem_id": "29f1b75a708a209c", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Two friends split $1738 in the ratio 6:5. How much does each receive?", "output": "Total ratio parts = 6+5 = 11.\nEach part = 1738 ÷ 11 = 158.\nFirst share = 6 × 158 = 948.\nSecond share = 5 × 158 = 790.\nCheck: 948+790=1738.\nFinal answer: $790$", "final_answer": "790", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9d1d06e78b74fbe7", "problem_id": "29f1b75a708a209c", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Two friends split $1738 in the ratio 6:5. How much does each receive?", "output": "1) Total ratio parts = 6+5 = 11.\n2) Each part = 1738 ÷ 11 = 158.\n3) First share = 6 × 158 = 948.\n4) Second share = 5 × 158 = 790.\nFinal answer: $790$", "final_answer": "790", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "049d30a011692869", "problem_id": "8c2b46771a27cbde", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} \\sin{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $x^{3} \\sin{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $x^{3}\\cdot \\cos{\\left(x \\right)} + 3\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - \\cos{\\left(x \\right)}$", "final_answer": "x^{3}\\cdot \\cos{\\left(x \\right)} + 3\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f32247928c4ba334", "problem_id": "8c2b46771a27cbde", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} \\sin{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $x^{3} \\sin{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.\nFinal answer: $x^{3}\\cdot \\cos{\\left(x \\right)} + 3\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - \\cos{\\left(x \\right)}$", "final_answer": "x^{3}\\cdot \\cos{\\left(x \\right)} + 3\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ecd0e29c66740086", "problem_id": "f6b341b9c2c7b6bc", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 14 times, what is the probability of exactly 11 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=14, k=11, p=$\\frac{1}{10}$ and simplify.\nHere C(14,11) = 364.\nCheck: expression simplifies to $\\frac{66339}{25000000000000}$.\nFinal answer: $\\frac{66339}{25000000000000}$", "final_answer": "\\frac{66339}{25000000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "97feb6e69eb567ff", "problem_id": "f6b341b9c2c7b6bc", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 14 times, what is the probability of exactly 11 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=14, k=11, p=$\\frac{1}{10}$ and simplify.\n4) Here C(14,11) = 364.\nFinal answer: $\\frac{66339}{25000000000000}$", "final_answer": "\\frac{66339}{25000000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "594bef4bd050a80c", "problem_id": "2e4ea81e4126e17c", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{14 x}{15} - 4$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$\\frac{14}{15}$, c=$-4$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{7396}{225}$, so √Δ = $\\frac{86}{15}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{6}{5}$.\nx2 = $- \\frac{5}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{6}{5}, - \\frac{5}{3}\\}$", "final_answer": "x \\in \\{\\frac{6}{5}, - \\frac{5}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3121f341ba00f6d4", "problem_id": "2e4ea81e4126e17c", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{14 x}{15} - 4$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$\\frac{14}{15}$, c=$-4$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{7396}{225}$, so √Δ = $\\frac{86}{15}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{6}{5}$.\n7) x2 = $- \\frac{5}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{6}{5}, - \\frac{5}{3}\\}$", "final_answer": "x \\in \\{\\frac{6}{5}, - \\frac{5}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8d2c31440a46382c", "problem_id": "ad425eaeb3daac3e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 2 x^{2} \\sin{\\left(x \\right)}) = - 2 x^{2} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 2 x^{2} \\sin{\\left(x \\right)}) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}\\right)$.\nFinal answer: $2 \\left(- x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}\\right)$", "final_answer": "2 \\left(- x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "99d2f81927745b97", "problem_id": "40f23482a5dc7a37", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$12 x$ = $72$\n$- x - 12 y$ = $18$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $- 12 x$ = $-72$.\nMultiply the second equation by $12$: $- 12 x - 144 y$ = $216$.\nSubtract to eliminate x: $144 y$ = $-288$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $6$.\nCheck: plug (x,y)=(6,-2) into both equations.\nFinal answer: $(x,y)=\\left(6, -2\\right)$", "final_answer": "(x,y)=\\left(6, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "48bb9458f4e99600", "problem_id": "e029f52b5a14feb3", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_4=12$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=12 and k=4.\nCompute C(15,3) = 455.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $455$", "final_answer": "455", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "512a8957af42f9a6", "problem_id": "e3000c6b13e01f35", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(- x^{3} - 3 x^{2} + x - 2\\right) \\cos{\\left(3 x - 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x^{3} \\cos{\\left(3 x - 1 \\right)}) = x^{3} \\sin{\\left(3 x - 1 \\right)}$.\n3) So $f'(x) = x^{3} \\sin{\\left(3 x - 1 \\right)} + 9 x^{2} \\sin{\\left(3 x - 1 \\right)} + 2 x e^{x} - 3 x \\sin{\\left(3 x - 1 \\right)} - 6 x \\cos{\\left(3 x - 1 \\right)} + 6 e^{x} + 6 \\sin{\\left(3 x - 1 \\right)} + \\cos{\\left(3 x - 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x^{3} \\cos{\\left(3 x - 1 \\right)}) = 3 x^{3} \\sin{\\left(3 x - 1 \\right)} - 3 x^{2} \\cos{\\left(3 x - 1 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x + 2\\right) e^{x} - 3 \\left(x \\left(x \\left(- x - 3\\right) + 1\\right) - 2\\right) \\sin{\\left(3 x - 1 \\right)} + \\left(- 3 x^{2} - 6 x + 1\\right) \\cos{\\left(3 x - 1 \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(x + 2\\right) e^{x} - 3 \\left(x \\left(x \\left(- x - 3\\right) + 1\\right) - 2\\right) \\sin{\\left(3 x - 1 \\right)} + \\left(- 3 x^{2} - 6 x + 1\\right) \\cos{\\left(3 x - 1 \\right)} + 2 e^{x}$", "final_answer": "2 \\left(x + 2\\right) e^{x} - 3 \\left(x \\left(x \\left(- x - 3\\right) + 1\\right) - 2\\right) \\sin{\\left(3 x - 1 \\right)} + \\left(- 3 x^{2} - 6 x + 1\\right) \\cos{\\left(3 x - 1 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f33c303189330636", "problem_id": "d51ceadba10f882c", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $187 in the ratio 3:8. How much does each receive?", "output": "Total ratio parts = 3+8 = 11.\nEach part = 187 ÷ 11 = 17.\nFirst share = 3 × 17 = 51.\nSecond share = 8 × 17 = 136.\nCheck: 51+136=187.\nFinal answer: $136$", "final_answer": "136", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1d284f9d204ef087", "problem_id": "d51ceadba10f882c", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Two friends split $187 in the ratio 3:8. How much does each receive?", "output": "1) Total ratio parts = 3+8 = 11.\n2) Each part = 187 ÷ 11 = 17.\n3) First share = 3 × 17 = 51.\n4) Second share = 8 × 17 = 136.\nFinal answer: $136$", "final_answer": "136", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f7598d946003f3ce", "problem_id": "f159eca792120ea4", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} 2 x^{3} + 11 x^{2} + 41 x + 162$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $2 x^{4} + 3 x^{3} - 3 x^{2} - 2 x - 3$ and a = 4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $8 x^{3} + 9 x^{2} - 6 x - 2$.\n5) Evaluate at a: p'(4) = $630$.\nFinal answer: $630$", "final_answer": "630", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "858b632766c84a24", "problem_id": "f159eca792120ea4", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} 2 x^{3} + 11 x^{2} + 41 x + 162$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{4} + 3 x^{3} - 3 x^{2} - 2 x - 3$ and a = 4.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $8 x^{3} + 9 x^{2} - 6 x - 2$.\nEvaluate at a: p'(4) = $630$.\nCheck: the identity gives $630$.\nFinal answer: $630$", "final_answer": "630", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0bf9a4f627375b9a", "problem_id": "4fb21e27be4d7362", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 9 red balls and 13 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(22,4).\n3) Favorable samples: choose k reds and n−k blues: C(9,2)·C(13,2).\n4) Compute totals: C(22,4) = 7315, and favorable = 2808.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{2808}{7315}$", "final_answer": "\\frac{2808}{7315}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5f58d0b62c532638", "problem_id": "4fb21e27be4d7362", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "An urn has 9 red balls and 13 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(22,4).\nFavorable samples: choose k reds and n−k blues: C(9,2)·C(13,2).\nFinal answer: $\\frac{2808}{7315}$", "final_answer": "\\frac{2808}{7315}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a14b5d839d8b83cc", "problem_id": "c096612af7ceede0", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_4=25$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=25 and k=4.\n4) Compute C(28,3) = 3276.\nFinal answer: $3276$", "final_answer": "3276", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "24c48f55613e6104", "problem_id": "c096612af7ceede0", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_4=25$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=25 and k=4.\nCompute C(28,3) = 3276.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $3276$", "final_answer": "3276", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "53f75fb4c2d12f00", "problem_id": "e5d73494ceba5075", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 3 x - 28$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} + 3 x - 28$ = $\\left(x - 4\\right) \\left(x + 7\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $4$.\n5) From the second factor: x = $-7$.\nFinal answer: $x \\in \\{4, -7\\}$", "final_answer": "x \\in \\{4, -7\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b320e7a5a8f5bb3d", "problem_id": "e5d73494ceba5075", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 3 x - 28$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} + 3 x - 28$ = $\\left(x - 4\\right) \\left(x + 7\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $4$.\nFrom the second factor: x = $-7$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{4, -7\\}$", "final_answer": "x \\in \\{4, -7\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "38b01d81192d0b10", "problem_id": "f4aa48561a6d5b70", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the linear system:\n$6 x - 3 y$ = $-42$\n$7 x + 2 y$ = $-71$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $7$: $42 x - 21 y$ = $-294$.\nMultiply the second equation by $6$: $42 x + 12 y$ = $-426$.\nFinal answer: $(x,y)=\\left(-9, -4\\right)$", "final_answer": "(x,y)=\\left(-9, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "31550ca2ebdddfbb", "problem_id": "f4aa48561a6d5b70", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$6 x - 3 y$ = $-42$\n$7 x + 2 y$ = $-71$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $7$: $42 x - 21 y$ = $-294$.\n3) Multiply the second equation by $6$: $42 x + 12 y$ = $-426$.\n4) Subtract to eliminate x: $- 33 y$ = $132$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $-9$.\nFinal answer: $(x,y)=\\left(-9, -4\\right)$", "final_answer": "(x,y)=\\left(-9, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5c129efed8532fa3", "problem_id": "ddc859565bacadda", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A store sells a backpack for $78. It is discounted by 30%. You buy 7 backpacks. What is the total cost?", "output": "1) Discounted price = 78 × (1 − 30/100) = 78 × 70/100 = 273/5.\n2) Subtotal = 7 × 273/5 = 1911/5.\nFinal answer: $\\frac{1911}{5}$", "final_answer": "\\frac{1911}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c81d1bdbae26a6aa", "problem_id": "4de46a39da1882c5", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(- 2 x^{3} - 3\\right) \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\n3) So $f'(x) = 4 x^{3} \\sin{\\left(2 x \\right)} - 6 x^{2} \\cos{\\left(2 x \\right)} + x e^{x} + 3 e^{x} + 6 \\sin{\\left(2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = - 6 x^{2} \\cos{\\left(2 x \\right)} + \\left(x + 2\\right) e^{x} - 2 \\left(- 2 x^{3} - 3\\right) \\sin{\\left(2 x \\right)} + e^{x}$.\nFinal answer: $- 6 x^{2} \\cos{\\left(2 x \\right)} + \\left(x + 2\\right) e^{x} - 2 \\left(- 2 x^{3} - 3\\right) \\sin{\\left(2 x \\right)} + e^{x}$", "final_answer": "- 6 x^{2} \\cos{\\left(2 x \\right)} + \\left(x + 2\\right) e^{x} - 2 \\left(- 2 x^{3} - 3\\right) \\sin{\\left(2 x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "89404d406878a267", "problem_id": "d6a97cad3d3c8553", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 13 x + 36$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} + 13 x + 36$ = $\\left(x + 4\\right) \\left(x + 9\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-4$.\n5) From the second factor: x = $-9$.\nFinal answer: $x \\in \\{-4, -9\\}$", "final_answer": "x \\in \\{-4, -9\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "55fd1cbf7d0457a3", "problem_id": "d6a97cad3d3c8553", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 13 x + 36$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} + 13 x + 36$ = $\\left(x + 4\\right) \\left(x + 9\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $-4$.\nFrom the second factor: x = $-9$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-4, -9\\}$", "final_answer": "x \\in \\{-4, -9\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "20a384736ea03dcd", "problem_id": "f7c77a289a6e0f6b", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $396 in the ratio 2:1. How much does each receive?", "output": "1) Total ratio parts = 2+1 = 3.\n2) Each part = 396 ÷ 3 = 132.\n3) First share = 2 × 132 = 264.\n4) Second share = 1 × 132 = 132.\nFinal answer: $132$", "final_answer": "132", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a120fc544756b4d8", "problem_id": "f7c77a289a6e0f6b", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Two friends split $396 in the ratio 2:1. How much does each receive?", "output": "$132$", "final_answer": "132", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a04a45e0a7ea46b0", "problem_id": "8c2b46771a27cbde", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} \\sin{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 3 x^{2} \\sin{\\left(x \\right)}) = - 3 x^{2} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 3 x^{2} \\cos{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 3 x^{2} \\sin{\\left(x \\right)}) = - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.\nFinal answer: $x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$", "final_answer": "x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e7059c7475a4ee12", "problem_id": "8042a8a505da3af9", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Two friends split $1680 in the ratio 9:7. How much does each receive?", "output": "Total ratio parts = 9+7 = 16.\nEach part = 1680 ÷ 16 = 105.\nFirst share = 9 × 105 = 945.\nSecond share = 7 × 105 = 735.\nCheck: 945+735=1680.\nFinal answer: $735$", "final_answer": "735", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "076e2791fd6fbca3", "problem_id": "00a37c4d069d9402", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A cyclist rides at 89 miles per hour for 1/2 hours, then at 56 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 89 × 1/2 = 89/2 miles.\n2) Second leg distance = 56 × 2 = 112 miles.\n3) Total distance = 89/2 + 112 = 313/2 miles.\nFinal answer: $\\frac{313}{2}$", "final_answer": "\\frac{313}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3955974bee3877f5", "problem_id": "00a37c4d069d9402", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "A cyclist rides at 89 miles per hour for 1/2 hours, then at 56 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 89 × 1/2 = 89/2 miles.\nSecond leg distance = 56 × 2 = 112 miles.\nTotal distance = 89/2 + 112 = 313/2 miles.\nFinal answer: $\\frac{313}{2}$", "final_answer": "\\frac{313}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1983d86d3a3dc569", "problem_id": "a514c4bdaf682579", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "You want 25 liters of a 45% salt solution. You have a 10% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 25−x liters is 50% solution.\n2) Salt equation: 10x + 50(25−x) = 45·25.\n3) (10-50)x = (45-50)·25 ⇒ x = 25/8.\n4) Then 25−x = 175/8.\nFinal answer: $25/8 L of 10% and 175/8 L of 50%$", "final_answer": "25/8 L of 10% and 175/8 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fcd8c21e4fac963d", "problem_id": "a514c4bdaf682579", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "You want 25 liters of a 45% salt solution. You have a 10% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 25−x liters is 50% solution.\nSalt equation: 10x + 50(25−x) = 45·25.\n(10-50)x = (45-50)·25 ⇒ x = 25/8.\nFinal answer: $25/8 L of 10% and 175/8 L of 50%$", "final_answer": "25/8 L of 10% and 175/8 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "133e996553e7d854", "problem_id": "c4c9a006aaef5c9e", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Two friends split $1620 in the ratio 1:9. How much does each receive?", "output": "1) Total ratio parts = 1+9 = 10.\n2) Each part = 1620 ÷ 10 = 162.\n3) First share = 1 × 162 = 162.\n4) Second share = 9 × 162 = 1458.\nFinal answer: $1458$", "final_answer": "1458", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e3ed4c798c65fa1a", "problem_id": "c4c9a006aaef5c9e", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $1620 in the ratio 1:9. How much does each receive?", "output": "Total ratio parts = 1+9 = 10.\nEach part = 1620 ÷ 10 = 162.\nFirst share = 1 × 162 = 162.\nSecond share = 9 × 162 = 1458.\nCheck: 162+1458=1620.\nFinal answer: $1458$", "final_answer": "1458", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "28b161d9f9561dbc", "problem_id": "78c2aa1972e48b26", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 16 red balls and 9 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(25,4).\n3) Favorable samples: choose k reds and n−k blues: C(16,1)·C(9,3).\n4) Compute totals: C(25,4) = 12650, and favorable = 1344.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{672}{6325}$", "final_answer": "\\frac{672}{6325}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "53975dcf850ff40c", "problem_id": "78c2aa1972e48b26", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "An urn has 16 red balls and 9 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(25,4).\nFavorable samples: choose k reds and n−k blues: C(16,1)·C(9,3).\nFinal answer: $\\frac{672}{6325}$", "final_answer": "\\frac{672}{6325}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "83bc4ff80dca67ba", "problem_id": "4b21f53d247eae1f", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Two friends split $1937 in the ratio 7:6. How much does each receive?", "output": "Total ratio parts = 7+6 = 13.\nEach part = 1937 ÷ 13 = 149.\nFirst share = 7 × 149 = 1043.\nFinal answer: $894$", "final_answer": "894", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "59fbb488ae4ae494", "problem_id": "4b21f53d247eae1f", "style": "answer_only", "instruction": "Final answer only.", "input": "Two friends split $1937 in the ratio 7:6. How much does each receive?", "output": "$894$", "final_answer": "894", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "678c1a45e71362e3", "problem_id": "e2307f4bde46588e", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$8 x + 9 y$ = $84$\n$9 x$ = $54$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $9$: $72 x + 81 y$ = $756$.\n3) Multiply the second equation by $8$: $72 x$ = $432$.\n4) Subtract to eliminate x: $81 y$ = $324$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $6$.\nFinal answer: $(x,y)=\\left(6, 4\\right)$", "final_answer": "(x,y)=\\left(6, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "12a22485bc7de0e5", "problem_id": "e2307f4bde46588e", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$8 x + 9 y$ = $84$\n$9 x$ = $54$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $9$: $72 x + 81 y$ = $756$.\nMultiply the second equation by $8$: $72 x$ = $432$.\nSubtract to eliminate x: $81 y$ = $324$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $6$.\nCheck: plug (x,y)=(6,4) into both equations.\nFinal answer: $(x,y)=\\left(6, 4\\right)$", "final_answer": "(x,y)=\\left(6, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8bc99959b59eb4e6", "problem_id": "d784827591fd6768", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many binary strings of length 15 contain exactly 2 ones with no two ones adjacent?", "output": "Reserve 1 mandatory zeros between the 2 ones.\nThis leaves n−k+1 = 14 valid slots for the ones.\nChoose the slots: C(14,2) = 91.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $91$", "final_answer": "91", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "514f43a27602f015", "problem_id": "d784827591fd6768", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many binary strings of length 15 contain exactly 2 ones with no two ones adjacent?", "output": "1) Reserve 1 mandatory zeros between the 2 ones.\n2) This leaves n−k+1 = 14 valid slots for the ones.\n3) Choose the slots: C(14,2) = 91.\nFinal answer: $91$", "final_answer": "91", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dde27d55759923ee", "problem_id": "4b52802224f7a376", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A disease has prevalence $\\frac{1}{200}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{109}{2000}$.\nCompute P(D|+) = $\\frac{19}{218}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{19}{218}$", "final_answer": "\\frac{19}{218}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8ae27d65b0d301f7", "problem_id": "4b52802224f7a376", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "A disease has prevalence $\\frac{1}{200}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{109}{2000}$.\nCompute P(D|+) = $\\frac{19}{218}$.\nFinal answer: $\\frac{19}{218}$", "final_answer": "\\frac{19}{218}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c78bb77c480342c6", "problem_id": "faf41fcc8b4425ee", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{13 x}{2} + \\frac{3}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$- \\frac{13}{2}$, c=$\\frac{3}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{121}{4}$, so √Δ = $\\frac{11}{2}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $3$.\nx2 = $\\frac{1}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{3, \\frac{1}{4}\\}$", "final_answer": "x \\in \\{3, \\frac{1}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fdc0158f4355893b", "problem_id": "a9836c54ff5c5377", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{3} + 3 x^{2} + 2 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{2 x - 3}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 3}) = \\frac{2 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 3}$.\n3) So $f'(x) = \\frac{8 x^{3}}{8 x^{2} - 24 x + 18} + \\frac{4 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{2} - 24 x + 18} + \\frac{8 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{2} - 24 x + 18} - \\frac{6 x^{2}}{8 x^{2} - 24 x + 18} - \\frac{12 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{2} - 24 x + 18} - \\frac{24 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{2} - 24 x + 18} - \\frac{36 x}{8 x^{2} - 24 x + 18} + \\frac{4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{4 x^{2} - 12 x + 9} + \\frac{2 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 3} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 3} - \\frac{2 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 3} + \\frac{9 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{2} - 24 x + 18} + \\frac{18 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{2} - 24 x + 18} - \\frac{20}{8 x^{2} - 24 x + 18} - \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 3}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 3}) = \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 3} + \\frac{2 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 3} - \\frac{4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x - 3\\right)^{2}} + \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 3}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 4 x^{3} - 12 x^{2} - 8 x + \\left(2 x - 3\\right) \\left(6 x^{2} + 12 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 4 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8}{2 \\left(2 x - 3\\right)^{2}}$.\nFinal answer: $\\frac{- 4 x^{3} - 12 x^{2} - 8 x + \\left(2 x - 3\\right) \\left(6 x^{2} + 12 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 4 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8}{2 \\left(2 x - 3\\right)^{2}}$", "final_answer": "\\frac{- 4 x^{3} - 12 x^{2} - 8 x + \\left(2 x - 3\\right) \\left(6 x^{2} + 12 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 4 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8}{2 \\left(2 x - 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "73fb7d7f48b869b5", "problem_id": "127c1fbaa42cc3a3", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Two friends split $420 in the ratio 4:8. How much does each receive?", "output": "1) Total ratio parts = 4+8 = 12.\n2) Each part = 420 ÷ 12 = 35.\n3) First share = 4 × 35 = 140.\n4) Second share = 8 × 35 = 280.\nFinal answer: $280$", "final_answer": "280", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c5256c64e8bbd8b9", "problem_id": "127c1fbaa42cc3a3", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Two friends split $420 in the ratio 4:8. How much does each receive?", "output": "Total ratio parts = 4+8 = 12.\nEach part = 420 ÷ 12 = 35.\nFirst share = 4 × 35 = 140.\nFinal answer: $280$", "final_answer": "280", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "83b7a5d851fea932", "problem_id": "390b661208526823", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Worker A finishes a job in 8 hours and Worker B finishes it in 6 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/8 job/hour. B's rate = 1/6 job/hour.\n2) Combined rate = 1/8 + 1/6 = 7/24 job/hour.\n3) Time = 1 ÷ (combined rate) = 24/7 hours.\nFinal answer: $\\frac{24}{7}$", "final_answer": "\\frac{24}{7}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "098ed8122381913c", "problem_id": "390b661208526823", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Worker A finishes a job in 8 hours and Worker B finishes it in 6 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/8 job/hour. B's rate = 1/6 job/hour.\nCombined rate = 1/8 + 1/6 = 7/24 job/hour.\nTime = 1 ÷ (combined rate) = 24/7 hours.\nCheck: (combined rate)×(time) = 7/24×24/7 = 1.\nFinal answer: $\\frac{24}{7}$", "final_answer": "\\frac{24}{7}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "82ccb43321d777dd", "problem_id": "6042d2c58fdc8d1d", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$- 4 y$ = $-16$\n$- 3 x - 3 y$ = $-24$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $12 y$ = $48$.\n3) Multiply the second equation by $0$: $0$ = $0$.\n4) Subtract to eliminate x: $12 y$ = $48$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, 4\\right)$", "final_answer": "(x,y)=\\left(4, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "51a0d60d6deb3d3c", "problem_id": "6042d2c58fdc8d1d", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- 4 y$ = $-16$\n$- 3 x - 3 y$ = $-24$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $12 y$ = $48$.\nMultiply the second equation by $0$: $0$ = $0$.\nSubtract to eliminate x: $12 y$ = $48$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $4$.\nCheck: plug (x,y)=(4,4) into both equations.\nFinal answer: $(x,y)=\\left(4, 4\\right)$", "final_answer": "(x,y)=\\left(4, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bf6d6a3f6c59a533", "problem_id": "6de48fafdaf1af3d", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A cyclist rides at 39 miles per hour for 1/2 hours, then at 42 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 39 × 1/2 = 39/2 miles.\n2) Second leg distance = 42 × 1 = 42 miles.\n3) Total distance = 39/2 + 42 = 123/2 miles.\nFinal answer: $\\frac{123}{2}$", "final_answer": "\\frac{123}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "31cac410d2e6e616", "problem_id": "6de48fafdaf1af3d", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A cyclist rides at 39 miles per hour for 1/2 hours, then at 42 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 39 × 1/2 = 39/2 miles.\nSecond leg distance = 42 × 1 = 42 miles.\nTotal distance = 39/2 + 42 = 123/2 miles.\nCheck: adding both legs gives 123/2.\nFinal answer: $\\frac{123}{2}$", "final_answer": "\\frac{123}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "df7adf9cbcc3846b", "problem_id": "68c789a421a9a47a", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - 2 x^{3} + 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{3} + 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{255 \\pi^{4}}{32} - 2 + \\frac{5 \\pi}{2} + \\frac{45 \\pi^{2}}{8} + \\frac{65 \\pi^{3}}{8}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{255\\cdot \\pi^{4}}{32} - 2 + \\frac{5\\cdot \\pi}{2} + \\frac{45\\cdot \\pi^{2}}{8} + \\frac{65\\cdot \\pi^{3}}{8}$", "final_answer": "- \\frac{255\\cdot \\pi^{4}}{32} - 2 + \\frac{5\\cdot \\pi}{2} + \\frac{45\\cdot \\pi^{2}}{8} + \\frac{65\\cdot \\pi^{3}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eee1ee98db20935d", "problem_id": "68c789a421a9a47a", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - 2 x^{3} + 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{3} + 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{255 \\pi^{4}}{32} - 2 + \\frac{5 \\pi}{2} + \\frac{45 \\pi^{2}}{8} + \\frac{65 \\pi^{3}}{8}$.\n4) Simplify the result.\nFinal answer: $- \\frac{255\\cdot \\pi^{4}}{32} - 2 + \\frac{5\\cdot \\pi}{2} + \\frac{45\\cdot \\pi^{2}}{8} + \\frac{65\\cdot \\pi^{3}}{8}$", "final_answer": "- \\frac{255\\cdot \\pi^{4}}{32} - 2 + \\frac{5\\cdot \\pi}{2} + \\frac{45\\cdot \\pi^{2}}{8} + \\frac{65\\cdot \\pi^{3}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "22c44e0ec1869452", "problem_id": "eb3dc200753b87c5", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 2 x - 2$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to x - 2.\n3) Then the limit is -3.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: -4.\nFinal answer: $-4$", "final_answer": "-4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5599678d8a1a50ec", "problem_id": "bf63ec3e0358a4fd", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{11} x + \\frac{5 x}{2} - \\frac{\\sqrt{11}}{2} + 1$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{5}{2} - \\sqrt{11}$, c=$1 - \\frac{\\sqrt{11}}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{53}{4} - 3 \\sqrt{11}$, so √Δ = $\\sqrt{\\frac{53}{4} - 3 \\sqrt{11}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{5}{4} + \\frac{\\sqrt{53 - 12 \\sqrt{11}}}{4} + \\frac{\\sqrt{11}}{2}$.\n7) x2 = $- \\frac{5}{4} - \\frac{\\sqrt{53 - 12 \\sqrt{11}}}{4} + \\frac{\\sqrt{11}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{5}{4} + \\frac{\\sqrt{53 - 12\\cdot \\sqrt{11}}}{4} + \\frac{\\sqrt{11}}{2}, - \\frac{5}{4} - \\frac{\\sqrt{53 - 12\\cdot \\sqrt{11}}}{4} + \\frac{\\sqrt{11}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{5}{4} + \\frac{\\sqrt{53 - 12\\cdot \\sqrt{11}}}{4} + \\frac{\\sqrt{11}}{2}, - \\frac{5}{4} - \\frac{\\sqrt{53 - 12\\cdot \\sqrt{11}}}{4} + \\frac{\\sqrt{11}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9a89bc75fb57e19c", "problem_id": "e604401ef4cee099", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$- 8 x + 3 y$ = $16$\n$x + 7 y$ = $-61$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $- 8 x + 3 y$ = $16$.\n3) Multiply the second equation by $-8$: $- 8 x - 56 y$ = $488$.\n4) Subtract to eliminate x: $59 y$ = $-472$.\n5) Solve for y: y = $-8$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, -8\\right)$", "final_answer": "(x,y)=\\left(-5, -8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0c5bdcbc25daf808", "problem_id": "e604401ef4cee099", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$- 8 x + 3 y$ = $16$\n$x + 7 y$ = $-61$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $- 8 x + 3 y$ = $16$.\nMultiply the second equation by $-8$: $- 8 x - 56 y$ = $488$.\nSubtract to eliminate x: $59 y$ = $-472$.\nSolve for y: y = $-8$.\nSubstitute back to find x: x = $-5$.\nCheck: plug (x,y)=(-5,-8) into both equations.\nFinal answer: $(x,y)=\\left(-5, -8\\right)$", "final_answer": "(x,y)=\\left(-5, -8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "548313bb9053bcd7", "problem_id": "66f6ad4459811a14", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$3 x + 3 y$ = $0$\n$- 3 x$ = $0$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $- 9 x - 9 y$ = $0$.\n3) Multiply the second equation by $3$: $- 9 x$ = $0$.\n4) Subtract to eliminate x: $- 9 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, 0\\right)$", "final_answer": "(x,y)=\\left(0, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b7c74ea439d8848", "problem_id": "66f6ad4459811a14", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the linear system:\n$3 x + 3 y$ = $0$\n$- 3 x$ = $0$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $- 9 x - 9 y$ = $0$.\nMultiply the second equation by $3$: $- 9 x$ = $0$.\nFinal answer: $(x,y)=\\left(0, 0\\right)$", "final_answer": "(x,y)=\\left(0, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c4838d4f4f3aaad4", "problem_id": "9cb2afc112a2fcaf", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 19 times, what is the probability of exactly 6 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=19, k=6, p=$\\frac{1}{2}$ and simplify.\nHere C(19,6) = 27132.\nCheck: expression simplifies to $\\frac{6783}{131072}$.\nFinal answer: $\\frac{6783}{131072}$", "final_answer": "\\frac{6783}{131072}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6ae12c17d1b0433c", "problem_id": "9cb2afc112a2fcaf", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 19 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=19, k=6, p=$\\frac{1}{2}$ and simplify.\n4) Here C(19,6) = 27132.\nFinal answer: $\\frac{6783}{131072}$", "final_answer": "\\frac{6783}{131072}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6bced24c7d5c65ff", "problem_id": "541eec98953cd631", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{10}$. If it is flipped 15 times, what is the probability of exactly 9 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=15, k=9, p=$\\frac{3}{10}$ and simplify.\n4) Here C(15,9) = 5005.\nFinal answer: $\\frac{2318000952267}{200000000000000}$", "final_answer": "\\frac{2318000952267}{200000000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9992a0938e36df94", "problem_id": "541eec98953cd631", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{10}$. If it is flipped 15 times, what is the probability of exactly 9 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=15, k=9, p=$\\frac{3}{10}$ and simplify.\nFinal answer: $\\frac{2318000952267}{200000000000000}$", "final_answer": "\\frac{2318000952267}{200000000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5008b31ab385cfae", "problem_id": "80b6a841a68bbb2b", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{4} + 4 x^{3} + x + 1$.", "output": "1) Differentiate term-by-term:\n2) d/dx(1) = 0.\n3) d/dx(x) = 1.\n4) d/dx(- 3 x^{4}) = - 12 x^{3}.\n5) d/dx(4 x^{3}) = 12 x^{2}.\n6) Combine: f'(x) = - 12 x^{3} + 12 x^{2} + 1.\nFinal answer: $- 12\\cdot x^{3} + 12\\cdot x^{2} + 1$", "final_answer": "- 12\\cdot x^{3} + 12\\cdot x^{2} + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cdd4c5ef3a04c70e", "problem_id": "80b6a841a68bbb2b", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{4} + 4 x^{3} + x + 1$.", "output": "Differentiate term-by-term:\nd/dx(1) = 0.\nd/dx(x) = 1.\nd/dx(- 3 x^{4}) = - 12 x^{3}.\nd/dx(4 x^{3}) = 12 x^{2}.\nCombine: f'(x) = - 12 x^{3} + 12 x^{2} + 1.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 12\\cdot x^{3} + 12\\cdot x^{2} + 1$", "final_answer": "- 12\\cdot x^{3} + 12\\cdot x^{2} + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d11435b715c90238", "problem_id": "688965eaa6745f79", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int \\left(x + 4\\right) e^{x} + \\left(2 x + 1\\right) \\cos{\\left(x \\right)}\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $\\left(x + 4\\right) e^{x} + \\left(2 x + 1\\right) \\cos{\\left(x \\right)}$.\n3) Compute an antiderivative: F(x) = $2 x \\sin{\\left(x \\right)} + \\left(x + 3\\right) e^{x} + \\sin{\\left(x \\right)} + 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\left(x + 3\\right)\\cdot e^{x} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\left(x + 3\\right)\\cdot e^{x} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fe20089022eeca69", "problem_id": "688965eaa6745f79", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int \\left(x + 4\\right) e^{x} + \\left(2 x + 1\\right) \\cos{\\left(x \\right)}\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $\\left(x + 4\\right) e^{x} + \\left(2 x + 1\\right) \\cos{\\left(x \\right)}$.\nCompute an antiderivative: F(x) = $2 x \\sin{\\left(x \\right)} + \\left(x + 3\\right) e^{x} + \\sin{\\left(x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $C + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\left(x + 3\\right)\\cdot e^{x} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\left(x + 3\\right)\\cdot e^{x} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "004890c085f714ac", "problem_id": "9fcdff8b97119065", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Worker A finishes a job in 12 hours and Worker B finishes it in 11 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/12 job/hour. B's rate = 1/11 job/hour.\n2) Combined rate = 1/12 + 1/11 = 23/132 job/hour.\n3) Time = 1 ÷ (combined rate) = 132/23 hours.\nFinal answer: $\\frac{132}{23}$", "final_answer": "\\frac{132}{23}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6ac0975732f4a76f", "problem_id": "9fcdff8b97119065", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Worker A finishes a job in 12 hours and Worker B finishes it in 11 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/12 job/hour. B's rate = 1/11 job/hour.\nCombined rate = 1/12 + 1/11 = 23/132 job/hour.\nTime = 1 ÷ (combined rate) = 132/23 hours.\nCheck: (combined rate)×(time) = 23/132×132/23 = 1.\nFinal answer: $\\frac{132}{23}$", "final_answer": "\\frac{132}{23}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "abfa52aaf8361795", "problem_id": "791f6e4c06c6c48c", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A store sells a backpack for $880. It is discounted by 40%. You buy 10 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 880 × (1 − 40/100) = 880 × 60/100 = 528.\n2) Subtotal = 10 × 528 = 5280.\n3) Total with tax = 5280 × 107/100 = 28248/5.\nFinal answer: $\\frac{28248}{5}$", "final_answer": "\\frac{28248}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "25ae1fc51a2287b9", "problem_id": "791f6e4c06c6c48c", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A store sells a backpack for $880. It is discounted by 40%. You buy 10 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 880 × (1 − 40/100) = 880 × 60/100 = 528.\nSubtotal = 10 × 528 = 5280.\nTotal with tax = 5280 × 107/100 = 28248/5.\nFinal answer: $\\frac{28248}{5}$", "final_answer": "\\frac{28248}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a5e9a9003320d0a5", "problem_id": "b9f6a80beb5a3db6", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many binary strings of length 22 contain exactly 3 ones with no two ones adjacent?", "output": "Reserve 2 mandatory zeros between the 3 ones.\nThis leaves n−k+1 = 20 valid slots for the ones.\nChoose the slots: C(20,3) = 1140.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $1140$", "final_answer": "1140", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b6f2d2469569bc69", "problem_id": "b9f6a80beb5a3db6", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "How many binary strings of length 22 contain exactly 3 ones with no two ones adjacent?", "output": "1) Reserve 2 mandatory zeros between the 3 ones.\n2) This leaves n−k+1 = 20 valid slots for the ones.\n3) Choose the slots: C(20,3) = 1140.\nFinal answer: $1140$", "final_answer": "1140", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "99e862f643a30277", "problem_id": "7d49362ac5c9b81e", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + \\frac{3 x}{4} - \\frac{135}{8}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$\\frac{3}{4}$, c=$- \\frac{135}{8}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{3249}{16}$, so √Δ = $\\frac{57}{4}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{9}{4}$.\nx2 = $- \\frac{5}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{9}{4}, - \\frac{5}{2}\\}$", "final_answer": "x \\in \\{\\frac{9}{4}, - \\frac{5}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "32321168b36c910a", "problem_id": "7d49362ac5c9b81e", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + \\frac{3 x}{4} - \\frac{135}{8}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$\\frac{3}{4}$, c=$- \\frac{135}{8}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{9}{4}, - \\frac{5}{2}\\}$", "final_answer": "x \\in \\{\\frac{9}{4}, - \\frac{5}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0855f37dc1eae6cc", "problem_id": "2b301ab1c3438b89", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡2 (mod 8), x≡4 (mod 5). Give the solution modulo 40.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 40.\n3) Compute the CRT solution to get x ≡ 34 (mod 40).\nFinal answer: $x\\equiv 34\\pmod{40}$", "final_answer": "x\\equiv 34\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "76b4524977f5256a", "problem_id": "30d6cf9a97495d00", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{x^{2} + 3}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 3}) = \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} - \\frac{2 x^{4}}{2 x^{4} + 12 x^{2} + 18} + \\frac{6 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} + \\frac{12 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} - \\frac{12 x^{2}}{2 x^{4} + 12 x^{2} + 18} + \\frac{36 x}{2 x^{4} + 12 x^{2} + 18} + \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 6 x^{2} + 9} + \\frac{9 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} + \\frac{18 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} - \\frac{18}{2 x^{4} + 12 x^{2} + 18} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 6} - \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3} + \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 3}) = - \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3\\right)^{2}} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x^{2} + 3\\right)} + \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 4 x \\left(- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} + 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x^{2} + 3\\right)^{2}}$.\nFinal answer: $\\frac{- 4 x \\left(- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} + 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x^{2} + 3\\right)^{2}}$", "final_answer": "\\frac{- 4 x \\left(- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} + 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x^{2} + 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6a7ce506021b6ef2", "problem_id": "a57fb78994e6da61", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "You want 44 liters of a 45% salt solution. You have a 15% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 44−x liters is 50% solution.\n2) Salt equation: 15x + 50(44−x) = 45·44.\n3) (15-50)x = (45-50)·44 ⇒ x = 44/7.\n4) Then 44−x = 264/7.\nFinal answer: $44/7 L of 15% and 264/7 L of 50%$", "final_answer": "44/7 L of 15% and 264/7 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "40e87bd123df2187", "problem_id": "a57fb78994e6da61", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "You want 44 liters of a 45% salt solution. You have a 15% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 44−x liters is 50% solution.\nSalt equation: 15x + 50(44−x) = 45·44.\n(15-50)x = (45-50)·44 ⇒ x = 44/7.\nFinal answer: $44/7 L of 15% and 264/7 L of 50%$", "final_answer": "44/7 L of 15% and 264/7 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "39d9ffea7257cce1", "problem_id": "b3bd1265184b05ad", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A store sells a backpack for $64. It is discounted by 25%. You buy 13 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 64 × (1 − 25/100) = 64 × 75/100 = 48.\n2) Subtotal = 13 × 48 = 624.\n3) Total with tax = 624 × 110/100 = 3432/5.\nFinal answer: $\\frac{3432}{5}$", "final_answer": "\\frac{3432}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5c3db938368c33f8", "problem_id": "b3bd1265184b05ad", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A store sells a backpack for $64. It is discounted by 25%. You buy 13 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 64 × (1 − 25/100) = 64 × 75/100 = 48.\nSubtotal = 13 × 48 = 624.\nTotal with tax = 624 × 110/100 = 3432/5.\nCheck: recomputing the formula yields 3432/5.\nFinal answer: $\\frac{3432}{5}$", "final_answer": "\\frac{3432}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "470ddc2d1a841657", "problem_id": "f6e3f005a7920f4e", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A store sells a backpack for $244. It is discounted by 20%. You buy 12 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 244 × (1 − 20/100) = 244 × 80/100 = 976/5.\n2) Subtotal = 12 × 976/5 = 11712/5.\n3) Total with tax = 11712/5 × 109/100 = 319152/125.\nFinal answer: $\\frac{319152}{125}$", "final_answer": "\\frac{319152}{125}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "61a317275d9cc5e1", "problem_id": "f6e3f005a7920f4e", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $244. It is discounted by 20%. You buy 12 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 244 × (1 − 20/100) = 244 × 80/100 = 976/5.\nSubtotal = 12 × 976/5 = 11712/5.\nTotal with tax = 11712/5 × 109/100 = 319152/125.\nCheck: recomputing the formula yields 319152/125.\nFinal answer: $\\frac{319152}{125}$", "final_answer": "\\frac{319152}{125}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3567165078e1a4c4", "problem_id": "13672728a0e80383", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- 8 x + 3 y$ = $10$\n$- 2 x + 8 y$ = $46$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $16 x - 6 y$ = $-20$.\nMultiply the second equation by $-8$: $16 x - 64 y$ = $-368$.\nSubtract to eliminate x: $58 y$ = $348$.\nSolve for y: y = $6$.\nSubstitute back to find x: x = $1$.\nCheck: plug (x,y)=(1,6) into both equations.\nFinal answer: $(x,y)=\\left(1, 6\\right)$", "final_answer": "(x,y)=\\left(1, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5c69821a788c2d0f", "problem_id": "888ac461b1afacb4", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} 2 x^{2} + 4 x + 1$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{2} + 3 x + 1.\n3) Then the limit is 6.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: 7.\nFinal answer: $7$", "final_answer": "7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "38b76f9c36cad1f1", "problem_id": "d37a8dd6d1d2c67d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{3} - 3 x^{2} - 2 x - 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x) = - 2 x$.\n3) So $f'(x) = - 3 x^{2} - 8 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x) = -2$.\nTherefore the correct derivative is $f'(x) = - 3 x^{2} - 6 x - 2$.\nFinal answer: $- 3 x^{2} - 6 x - 2$", "final_answer": "- 3 x^{2} - 6 x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b8d84c03b2db1ef3", "problem_id": "d6892d7b5e4db7f0", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} - 2 x^{2} + 11 x - 41$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 2 x^{3} + 3 x^{2} + 3 x - 3$ and a = -4.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 6 x^{2} + 6 x + 3$.\nEvaluate at a: p'(-4) = $-117$.\nCheck: the identity gives $-117$.\nFinal answer: $-117$", "final_answer": "-117", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "67d34ef8f464a872", "problem_id": "69bd12b547be01b6", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A store sells a backpack for $685. It is discounted by 30%. You buy 3 backpacks. What is the total cost?", "output": "1) Discounted price = 685 × (1 − 30/100) = 685 × 70/100 = 959/2.\n2) Subtotal = 3 × 959/2 = 2877/2.\nFinal answer: $\\frac{2877}{2}$", "final_answer": "\\frac{2877}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0663a3000512ad81", "problem_id": "69bd12b547be01b6", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A store sells a backpack for $685. It is discounted by 30%. You buy 3 backpacks. What is the total cost?", "output": "Discounted price = 685 × (1 − 30/100) = 685 × 70/100 = 959/2.\nSubtotal = 3 × 959/2 = 2877/2.\nFinal answer: $\\frac{2877}{2}$", "final_answer": "\\frac{2877}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "234d27c4a9879c01", "problem_id": "998deb60cf52cacd", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Worker A finishes a job in 15 hours and Worker B finishes it in 13 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/15 job/hour. B's rate = 1/13 job/hour.\n2) Combined rate = 1/15 + 1/13 = 28/195 job/hour.\n3) Time = 1 ÷ (combined rate) = 195/28 hours.\nFinal answer: $\\frac{195}{28}$", "final_answer": "\\frac{195}{28}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e5484cef8d48bcc1", "problem_id": "ec402b48406a19af", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - x^{3} - 6 x^{2} - 15 x - 45$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to - x^{3} - 6 x^{2} - 16 x - 45.\n3) Then the limit is -174.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: -171.\nFinal answer: $-171$", "final_answer": "-171", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "fb9e87de4682c09f", "problem_id": "e48f8f71d67997ff", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many binary strings of length 12 contain exactly 3 ones with no two ones adjacent?", "output": "Reserve 2 mandatory zeros between the 3 ones.\nThis leaves n−k+1 = 10 valid slots for the ones.\nChoose the slots: C(10,3) = 120.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $120$", "final_answer": "120", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "16a43cd70c1f38b6", "problem_id": "e48f8f71d67997ff", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "How many binary strings of length 12 contain exactly 3 ones with no two ones adjacent?", "output": "1) Reserve 2 mandatory zeros between the 3 ones.\n2) This leaves n−k+1 = 10 valid slots for the ones.\n3) Choose the slots: C(10,3) = 120.\nFinal answer: $120$", "final_answer": "120", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d929c22f7440532", "problem_id": "ac7aa19d28ca3e0c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 2 x^{3} + 8 x^{2} - 22 x + 68$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{3} + 8 x^{2} - 23 x + 68.\n3) Then the limit is 263.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: 260.\nFinal answer: $260$", "final_answer": "260", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "021da552c6a51234", "problem_id": "0ed25460d9e625a8", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{7} x - \\frac{x}{4} - \\frac{5 \\sqrt{7}}{4} - \\frac{15}{8}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\sqrt{7} - \\frac{1}{4}$, c=$- \\frac{5 \\sqrt{7}}{4} - \\frac{15}{8}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{11 \\sqrt{7}}{2} + \\frac{233}{16}$, so √Δ = $\\sqrt{\\frac{11 \\sqrt{7}}{2} + \\frac{233}{16}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{1}{8} + \\frac{\\sqrt{7}}{2} + \\frac{\\sqrt{88 \\sqrt{7} + 233}}{8}$.\nx2 = $- \\frac{\\sqrt{88 \\sqrt{7} + 233}}{8} + \\frac{1}{8} + \\frac{\\sqrt{7}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{1}{8} + \\frac{\\sqrt{7}}{2} + \\frac{\\sqrt{88\\cdot \\sqrt{7} + 233}}{8}, - \\frac{\\sqrt{88\\cdot \\sqrt{7} + 233}}{8} + \\frac{1}{8} + \\frac{\\sqrt{7}}{2}\\}$", "final_answer": "x \\in \\{\\frac{1}{8} + \\frac{\\sqrt{7}}{2} + \\frac{\\sqrt{88\\cdot \\sqrt{7} + 233}}{8}, - \\frac{\\sqrt{88\\cdot \\sqrt{7} + 233}}{8} + \\frac{1}{8} + \\frac{\\sqrt{7}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "efa6753a152ae9b2", "problem_id": "88e4176b19993546", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_5=44$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_5=39.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=39 and k=5.\n4) Compute C(43,4) = 123410.\nFinal answer: $123410$", "final_answer": "123410", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3020662c59d0a3ef", "problem_id": "88e4176b19993546", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_5=44$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_5=39.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=39 and k=5.\nCompute C(43,4) = 123410.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $123410$", "final_answer": "123410", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a42a2a9427a30df2", "problem_id": "94f7b64c30dd4eca", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A cyclist rides at 78 miles per hour for 1/2 hours, then at 37 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 78 × 1/2 = 39 miles.\n2) Second leg distance = 37 × 5/2 = 185/2 miles.\n3) Total distance = 39 + 185/2 = 263/2 miles.\nFinal answer: $\\frac{263}{2}$", "final_answer": "\\frac{263}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "19829cf5cab2dcc8", "problem_id": "94f7b64c30dd4eca", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 78 miles per hour for 1/2 hours, then at 37 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 78 × 1/2 = 39 miles.\nSecond leg distance = 37 × 5/2 = 185/2 miles.\nTotal distance = 39 + 185/2 = 263/2 miles.\nCheck: adding both legs gives 263/2.\nFinal answer: $\\frac{263}{2}$", "final_answer": "\\frac{263}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "286a63df8e490efc", "problem_id": "8f18b2f04cac390b", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A disease has prevalence $\\frac{1}{50}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{67}{1000}$.\n3) Compute P(D|+) = $\\frac{18}{67}$.\nFinal answer: $\\frac{18}{67}$", "final_answer": "\\frac{18}{67}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "127717dcf1a9e993", "problem_id": "8f18b2f04cac390b", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A disease has prevalence $\\frac{1}{50}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{67}{1000}$.\nCompute P(D|+) = $\\frac{18}{67}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{18}{67}$", "final_answer": "\\frac{18}{67}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d3d99dd176cfc15a", "problem_id": "275a1ba56da6927e", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{2} - x + 3$ at $x=1$.", "output": "Differentiate: f'(x) = $- 6 x - 1$.\nSlope at x=1: m = f'(1) = $-7$.\nPoint on curve: (1, f(1)) = (1, $-1$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -1 = -7(x − 1).\nOptionally expand: y = $6 - 7 x$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 6 - 7\\cdot x$", "final_answer": "y = 6 - 7\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "95dac1e4dbb2351f", "problem_id": "275a1ba56da6927e", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Find the equation of the tangent line to $y=- 3 x^{2} - x + 3$ at $x=1$.", "output": "1) Differentiate: f'(x) = $- 6 x - 1$.\n2) Slope at x=1: m = f'(1) = $-7$.\n3) Point on curve: (1, f(1)) = (1, $-1$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -1 = -7(x − 1).\n6) Optionally expand: y = $6 - 7 x$.\nFinal answer: $y = 6 - 7\\cdot x$", "final_answer": "y = 6 - 7\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "57723ecbe2d8ea37", "problem_id": "11ada23bf0f077e6", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} - x^{4} - 3 x^{3} + 2 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{4} - 3 x^{3} + 2 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $\\frac{3 \\pi \\left(- 44 \\pi^{4} - 75 \\pi^{3} + 40 + 30 \\pi + 40 \\pi^{2}\\right)}{20}$.\n4) Simplify the result.\nFinal answer: $\\frac{3\\cdot \\pi\\cdot \\left(- 44\\cdot \\pi^{4} - 75\\cdot \\pi^{3} + 40 + 30\\cdot \\pi + 40\\cdot \\pi^{2}\\right)}{20}$", "final_answer": "\\frac{3\\cdot \\pi\\cdot \\left(- 44\\cdot \\pi^{4} - 75\\cdot \\pi^{3} + 40 + 30\\cdot \\pi + 40\\cdot \\pi^{2}\\right)}{20}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ca1923b9f936b9f", "problem_id": "11ada23bf0f077e6", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} - x^{4} - 3 x^{3} + 2 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{4} - 3 x^{3} + 2 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $\\frac{3 \\pi \\left(- 44 \\pi^{4} - 75 \\pi^{3} + 40 + 30 \\pi + 40 \\pi^{2}\\right)}{20}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{3\\cdot \\pi\\cdot \\left(- 44\\cdot \\pi^{4} - 75\\cdot \\pi^{3} + 40 + 30\\cdot \\pi + 40\\cdot \\pi^{2}\\right)}{20}$", "final_answer": "\\frac{3\\cdot \\pi\\cdot \\left(- 44\\cdot \\pi^{4} - 75\\cdot \\pi^{3} + 40 + 30\\cdot \\pi + 40\\cdot \\pi^{2}\\right)}{20}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "56ea21555b4fcf48", "problem_id": "5229a7ff8f8afbe7", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$- x + 8 y$ = $-24$\n$x + 5 y$ = $-2$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $- x + 8 y$ = $-24$.\n3) Multiply the second equation by $-1$: $- x - 5 y$ = $2$.\n4) Subtract to eliminate x: $13 y$ = $-26$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $8$.\nFinal answer: $(x,y)=\\left(8, -2\\right)$", "final_answer": "(x,y)=\\left(8, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4dab0cfb02ccbb6d", "problem_id": "86ad5e3e1f9c98d9", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - x^{2} + 5 x - 10$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} + 4 x - 10.\n3) Then the limit is -22.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -24.\nFinal answer: $-24$", "final_answer": "-24", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2be7e5b8a3f829f7", "problem_id": "0cb8418a7be08dd4", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the multiplicative inverse of 2 modulo 11.", "output": "1) We want x such that 2x ≡ 1 (mod 11).\n2) Use extended Euclid to find s,t with 2s + 11t = 1.\n3) One Bezout identity is: 2·(-5) + 11·(1) = 1.\n4) Reduce s modulo 11: x ≡ 6 (mod 11).\nFinal answer: $x\\equiv 6\\pmod{11}$", "final_answer": "x\\equiv 6\\pmod{11}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "02682f0d408d922a", "problem_id": "0cb8418a7be08dd4", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the multiplicative inverse of 2 modulo 11.", "output": "We want x such that 2x ≡ 1 (mod 11).\nUse extended Euclid to find s,t with 2s + 11t = 1.\nOne Bezout identity is: 2·(-5) + 11·(1) = 1.\nReduce s modulo 11: x ≡ 6 (mod 11).\nCheck: 2·6 mod 11 = 1.\nFinal answer: $x\\equiv 6\\pmod{11}$", "final_answer": "x\\equiv 6\\pmod{11}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "131982ba48a8d24d", "problem_id": "9047de539922cc99", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(- 3 x^{2} - x - 1\\right) \\cos{\\left(x - 1 \\right)}$.", "output": "Let f(x) = $\\left(x - 3\\right) e^{x} + \\left(- 3 x^{2} - x - 1\\right) \\cos{\\left(x - 1 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\left(- 6 x - 1\\right) \\cos{\\left(x - 1 \\right)} + \\left(x - 3\\right) e^{x} - \\left(- 3 x^{2} - x - 1\\right) \\sin{\\left(x - 1 \\right)} + e^{x}$.\nFinal answer: $\\left(- 6\\cdot x - 1\\right)\\cdot \\cos{\\left(x - 1 \\right)} + \\left(x - 3\\right)\\cdot e^{x} - \\left(- 3\\cdot x^{2} - x - 1\\right)\\cdot \\sin{\\left(x - 1 \\right)} + e^{x}$", "final_answer": "\\left(- 6\\cdot x - 1\\right)\\cdot \\cos{\\left(x - 1 \\right)} + \\left(x - 3\\right)\\cdot e^{x} - \\left(- 3\\cdot x^{2} - x - 1\\right)\\cdot \\sin{\\left(x - 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a0a9802eeaf668bf", "problem_id": "9047de539922cc99", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(- 3 x^{2} - x - 1\\right) \\cos{\\left(x - 1 \\right)}$.", "output": "1) Let f(x) = $\\left(x - 3\\right) e^{x} + \\left(- 3 x^{2} - x - 1\\right) \\cos{\\left(x - 1 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\left(- 6 x - 1\\right) \\cos{\\left(x - 1 \\right)} + \\left(x - 3\\right) e^{x} - \\left(- 3 x^{2} - x - 1\\right) \\sin{\\left(x - 1 \\right)} + e^{x}$.\nFinal answer: $\\left(- 6\\cdot x - 1\\right)\\cdot \\cos{\\left(x - 1 \\right)} + \\left(x - 3\\right)\\cdot e^{x} - \\left(- 3\\cdot x^{2} - x - 1\\right)\\cdot \\sin{\\left(x - 1 \\right)} + e^{x}$", "final_answer": "\\left(- 6\\cdot x - 1\\right)\\cdot \\cos{\\left(x - 1 \\right)} + \\left(x - 3\\right)\\cdot e^{x} - \\left(- 3\\cdot x^{2} - x - 1\\right)\\cdot \\sin{\\left(x - 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bf9e52713ec85075", "problem_id": "624f2627104826be", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{2} + x - 4$ at $x=0$.", "output": "Differentiate: f'(x) = $4 x + 1$.\nSlope at x=0: m = f'(0) = $1$.\nPoint on curve: (0, f(0)) = (0, $-4$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -4 = 1(x − 0).\nOptionally expand: y = $x - 4$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = x - 4$", "final_answer": "y = x - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "68e2dbb9b77b54d8", "problem_id": "624f2627104826be", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the equation of the tangent line to $y=2 x^{2} + x - 4$ at $x=0$.", "output": "1) Differentiate: f'(x) = $4 x + 1$.\n2) Slope at x=0: m = f'(0) = $1$.\n3) Point on curve: (0, f(0)) = (0, $-4$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -4 = 1(x − 0).\n6) Optionally expand: y = $x - 4$.\nFinal answer: $y = x - 4$", "final_answer": "y = x - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "757f07390afbd85e", "problem_id": "7b3d918608903f58", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{4} - x^{3} - 4 x^{2} + x + 3\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 3 x^{4} - x^{3} - 4 x^{2} + x + 3$.\nCompute an antiderivative: F(x) = $- \\frac{3 x^{5}}{5} - \\frac{x^{4}}{4} - \\frac{4 x^{3}}{3} + \\frac{x^{2}}{2} + 3 x$.\nFinal answer: $C - \\frac{3\\cdot x^{5}}{5} - \\frac{x^{4}}{4} - \\frac{4\\cdot x^{3}}{3} + \\frac{x^{2}}{2} + 3\\cdot x$", "final_answer": "C - \\frac{3\\cdot x^{5}}{5} - \\frac{x^{4}}{4} - \\frac{4\\cdot x^{3}}{3} + \\frac{x^{2}}{2} + 3\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "04d0cc734c62550a", "problem_id": "7b3d918608903f58", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{4} - x^{3} - 4 x^{2} + x + 3\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 3 x^{4} - x^{3} - 4 x^{2} + x + 3$.\n3) Compute an antiderivative: F(x) = $- \\frac{3 x^{5}}{5} - \\frac{x^{4}}{4} - \\frac{4 x^{3}}{3} + \\frac{x^{2}}{2} + 3 x$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{3\\cdot x^{5}}{5} - \\frac{x^{4}}{4} - \\frac{4\\cdot x^{3}}{3} + \\frac{x^{2}}{2} + 3\\cdot x$", "final_answer": "C - \\frac{3\\cdot x^{5}}{5} - \\frac{x^{4}}{4} - \\frac{4\\cdot x^{3}}{3} + \\frac{x^{2}}{2} + 3\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "26cdb355239a1992", "problem_id": "327ba6424f30ba9b", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Worker A finishes a job in 8 hours and Worker B finishes it in 5 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/8 job/hour. B's rate = 1/5 job/hour.\n2) Combined rate = 1/8 + 1/5 = 13/40 job/hour.\n3) Time = 1 ÷ (combined rate) = 40/13 hours.\nFinal answer: $\\frac{40}{13}$", "final_answer": "\\frac{40}{13}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f8a16f26b62e3c6c", "problem_id": "327ba6424f30ba9b", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Worker A finishes a job in 8 hours and Worker B finishes it in 5 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/8 job/hour. B's rate = 1/5 job/hour.\nCombined rate = 1/8 + 1/5 = 13/40 job/hour.\nTime = 1 ÷ (combined rate) = 40/13 hours.\nCheck: (combined rate)×(time) = 13/40×40/13 = 1.\nFinal answer: $\\frac{40}{13}$", "final_answer": "\\frac{40}{13}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8a5bbdd28ac60311", "problem_id": "8f0c1b1edfd5fdad", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{3} - x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $2 x^{3} - x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 2$.\n3) Compute an antiderivative: F(x) = $\\frac{x^{4}}{2} - \\frac{x^{2}}{2} - 2 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{x^{4}}{2} - \\frac{x^{2}}{2} - 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{4}}{2} - \\frac{x^{2}}{2} - 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6dba030ba8a1841c", "problem_id": "8f0c1b1edfd5fdad", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{3} - x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $2 x^{3} - x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 2$.\nCompute an antiderivative: F(x) = $\\frac{x^{4}}{2} - \\frac{x^{2}}{2} - 2 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{x^{4}}{2} - \\frac{x^{2}}{2} - 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{4}}{2} - \\frac{x^{2}}{2} - 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "38072ba4f0dd4048", "problem_id": "26e9853dcb83fb74", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{4} - 2 x^{3} - x^{2} + x - 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(3 x^{4}) = 12 x^{4}$.\n3) So $f'(x) = 12 x^{4} - 6 x^{2} - 2 x + 1$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(3 x^{4}) = 12 x^{3}$.\nTherefore the correct derivative is $f'(x) = 12 x^{3} - 6 x^{2} - 2 x + 1$.\nFinal answer: $12 x^{3} - 6 x^{2} - 2 x + 1$", "final_answer": "12 x^{3} - 6 x^{2} - 2 x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "4a41456a282c8000", "problem_id": "5acd98b3b964d31d", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡0 (mod 5), x≡0 (mod 8). Give the solution modulo 40.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 40.\n3) Compute the CRT solution to get x ≡ 0 (mod 40).\nFinal answer: $x\\equiv 0\\pmod{40}$", "final_answer": "x\\equiv 0\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c90a72f64625eccb", "problem_id": "dbcc98461e533aea", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A cyclist rides at 72 miles per hour for 2 hours, then at 88 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 72 × 2 = 144 miles.\n2) Second leg distance = 88 × 1 = 88 miles.\n3) Total distance = 144 + 88 = 232 miles.\nFinal answer: $232$", "final_answer": "232", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e4fcc63f4334700a", "problem_id": "dbcc98461e533aea", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 72 miles per hour for 2 hours, then at 88 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 72 × 2 = 144 miles.\nSecond leg distance = 88 × 1 = 88 miles.\nTotal distance = 144 + 88 = 232 miles.\nCheck: adding both legs gives 232.\nFinal answer: $232$", "final_answer": "232", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6f622a67fd746d69", "problem_id": "40056030801488e7", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{4} - 2 x^{3} + 4 x^{2} + 3$.", "output": "1) Differentiate term-by-term:\n2) d/dx(3) = 0.\n3) d/dx(x^{4}) = 4 x^{3}.\n4) d/dx(- 2 x^{3}) = - 6 x^{2}.\n5) d/dx(4 x^{2}) = 8 x.\n6) Combine: f'(x) = 2 x \\left(2 x^{2} - 3 x + 4\\right).\nFinal answer: $2\\cdot x\\cdot \\left(2\\cdot x^{2} - 3\\cdot x + 4\\right)$", "final_answer": "2\\cdot x\\cdot \\left(2\\cdot x^{2} - 3\\cdot x + 4\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9b7c068b6f6882a0", "problem_id": "40056030801488e7", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{4} - 2 x^{3} + 4 x^{2} + 3$.", "output": "Differentiate term-by-term:\nd/dx(3) = 0.\nd/dx(x^{4}) = 4 x^{3}.\nFinal answer: $2\\cdot x\\cdot \\left(2\\cdot x^{2} - 3\\cdot x + 4\\right)$", "final_answer": "2\\cdot x\\cdot \\left(2\\cdot x^{2} - 3\\cdot x + 4\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "11cc1a1d17da4d69", "problem_id": "fc1d561955844032", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Find the multiplicative inverse of 34 modulo 41.", "output": "We want x such that 34x ≡ 1 (mod 41).\nUse extended Euclid to find s,t with 34s + 41t = 1.\nOne Bezout identity is: 34·(-6) + 41·(5) = 1.\nFinal answer: $x\\equiv 35\\pmod{41}$", "final_answer": "x\\equiv 35\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72294f94e885ce43", "problem_id": "fc1d561955844032", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the multiplicative inverse of 34 modulo 41.", "output": "1) We want x such that 34x ≡ 1 (mod 41).\n2) Use extended Euclid to find s,t with 34s + 41t = 1.\n3) One Bezout identity is: 34·(-6) + 41·(5) = 1.\n4) Reduce s modulo 41: x ≡ 35 (mod 41).\nFinal answer: $x\\equiv 35\\pmod{41}$", "final_answer": "x\\equiv 35\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8237f042ae0a1392", "problem_id": "3559781b87ede562", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$4 x + 2 y$ = $-16$\n$8 x + 5 y$ = $-28$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $8$: $32 x + 16 y$ = $-128$.\n3) Multiply the second equation by $4$: $32 x + 20 y$ = $-112$.\n4) Subtract to eliminate x: $- 4 y$ = $-16$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $-6$.\nFinal answer: $(x,y)=\\left(-6, 4\\right)$", "final_answer": "(x,y)=\\left(-6, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5f2a114fe84e3d91", "problem_id": "3559781b87ede562", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the linear system:\n$4 x + 2 y$ = $-16$\n$8 x + 5 y$ = $-28$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $8$: $32 x + 16 y$ = $-128$.\nMultiply the second equation by $4$: $32 x + 20 y$ = $-112$.\nFinal answer: $(x,y)=\\left(-6, 4\\right)$", "final_answer": "(x,y)=\\left(-6, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "55f2fc1fefb15ea0", "problem_id": "605527daa4ddfdea", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 4), x≡8 (mod 11). Give the solution modulo 44.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 44.\nCompute the CRT solution to get x ≡ 30 (mod 44).\nCheck: 30 mod 4=2 and mod 11=8.\nFinal answer: $x\\equiv 30\\pmod{44}$", "final_answer": "x\\equiv 30\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "55742f2f50fc41fd", "problem_id": "99b06a6bb87dc2b6", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 0.\n3) Then the limit is 0.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 2.\nFinal answer: $2$", "final_answer": "2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f0cd88002393d491", "problem_id": "561de89f8d614f10", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{2} + 3 x + \\left(x^{2} + 3 x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 3 x - 6}$.", "output": "1) Let f(x) = $\\frac{x^{2} + 3 x + \\left(x^{2} + 3 x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 3 x - 6}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{2 \\left(- 2 x - 3\\right) \\left(x^{2} + 3 x + \\left(x^{2} + 3 x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} + 3 x - 6\\right) \\left(4 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} + 3 x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 3 x - 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6\\right)}{2 \\left(x^{2} + 3 x - 6\\right)^{2}}$.\nFinal answer: $\\frac{\\left(- 4\\cdot x - 6\\right)\\cdot \\left(x^{2} + 3\\cdot x + \\left(x^{2} + 3\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)}\\right) + \\left(x^{2} + 3\\cdot x - 6\\right)\\cdot \\left(4\\cdot x + \\left(4\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(x^{2} + 3\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 6\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\right)}{2\\cdot \\left(x^{2} + 3\\cdot x - 6\\right)^{2}}$", "final_answer": "\\frac{\\left(- 4\\cdot x - 6\\right)\\cdot \\left(x^{2} + 3\\cdot x + \\left(x^{2} + 3\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)}\\right) + \\left(x^{2} + 3\\cdot x - 6\\right)\\cdot \\left(4\\cdot x + \\left(4\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(x^{2} + 3\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 6\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\right)}{2\\cdot \\left(x^{2} + 3\\cdot x - 6\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e9090cddb986feba", "problem_id": "32d8d20afd531b76", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{3} + x^{2} - \\frac{x}{2} + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - \\frac{1}{2}}{x - 4}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}) = \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4}$.\n3) So $f'(x) = \\frac{4 x^{3}}{2 x^{2} - 16 x + 32} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{22 x^{2}}{2 x^{2} - 16 x + 32} - \\frac{8 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{16 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{16 x}{2 x^{2} - 16 x + 32} + \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 8 x + 16} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 8} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} - \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} + \\frac{16 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{32 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{5}{2 x^{2} - 16 x + 32} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}) = \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x - 4\\right)} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x - 4\\right)^{2}} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 2 x^{3} - 2 x^{2} + x + \\left(x - 4\\right) \\left(6 x^{2} + 4 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{2 \\left(x - 4\\right)^{2}}$.\nFinal answer: $\\frac{- 2 x^{3} - 2 x^{2} + x + \\left(x - 4\\right) \\left(6 x^{2} + 4 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{2 \\left(x - 4\\right)^{2}}$", "final_answer": "\\frac{- 2 x^{3} - 2 x^{2} + x + \\left(x - 4\\right) \\left(6 x^{2} + 4 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{2 \\left(x - 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f401702c215efe95", "problem_id": "e699595f5fb50c1a", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A store sells a backpack for $915. It is discounted by 15%. You buy 12 backpacks. What is the total cost?", "output": "1) Discounted price = 915 × (1 − 15/100) = 915 × 85/100 = 3111/4.\n2) Subtotal = 12 × 3111/4 = 9333.\nFinal answer: $9333$", "final_answer": "9333", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "34a132ab3c7159f1", "problem_id": "e699595f5fb50c1a", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $915. It is discounted by 15%. You buy 12 backpacks. What is the total cost?", "output": "Discounted price = 915 × (1 − 15/100) = 915 × 85/100 = 3111/4.\nSubtotal = 12 × 3111/4 = 9333.\nCheck: recomputing the formula yields 9333.\nFinal answer: $9333$", "final_answer": "9333", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4b2bd97e966cd61b", "problem_id": "7568d7a5d7ee76cc", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the multiplicative inverse of 36 modulo 43.", "output": "We want x such that 36x ≡ 1 (mod 43).\nUse extended Euclid to find s,t with 36s + 43t = 1.\nOne Bezout identity is: 36·(6) + 43·(-5) = 1.\nReduce s modulo 43: x ≡ 6 (mod 43).\nCheck: 36·6 mod 43 = 1.\nFinal answer: $x\\equiv 6\\pmod{43}$", "final_answer": "x\\equiv 6\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e1aa9fe04492b88c", "problem_id": "7568d7a5d7ee76cc", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Find the multiplicative inverse of 36 modulo 43.", "output": "1) We want x such that 36x ≡ 1 (mod 43).\n2) Use extended Euclid to find s,t with 36s + 43t = 1.\n3) One Bezout identity is: 36·(6) + 43·(-5) = 1.\n4) Reduce s modulo 43: x ≡ 6 (mod 43).\nFinal answer: $x\\equiv 6\\pmod{43}$", "final_answer": "x\\equiv 6\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dd2bf21dbe5af350", "problem_id": "7116f9a1aa900e52", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$2 y$ = $8$\n$- 4 x + 4 y$ = $4$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $- 8 y$ = $-32$.\n3) Multiply the second equation by $0$: $0$ = $0$.\n4) Subtract to eliminate x: $- 8 y$ = $-32$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, 4\\right)$", "final_answer": "(x,y)=\\left(3, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a6c263ffede1a8b0", "problem_id": "7116f9a1aa900e52", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$2 y$ = $8$\n$- 4 x + 4 y$ = $4$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $- 8 y$ = $-32$.\nMultiply the second equation by $0$: $0$ = $0$.\nSubtract to eliminate x: $- 8 y$ = $-32$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $3$.\nCheck: plug (x,y)=(3,4) into both equations.\nFinal answer: $(x,y)=\\left(3, 4\\right)$", "final_answer": "(x,y)=\\left(3, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0f6748c2daffc1b4", "problem_id": "0ab174b73dd00b99", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$3 x + 2 y$ = $6$\n$7 x + 5 y$ = $12$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $7$: $21 x + 14 y$ = $42$.\n3) Multiply the second equation by $3$: $21 x + 15 y$ = $36$.\n4) Subtract to eliminate x: $- y$ = $6$.\n5) Solve for y: y = $-6$.\n6) Substitute back to find x: x = $6$.\nFinal answer: $(x,y)=\\left(6, -6\\right)$", "final_answer": "(x,y)=\\left(6, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "71d582bf73f35bdf", "problem_id": "0ab174b73dd00b99", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$3 x + 2 y$ = $6$\n$7 x + 5 y$ = $12$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $7$: $21 x + 14 y$ = $42$.\nMultiply the second equation by $3$: $21 x + 15 y$ = $36$.\nSubtract to eliminate x: $- y$ = $6$.\nSolve for y: y = $-6$.\nSubstitute back to find x: x = $6$.\nCheck: plug (x,y)=(6,-6) into both equations.\nFinal answer: $(x,y)=\\left(6, -6\\right)$", "final_answer": "(x,y)=\\left(6, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "28b5c35a46300a52", "problem_id": "b47130c8bafbf587", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "How many binary strings of length 14 contain exactly 3 ones with no two ones adjacent?", "output": "1) Reserve 2 mandatory zeros between the 3 ones.\n2) This leaves n−k+1 = 12 valid slots for the ones.\n3) Choose the slots: C(12,3) = 220.\nFinal answer: $220$", "final_answer": "220", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9a492633dd8d0a63", "problem_id": "b47130c8bafbf587", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many binary strings of length 14 contain exactly 3 ones with no two ones adjacent?", "output": "Reserve 2 mandatory zeros between the 3 ones.\nThis leaves n−k+1 = 12 valid slots for the ones.\nChoose the slots: C(12,3) = 220.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $220$", "final_answer": "220", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "88f294cb7d317f9a", "problem_id": "e030f7ded40c24f7", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 28 red balls and 15 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(43,11).\nFavorable samples: choose k reds and n−k blues: C(28,3)·C(15,8).\nCompute totals: C(43,11) = 5752004349, and favorable = 21081060.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{77220}{21069613}$.\nFinal answer: $\\frac{77220}{21069613}$", "final_answer": "\\frac{77220}{21069613}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "208d7480018c562a", "problem_id": "e030f7ded40c24f7", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 28 red balls and 15 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(43,11).\n3) Favorable samples: choose k reds and n−k blues: C(28,3)·C(15,8).\n4) Compute totals: C(43,11) = 5752004349, and favorable = 21081060.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{77220}{21069613}$", "final_answer": "\\frac{77220}{21069613}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c5f9f0db456d9853", "problem_id": "5b8441e1da5e27af", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A cyclist rides at 48 miles per hour for 1 hours, then at 82 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 48 × 1 = 48 miles.\nSecond leg distance = 82 × 3 = 246 miles.\nTotal distance = 48 + 246 = 294 miles.\nCheck: adding both legs gives 294.\nFinal answer: $294$", "final_answer": "294", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8c1741e72b085b72", "problem_id": "5b8441e1da5e27af", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A cyclist rides at 48 miles per hour for 1 hours, then at 82 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 48 × 1 = 48 miles.\n2) Second leg distance = 82 × 3 = 246 miles.\n3) Total distance = 48 + 246 = 294 miles.\nFinal answer: $294$", "final_answer": "294", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "60588b03b3be929f", "problem_id": "5baffeeac5a4af65", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{2} - 4$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- x^{2}) = - 2 x^{2}$.\n3) So $f'(x) = - 2 x^{2}$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- x^{2}) = - 2 x$.\nTherefore the correct derivative is $f'(x) = - 2 x$.\nFinal answer: $- 2 x$", "final_answer": "- 2 x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7c6b8d76f63f0235", "problem_id": "b272c3092de81801", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the multiplicative inverse of 13 modulo 31.", "output": "1) We want x such that 13x ≡ 1 (mod 31).\n2) Use extended Euclid to find s,t with 13s + 31t = 1.\n3) One Bezout identity is: 13·(12) + 31·(-5) = 1.\n4) Reduce s modulo 31: x ≡ 12 (mod 31).\nFinal answer: $x\\equiv 12\\pmod{31}$", "final_answer": "x\\equiv 12\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "45f5e0ac83db0d46", "problem_id": "aaa146a38fe8ea5e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{2} + 2 x - 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(4 x^{2}) = 8 x^{2}$.\n3) So $f'(x) = 8 x^{2} + 2$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(4 x^{2}) = 8 x$.\nTherefore the correct derivative is $f'(x) = 2 \\left(4 x + 1\\right)$.\nFinal answer: $2 \\left(4 x + 1\\right)$", "final_answer": "2 \\left(4 x + 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b00cd831a22ba827", "problem_id": "a65d829e64038e35", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the equation of the tangent line to $y=- x^{2} - 2 x + 2 \\sin{\\left(x \\right)} + 1$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $- 2 x + 2 \\cos{\\left(x \\right)} - 2$.\n2) Slope at x=-2: m = f'(-2) = $2 \\cos{\\left(2 \\right)} + 2$.\n3) Point on curve: (-2, f(-2)) = (-2, $1 - 2 \\sin{\\left(2 \\right)}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 1 - 2 \\sin{\\left(2 \\right)} = 2 \\cos{\\left(2 \\right)} + 2(x − -2).\n6) Optionally expand: y = $2 x \\cos{\\left(2 \\right)} + 2 x - 2 \\sin{\\left(2 \\right)} + 4 \\cos{\\left(2 \\right)} + 5$.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 2\\cdot x - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)} + 5$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 2\\cdot x - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)} + 5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2100ac856e6d7cbb", "problem_id": "1672af11a528d223", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Find the multiplicative inverse of 21 modulo 23.", "output": "We want x such that 21x ≡ 1 (mod 23).\nUse extended Euclid to find s,t with 21s + 23t = 1.\nOne Bezout identity is: 21·(11) + 23·(-10) = 1.\nFinal answer: $x\\equiv 11\\pmod{23}$", "final_answer": "x\\equiv 11\\pmod{23}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7a2cdb2fab58f252", "problem_id": "1672af11a528d223", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the multiplicative inverse of 21 modulo 23.", "output": "1) We want x such that 21x ≡ 1 (mod 23).\n2) Use extended Euclid to find s,t with 21s + 23t = 1.\n3) One Bezout identity is: 21·(11) + 23·(-10) = 1.\n4) Reduce s modulo 23: x ≡ 11 (mod 23).\nFinal answer: $x\\equiv 11\\pmod{23}$", "final_answer": "x\\equiv 11\\pmod{23}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d78b1208cfb0ff4", "problem_id": "28decaa2fac17fc6", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Two friends split $248 in the ratio 6:2. How much does each receive?", "output": "1) Total ratio parts = 6+2 = 8.\n2) Each part = 248 ÷ 8 = 31.\n3) First share = 6 × 31 = 186.\n4) Second share = 2 × 31 = 62.\nFinal answer: $62$", "final_answer": "62", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aec4d14bbce6b00b", "problem_id": "28decaa2fac17fc6", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Two friends split $248 in the ratio 6:2. How much does each receive?", "output": "Total ratio parts = 6+2 = 8.\nEach part = 248 ÷ 8 = 31.\nFirst share = 6 × 31 = 186.\nFinal answer: $62$", "final_answer": "62", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "89cb414e327f88f3", "problem_id": "4cf4c136e5c984ce", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $- 2 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - \\cos{\\left(x \\right)}$", "final_answer": "- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bc114fc87ad42454", "problem_id": "4cf4c136e5c984ce", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $- 2 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.\nFinal answer: $- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - \\cos{\\left(x \\right)}$", "final_answer": "- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "919ddace0bcf2df9", "problem_id": "a43ff5a1abcb3ae9", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A cyclist rides at 23 miles per hour for 1 hours, then at 74 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 23 × 1 = 23 miles.\n2) Second leg distance = 74 × 2 = 148 miles.\n3) Total distance = 23 + 148 = 171 miles.\nFinal answer: $171$", "final_answer": "171", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aa7bc4932e1335b2", "problem_id": "a43ff5a1abcb3ae9", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A cyclist rides at 23 miles per hour for 1 hours, then at 74 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 23 × 1 = 23 miles.\nSecond leg distance = 74 × 2 = 148 miles.\nTotal distance = 23 + 148 = 171 miles.\nFinal answer: $171$", "final_answer": "171", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c2ecd699b7cd336c", "problem_id": "c694405f4531f96f", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 10 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=5, p=$\\frac{3}{8}$ and simplify.\n4) Here C(10,5) = 252.\nFinal answer: $\\frac{47840625}{268435456}$", "final_answer": "\\frac{47840625}{268435456}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d8ef3f221d929c7d", "problem_id": "c694405f4531f96f", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 10 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=5, p=$\\frac{3}{8}$ and simplify.\nHere C(10,5) = 252.\nCheck: expression simplifies to $\\frac{47840625}{268435456}$.\nFinal answer: $\\frac{47840625}{268435456}$", "final_answer": "\\frac{47840625}{268435456}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b704848c0a4a4b0c", "problem_id": "967f0aa98a9862e8", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 30 red balls and 13 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(43,8).\n3) Favorable samples: choose k reds and n−k blues: C(30,3)·C(13,5).\n4) Compute totals: C(43,8) = 145008513, and favorable = 5225220.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{44660}{1239389}$", "final_answer": "\\frac{44660}{1239389}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7f4b9101275648fe", "problem_id": "967f0aa98a9862e8", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 30 red balls and 13 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(43,8).\nFavorable samples: choose k reds and n−k blues: C(30,3)·C(13,5).\nCompute totals: C(43,8) = 145008513, and favorable = 5225220.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{44660}{1239389}$.\nFinal answer: $\\frac{44660}{1239389}$", "final_answer": "\\frac{44660}{1239389}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dfe5dde980b7dafe", "problem_id": "47fd20c21c33f5eb", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A cyclist rides at 71 miles per hour for 2 hours, then at 40 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 71 × 2 = 142 miles.\n2) Second leg distance = 40 × 3/2 = 60 miles.\n3) Total distance = 142 + 60 = 202 miles.\nFinal answer: $202$", "final_answer": "202", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "36a13497d9afa76d", "problem_id": "47fd20c21c33f5eb", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A cyclist rides at 71 miles per hour for 2 hours, then at 40 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 71 × 2 = 142 miles.\nSecond leg distance = 40 × 3/2 = 60 miles.\nTotal distance = 142 + 60 = 202 miles.\nFinal answer: $202$", "final_answer": "202", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "915502d282edaa21", "problem_id": "432a63160d2e9794", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the multiplicative inverse of 16 modulo 43.", "output": "1) We want x such that 16x ≡ 1 (mod 43).\n2) Use extended Euclid to find s,t with 16s + 43t = 1.\n3) One Bezout identity is: 16·(-8) + 43·(3) = 1.\n4) Reduce s modulo 43: x ≡ 35 (mod 43).\nFinal answer: $x\\equiv 35\\pmod{43}$", "final_answer": "x\\equiv 35\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f16c8f58ddf9205a", "problem_id": "dbe34a8b224711d0", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 x^{3} - 4 x^{2} + 11 x - 19$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} - 4 x^{2} + 10 x - 19.\n3) Then the limit is -79.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -81.\nFinal answer: $-81$", "final_answer": "-81", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "246586ee5c25b49b", "problem_id": "ed467bf02e4da0e5", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many binary strings of length 8 contain exactly 3 ones with no two ones adjacent?", "output": "1) Reserve 2 mandatory zeros between the 3 ones.\n2) This leaves n−k+1 = 6 valid slots for the ones.\n3) Choose the slots: C(6,3) = 20.\nFinal answer: $20$", "final_answer": "20", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5195b502c035fc5b", "problem_id": "ed467bf02e4da0e5", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many binary strings of length 8 contain exactly 3 ones with no two ones adjacent?", "output": "Reserve 2 mandatory zeros between the 3 ones.\nThis leaves n−k+1 = 6 valid slots for the ones.\nChoose the slots: C(6,3) = 20.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $20$", "final_answer": "20", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c474b800d0c4ce6b", "problem_id": "0c1aab26a264af91", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 4 x^{3} + x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $4 x^{3} + x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $\\frac{3 \\pi \\left(-4 - \\pi + 2 \\pi^{2} + 10 \\pi^{3}\\right)}{2}$.\n4) Simplify the result.\nFinal answer: $\\frac{3\\cdot \\pi\\cdot \\left(-4 - \\pi + 2\\cdot \\pi^{2} + 10\\cdot \\pi^{3}\\right)}{2}$", "final_answer": "\\frac{3\\cdot \\pi\\cdot \\left(-4 - \\pi + 2\\cdot \\pi^{2} + 10\\cdot \\pi^{3}\\right)}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e67c3712e407c91c", "problem_id": "0c1aab26a264af91", "style": "answer_only", "instruction": "Final answer only.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 4 x^{3} + x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2\\,dx$.", "output": "$\\frac{3\\cdot \\pi\\cdot \\left(-4 - \\pi + 2\\cdot \\pi^{2} + 10\\cdot \\pi^{3}\\right)}{2}$", "final_answer": "\\frac{3\\cdot \\pi\\cdot \\left(-4 - \\pi + 2\\cdot \\pi^{2} + 10\\cdot \\pi^{3}\\right)}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "17b9c09f485762e4", "problem_id": "fb071099e67e8336", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the congruences: x≡1 (mod 4), x≡4 (mod 7). Give the solution modulo 28.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 28.\n3) Compute the CRT solution to get x ≡ 25 (mod 28).\nFinal answer: $x\\equiv 25\\pmod{28}$", "final_answer": "x\\equiv 25\\pmod{28}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7440910df1942513", "problem_id": "fb071099e67e8336", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡1 (mod 4), x≡4 (mod 7). Give the solution modulo 28.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 28.\nCompute the CRT solution to get x ≡ 25 (mod 28).\nCheck: 25 mod 4=1 and mod 7=4.\nFinal answer: $x\\equiv 25\\pmod{28}$", "final_answer": "x\\equiv 25\\pmod{28}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0c0dc00dd78c36b7", "problem_id": "4be3b780a389b38a", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_6=39$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_6=21.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=21 and k=6.\n4) Compute C(26,5) = 65780.\nFinal answer: $65780$", "final_answer": "65780", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f5fe8ef0337286ac", "problem_id": "4be3b780a389b38a", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_6=39$?", "output": "Let y_i=x_i-3, then y_i≥0 and y_1+…+y_6=21.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=21 and k=6.\nCompute C(26,5) = 65780.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $65780$", "final_answer": "65780", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "055721d27b21d1e9", "problem_id": "ca1db2df9672f21b", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $- 2 x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$.\nFinal answer: $- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)}$", "final_answer": "- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ea0668e1d4573eac", "problem_id": "ca1db2df9672f21b", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $- 2 x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)}$", "final_answer": "- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b3df89ef9827f9fa", "problem_id": "63863301abd5597b", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(2 x^{2} - 3 x + 2\\right) \\cos{\\left(x - 1 \\right)}$.", "output": "Let f(x) = $2 \\left(x + 1\\right) e^{x} + \\left(2 x^{2} - 3 x + 2\\right) \\cos{\\left(x - 1 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $2 \\left(x + 1\\right) e^{x} + \\left(4 x - 3\\right) \\cos{\\left(x - 1 \\right)} - \\left(2 x^{2} - 3 x + 2\\right) \\sin{\\left(x - 1 \\right)} + 2 e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\left(2\\cdot x + 2\\right)\\cdot e^{x} + \\left(4\\cdot x - 3\\right)\\cdot \\cos{\\left(x - 1 \\right)} - \\left(2\\cdot x^{2} - 3\\cdot x + 2\\right)\\cdot \\sin{\\left(x - 1 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x + 2\\right)\\cdot e^{x} + \\left(4\\cdot x - 3\\right)\\cdot \\cos{\\left(x - 1 \\right)} - \\left(2\\cdot x^{2} - 3\\cdot x + 2\\right)\\cdot \\sin{\\left(x - 1 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a40ac0056f0fc6f6", "problem_id": "63863301abd5597b", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(2 x^{2} - 3 x + 2\\right) \\cos{\\left(x - 1 \\right)}$.", "output": "1) Let f(x) = $2 \\left(x + 1\\right) e^{x} + \\left(2 x^{2} - 3 x + 2\\right) \\cos{\\left(x - 1 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $2 \\left(x + 1\\right) e^{x} + \\left(4 x - 3\\right) \\cos{\\left(x - 1 \\right)} - \\left(2 x^{2} - 3 x + 2\\right) \\sin{\\left(x - 1 \\right)} + 2 e^{x}$.\nFinal answer: $\\left(2\\cdot x + 2\\right)\\cdot e^{x} + \\left(4\\cdot x - 3\\right)\\cdot \\cos{\\left(x - 1 \\right)} - \\left(2\\cdot x^{2} - 3\\cdot x + 2\\right)\\cdot \\sin{\\left(x - 1 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x + 2\\right)\\cdot e^{x} + \\left(4\\cdot x - 3\\right)\\cdot \\cos{\\left(x - 1 \\right)} - \\left(2\\cdot x^{2} - 3\\cdot x + 2\\right)\\cdot \\sin{\\left(x - 1 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6cdc6c4d536b55ed", "problem_id": "d0b2387f7eb85125", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} - 3 x^{3} + x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{3} + x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $\\frac{\\pi \\left(-24 + \\pi^{2}\\right)}{12}$.\n4) Simplify the result.\nFinal answer: $\\frac{\\pi\\cdot \\left(-24 + \\pi^{2}\\right)}{12}$", "final_answer": "\\frac{\\pi\\cdot \\left(-24 + \\pi^{2}\\right)}{12}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "157614c04afcf06c", "problem_id": "d0b2387f7eb85125", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} - 3 x^{3} + x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{3} + x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $\\frac{\\pi \\left(-24 + \\pi^{2}\\right)}{12}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{\\pi\\cdot \\left(-24 + \\pi^{2}\\right)}{12}$", "final_answer": "\\frac{\\pi\\cdot \\left(-24 + \\pi^{2}\\right)}{12}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "71c24fee1c9c06c8", "problem_id": "d717be4189afd20f", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_9=31$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_9=13.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=13 and k=9.\nCompute C(21,8) = 203490.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $203490$", "final_answer": "203490", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7722fe8ea65d7397", "problem_id": "d717be4189afd20f", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_9=31$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_9=13.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=13 and k=9.\n4) Compute C(21,8) = 203490.\nFinal answer: $203490$", "final_answer": "203490", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "721183e4f3e8da5a", "problem_id": "5450648b840e1415", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + \\frac{23 x}{20} - \\frac{21}{20}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{23}{20}$, c=$- \\frac{21}{20}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{2209}{400}$, so √Δ = $\\frac{47}{20}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{3}{5}$.\n7) x2 = $- \\frac{7}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{3}{5}, - \\frac{7}{4}\\}$", "final_answer": "x \\in \\{\\frac{3}{5}, - \\frac{7}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "76b4524977f5256a", "problem_id": "30d6cf9a97495d00", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{x^{2} + 3}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 3}) = \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} - \\frac{2 x^{4}}{2 x^{4} + 12 x^{2} + 18} + \\frac{6 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} + \\frac{12 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} - \\frac{12 x^{2}}{2 x^{4} + 12 x^{2} + 18} + \\frac{36 x}{2 x^{4} + 12 x^{2} + 18} + \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 6 x^{2} + 9} + \\frac{9 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} + \\frac{18 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} - \\frac{18}{2 x^{4} + 12 x^{2} + 18} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 6} - \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3} + \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 3}) = - \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3\\right)^{2}} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x^{2} + 3\\right)} + \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 4 x \\left(- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} + 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x^{2} + 3\\right)^{2}}$.\nFinal answer: $\\frac{- 4 x \\left(- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} + 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x^{2} + 3\\right)^{2}}$", "final_answer": "\\frac{- 4 x \\left(- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} + 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x^{2} + 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "52c8b2881f9947f6", "problem_id": "bf0125360ad8138d", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A store sells a backpack for $105. It is discounted by 25%. You buy 5 backpacks. What is the total cost?", "output": "1) Discounted price = 105 × (1 − 25/100) = 105 × 75/100 = 315/4.\n2) Subtotal = 5 × 315/4 = 1575/4.\nFinal answer: $\\frac{1575}{4}$", "final_answer": "\\frac{1575}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "945ea6b05e835d89", "problem_id": "bf0125360ad8138d", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "A store sells a backpack for $105. It is discounted by 25%. You buy 5 backpacks. What is the total cost?", "output": "Discounted price = 105 × (1 − 25/100) = 105 × 75/100 = 315/4.\nSubtotal = 5 × 315/4 = 1575/4.\nFinal answer: $\\frac{1575}{4}$", "final_answer": "\\frac{1575}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5ef7501869de323a", "problem_id": "0cbe64e238666cb2", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many binary strings of length 12 contain exactly 5 ones with no two ones adjacent?", "output": "Reserve 4 mandatory zeros between the 5 ones.\nThis leaves n−k+1 = 8 valid slots for the ones.\nChoose the slots: C(8,5) = 56.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $56$", "final_answer": "56", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bef5af9a61bcc018", "problem_id": "0cbe64e238666cb2", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "How many binary strings of length 12 contain exactly 5 ones with no two ones adjacent?", "output": "1) Reserve 4 mandatory zeros between the 5 ones.\n2) This leaves n−k+1 = 8 valid slots for the ones.\n3) Choose the slots: C(8,5) = 56.\nFinal answer: $56$", "final_answer": "56", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0484436461a06716", "problem_id": "11b03a588fb97e9a", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many binary strings of length 24 contain exactly 9 ones with no two ones adjacent?", "output": "Reserve 8 mandatory zeros between the 9 ones.\nThis leaves n−k+1 = 16 valid slots for the ones.\nChoose the slots: C(16,9) = 11440.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $11440$", "final_answer": "11440", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4f80425d0f04e70d", "problem_id": "11b03a588fb97e9a", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many binary strings of length 24 contain exactly 9 ones with no two ones adjacent?", "output": "1) Reserve 8 mandatory zeros between the 9 ones.\n2) This leaves n−k+1 = 16 valid slots for the ones.\n3) Choose the slots: C(16,9) = 11440.\nFinal answer: $11440$", "final_answer": "11440", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c7747e4310c305e1", "problem_id": "2514bd1c29c0b8a0", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A cyclist rides at 47 miles per hour for 1/2 hours, then at 22 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 47 × 1/2 = 47/2 miles.\n2) Second leg distance = 22 × 2 = 44 miles.\n3) Total distance = 47/2 + 44 = 135/2 miles.\nFinal answer: $\\frac{135}{2}$", "final_answer": "\\frac{135}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1eacd3743f542f8d", "problem_id": "2514bd1c29c0b8a0", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A cyclist rides at 47 miles per hour for 1/2 hours, then at 22 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 47 × 1/2 = 47/2 miles.\nSecond leg distance = 22 × 2 = 44 miles.\nTotal distance = 47/2 + 44 = 135/2 miles.\nCheck: adding both legs gives 135/2.\nFinal answer: $\\frac{135}{2}$", "final_answer": "\\frac{135}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0c2914eed7e6b08e", "problem_id": "7806e61eda5cf76c", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Two friends split $936 in the ratio 2:6. How much does each receive?", "output": "1) Total ratio parts = 2+6 = 8.\n2) Each part = 936 ÷ 8 = 117.\n3) First share = 2 × 117 = 234.\n4) Second share = 6 × 117 = 702.\nFinal answer: $702$", "final_answer": "702", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8dcdd39d8ded68c2", "problem_id": "7806e61eda5cf76c", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $936 in the ratio 2:6. How much does each receive?", "output": "Total ratio parts = 2+6 = 8.\nEach part = 936 ÷ 8 = 117.\nFirst share = 2 × 117 = 234.\nSecond share = 6 × 117 = 702.\nCheck: 234+702=936.\nFinal answer: $702$", "final_answer": "702", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "69545c80ac2eb5ec", "problem_id": "dedb59cecad1c87e", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "A cyclist rides at 58 miles per hour for 1/2 hours, then at 62 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 58 × 1/2 = 29 miles.\nSecond leg distance = 62 × 2 = 124 miles.\nTotal distance = 29 + 124 = 153 miles.\nFinal answer: $153$", "final_answer": "153", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5e27881dace3d41d", "problem_id": "dedb59cecad1c87e", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A cyclist rides at 58 miles per hour for 1/2 hours, then at 62 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 58 × 1/2 = 29 miles.\n2) Second leg distance = 62 × 2 = 124 miles.\n3) Total distance = 29 + 124 = 153 miles.\nFinal answer: $153$", "final_answer": "153", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1e9bfffd48fc77de", "problem_id": "6468fd3fe735b110", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Worker A finishes a job in 13 hours and Worker B finishes it in 12 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/13 job/hour. B's rate = 1/12 job/hour.\n2) Combined rate = 1/13 + 1/12 = 25/156 job/hour.\n3) Time = 1 ÷ (combined rate) = 156/25 hours.\nFinal answer: $\\frac{156}{25}$", "final_answer": "\\frac{156}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1c852b2d5320caa2", "problem_id": "e4305ec474fba739", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x^{3} \\sin{\\left(x \\right)}) = x^{3} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = x^{3} \\cos{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x^{3} \\sin{\\left(x \\right)}) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.\nFinal answer: $x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$", "final_answer": "x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "552e2e223471102c", "problem_id": "751c962d16ac3c7c", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 10 red balls and 14 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(24,7).\n3) Favorable samples: choose k reds and n−k blues: C(10,2)·C(14,5).\n4) Compute totals: C(24,7) = 346104, and favorable = 90090.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{455}{1748}$", "final_answer": "\\frac{455}{1748}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0f2c4ad7a9a4afe3", "problem_id": "751c962d16ac3c7c", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "An urn has 10 red balls and 14 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(24,7).\nFavorable samples: choose k reds and n−k blues: C(10,2)·C(14,5).\nFinal answer: $\\frac{455}{1748}$", "final_answer": "\\frac{455}{1748}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d991e08e6412388", "problem_id": "af1c0d13947067c9", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int x^{3} - 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $x^{3} - 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1$.\n3) Compute an antiderivative: F(x) = $\\frac{x^{4}}{4} - x^{3} + x^{2} - x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{x^{4}}{4} - x^{3} + x^{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{4}}{4} - x^{3} + x^{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a97bfd773d98673b", "problem_id": "8ca33e415b9c53cf", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} + 3 x + 4$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x^{2}) = - 4 x^{2}$.\n3) So $f'(x) = 3 - 4 x^{2}$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x^{2}) = - 4 x$.\nTherefore the correct derivative is $f'(x) = 3 - 4 x$.\nFinal answer: $3 - 4 x$", "final_answer": "3 - 4 x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "4122351ca53c5d8e", "problem_id": "fea128ddbdd7c789", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡8 (mod 11), x≡4 (mod 5). Give the solution modulo 55.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 55.\n3) Compute the CRT solution to get x ≡ 19 (mod 55).\nFinal answer: $x\\equiv 19\\pmod{55}$", "final_answer": "x\\equiv 19\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fa896d5ab72a316c", "problem_id": "24d123f4406e305d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{3} - x^{2} - \\frac{3 x}{2} + \\left(x^{2} - x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\frac{3}{2}}{x^{2} - x + 3}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 3}) = \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 3}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18} + \\frac{2 x^{4}}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18} - \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18} - \\frac{4 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18} - \\frac{4 x^{3}}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18} + \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 7 x^{2} - 6 x + 9} + \\frac{7 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18} + \\frac{14 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18} + \\frac{23 x^{2}}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 7 x^{2} - 6 x + 9} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 2 x + 6} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 3} - \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 3} - \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18} - \\frac{12 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18} - \\frac{18 x}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18} - \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 3} + \\frac{9 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18} + \\frac{18 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18} - \\frac{6}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 3}) = \\frac{x^{2} \\left(1 - 2 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(x^{2} - x\\right) + 3\\right)^{2}} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(x^{2} - x\\right) + 3\\right)} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 3} + \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 3}$.\nTherefore the correct derivative is $f'(x) = \\frac{\\left(1 - 2 x\\right) \\left(2 x^{3} - 2 x^{2} - 3 x + 2 \\left(x \\left(x - 1\\right) + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(x - 1\\right) + 3\\right) \\left(6 x^{2} - 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 3\\right)}{2 \\left(x \\left(x - 1\\right) + 3\\right)^{2}}$.\nFinal answer: $\\frac{\\left(1 - 2 x\\right) \\left(2 x^{3} - 2 x^{2} - 3 x + 2 \\left(x \\left(x - 1\\right) + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(x - 1\\right) + 3\\right) \\left(6 x^{2} - 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 3\\right)}{2 \\left(x \\left(x - 1\\right) + 3\\right)^{2}}$", "final_answer": "\\frac{\\left(1 - 2 x\\right) \\left(2 x^{3} - 2 x^{2} - 3 x + 2 \\left(x \\left(x - 1\\right) + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(x - 1\\right) + 3\\right) \\left(6 x^{2} - 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 3\\right)}{2 \\left(x \\left(x - 1\\right) + 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a29bc056ae16d0a2", "problem_id": "0c3b9762dedf7c90", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} - 3 x^{3} - x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{3} - x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1$.\n2) Evaluate at the bounds: F(2 \\pi) and F(0).\n3) Subtract: F(2 \\pi) − F(0) = $- 12 \\pi^{4} - \\frac{8 \\pi^{3}}{3} + 2 \\pi$.\n4) Simplify the result.\nFinal answer: $- 12\\cdot \\pi^{4} - \\frac{8\\cdot \\pi^{3}}{3} + 2\\cdot \\pi$", "final_answer": "- 12\\cdot \\pi^{4} - \\frac{8\\cdot \\pi^{3}}{3} + 2\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aa8a9dd9e9ed2c92", "problem_id": "0c3b9762dedf7c90", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} - 3 x^{3} - x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{3} - x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1$.\nEvaluate at the bounds: F(2 \\pi) and F(0).\nSubtract: F(2 \\pi) − F(0) = $- 12 \\pi^{4} - \\frac{8 \\pi^{3}}{3} + 2 \\pi$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- 12\\cdot \\pi^{4} - \\frac{8\\cdot \\pi^{3}}{3} + 2\\cdot \\pi$", "final_answer": "- 12\\cdot \\pi^{4} - \\frac{8\\cdot \\pi^{3}}{3} + 2\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7c1eb5f2f47562dd", "problem_id": "73277a04bb1ce86d", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$- x - y$ = $11$\n$4 x + 5 y$ = $-46$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $- 4 x - 4 y$ = $44$.\nMultiply the second equation by $-1$: $- 4 x - 5 y$ = $46$.\nSubtract to eliminate x: $y$ = $-2$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $-9$.\nCheck: plug (x,y)=(-9,-2) into both equations.\nFinal answer: $(x,y)=\\left(-9, -2\\right)$", "final_answer": "(x,y)=\\left(-9, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eb5b7488d82c6b4f", "problem_id": "73277a04bb1ce86d", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the linear system:\n$- x - y$ = $11$\n$4 x + 5 y$ = $-46$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $- 4 x - 4 y$ = $44$.\nMultiply the second equation by $-1$: $- 4 x - 5 y$ = $46$.\nFinal answer: $(x,y)=\\left(-9, -2\\right)$", "final_answer": "(x,y)=\\left(-9, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec27992e39dde273", "problem_id": "32e5e5c220f987d1", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$x + 11 y$ = $-91$\n$12 x - 2 y$ = $114$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $12$: $12 x + 132 y$ = $-1092$.\n3) Multiply the second equation by $1$: $12 x - 2 y$ = $114$.\n4) Subtract to eliminate x: $134 y$ = $-1206$.\n5) Solve for y: y = $-9$.\n6) Substitute back to find x: x = $8$.\nFinal answer: $(x,y)=\\left(8, -9\\right)$", "final_answer": "(x,y)=\\left(8, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5c5af452a91628b7", "problem_id": "32e5e5c220f987d1", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$x + 11 y$ = $-91$\n$12 x - 2 y$ = $114$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $12$: $12 x + 132 y$ = $-1092$.\nMultiply the second equation by $1$: $12 x - 2 y$ = $114$.\nSubtract to eliminate x: $134 y$ = $-1206$.\nSolve for y: y = $-9$.\nSubstitute back to find x: x = $8$.\nCheck: plug (x,y)=(8,-9) into both equations.\nFinal answer: $(x,y)=\\left(8, -9\\right)$", "final_answer": "(x,y)=\\left(8, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4452279267d6998c", "problem_id": "3f1bf34f2bb0cd90", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$- x + y$ = $-14$\n$3 x + 7 y$ = $-38$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $3$: $- 3 x + 3 y$ = $-42$.\n3) Multiply the second equation by $-1$: $- 3 x - 7 y$ = $38$.\n4) Subtract to eliminate x: $10 y$ = $-80$.\n5) Solve for y: y = $-8$.\n6) Substitute back to find x: x = $6$.\nFinal answer: $(x,y)=\\left(6, -8\\right)$", "final_answer": "(x,y)=\\left(6, -8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "142ba251464d1daf", "problem_id": "3f1bf34f2bb0cd90", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the linear system:\n$- x + y$ = $-14$\n$3 x + 7 y$ = $-38$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $- 3 x + 3 y$ = $-42$.\nMultiply the second equation by $-1$: $- 3 x - 7 y$ = $38$.\nFinal answer: $(x,y)=\\left(6, -8\\right)$", "final_answer": "(x,y)=\\left(6, -8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cc778d508bf005aa", "problem_id": "f27190461ad8880c", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many binary strings of length 17 contain exactly 5 ones with no two ones adjacent?", "output": "1) Reserve 4 mandatory zeros between the 5 ones.\n2) This leaves n−k+1 = 13 valid slots for the ones.\n3) Choose the slots: C(13,5) = 1287.\nFinal answer: $1287$", "final_answer": "1287", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "524ed71c6578954c", "problem_id": "f27190461ad8880c", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many binary strings of length 17 contain exactly 5 ones with no two ones adjacent?", "output": "Reserve 4 mandatory zeros between the 5 ones.\nThis leaves n−k+1 = 13 valid slots for the ones.\nChoose the slots: C(13,5) = 1287.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $1287$", "final_answer": "1287", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f9c7c3c62c1a58c7", "problem_id": "bfa78aba0a465e09", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$- 3 x - 8 y$ = $-8$\n$8 x - 5 y$ = $74$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $8$: $- 24 x - 64 y$ = $-64$.\n3) Multiply the second equation by $-3$: $- 24 x + 15 y$ = $-222$.\n4) Subtract to eliminate x: $- 79 y$ = $158$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $8$.\nFinal answer: $(x,y)=\\left(8, -2\\right)$", "final_answer": "(x,y)=\\left(8, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a1ddc187526e60da", "problem_id": "bfa78aba0a465e09", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$- 3 x - 8 y$ = $-8$\n$8 x - 5 y$ = $74$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $8$: $- 24 x - 64 y$ = $-64$.\nMultiply the second equation by $-3$: $- 24 x + 15 y$ = $-222$.\nSubtract to eliminate x: $- 79 y$ = $158$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $8$.\nCheck: plug (x,y)=(8,-2) into both equations.\nFinal answer: $(x,y)=\\left(8, -2\\right)$", "final_answer": "(x,y)=\\left(8, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d994ccb35e2f645c", "problem_id": "338bcb8a757a8d17", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} - 3 x^{2} - 4 x + 4$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 3 x^{2} - 6 x - 4$.\n2) Slope at x=2: m = f'(2) = $-28$.\n3) Point on curve: (2, f(2)) = (2, $-24$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -24 = -28(x − 2).\n6) Optionally expand: y = $32 - 28 x$.\nFinal answer: $y = 32 - 28\\cdot x$", "final_answer": "y = 32 - 28\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1ce911ce74aad72d", "problem_id": "338bcb8a757a8d17", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} - 3 x^{2} - 4 x + 4$ at $x=2$.", "output": "Differentiate: f'(x) = $- 3 x^{2} - 6 x - 4$.\nSlope at x=2: m = f'(2) = $-28$.\nPoint on curve: (2, f(2)) = (2, $-24$).\nFinal answer: $y = 32 - 28\\cdot x$", "final_answer": "y = 32 - 28\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a746ee3ab1639bc", "problem_id": "8e4c36b40a7bcc0b", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{7 x}{3} - \\frac{20}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{7}{3}$, c=$- \\frac{20}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{289}{9}$, so √Δ = $\\frac{17}{3}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $4$.\n7) x2 = $- \\frac{5}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{4, - \\frac{5}{3}\\}$", "final_answer": "x \\in \\{4, - \\frac{5}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "28d93389bc90b5c6", "problem_id": "8e4c36b40a7bcc0b", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{7 x}{3} - \\frac{20}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{7}{3}$, c=$- \\frac{20}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{4, - \\frac{5}{3}\\}$", "final_answer": "x \\in \\{4, - \\frac{5}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a2d7a40c6e758a83", "problem_id": "4de38a78c0f23b79", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the multiplicative inverse of 4 modulo 11.", "output": "We want x such that 4x ≡ 1 (mod 11).\nUse extended Euclid to find s,t with 4s + 11t = 1.\nOne Bezout identity is: 4·(3) + 11·(-1) = 1.\nReduce s modulo 11: x ≡ 3 (mod 11).\nCheck: 4·3 mod 11 = 1.\nFinal answer: $x\\equiv 3\\pmod{11}$", "final_answer": "x\\equiv 3\\pmod{11}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2e5a2aba34b088fb", "problem_id": "4de38a78c0f23b79", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the multiplicative inverse of 4 modulo 11.", "output": "1) We want x such that 4x ≡ 1 (mod 11).\n2) Use extended Euclid to find s,t with 4s + 11t = 1.\n3) One Bezout identity is: 4·(3) + 11·(-1) = 1.\n4) Reduce s modulo 11: x ≡ 3 (mod 11).\nFinal answer: $x\\equiv 3\\pmod{11}$", "final_answer": "x\\equiv 3\\pmod{11}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7006eb8f45dbe08d", "problem_id": "b33f007278b4a1ff", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$3 x - 3 y$ = $-33$\n$8 y$ = $16$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $3$: $24 y$ = $48$.\n4) Subtract to eliminate x: $- 24 y$ = $-48$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $-9$.\nFinal answer: $(x,y)=\\left(-9, 2\\right)$", "final_answer": "(x,y)=\\left(-9, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4c79db24a4647ed3", "problem_id": "c2beab8a129c5d2e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} 3 x + 9$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to 2 x + 9.\n3) Then the limit is 15.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: 18.\nFinal answer: $18$", "final_answer": "18", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "369f318540c4e0c9", "problem_id": "d82fc3c2cc7e3d48", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$- 7 x + 5 y$ = $-8$\n$4 x + 4 y$ = $-16$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $- 28 x + 20 y$ = $-32$.\nMultiply the second equation by $-7$: $- 28 x - 28 y$ = $112$.\nSubtract to eliminate x: $48 y$ = $-144$.\nSolve for y: y = $-3$.\nSubstitute back to find x: x = $-1$.\nCheck: plug (x,y)=(-1,-3) into both equations.\nFinal answer: $(x,y)=\\left(-1, -3\\right)$", "final_answer": "(x,y)=\\left(-1, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "118761f48acccb64", "problem_id": "d82fc3c2cc7e3d48", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the linear system:\n$- 7 x + 5 y$ = $-8$\n$4 x + 4 y$ = $-16$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $- 28 x + 20 y$ = $-32$.\n3) Multiply the second equation by $-7$: $- 28 x - 28 y$ = $112$.\n4) Subtract to eliminate x: $48 y$ = $-144$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, -3\\right)$", "final_answer": "(x,y)=\\left(-1, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f62021f0b7ab59b4", "problem_id": "5b4845ccc9836194", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 4 x^{3} - 2 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $4 x^{3} - 2 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $- 6 \\pi^{3} + 3 \\pi + 15 \\pi^{4}$.\n4) Simplify the result.\nFinal answer: $- 6\\cdot \\pi^{3} + 3\\cdot \\pi + 15\\cdot \\pi^{4}$", "final_answer": "- 6\\cdot \\pi^{3} + 3\\cdot \\pi + 15\\cdot \\pi^{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "20d74a4e0b95cac4", "problem_id": "5b4845ccc9836194", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 4 x^{3} - 2 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $4 x^{3} - 2 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $- 6 \\pi^{3} + 3 \\pi + 15 \\pi^{4}$.\nFinal answer: $- 6\\cdot \\pi^{3} + 3\\cdot \\pi + 15\\cdot \\pi^{4}$", "final_answer": "- 6\\cdot \\pi^{3} + 3\\cdot \\pi + 15\\cdot \\pi^{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "830c8f829215bfd0", "problem_id": "e3962e367ed3f02a", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{2} + x - 1$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-1) = 0.\n3) d/dx(x) = 1.\n4) d/dx(4 x^{2}) = 8 x.\n5) Combine: f'(x) = 8 x + 1.\nFinal answer: $8\\cdot x + 1$", "final_answer": "8\\cdot x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1777ab8c76fe0c65", "problem_id": "e3962e367ed3f02a", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{2} + x - 1$.", "output": "Differentiate term-by-term:\nd/dx(-1) = 0.\nd/dx(x) = 1.\nFinal answer: $8\\cdot x + 1$", "final_answer": "8\\cdot x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "47f50f90f4bbe8f4", "problem_id": "f1a92a86f6b6b8e4", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 2 x^{2} + x - 4\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 2 x^{2} + x - 4$.\n3) Compute an antiderivative: F(x) = $- \\frac{2 x^{3}}{3} + \\frac{x^{2}}{2} - 4 x$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{2\\cdot x^{3}}{3} + \\frac{x^{2}}{2} - 4\\cdot x$", "final_answer": "C - \\frac{2\\cdot x^{3}}{3} + \\frac{x^{2}}{2} - 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "075d71604ee44661", "problem_id": "f1a92a86f6b6b8e4", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 2 x^{2} + x - 4\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 2 x^{2} + x - 4$.\nCompute an antiderivative: F(x) = $- \\frac{2 x^{3}}{3} + \\frac{x^{2}}{2} - 4 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - \\frac{2\\cdot x^{3}}{3} + \\frac{x^{2}}{2} - 4\\cdot x$", "final_answer": "C - \\frac{2\\cdot x^{3}}{3} + \\frac{x^{2}}{2} - 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5efd89573711d890", "problem_id": "70035f97addac3fe", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $61. It is discounted by 15%. You buy 11 backpacks. What is the total cost?", "output": "Discounted price = 61 × (1 − 15/100) = 61 × 85/100 = 1037/20.\nSubtotal = 11 × 1037/20 = 11407/20.\nCheck: recomputing the formula yields 11407/20.\nFinal answer: $\\frac{11407}{20}$", "final_answer": "\\frac{11407}{20}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f49f0e18c612768", "problem_id": "70035f97addac3fe", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A store sells a backpack for $61. It is discounted by 15%. You buy 11 backpacks. What is the total cost?", "output": "Discounted price = 61 × (1 − 15/100) = 61 × 85/100 = 1037/20.\nSubtotal = 11 × 1037/20 = 11407/20.\nFinal answer: $\\frac{11407}{20}$", "final_answer": "\\frac{11407}{20}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b5cc8895ad279e58", "problem_id": "96c9e315852ffdbc", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - x^{4} + 2 x^{3} - x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{4} + 2 x^{3} - x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{205 \\pi^{5}}{32} - \\frac{65 \\pi^{3}}{24} - \\frac{15 \\pi^{2}}{2} - 1 + \\frac{15 \\pi}{2} + \\frac{255 \\pi^{4}}{32}$.\n4) Simplify the result.\nFinal answer: $- \\frac{205\\cdot \\pi^{5}}{32} - \\frac{65\\cdot \\pi^{3}}{24} - \\frac{15\\cdot \\pi^{2}}{2} - 1 + \\frac{15\\cdot \\pi}{2} + \\frac{255\\cdot \\pi^{4}}{32}$", "final_answer": "- \\frac{205\\cdot \\pi^{5}}{32} - \\frac{65\\cdot \\pi^{3}}{24} - \\frac{15\\cdot \\pi^{2}}{2} - 1 + \\frac{15\\cdot \\pi}{2} + \\frac{255\\cdot \\pi^{4}}{32}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "100dd60218d325e4", "problem_id": "96c9e315852ffdbc", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - x^{4} + 2 x^{3} - x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{4} + 2 x^{3} - x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{205 \\pi^{5}}{32} - \\frac{65 \\pi^{3}}{24} - \\frac{15 \\pi^{2}}{2} - 1 + \\frac{15 \\pi}{2} + \\frac{255 \\pi^{4}}{32}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{205\\cdot \\pi^{5}}{32} - \\frac{65\\cdot \\pi^{3}}{24} - \\frac{15\\cdot \\pi^{2}}{2} - 1 + \\frac{15\\cdot \\pi}{2} + \\frac{255\\cdot \\pi^{4}}{32}$", "final_answer": "- \\frac{205\\cdot \\pi^{5}}{32} - \\frac{65\\cdot \\pi^{3}}{24} - \\frac{15\\cdot \\pi^{2}}{2} - 1 + \\frac{15\\cdot \\pi}{2} + \\frac{255\\cdot \\pi^{4}}{32}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1bae46de52a0ae81", "problem_id": "3bf5e437c1d71c8c", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - \\frac{37 x}{5} - \\frac{8}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$- \\frac{37}{5}$, c=$- \\frac{8}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{1849}{25}$, so √Δ = $\\frac{43}{5}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{8}{3}$.\n7) x2 = $- \\frac{1}{5}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{8}{3}, - \\frac{1}{5}\\}$", "final_answer": "x \\in \\{\\frac{8}{3}, - \\frac{1}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "70c5be5b58107d17", "problem_id": "3bf5e437c1d71c8c", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - \\frac{37 x}{5} - \\frac{8}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$- \\frac{37}{5}$, c=$- \\frac{8}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{1849}{25}$, so √Δ = $\\frac{43}{5}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{8}{3}$.\nx2 = $- \\frac{1}{5}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{8}{3}, - \\frac{1}{5}\\}$", "final_answer": "x \\in \\{\\frac{8}{3}, - \\frac{1}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "384fd1c3d4b7caa4", "problem_id": "df768cff59b9db9c", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(3 x^{2} - x + 1\\right) \\cos{\\left(x^{2} + x + 2 \\right)}$.", "output": "1) Let f(x) = $\\left(x - 3\\right) e^{x} + \\left(3 x^{2} - x + 1\\right) \\cos{\\left(x^{2} + x + 2 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\left(x - 3\\right) e^{x} - \\left(2 x + 1\\right) \\left(3 x^{2} - x + 1\\right) \\sin{\\left(x^{2} + x + 2 \\right)} + \\left(6 x - 1\\right) \\cos{\\left(x^{2} + x + 2 \\right)} + e^{x}$.\nFinal answer: $\\left(x - 3\\right)\\cdot e^{x} - \\left(2\\cdot x + 1\\right)\\cdot \\left(3\\cdot x^{2} - x + 1\\right)\\cdot \\sin{\\left(x^{2} + x + 2 \\right)} + \\left(6\\cdot x - 1\\right)\\cdot \\cos{\\left(x^{2} + x + 2 \\right)} + e^{x}$", "final_answer": "\\left(x - 3\\right)\\cdot e^{x} - \\left(2\\cdot x + 1\\right)\\cdot \\left(3\\cdot x^{2} - x + 1\\right)\\cdot \\sin{\\left(x^{2} + x + 2 \\right)} + \\left(6\\cdot x - 1\\right)\\cdot \\cos{\\left(x^{2} + x + 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c652d8eae55fd72b", "problem_id": "e1636336dac3aceb", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{4} - 2 x^{3} + 2 x^{2} + 4 x + 2$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(2 x^{2}) = 4 x^{2}$.\n3) So $f'(x) = 8 x^{3} - 2 x^{2} + 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(2 x^{2}) = 4 x$.\nTherefore the correct derivative is $f'(x) = 2 \\left(4 x^{3} - 3 x^{2} + 2 x + 2\\right)$.\nFinal answer: $2 \\left(4 x^{3} - 3 x^{2} + 2 x + 2\\right)$", "final_answer": "2 \\left(4 x^{3} - 3 x^{2} + 2 x + 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3be18bc09b390a09", "problem_id": "1698b0d61c6a73d0", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - 2 x - 9$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 2 x^{2} - 3 x + 3$ and a = 3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 4 x - 3$.\n5) Evaluate at a: p'(3) = $-15$.\nFinal answer: $-15$", "final_answer": "-15", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "512011df10dc5467", "problem_id": "1698b0d61c6a73d0", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - 2 x - 9$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 2 x^{2} - 3 x + 3$ and a = 3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-15$", "final_answer": "-15", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f8a2d95a02ba1c67", "problem_id": "d9fc0d985cc3b876", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{13} x - x - 6 \\sqrt{13} - 6$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- 3 \\sqrt{13} - 1$, c=$- 6 \\sqrt{13} - 6$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $30 \\sqrt{13} + 142$, so √Δ = $\\sqrt{30 \\sqrt{13} + 142}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{1}{2} + \\frac{3 \\sqrt{13}}{2} + \\frac{\\sqrt{30 \\sqrt{13} + 142}}{2}$.\n7) x2 = $- \\frac{\\sqrt{30 \\sqrt{13} + 142}}{2} + \\frac{1}{2} + \\frac{3 \\sqrt{13}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{1}{2} + \\frac{3\\cdot \\sqrt{13}}{2} + \\frac{\\sqrt{30\\cdot \\sqrt{13} + 142}}{2}, - \\frac{\\sqrt{30\\cdot \\sqrt{13} + 142}}{2} + \\frac{1}{2} + \\frac{3\\cdot \\sqrt{13}}{2}\\}$", "final_answer": "x \\in \\{\\frac{1}{2} + \\frac{3\\cdot \\sqrt{13}}{2} + \\frac{\\sqrt{30\\cdot \\sqrt{13} + 142}}{2}, - \\frac{\\sqrt{30\\cdot \\sqrt{13} + 142}}{2} + \\frac{1}{2} + \\frac{3\\cdot \\sqrt{13}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "71d419f611628806", "problem_id": "d9fc0d985cc3b876", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{13} x - x - 6 \\sqrt{13} - 6$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- 3 \\sqrt{13} - 1$, c=$- 6 \\sqrt{13} - 6$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $30 \\sqrt{13} + 142$, so √Δ = $\\sqrt{30 \\sqrt{13} + 142}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{1}{2} + \\frac{3 \\sqrt{13}}{2} + \\frac{\\sqrt{30 \\sqrt{13} + 142}}{2}$.\nx2 = $- \\frac{\\sqrt{30 \\sqrt{13} + 142}}{2} + \\frac{1}{2} + \\frac{3 \\sqrt{13}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{1}{2} + \\frac{3\\cdot \\sqrt{13}}{2} + \\frac{\\sqrt{30\\cdot \\sqrt{13} + 142}}{2}, - \\frac{\\sqrt{30\\cdot \\sqrt{13} + 142}}{2} + \\frac{1}{2} + \\frac{3\\cdot \\sqrt{13}}{2}\\}$", "final_answer": "x \\in \\{\\frac{1}{2} + \\frac{3\\cdot \\sqrt{13}}{2} + \\frac{\\sqrt{30\\cdot \\sqrt{13} + 142}}{2}, - \\frac{\\sqrt{30\\cdot \\sqrt{13} + 142}}{2} + \\frac{1}{2} + \\frac{3\\cdot \\sqrt{13}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d99373090c1439c1", "problem_id": "52f83643d177c4a4", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} - 4 x^{2} - 3 x + 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 3 x) = - 3 x$.\n3) So $f'(x) = 3 x^{2} - 11 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 3 x) = -3$.\nTherefore the correct derivative is $f'(x) = 3 x^{2} - 8 x - 3$.\nFinal answer: $3 x^{2} - 8 x - 3$", "final_answer": "3 x^{2} - 8 x - 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c7cf02f71d06f123", "problem_id": "322bbf04484b6a3b", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_7=28$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_7=7.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=7 and k=7.\n4) Compute C(13,6) = 1716.\nFinal answer: $1716$", "final_answer": "1716", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "697cdb7ea9318059", "problem_id": "81fb95491388fd68", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A cyclist rides at 39 miles per hour for 1 hours, then at 35 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 39 × 1 = 39 miles.\nSecond leg distance = 35 × 2 = 70 miles.\nTotal distance = 39 + 70 = 109 miles.\nFinal answer: $109$", "final_answer": "109", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a56a1c5bd3c263a5", "problem_id": "81fb95491388fd68", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A cyclist rides at 39 miles per hour for 1 hours, then at 35 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 39 × 1 = 39 miles.\n2) Second leg distance = 35 × 2 = 70 miles.\n3) Total distance = 39 + 70 = 109 miles.\nFinal answer: $109$", "final_answer": "109", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0379333b54bafac9", "problem_id": "0d94f39c271a4b7b", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $2640 in the ratio 8:8. How much does each receive?", "output": "Total ratio parts = 8+8 = 16.\nEach part = 2640 ÷ 16 = 165.\nFirst share = 8 × 165 = 1320.\nSecond share = 8 × 165 = 1320.\nCheck: 1320+1320=2640.\nFinal answer: $1320$", "final_answer": "1320", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7150b1c3ae968135", "problem_id": "0d94f39c271a4b7b", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Two friends split $2640 in the ratio 8:8. How much does each receive?", "output": "1) Total ratio parts = 8+8 = 16.\n2) Each part = 2640 ÷ 16 = 165.\n3) First share = 8 × 165 = 1320.\n4) Second share = 8 × 165 = 1320.\nFinal answer: $1320$", "final_answer": "1320", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b0c0d827ac846758", "problem_id": "2884714a5ec2f63f", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $154 in the ratio 2:9. How much does each receive?", "output": "Total ratio parts = 2+9 = 11.\nEach part = 154 ÷ 11 = 14.\nFirst share = 2 × 14 = 28.\nSecond share = 9 × 14 = 126.\nCheck: 28+126=154.\nFinal answer: $126$", "final_answer": "126", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "44f91e9bf8a1b7df", "problem_id": "2884714a5ec2f63f", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Two friends split $154 in the ratio 2:9. How much does each receive?", "output": "1) Total ratio parts = 2+9 = 11.\n2) Each part = 154 ÷ 11 = 14.\n3) First share = 2 × 14 = 28.\n4) Second share = 9 × 14 = 126.\nFinal answer: $126$", "final_answer": "126", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ae037afbf030e72e", "problem_id": "07902f1cfdc58260", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - \\frac{7 x}{2} - 12$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$- \\frac{7}{2}$, c=$-12$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{625}{4}$, so √Δ = $\\frac{25}{2}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{8}{3}$.\nx2 = $- \\frac{3}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{8}{3}, - \\frac{3}{2}\\}$", "final_answer": "x \\in \\{\\frac{8}{3}, - \\frac{3}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "00ab81a7ffd275ab", "problem_id": "07902f1cfdc58260", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - \\frac{7 x}{2} - 12$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$- \\frac{7}{2}$, c=$-12$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{625}{4}$, so √Δ = $\\frac{25}{2}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{8}{3}$.\n7) x2 = $- \\frac{3}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{8}{3}, - \\frac{3}{2}\\}$", "final_answer": "x \\in \\{\\frac{8}{3}, - \\frac{3}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "56920dceac5ae5ec", "problem_id": "f736f3729c088664", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 50 miles per hour for 1 hours, then at 67 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 50 × 1 = 50 miles.\nSecond leg distance = 67 × 1 = 67 miles.\nTotal distance = 50 + 67 = 117 miles.\nCheck: adding both legs gives 117.\nFinal answer: $117$", "final_answer": "117", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c9cb5d687971f39f", "problem_id": "f736f3729c088664", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A cyclist rides at 50 miles per hour for 1 hours, then at 67 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 50 × 1 = 50 miles.\n2) Second leg distance = 67 × 1 = 67 miles.\n3) Total distance = 50 + 67 = 117 miles.\nFinal answer: $117$", "final_answer": "117", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bf905abddce2badc", "problem_id": "a979dde2c326c3e0", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the multiplicative inverse of 31 modulo 41.", "output": "1) We want x such that 31x ≡ 1 (mod 41).\n2) Use extended Euclid to find s,t with 31s + 41t = 1.\n3) One Bezout identity is: 31·(4) + 41·(-3) = 1.\n4) Reduce s modulo 41: x ≡ 4 (mod 41).\nFinal answer: $x\\equiv 4\\pmod{41}$", "final_answer": "x\\equiv 4\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "09c575715f8ee4f5", "problem_id": "a979dde2c326c3e0", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the multiplicative inverse of 31 modulo 41.", "output": "We want x such that 31x ≡ 1 (mod 41).\nUse extended Euclid to find s,t with 31s + 41t = 1.\nOne Bezout identity is: 31·(4) + 41·(-3) = 1.\nReduce s modulo 41: x ≡ 4 (mod 41).\nCheck: 31·4 mod 41 = 1.\nFinal answer: $x\\equiv 4\\pmod{41}$", "final_answer": "x\\equiv 4\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8665254cfe8c1a1f", "problem_id": "3b7205ef678719a8", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$- x$ = $3$\n$x - 2 y$ = $-7$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $- x$ = $3$.\nMultiply the second equation by $-1$: $- x + 2 y$ = $7$.\nSubtract to eliminate x: $- 2 y$ = $-4$.\nSolve for y: y = $2$.\nSubstitute back to find x: x = $-3$.\nCheck: plug (x,y)=(-3,2) into both equations.\nFinal answer: $(x,y)=\\left(-3, 2\\right)$", "final_answer": "(x,y)=\\left(-3, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "27323e978c03d2aa", "problem_id": "3b7205ef678719a8", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$- x$ = $3$\n$x - 2 y$ = $-7$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $- x$ = $3$.\n3) Multiply the second equation by $-1$: $- x + 2 y$ = $7$.\n4) Subtract to eliminate x: $- 2 y$ = $-4$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, 2\\right)$", "final_answer": "(x,y)=\\left(-3, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "88cbfdf0936baf31", "problem_id": "052a317d87cb6202", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many binary strings of length 9 contain exactly 2 ones with no two ones adjacent?", "output": "1) Reserve 1 mandatory zeros between the 2 ones.\n2) This leaves n−k+1 = 8 valid slots for the ones.\n3) Choose the slots: C(8,2) = 28.\nFinal answer: $28$", "final_answer": "28", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "99e28f554d9353ae", "problem_id": "052a317d87cb6202", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many binary strings of length 9 contain exactly 2 ones with no two ones adjacent?", "output": "Reserve 1 mandatory zeros between the 2 ones.\nThis leaves n−k+1 = 8 valid slots for the ones.\nChoose the slots: C(8,2) = 28.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $28$", "final_answer": "28", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "723c598e4f4edcda", "problem_id": "99c48b2bec20a268", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡0 (mod 11), x≡5 (mod 8). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 77 (mod 88).\nFinal answer: $x\\equiv 77\\pmod{88}$", "final_answer": "x\\equiv 77\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "27674d9510fc7976", "problem_id": "99c48b2bec20a268", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the congruences: x≡0 (mod 11), x≡5 (mod 8). Give the solution modulo 88.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 88.\nCompute the CRT solution to get x ≡ 77 (mod 88).\nFinal answer: $x\\equiv 77\\pmod{88}$", "final_answer": "x\\equiv 77\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0001c9f529460101", "problem_id": "10208cd40cdd6a03", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A store sells a backpack for $400. It is discounted by 30%. You buy 4 backpacks. What is the total cost?", "output": "Discounted price = 400 × (1 − 30/100) = 400 × 70/100 = 280.\nSubtotal = 4 × 280 = 1120.\nCheck: recomputing the formula yields 1120.\nFinal answer: $1120$", "final_answer": "1120", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "efc31508b4c2f4ce", "problem_id": "10208cd40cdd6a03", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A store sells a backpack for $400. It is discounted by 30%. You buy 4 backpacks. What is the total cost?", "output": "1) Discounted price = 400 × (1 − 30/100) = 400 × 70/100 = 280.\n2) Subtotal = 4 × 280 = 1120.\nFinal answer: $1120$", "final_answer": "1120", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c202039c7bd866db", "problem_id": "963b98c6cda549b0", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the congruences: x≡6 (mod 11), x≡7 (mod 8). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 39 (mod 88).\nFinal answer: $x\\equiv 39\\pmod{88}$", "final_answer": "x\\equiv 39\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "19495e6b92e6d425", "problem_id": "963b98c6cda549b0", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡6 (mod 11), x≡7 (mod 8). Give the solution modulo 88.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 88.\nCompute the CRT solution to get x ≡ 39 (mod 88).\nCheck: 39 mod 11=6 and mod 8=7.\nFinal answer: $x\\equiv 39\\pmod{88}$", "final_answer": "x\\equiv 39\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "12311e072bd4d591", "problem_id": "a771bbaa29e13e6f", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "You want 47 liters of a 35% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 47−x liters is 50% solution.\nSalt equation: 25x + 50(47−x) = 35·47.\n(25-50)x = (35-50)·47 ⇒ x = 141/5.\nFinal answer: $141/5 L of 25% and 94/5 L of 50%$", "final_answer": "141/5 L of 25% and 94/5 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6ee2e7a0a37575c3", "problem_id": "a771bbaa29e13e6f", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "You want 47 liters of a 35% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 47−x liters is 50% solution.\n2) Salt equation: 25x + 50(47−x) = 35·47.\n3) (25-50)x = (35-50)·47 ⇒ x = 141/5.\n4) Then 47−x = 94/5.\nFinal answer: $141/5 L of 25% and 94/5 L of 50%$", "final_answer": "141/5 L of 25% and 94/5 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "41aaf4d5a18b9a39", "problem_id": "8c66a3bdbdae0d60", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Two friends split $198 in the ratio 3:3. How much does each receive?", "output": "1) Total ratio parts = 3+3 = 6.\n2) Each part = 198 ÷ 6 = 33.\n3) First share = 3 × 33 = 99.\n4) Second share = 3 × 33 = 99.\nFinal answer: $99$", "final_answer": "99", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "92159b31d290a96c", "problem_id": "8c66a3bdbdae0d60", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Two friends split $198 in the ratio 3:3. How much does each receive?", "output": "Total ratio parts = 3+3 = 6.\nEach part = 198 ÷ 6 = 33.\nFirst share = 3 × 33 = 99.\nFinal answer: $99$", "final_answer": "99", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "29c360a9a62e355b", "problem_id": "86b258ce2d1631ea", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(- x^{3} - 2 x^{2} + 3\\right) \\cos{\\left(x + 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x^{3} \\cos{\\left(x + 1 \\right)}) = x^{3} \\sin{\\left(x + 1 \\right)}$.\n3) So $f'(x) = x^{3} \\sin{\\left(x + 1 \\right)} + 2 x^{2} \\sin{\\left(x + 1 \\right)} + x e^{x} - 4 x \\cos{\\left(x + 1 \\right)} - 2 e^{x} - 3 \\sin{\\left(x + 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x^{3} \\cos{\\left(x + 1 \\right)}) = x^{3} \\sin{\\left(x + 1 \\right)} - 3 x^{2} \\cos{\\left(x + 1 \\right)}$.\nTherefore the correct derivative is $f'(x) = x \\left(- 3 x - 4\\right) \\cos{\\left(x + 1 \\right)} + \\left(x - 3\\right) e^{x} - \\left(x^{2} \\left(- x - 2\\right) + 3\\right) \\sin{\\left(x + 1 \\right)} + e^{x}$.\nFinal answer: $x \\left(- 3 x - 4\\right) \\cos{\\left(x + 1 \\right)} + \\left(x - 3\\right) e^{x} - \\left(x^{2} \\left(- x - 2\\right) + 3\\right) \\sin{\\left(x + 1 \\right)} + e^{x}$", "final_answer": "x \\left(- 3 x - 4\\right) \\cos{\\left(x + 1 \\right)} + \\left(x - 3\\right) e^{x} - \\left(x^{2} \\left(- x - 2\\right) + 3\\right) \\sin{\\left(x + 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "dfe5c313660cea3a", "problem_id": "52119ff68cb17c7e", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(- 3 x^{3} + x^{2} + 2 x - 2\\right) \\cos{\\left(x - 3 \\right)}$.", "output": "1) Let f(x) = $\\left(x + 2\\right) e^{x} + \\left(- 3 x^{3} + x^{2} + 2 x - 2\\right) \\cos{\\left(x - 3 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\left(x + 2\\right) e^{x} + \\left(- 9 x^{2} + 2 x + 2\\right) \\cos{\\left(x - 3 \\right)} - \\left(- 3 x^{3} + x^{2} + 2 x - 2\\right) \\sin{\\left(x - 3 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 2\\right)\\cdot e^{x} + \\left(- 9\\cdot x^{2} + 2\\cdot x + 2\\right)\\cdot \\cos{\\left(x - 3 \\right)} - \\left(- 3\\cdot x^{3} + x^{2} + 2\\cdot x - 2\\right)\\cdot \\sin{\\left(x - 3 \\right)} + e^{x}$", "final_answer": "\\left(x + 2\\right)\\cdot e^{x} + \\left(- 9\\cdot x^{2} + 2\\cdot x + 2\\right)\\cdot \\cos{\\left(x - 3 \\right)} - \\left(- 3\\cdot x^{3} + x^{2} + 2\\cdot x - 2\\right)\\cdot \\sin{\\left(x - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cd4a80cf9d54a361", "problem_id": "2cea81c0c13916ca", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Worker A finishes a job in 14 hours and Worker B finishes it in 15 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/14 job/hour. B's rate = 1/15 job/hour.\n2) Combined rate = 1/14 + 1/15 = 29/210 job/hour.\n3) Time = 1 ÷ (combined rate) = 210/29 hours.\nFinal answer: $\\frac{210}{29}$", "final_answer": "\\frac{210}{29}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4d41846482a1b2bf", "problem_id": "2cea81c0c13916ca", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Worker A finishes a job in 14 hours and Worker B finishes it in 15 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/14 job/hour. B's rate = 1/15 job/hour.\nCombined rate = 1/14 + 1/15 = 29/210 job/hour.\nTime = 1 ÷ (combined rate) = 210/29 hours.\nCheck: (combined rate)×(time) = 29/210×210/29 = 1.\nFinal answer: $\\frac{210}{29}$", "final_answer": "\\frac{210}{29}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cbd1fda97f630b2e", "problem_id": "b38bd3c2dc4e2fbb", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 6 x + 4$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} + 6 x + 4$ = $2 \\left(x + 1\\right) \\left(x + 2\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-1$.\n5) From the second factor: x = $-2$.\nFinal answer: $x \\in \\{-1, -2\\}$", "final_answer": "x \\in \\{-1, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8d49b5c4049fef56", "problem_id": "b38bd3c2dc4e2fbb", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 6 x + 4$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} + 6 x + 4$ = $2 \\left(x + 1\\right) \\left(x + 2\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $-1$.\nFrom the second factor: x = $-2$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-1, -2\\}$", "final_answer": "x \\in \\{-1, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3bf7e8f633a4a0f9", "problem_id": "697933b225d97805", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the multiplicative inverse of 34 modulo 43.", "output": "1) We want x such that 34x ≡ 1 (mod 43).\n2) Use extended Euclid to find s,t with 34s + 43t = 1.\n3) One Bezout identity is: 34·(19) + 43·(-15) = 1.\n4) Reduce s modulo 43: x ≡ 19 (mod 43).\nFinal answer: $x\\equiv 19\\pmod{43}$", "final_answer": "x\\equiv 19\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8bdc46d49e3b394a", "problem_id": "232140c6d24e0324", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} x^{2} - 6 x + 22$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to x^{2} - 7 x + 22.\n3) Then the limit is 66.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: 62.\nFinal answer: $62$", "final_answer": "62", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3a14d261808a98a0", "problem_id": "5757ebbadcc704fc", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$6 x - 4 y$ = $6$\n$- 2 x + y$ = $-4$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $- 12 x + 8 y$ = $-12$.\n3) Multiply the second equation by $6$: $- 12 x + 6 y$ = $-24$.\n4) Subtract to eliminate x: $2 y$ = $12$.\n5) Solve for y: y = $6$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, 6\\right)$", "final_answer": "(x,y)=\\left(5, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4acdf88068098d69", "problem_id": "5757ebbadcc704fc", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$6 x - 4 y$ = $6$\n$- 2 x + y$ = $-4$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $- 12 x + 8 y$ = $-12$.\nMultiply the second equation by $6$: $- 12 x + 6 y$ = $-24$.\nSubtract to eliminate x: $2 y$ = $12$.\nSolve for y: y = $6$.\nSubstitute back to find x: x = $5$.\nCheck: plug (x,y)=(5,6) into both equations.\nFinal answer: $(x,y)=\\left(5, 6\\right)$", "final_answer": "(x,y)=\\left(5, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "18000be51f8c0916", "problem_id": "c43e57bfc44c829c", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} + 2 \\sin{\\left(x \\right)} + 1$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $- 6 x^{2} + 2 \\cos{\\left(x \\right)}$.\n2) Slope at x=-2: m = f'(-2) = $-24 + 2 \\cos{\\left(2 \\right)}$.\n3) Point on curve: (-2, f(-2)) = (-2, $17 - 2 \\sin{\\left(2 \\right)}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 17 - 2 \\sin{\\left(2 \\right)} = -24 + 2 \\cos{\\left(2 \\right)}(x − -2).\n6) Optionally expand: y = $- 24 x + 2 x \\cos{\\left(2 \\right)} - 31 - 2 \\sin{\\left(2 \\right)} + 4 \\cos{\\left(2 \\right)}$.\nFinal answer: $y = - 24\\cdot x + 2\\cdot x\\cdot \\cos{\\left(2 \\right)} - 31 - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)}$", "final_answer": "y = - 24\\cdot x + 2\\cdot x\\cdot \\cos{\\left(2 \\right)} - 31 - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ed7cdfc56268eaca", "problem_id": "c43e57bfc44c829c", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Find the equation of the tangent line to $y=- 2 x^{3} + 2 \\sin{\\left(x \\right)} + 1$ at $x=-2$.", "output": "$y = - 24\\cdot x + 2\\cdot x\\cdot \\cos{\\left(2 \\right)} - 31 - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)}$", "final_answer": "y = - 24\\cdot x + 2\\cdot x\\cdot \\cos{\\left(2 \\right)} - 31 - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6183e616031eea85", "problem_id": "6072c8c79b93a8e2", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_8=31$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_8=23.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=23 and k=8.\n4) Compute C(30,7) = 2035800.\nFinal answer: $2035800$", "final_answer": "2035800", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "65a227fe7ba44ee0", "problem_id": "4154d331602d7bf5", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} 3 x^{2} + 4 x + 3$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{2} + 3 x + 3.\n3) Then the limit is 9.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: 10.\nFinal answer: $10$", "final_answer": "10", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "bb944ca7f4e83f66", "problem_id": "9540a7ce2de5a3e2", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A cyclist rides at 60 miles per hour for 1 hours, then at 75 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 60 × 1 = 60 miles.\nSecond leg distance = 75 × 2 = 150 miles.\nTotal distance = 60 + 150 = 210 miles.\nCheck: adding both legs gives 210.\nFinal answer: $210$", "final_answer": "210", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2df97408e415566e", "problem_id": "9540a7ce2de5a3e2", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A cyclist rides at 60 miles per hour for 1 hours, then at 75 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 60 × 1 = 60 miles.\n2) Second leg distance = 75 × 2 = 150 miles.\n3) Total distance = 60 + 150 = 210 miles.\nFinal answer: $210$", "final_answer": "210", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b392af791ead82f9", "problem_id": "8bf51cbdef03ea4c", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Two friends split $320 in the ratio 2:6. How much does each receive?", "output": "1) Total ratio parts = 2+6 = 8.\n2) Each part = 320 ÷ 8 = 40.\n3) First share = 2 × 40 = 80.\n4) Second share = 6 × 40 = 240.\nFinal answer: $240$", "final_answer": "240", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "75761607d58f9de4", "problem_id": "8bf51cbdef03ea4c", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $320 in the ratio 2:6. How much does each receive?", "output": "Total ratio parts = 2+6 = 8.\nEach part = 320 ÷ 8 = 40.\nFirst share = 2 × 40 = 80.\nSecond share = 6 × 40 = 240.\nCheck: 80+240=320.\nFinal answer: $240$", "final_answer": "240", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c31c6a67e580f603", "problem_id": "4f51c89a2acbf2f5", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 16 red balls and 25 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(41,10).\n3) Favorable samples: choose k reds and n−k blues: C(16,2)·C(25,8).\n4) Compute totals: C(41,10) = 1121099408, and favorable = 129789000.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{77625}{670514}$", "final_answer": "\\frac{77625}{670514}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8af714874d4d3567", "problem_id": "e8ea7e17163d28ac", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(3 x^{3} + 2 x^{2} + 2 x - 3\\right) \\cos{\\left(3 x^{2} + x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 3 \\cos{\\left(3 x^{2} + x \\right)}) = 3 \\sin{\\left(x \\left(3 x + 1\\right) \\right)}$.\n3) So $f'(x) = - 18 x^{4} \\sin{\\left(3 x^{2} + x \\right)} - 15 x^{3} \\sin{\\left(3 x^{2} + x \\right)} - 14 x^{2} \\sin{\\left(3 x^{2} + x \\right)} + 9 x^{2} \\cos{\\left(3 x^{2} + x \\right)} + 2 x e^{x} - 2 x \\sin{\\left(3 x^{2} + x \\right)} + 4 x \\cos{\\left(3 x^{2} + x \\right)} + 4 e^{x} + 3 \\sin{\\left(3 x^{2} + x \\right)} + 2 \\cos{\\left(3 x^{2} + x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 3 \\cos{\\left(3 x^{2} + x \\right)}) = 3 \\left(6 x + 1\\right) \\sin{\\left(3 x^{2} + x \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x + 1\\right) e^{x} - \\left(6 x + 1\\right) \\left(x \\left(x \\left(3 x + 2\\right) + 2\\right) - 3\\right) \\sin{\\left(3 x^{2} + x \\right)} + \\left(9 x^{2} + 4 x + 2\\right) \\cos{\\left(3 x^{2} + x \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(x + 1\\right) e^{x} - \\left(6 x + 1\\right) \\left(x \\left(x \\left(3 x + 2\\right) + 2\\right) - 3\\right) \\sin{\\left(3 x^{2} + x \\right)} + \\left(9 x^{2} + 4 x + 2\\right) \\cos{\\left(3 x^{2} + x \\right)} + 2 e^{x}$", "final_answer": "2 \\left(x + 1\\right) e^{x} - \\left(6 x + 1\\right) \\left(x \\left(x \\left(3 x + 2\\right) + 2\\right) - 3\\right) \\sin{\\left(3 x^{2} + x \\right)} + \\left(9 x^{2} + 4 x + 2\\right) \\cos{\\left(3 x^{2} + x \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "1011861de1cc06aa", "problem_id": "9f1ce89ed674d897", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$x - 2 y$ = $4$\n$4 x + 3 y$ = $-17$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $4 x - 8 y$ = $16$.\nMultiply the second equation by $1$: $4 x + 3 y$ = $-17$.\nSubtract to eliminate x: $- 11 y$ = $33$.\nSolve for y: y = $-3$.\nSubstitute back to find x: x = $-2$.\nCheck: plug (x,y)=(-2,-3) into both equations.\nFinal answer: $(x,y)=\\left(-2, -3\\right)$", "final_answer": "(x,y)=\\left(-2, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d7be54f5b64afeb1", "problem_id": "9f1ce89ed674d897", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$x - 2 y$ = $4$\n$4 x + 3 y$ = $-17$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $4 x - 8 y$ = $16$.\n3) Multiply the second equation by $1$: $4 x + 3 y$ = $-17$.\n4) Subtract to eliminate x: $- 11 y$ = $33$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, -3\\right)$", "final_answer": "(x,y)=\\left(-2, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0413c4fedcd3d0a2", "problem_id": "5196868997c8b2a9", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{3} - x^{2} - 4$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-4) = 0.\n3) d/dx(- x^{2}) = - 2 x.\n4) d/dx(- 3 x^{3}) = - 9 x^{2}.\n5) Combine: f'(x) = x \\left(- 9 x - 2\\right).\nFinal answer: $x\\cdot \\left(- 9\\cdot x - 2\\right)$", "final_answer": "x\\cdot \\left(- 9\\cdot x - 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d419b854cad056a0", "problem_id": "5196868997c8b2a9", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{3} - x^{2} - 4$.", "output": "Differentiate term-by-term:\nd/dx(-4) = 0.\nd/dx(- x^{2}) = - 2 x.\nd/dx(- 3 x^{3}) = - 9 x^{2}.\nCombine: f'(x) = x \\left(- 9 x - 2\\right).\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $x\\cdot \\left(- 9\\cdot x - 2\\right)$", "final_answer": "x\\cdot \\left(- 9\\cdot x - 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "73c19b5ef006901c", "problem_id": "be5712ae74c7a89d", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Worker A finishes a job in 8 hours and Worker B finishes it in 20 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/8 job/hour. B's rate = 1/20 job/hour.\n2) Combined rate = 1/8 + 1/20 = 7/40 job/hour.\n3) Time = 1 ÷ (combined rate) = 40/7 hours.\nFinal answer: $\\frac{40}{7}$", "final_answer": "\\frac{40}{7}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "381fa7c41856e369", "problem_id": "be5712ae74c7a89d", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Worker A finishes a job in 8 hours and Worker B finishes it in 20 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/8 job/hour. B's rate = 1/20 job/hour.\nCombined rate = 1/8 + 1/20 = 7/40 job/hour.\nTime = 1 ÷ (combined rate) = 40/7 hours.\nCheck: (combined rate)×(time) = 7/40×40/7 = 1.\nFinal answer: $\\frac{40}{7}$", "final_answer": "\\frac{40}{7}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "067894dd692eca45", "problem_id": "e97e385bd07df771", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many binary strings of length 17 contain exactly 7 ones with no two ones adjacent?", "output": "1) Reserve 6 mandatory zeros between the 7 ones.\n2) This leaves n−k+1 = 11 valid slots for the ones.\n3) Choose the slots: C(11,7) = 330.\nFinal answer: $330$", "final_answer": "330", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "763620f93299c718", "problem_id": "e97e385bd07df771", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "How many binary strings of length 17 contain exactly 7 ones with no two ones adjacent?", "output": "Reserve 6 mandatory zeros between the 7 ones.\nThis leaves n−k+1 = 11 valid slots for the ones.\nChoose the slots: C(11,7) = 330.\nFinal answer: $330$", "final_answer": "330", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "78f562358a990c44", "problem_id": "b045a3e215d17ab1", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A cyclist rides at 46 miles per hour for 1 hours, then at 66 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 46 × 1 = 46 miles.\nSecond leg distance = 66 × 2 = 132 miles.\nTotal distance = 46 + 132 = 178 miles.\nFinal answer: $178$", "final_answer": "178", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cb43249b603bccd4", "problem_id": "b045a3e215d17ab1", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A cyclist rides at 46 miles per hour for 1 hours, then at 66 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 46 × 1 = 46 miles.\n2) Second leg distance = 66 × 2 = 132 miles.\n3) Total distance = 46 + 132 = 178 miles.\nFinal answer: $178$", "final_answer": "178", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7cd22f61b4deb6ac", "problem_id": "57e16f5e1cfb858d", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $-4 + 6 \\pi + 3 \\pi^{2} + 9 \\pi^{3}$.\n4) Simplify the result.\nFinal answer: $-4 + 6\\cdot \\pi + 3\\cdot \\pi^{2} + 9\\cdot \\pi^{3}$", "final_answer": "-4 + 6\\cdot \\pi + 3\\cdot \\pi^{2} + 9\\cdot \\pi^{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4dab0cfb02ccbb6d", "problem_id": "86ad5e3e1f9c98d9", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - x^{2} + 5 x - 10$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} + 4 x - 10.\n3) Then the limit is -22.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -24.\nFinal answer: $-24$", "final_answer": "-24", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a898a0bbd40fe1e3", "problem_id": "06b245128810688c", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the multiplicative inverse of 26 modulo 41.", "output": "We want x such that 26x ≡ 1 (mod 41).\nUse extended Euclid to find s,t with 26s + 41t = 1.\nOne Bezout identity is: 26·(-11) + 41·(7) = 1.\nReduce s modulo 41: x ≡ 30 (mod 41).\nCheck: 26·30 mod 41 = 1.\nFinal answer: $x\\equiv 30\\pmod{41}$", "final_answer": "x\\equiv 30\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "991a6f2cb315998c", "problem_id": "06b245128810688c", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the multiplicative inverse of 26 modulo 41.", "output": "1) We want x such that 26x ≡ 1 (mod 41).\n2) Use extended Euclid to find s,t with 26s + 41t = 1.\n3) One Bezout identity is: 26·(-11) + 41·(7) = 1.\n4) Reduce s modulo 41: x ≡ 30 (mod 41).\nFinal answer: $x\\equiv 30\\pmod{41}$", "final_answer": "x\\equiv 30\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f3c226626891ab5a", "problem_id": "954dc9b1336df2a1", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(2 x^{2} - 1\\right) \\cos{\\left(3 x^{2} - 3 x \\right)}$.", "output": "1) Let f(x) = $\\left(x + 2\\right) e^{x} + \\left(2 x^{2} - 1\\right) \\cos{\\left(3 x^{2} - 3 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $4 x \\cos{\\left(3 x^{2} - 3 x \\right)} + \\left(x + 2\\right) e^{x} - 3 \\left(2 x - 1\\right) \\left(2 x^{2} - 1\\right) \\sin{\\left(3 x^{2} - 3 x \\right)} + e^{x}$.\nFinal answer: $4\\cdot x\\cdot \\cos{\\left(3\\cdot x^{2} - 3\\cdot x \\right)} + \\left(x + 2\\right)\\cdot e^{x} - \\left(6\\cdot x - 3\\right)\\cdot \\left(2\\cdot x^{2} - 1\\right)\\cdot \\sin{\\left(3\\cdot x^{2} - 3\\cdot x \\right)} + e^{x}$", "final_answer": "4\\cdot x\\cdot \\cos{\\left(3\\cdot x^{2} - 3\\cdot x \\right)} + \\left(x + 2\\right)\\cdot e^{x} - \\left(6\\cdot x - 3\\right)\\cdot \\left(2\\cdot x^{2} - 1\\right)\\cdot \\sin{\\left(3\\cdot x^{2} - 3\\cdot x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f3e3d69d56c39e43", "problem_id": "954dc9b1336df2a1", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(2 x^{2} - 1\\right) \\cos{\\left(3 x^{2} - 3 x \\right)}$.", "output": "Let f(x) = $\\left(x + 2\\right) e^{x} + \\left(2 x^{2} - 1\\right) \\cos{\\left(3 x^{2} - 3 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $4 x \\cos{\\left(3 x^{2} - 3 x \\right)} + \\left(x + 2\\right) e^{x} - 3 \\left(2 x - 1\\right) \\left(2 x^{2} - 1\\right) \\sin{\\left(3 x^{2} - 3 x \\right)} + e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $4\\cdot x\\cdot \\cos{\\left(3\\cdot x^{2} - 3\\cdot x \\right)} + \\left(x + 2\\right)\\cdot e^{x} - \\left(6\\cdot x - 3\\right)\\cdot \\left(2\\cdot x^{2} - 1\\right)\\cdot \\sin{\\left(3\\cdot x^{2} - 3\\cdot x \\right)} + e^{x}$", "final_answer": "4\\cdot x\\cdot \\cos{\\left(3\\cdot x^{2} - 3\\cdot x \\right)} + \\left(x + 2\\right)\\cdot e^{x} - \\left(6\\cdot x - 3\\right)\\cdot \\left(2\\cdot x^{2} - 1\\right)\\cdot \\sin{\\left(3\\cdot x^{2} - 3\\cdot x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f66fa11497d31c54", "problem_id": "16c6d08ead909859", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=32$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=32 and k=7.\n4) Compute C(38,6) = 2760681.\nFinal answer: $2760681$", "final_answer": "2760681", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b5093b13d91a4f38", "problem_id": "16c6d08ead909859", "style": "answer_only", "instruction": "Final answer only.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=32$?", "output": "$2760681$", "final_answer": "2760681", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9123b2805a819e52", "problem_id": "410a3847b5d2e0ba", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3$.\n3) Compute an antiderivative: F(x) = $x^{3} + \\frac{3 x^{2}}{2} - 3 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + x^{3} + \\frac{3\\cdot x^{2}}{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + x^{3} + \\frac{3\\cdot x^{2}}{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c190eb3646f17a3b", "problem_id": "410a3847b5d2e0ba", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3$.\nCompute an antiderivative: F(x) = $x^{3} + \\frac{3 x^{2}}{2} - 3 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $C + x^{3} + \\frac{3\\cdot x^{2}}{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + x^{3} + \\frac{3\\cdot x^{2}}{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d40586f0b370b530", "problem_id": "62fa662b5bc6ff93", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 22 x + 56$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 22 x + 56$ = $2 \\left(x - 7\\right) \\left(x - 4\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $7$.\n5) From the second factor: x = $4$.\nFinal answer: $x \\in \\{7, 4\\}$", "final_answer": "x \\in \\{7, 4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7c72ac8055f2e2db", "problem_id": "62fa662b5bc6ff93", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 22 x + 56$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} - 22 x + 56$ = $2 \\left(x - 7\\right) \\left(x - 4\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $7$.\nFrom the second factor: x = $4$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{7, 4\\}$", "final_answer": "x \\in \\{7, 4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "28e063d88d91eabd", "problem_id": "1c639cdb1ee993a4", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 x - \\frac{\\sqrt{11} x}{2} + \\frac{\\sqrt{11}}{2} + 2$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$-3 - \\frac{\\sqrt{11}}{2}$, c=$\\frac{\\sqrt{11}}{2} + 2$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\sqrt{11} + \\frac{15}{4}$, so √Δ = $\\sqrt{\\sqrt{11} + \\frac{15}{4}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{11}}{4} + \\frac{\\sqrt{4 \\sqrt{11} + 15}}{4} + \\frac{3}{2}$.\nx2 = $- \\frac{\\sqrt{4 \\sqrt{11} + 15}}{4} + \\frac{\\sqrt{11}}{4} + \\frac{3}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{11}}{4} + \\frac{\\sqrt{4\\cdot \\sqrt{11} + 15}}{4} + \\frac{3}{2}, - \\frac{\\sqrt{4\\cdot \\sqrt{11} + 15}}{4} + \\frac{\\sqrt{11}}{4} + \\frac{3}{2}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{11}}{4} + \\frac{\\sqrt{4\\cdot \\sqrt{11} + 15}}{4} + \\frac{3}{2}, - \\frac{\\sqrt{4\\cdot \\sqrt{11} + 15}}{4} + \\frac{\\sqrt{11}}{4} + \\frac{3}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b72544273b21af6", "problem_id": "1c639cdb1ee993a4", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 x - \\frac{\\sqrt{11} x}{2} + \\frac{\\sqrt{11}}{2} + 2$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$-3 - \\frac{\\sqrt{11}}{2}$, c=$\\frac{\\sqrt{11}}{2} + 2$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\sqrt{11} + \\frac{15}{4}$, so √Δ = $\\sqrt{\\sqrt{11} + \\frac{15}{4}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{11}}{4} + \\frac{\\sqrt{4 \\sqrt{11} + 15}}{4} + \\frac{3}{2}$.\n7) x2 = $- \\frac{\\sqrt{4 \\sqrt{11} + 15}}{4} + \\frac{\\sqrt{11}}{4} + \\frac{3}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{11}}{4} + \\frac{\\sqrt{4\\cdot \\sqrt{11} + 15}}{4} + \\frac{3}{2}, - \\frac{\\sqrt{4\\cdot \\sqrt{11} + 15}}{4} + \\frac{\\sqrt{11}}{4} + \\frac{3}{2}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{11}}{4} + \\frac{\\sqrt{4\\cdot \\sqrt{11} + 15}}{4} + \\frac{3}{2}, - \\frac{\\sqrt{4\\cdot \\sqrt{11} + 15}}{4} + \\frac{\\sqrt{11}}{4} + \\frac{3}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ccfa7e3932ba2d46", "problem_id": "97e8884caa76d227", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Two friends split $1095 in the ratio 6:9. How much does each receive?", "output": "1) Total ratio parts = 6+9 = 15.\n2) Each part = 1095 ÷ 15 = 73.\n3) First share = 6 × 73 = 438.\n4) Second share = 9 × 73 = 657.\nFinal answer: $657$", "final_answer": "657", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6b225b46b6f05a19", "problem_id": "97e8884caa76d227", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Two friends split $1095 in the ratio 6:9. How much does each receive?", "output": "Total ratio parts = 6+9 = 15.\nEach part = 1095 ÷ 15 = 73.\nFirst share = 6 × 73 = 438.\nSecond share = 9 × 73 = 657.\nCheck: 438+657=1095.\nFinal answer: $657$", "final_answer": "657", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b7a9878fe54f1003", "problem_id": "29d3a0238175edbd", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- x + 3 y$ = $-12$\n$3 x + 2 y$ = $-19$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $- 3 x + 9 y$ = $-36$.\nMultiply the second equation by $-1$: $- 3 x - 2 y$ = $19$.\nSubtract to eliminate x: $11 y$ = $-55$.\nSolve for y: y = $-5$.\nSubstitute back to find x: x = $-3$.\nCheck: plug (x,y)=(-3,-5) into both equations.\nFinal answer: $(x,y)=\\left(-3, -5\\right)$", "final_answer": "(x,y)=\\left(-3, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a79a412d50088f3", "problem_id": "29d3a0238175edbd", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the linear system:\n$- x + 3 y$ = $-12$\n$3 x + 2 y$ = $-19$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $- 3 x + 9 y$ = $-36$.\nMultiply the second equation by $-1$: $- 3 x - 2 y$ = $19$.\nFinal answer: $(x,y)=\\left(-3, -5\\right)$", "final_answer": "(x,y)=\\left(-3, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "df946330a981b015", "problem_id": "902e43c969ef47bf", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - \\frac{3 \\sqrt{5} x}{2} - 2 x + \\sqrt{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$- \\frac{3 \\sqrt{5}}{2} - 2$, c=$\\sqrt{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{61}{4} - 6 \\sqrt{5}$, so √Δ = $\\sqrt{\\frac{61}{4} - 6 \\sqrt{5}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{61 - 24 \\sqrt{5}}}{12} + \\frac{1}{3} + \\frac{\\sqrt{5}}{4}$.\nx2 = $- \\frac{\\sqrt{61 - 24 \\sqrt{5}}}{12} + \\frac{1}{3} + \\frac{\\sqrt{5}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{61 - 24\\cdot \\sqrt{5}}}{12} + \\frac{1}{3} + \\frac{\\sqrt{5}}{4}, - \\frac{\\sqrt{61 - 24\\cdot \\sqrt{5}}}{12} + \\frac{1}{3} + \\frac{\\sqrt{5}}{4}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{61 - 24\\cdot \\sqrt{5}}}{12} + \\frac{1}{3} + \\frac{\\sqrt{5}}{4}, - \\frac{\\sqrt{61 - 24\\cdot \\sqrt{5}}}{12} + \\frac{1}{3} + \\frac{\\sqrt{5}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e83eb686ab6a911", "problem_id": "902e43c969ef47bf", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - \\frac{3 \\sqrt{5} x}{2} - 2 x + \\sqrt{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$- \\frac{3 \\sqrt{5}}{2} - 2$, c=$\\sqrt{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{61}{4} - 6 \\sqrt{5}$, so √Δ = $\\sqrt{\\frac{61}{4} - 6 \\sqrt{5}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{61 - 24 \\sqrt{5}}}{12} + \\frac{1}{3} + \\frac{\\sqrt{5}}{4}$.\n7) x2 = $- \\frac{\\sqrt{61 - 24 \\sqrt{5}}}{12} + \\frac{1}{3} + \\frac{\\sqrt{5}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{61 - 24\\cdot \\sqrt{5}}}{12} + \\frac{1}{3} + \\frac{\\sqrt{5}}{4}, - \\frac{\\sqrt{61 - 24\\cdot \\sqrt{5}}}{12} + \\frac{1}{3} + \\frac{\\sqrt{5}}{4}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{61 - 24\\cdot \\sqrt{5}}}{12} + \\frac{1}{3} + \\frac{\\sqrt{5}}{4}, - \\frac{\\sqrt{61 - 24\\cdot \\sqrt{5}}}{12} + \\frac{1}{3} + \\frac{\\sqrt{5}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "37dc75267d886dcb", "problem_id": "9f7302499d1f48f0", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the multiplicative inverse of 4 modulo 23.", "output": "1) We want x such that 4x ≡ 1 (mod 23).\n2) Use extended Euclid to find s,t with 4s + 23t = 1.\n3) One Bezout identity is: 4·(6) + 23·(-1) = 1.\n4) Reduce s modulo 23: x ≡ 6 (mod 23).\nFinal answer: $x\\equiv 6\\pmod{23}$", "final_answer": "x\\equiv 6\\pmod{23}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bd3b237e9b63a64d", "problem_id": "9f7302499d1f48f0", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Find the multiplicative inverse of 4 modulo 23.", "output": "We want x such that 4x ≡ 1 (mod 23).\nUse extended Euclid to find s,t with 4s + 23t = 1.\nOne Bezout identity is: 4·(6) + 23·(-1) = 1.\nFinal answer: $x\\equiv 6\\pmod{23}$", "final_answer": "x\\equiv 6\\pmod{23}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "480c3d03b0354c19", "problem_id": "1beb525a5a1105c4", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{11} x}{2} - \\frac{31 x}{10} + \\frac{12}{5} + \\frac{12 \\sqrt{11}}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{3 \\sqrt{11}}{2} - \\frac{31}{10}$, c=$\\frac{12}{5} + \\frac{12 \\sqrt{11}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{619}{25} - \\frac{3 \\sqrt{11}}{10}$, so √Δ = $\\sqrt{\\frac{619}{25} - \\frac{3 \\sqrt{11}}{10}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{31}{20} + \\frac{\\sqrt{2476 - 30 \\sqrt{11}}}{20} + \\frac{3 \\sqrt{11}}{4}$.\n7) x2 = $- \\frac{\\sqrt{2476 - 30 \\sqrt{11}}}{20} + \\frac{31}{20} + \\frac{3 \\sqrt{11}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{31}{20} + \\frac{\\sqrt{2476 - 30\\cdot \\sqrt{11}}}{20} + \\frac{3\\cdot \\sqrt{11}}{4}, - \\frac{\\sqrt{2476 - 30\\cdot \\sqrt{11}}}{20} + \\frac{31}{20} + \\frac{3\\cdot \\sqrt{11}}{4}\\}$", "final_answer": "x \\in \\{\\frac{31}{20} + \\frac{\\sqrt{2476 - 30\\cdot \\sqrt{11}}}{20} + \\frac{3\\cdot \\sqrt{11}}{4}, - \\frac{\\sqrt{2476 - 30\\cdot \\sqrt{11}}}{20} + \\frac{31}{20} + \\frac{3\\cdot \\sqrt{11}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "22c44e0ec1869452", "problem_id": "eb3dc200753b87c5", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 2 x - 2$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to x - 2.\n3) Then the limit is -3.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: -4.\nFinal answer: $-4$", "final_answer": "-4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "71178f081541711d", "problem_id": "3385dd446c2f6b74", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A store sells a backpack for $350. It is discounted by 10%. You buy 10 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 350 × (1 − 10/100) = 350 × 90/100 = 315.\n2) Subtotal = 10 × 315 = 3150.\n3) Total with tax = 3150 × 108/100 = 3402.\nFinal answer: $3402$", "final_answer": "3402", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "70ba9a22b621ee74", "problem_id": "53e2d97f7f5d58dd", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{4} + x^{3} - 2 x^{2} + 4 x + 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x^{3}) = 3 x^{3}$.\n3) So $f'(x) = 15 x^{3} - 4 x + 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x^{3}) = 3 x^{2}$.\nTherefore the correct derivative is $f'(x) = 12 x^{3} + 3 x^{2} - 4 x + 4$.\nFinal answer: $12 x^{3} + 3 x^{2} - 4 x + 4$", "final_answer": "12 x^{3} + 3 x^{2} - 4 x + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8ded20180a544dae", "problem_id": "35d62a9e24760d0a", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Two friends split $98 in the ratio 1:1. How much does each receive?", "output": "1) Total ratio parts = 1+1 = 2.\n2) Each part = 98 ÷ 2 = 49.\n3) First share = 1 × 49 = 49.\n4) Second share = 1 × 49 = 49.\nFinal answer: $49$", "final_answer": "49", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "98ae8481027da175", "problem_id": "35d62a9e24760d0a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $98 in the ratio 1:1. How much does each receive?", "output": "Total ratio parts = 1+1 = 2.\nEach part = 98 ÷ 2 = 49.\nFirst share = 1 × 49 = 49.\nSecond share = 1 × 49 = 49.\nCheck: 49+49=98.\nFinal answer: $49$", "final_answer": "49", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e6f996d21390f394", "problem_id": "7aefcd4063734339", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A disease has prevalence $\\frac{2}{25}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{21}{125}$.\nCompute P(D|+) = $\\frac{19}{42}$.\nFinal answer: $\\frac{19}{42}$", "final_answer": "\\frac{19}{42}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bfd9db45ad109a66", "problem_id": "7aefcd4063734339", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A disease has prevalence $\\frac{2}{25}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{21}{125}$.\nCompute P(D|+) = $\\frac{19}{42}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{19}{42}$", "final_answer": "\\frac{19}{42}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ad988676f419efbe", "problem_id": "241dbe78254603c5", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡2 (mod 4), x≡4 (mod 5). Give the solution modulo 20.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 20.\n3) Compute the CRT solution to get x ≡ 14 (mod 20).\nFinal answer: $x\\equiv 14\\pmod{20}$", "final_answer": "x\\equiv 14\\pmod{20}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "16024ebd6edbfae8", "problem_id": "6f207e9759e1c11d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{3} - \\frac{3 x}{2} + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}) = \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} - \\frac{2 x^{4}}{2 x^{4} - 12 x^{2} + 18} + \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 6 x^{2} + 9} - \\frac{6 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} - \\frac{12 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} + \\frac{21 x^{2}}{2 x^{4} - 12 x^{2} + 18} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 6} - \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3} - \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3} + \\frac{9 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} + \\frac{18 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} + \\frac{9}{2 x^{4} - 12 x^{2} + 18}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}) = - \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - 3\\right)^{2}} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x^{2} - 3\\right)} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3} + \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 2 x \\left(- 2 x^{3} - 3 x + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 3\\right)}{2 \\left(x^{2} - 3\\right)^{2}}$.\nFinal answer: $\\frac{- 2 x \\left(- 2 x^{3} - 3 x + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 3\\right)}{2 \\left(x^{2} - 3\\right)^{2}}$", "final_answer": "\\frac{- 2 x \\left(- 2 x^{3} - 3 x + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 3\\right)}{2 \\left(x^{2} - 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "337392f15eb4373b", "problem_id": "8a6217f9111d785d", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 - 3 x$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 3 x^{2} - 3 x + 2$ and a = -2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 6 x - 3$.\n5) Evaluate at a: p'(-2) = $9$.\nFinal answer: $9$", "final_answer": "9", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0eaa0f9517aac17c", "problem_id": "8a6217f9111d785d", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 - 3 x$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 3 x^{2} - 3 x + 2$ and a = -2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 6 x - 3$.\nEvaluate at a: p'(-2) = $9$.\nCheck: the identity gives $9$.\nFinal answer: $9$", "final_answer": "9", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "afc4b62bcd23bdea", "problem_id": "be0c6034c8b6bdd3", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{3} - 4 x^{2} + 2 x$.", "output": "1) Differentiate term-by-term:\n2) d/dx(- 4 x^{2}) = - 8 x.\n3) d/dx(- 3 x^{3}) = - 9 x^{2}.\n4) d/dx(2 x) = 2.\n5) Combine: f'(x) = - 9 x^{2} - 8 x + 2.\nFinal answer: $- 9\\cdot x^{2} - 8\\cdot x + 2$", "final_answer": "- 9\\cdot x^{2} - 8\\cdot x + 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "31da4703e3883af6", "problem_id": "be0c6034c8b6bdd3", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{3} - 4 x^{2} + 2 x$.", "output": "Differentiate term-by-term:\nd/dx(- 4 x^{2}) = - 8 x.\nd/dx(- 3 x^{3}) = - 9 x^{2}.\nd/dx(2 x) = 2.\nCombine: f'(x) = - 9 x^{2} - 8 x + 2.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 9\\cdot x^{2} - 8\\cdot x + 2$", "final_answer": "- 9\\cdot x^{2} - 8\\cdot x + 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1287676fc7b1befa", "problem_id": "6a546f7e07ed8210", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 11 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=11, k=3, p=$\\frac{3}{7}$ and simplify.\nHere C(11,3) = 165.\nCheck: expression simplifies to $\\frac{291962880}{1977326743}$.\nFinal answer: $\\frac{291962880}{1977326743}$", "final_answer": "\\frac{291962880}{1977326743}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8925efeef454020a", "problem_id": "6a546f7e07ed8210", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 11 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=11, k=3, p=$\\frac{3}{7}$ and simplify.\nFinal answer: $\\frac{291962880}{1977326743}$", "final_answer": "\\frac{291962880}{1977326743}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c8a7f50d3e45710e", "problem_id": "5f196b2ba1af6b9c", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{2} - 4$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $6 x$.\n2) Slope at x=-1: m = f'(-1) = $-6$.\n3) Point on curve: (-1, f(-1)) = (-1, $-1$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -1 = -6(x − -1).\n6) Optionally expand: y = $- 6 x - 7$.\nFinal answer: $y = - 6\\cdot x - 7$", "final_answer": "y = - 6\\cdot x - 7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "faccf687ad4737ba", "problem_id": "4515d91ae660cecc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - 2 x^{2} + 3 x - 8$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{2} + 2 x - 8.\n3) Then the limit is -20.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -22.\nFinal answer: $-22$", "final_answer": "-22", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "79285d80b317573e", "problem_id": "e9ea8694d5d5651d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - x^{3} + 2 x^{2} - 3 x + 4$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to - x^{3} + 2 x^{2} - 4 x + 4.\n3) Then the limit is 28.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 26.\nFinal answer: $26$", "final_answer": "26", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "59f8f997164f8f5a", "problem_id": "3a014ecf3128aee6", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$2 x - y$ = $-5$\n$x - y$ = $-1$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $2 x - y$ = $-5$.\n3) Multiply the second equation by $2$: $2 x - 2 y$ = $-2$.\n4) Subtract to eliminate x: $y$ = $-3$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $-4$.\nFinal answer: $(x,y)=\\left(-4, -3\\right)$", "final_answer": "(x,y)=\\left(-4, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f385fac60c54afbd", "problem_id": "02e8074d46070dff", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 3 x^{2} + 6 x + 14$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{3} + 2 x + 3$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $9 x^{2} + 2$.\n5) Evaluate at a: p'(2) = $38$.\nFinal answer: $38$", "final_answer": "38", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "90a547cb27597939", "problem_id": "d263b8f0be913e6d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - x - 4$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to - 2 x - 4.\n3) Then the limit is -10.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: -7.\nFinal answer: $-7$", "final_answer": "-7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "91afb0eb08c7446d", "problem_id": "45b75b2dbf39bc74", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} - 3 x^{4} + x^{3} - 2 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{4} + x^{3} - 2 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $\\frac{3 \\pi \\left(- 132 \\pi^{4} - 40 \\pi^{2} - 60 + 10 \\pi + 25 \\pi^{3}\\right)}{20}$.\n4) Simplify the result.\nFinal answer: $\\frac{3\\cdot \\pi\\cdot \\left(- 132\\cdot \\pi^{4} - 40\\cdot \\pi^{2} - 60 + 10\\cdot \\pi + 25\\cdot \\pi^{3}\\right)}{20}$", "final_answer": "\\frac{3\\cdot \\pi\\cdot \\left(- 132\\cdot \\pi^{4} - 40\\cdot \\pi^{2} - 60 + 10\\cdot \\pi + 25\\cdot \\pi^{3}\\right)}{20}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aa4ae075c7594e55", "problem_id": "45b75b2dbf39bc74", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} - 3 x^{4} + x^{3} - 2 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{4} + x^{3} - 2 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $\\frac{3 \\pi \\left(- 132 \\pi^{4} - 40 \\pi^{2} - 60 + 10 \\pi + 25 \\pi^{3}\\right)}{20}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{3\\cdot \\pi\\cdot \\left(- 132\\cdot \\pi^{4} - 40\\cdot \\pi^{2} - 60 + 10\\cdot \\pi + 25\\cdot \\pi^{3}\\right)}{20}$", "final_answer": "\\frac{3\\cdot \\pi\\cdot \\left(- 132\\cdot \\pi^{4} - 40\\cdot \\pi^{2} - 60 + 10\\cdot \\pi + 25\\cdot \\pi^{3}\\right)}{20}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1ba65d86d7772422", "problem_id": "0daf297be28226da", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 1$.\nCompute an antiderivative: F(x) = $\\frac{2 x^{3}}{3} + x^{2} + x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{2\\cdot x^{3}}{3} + x^{2} + x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{2\\cdot x^{3}}{3} + x^{2} + x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "385b44a8fed6f2a2", "problem_id": "0daf297be28226da", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 1$.\n3) Compute an antiderivative: F(x) = $\\frac{2 x^{3}}{3} + x^{2} + x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{2\\cdot x^{3}}{3} + x^{2} + x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{2\\cdot x^{3}}{3} + x^{2} + x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a1deeaa4e0ebb232", "problem_id": "b1ac84a4e4933a08", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 22 red balls and 19 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(41,8).\n3) Favorable samples: choose k reds and n−k blues: C(22,2)·C(19,6).\n4) Compute totals: C(41,8) = 95548245, and favorable = 6267492.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{6468}{98605}$", "final_answer": "\\frac{6468}{98605}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "af75e3ef5c2b88f6", "problem_id": "755a7e23efb2939a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} - x^{2} + 4 x - 16$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} + 3 x - 16.\n3) Then the limit is -44.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: -48.\nFinal answer: $-48$", "final_answer": "-48", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "394b0459907a25e1", "problem_id": "a9a355563f7f5f54", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡3 (mod 8), x≡1 (mod 7). Give the solution modulo 56.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 56.\nCompute the CRT solution to get x ≡ 43 (mod 56).\nCheck: 43 mod 8=3 and mod 7=1.\nFinal answer: $x\\equiv 43\\pmod{56}$", "final_answer": "x\\equiv 43\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "88a0fb6b195d7156", "problem_id": "a9a355563f7f5f54", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the congruences: x≡3 (mod 8), x≡1 (mod 7). Give the solution modulo 56.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 56.\n3) Compute the CRT solution to get x ≡ 43 (mod 56).\nFinal answer: $x\\equiv 43\\pmod{56}$", "final_answer": "x\\equiv 43\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "801555aa9bdabd92", "problem_id": "34640ab62238606d", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 2 x - 5$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{2} - x + 1$ and a = -2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $4 x - 1$.\nEvaluate at a: p'(-2) = $-9$.\nCheck: the identity gives $-9$.\nFinal answer: $-9$", "final_answer": "-9", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a783029716582c39", "problem_id": "34640ab62238606d", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 2 x - 5$.", "output": "$-9$", "final_answer": "-9", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e8b1b312457a77f0", "problem_id": "cf612f077c3a8aca", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "How many binary strings of length 24 contain exactly 8 ones with no two ones adjacent?", "output": "1) Reserve 7 mandatory zeros between the 8 ones.\n2) This leaves n−k+1 = 17 valid slots for the ones.\n3) Choose the slots: C(17,8) = 24310.\nFinal answer: $24310$", "final_answer": "24310", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "12e8d070741c8d1d", "problem_id": "7dd28420924931e1", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{3} + \\frac{2 x^{2}}{3} + \\frac{x}{3} + \\left(x^{2} + x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - \\frac{1}{3}}{x^{2} + x + 1}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + x\\right) + 1}) = \\frac{e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + x + 1}$.\n3) So $f'(x) = \\frac{3 x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{6 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{6 x^{4}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{6 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{12 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{12 x^{3}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{9 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{18 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{20 x^{2}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{12 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{12 x}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 1} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{4}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 1} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 2 x + 2} + \\frac{e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + x + 1} - \\frac{e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} + x\\right) + 1}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + x\\right) + 1}) = \\frac{\\left(- 2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(x^{2} + x\\right) + 1\\right)^{2}} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(x^{2} + x\\right) + 1\\right)} + \\frac{e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} + x\\right) + 1}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(- 2 x - 1\\right) \\left(3 x^{3} + 2 x^{2} + x + 3 \\left(x \\left(x + 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) + \\left(x \\left(x + 1\\right) + 1\\right) \\left(18 x^{2} + 8 x + 6 \\left(2 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3 \\left(x \\left(x + 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x \\left(x + 1\\right) + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{6 \\left(x \\left(x + 1\\right) + 1\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(- 2 x - 1\\right) \\left(3 x^{3} + 2 x^{2} + x + 3 \\left(x \\left(x + 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) + \\left(x \\left(x + 1\\right) + 1\\right) \\left(18 x^{2} + 8 x + 6 \\left(2 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3 \\left(x \\left(x + 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x \\left(x + 1\\right) + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{6 \\left(x \\left(x + 1\\right) + 1\\right)^{2}}$", "final_answer": "\\frac{2 \\left(- 2 x - 1\\right) \\left(3 x^{3} + 2 x^{2} + x + 3 \\left(x \\left(x + 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) + \\left(x \\left(x + 1\\right) + 1\\right) \\left(18 x^{2} + 8 x + 6 \\left(2 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3 \\left(x \\left(x + 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x \\left(x + 1\\right) + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{6 \\left(x \\left(x + 1\\right) + 1\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e9c47110800403b1", "problem_id": "dbf5f4a345f1fcb3", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the congruences: x≡5 (mod 9), x≡2 (mod 7). Give the solution modulo 63.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 63.\n3) Compute the CRT solution to get x ≡ 23 (mod 63).\nFinal answer: $x\\equiv 23\\pmod{63}$", "final_answer": "x\\equiv 23\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1534d360873ae42e", "problem_id": "dbf5f4a345f1fcb3", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡5 (mod 9), x≡2 (mod 7). Give the solution modulo 63.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 63.\nCompute the CRT solution to get x ≡ 23 (mod 63).\nCheck: 23 mod 9=5 and mod 7=2.\nFinal answer: $x\\equiv 23\\pmod{63}$", "final_answer": "x\\equiv 23\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "223183071977ecce", "problem_id": "3af3ebdec26fc6e3", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} - 2 x^{2} - x + 2 \\sin{\\left(x \\right)} + 2$ at $x=2$.", "output": "1) Differentiate: f'(x) = $9 x^{2} - 4 x + 2 \\cos{\\left(x \\right)} - 1$.\n2) Slope at x=2: m = f'(2) = $2 \\cos{\\left(2 \\right)} + 27$.\n3) Point on curve: (2, f(2)) = (2, $2 \\sin{\\left(2 \\right)} + 16$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 2 \\sin{\\left(2 \\right)} + 16 = 2 \\cos{\\left(2 \\right)} + 27(x − 2).\n6) Optionally expand: y = $2 x \\cos{\\left(2 \\right)} + 27 x - 38 - 4 \\cos{\\left(2 \\right)} + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 27\\cdot x - 38 - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 27\\cdot x - 38 - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7b95effff38bacc3", "problem_id": "3af3ebdec26fc6e3", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} - 2 x^{2} - x + 2 \\sin{\\left(x \\right)} + 2$ at $x=2$.", "output": "Differentiate: f'(x) = $9 x^{2} - 4 x + 2 \\cos{\\left(x \\right)} - 1$.\nSlope at x=2: m = f'(2) = $2 \\cos{\\left(2 \\right)} + 27$.\nPoint on curve: (2, f(2)) = (2, $2 \\sin{\\left(2 \\right)} + 16$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 2 \\sin{\\left(2 \\right)} + 16 = 2 \\cos{\\left(2 \\right)} + 27(x − 2).\nOptionally expand: y = $2 x \\cos{\\left(2 \\right)} + 27 x - 38 - 4 \\cos{\\left(2 \\right)} + 2 \\sin{\\left(2 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 27\\cdot x - 38 - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 27\\cdot x - 38 - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "04d82a16b7be5360", "problem_id": "49882216d0f286e6", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - x^{3} - 7 x^{2} - 25 x - 99$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to - x^{3} - 7 x^{2} - 26 x - 99.\n3) Then the limit is -379.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: -375.\nFinal answer: $-375$", "final_answer": "-375", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7bf9d8a540921cab", "problem_id": "58398c98c0b5e362", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_8=22$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_8=6.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=6 and k=8.\n4) Compute C(13,7) = 1716.\nFinal answer: $1716$", "final_answer": "1716", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "594bc008e6c55154", "problem_id": "31cb178dee94c128", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A disease has prevalence $\\frac{1}{50}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{237}{5000}$.\n3) Compute P(D|+) = $\\frac{30}{79}$.\nFinal answer: $\\frac{30}{79}$", "final_answer": "\\frac{30}{79}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "11d4902441bc4317", "problem_id": "31cb178dee94c128", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A disease has prevalence $\\frac{1}{50}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{237}{5000}$.\nCompute P(D|+) = $\\frac{30}{79}$.\nFinal answer: $\\frac{30}{79}$", "final_answer": "\\frac{30}{79}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "710cabfd2b751c12", "problem_id": "cfaf8bc399f86785", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "You want 28 liters of a 30% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 28−x liters is 70% solution.\n2) Salt equation: 25x + 70(28−x) = 30·28.\n3) (25-70)x = (30-70)·28 ⇒ x = 224/9.\n4) Then 28−x = 28/9.\nFinal answer: $224/9 L of 25% and 28/9 L of 70%$", "final_answer": "224/9 L of 25% and 28/9 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d68a959fd3c9d57f", "problem_id": "d3b067b3b14a6674", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{3} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x \\sin{\\left(x \\right)}) = 2 x \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x \\sin{\\left(x \\right)}) = 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.\nFinal answer: $- 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$", "final_answer": "- 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "68819932bdd079d1", "problem_id": "ece3e50526976743", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A store sells a backpack for $385. It is discounted by 5%. You buy 5 backpacks. What is the total cost?", "output": "1) Discounted price = 385 × (1 − 5/100) = 385 × 95/100 = 1463/4.\n2) Subtotal = 5 × 1463/4 = 7315/4.\nFinal answer: $\\frac{7315}{4}$", "final_answer": "\\frac{7315}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8f8dbd140d583823", "problem_id": "ece3e50526976743", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $385. It is discounted by 5%. You buy 5 backpacks. What is the total cost?", "output": "Discounted price = 385 × (1 − 5/100) = 385 × 95/100 = 1463/4.\nSubtotal = 5 × 1463/4 = 7315/4.\nCheck: recomputing the formula yields 7315/4.\nFinal answer: $\\frac{7315}{4}$", "final_answer": "\\frac{7315}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6f23f9bf0bac27c9", "problem_id": "ff08698d81a168ec", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A disease has prevalence $\\frac{2}{25}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{249}{2500}$.\n3) Compute P(D|+) = $\\frac{60}{83}$.\nFinal answer: $\\frac{60}{83}$", "final_answer": "\\frac{60}{83}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "37042a4ced05b2a5", "problem_id": "ff08698d81a168ec", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A disease has prevalence $\\frac{2}{25}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{249}{2500}$.\nCompute P(D|+) = $\\frac{60}{83}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{60}{83}$", "final_answer": "\\frac{60}{83}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "75f4393f40528cf1", "problem_id": "29f77c986c021347", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A store sells a backpack for $346. It is discounted by 10%. You buy 11 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 346 × (1 − 10/100) = 346 × 90/100 = 1557/5.\n2) Subtotal = 11 × 1557/5 = 17127/5.\n3) Total with tax = 17127/5 × 110/100 = 188397/50.\nFinal answer: $\\frac{188397}{50}$", "final_answer": "\\frac{188397}{50}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b674be46c8b30bb5", "problem_id": "29f77c986c021347", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $346. It is discounted by 10%. You buy 11 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 346 × (1 − 10/100) = 346 × 90/100 = 1557/5.\nSubtotal = 11 × 1557/5 = 17127/5.\nTotal with tax = 17127/5 × 110/100 = 188397/50.\nCheck: recomputing the formula yields 188397/50.\nFinal answer: $\\frac{188397}{50}$", "final_answer": "\\frac{188397}{50}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f4bbff99915d228d", "problem_id": "d49dd7bb7a5e1f53", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{3} + 2 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $3 x^{3} + 2 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1$.\n3) Compute an antiderivative: F(x) = $\\frac{3 x^{4}}{4} + \\frac{2 x^{3}}{3} - \\frac{3 x^{2}}{2} - x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{3\\cdot x^{4}}{4} + \\frac{2\\cdot x^{3}}{3} - \\frac{3\\cdot x^{2}}{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{3\\cdot x^{4}}{4} + \\frac{2\\cdot x^{3}}{3} - \\frac{3\\cdot x^{2}}{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5aa9a10b5ace72d4", "problem_id": "d49dd7bb7a5e1f53", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{3} + 2 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $3 x^{3} + 2 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1$.\nCompute an antiderivative: F(x) = $\\frac{3 x^{4}}{4} + \\frac{2 x^{3}}{3} - \\frac{3 x^{2}}{2} - x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{3\\cdot x^{4}}{4} + \\frac{2\\cdot x^{3}}{3} - \\frac{3\\cdot x^{2}}{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{3\\cdot x^{4}}{4} + \\frac{2\\cdot x^{3}}{3} - \\frac{3\\cdot x^{2}}{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "70c32ec9c702aee1", "problem_id": "0ed08109ffb9e88e", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_8=32$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=32 and k=8.\nFinal answer: $15380937$", "final_answer": "15380937", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c64bc8284fcb1cb7", "problem_id": "0ed08109ffb9e88e", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_8=32$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=32 and k=8.\n4) Compute C(39,7) = 15380937.\nFinal answer: $15380937$", "final_answer": "15380937", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0202b31078ccb421", "problem_id": "9f2f9f5d1ea6a989", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_5=32$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_5=22.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=22 and k=5.\n4) Compute C(26,4) = 14950.\nFinal answer: $14950$", "final_answer": "14950", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bcc38152e46460bf", "problem_id": "9f2f9f5d1ea6a989", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_5=32$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_5=22.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=22 and k=5.\nCompute C(26,4) = 14950.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $14950$", "final_answer": "14950", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f4b7288a579632b4", "problem_id": "b4b5cd14b1cacc29", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A store sells a backpack for $86. It is discounted by 40%. You buy 11 backpacks. What is the total cost?", "output": "1) Discounted price = 86 × (1 − 40/100) = 86 × 60/100 = 258/5.\n2) Subtotal = 11 × 258/5 = 2838/5.\nFinal answer: $\\frac{2838}{5}$", "final_answer": "\\frac{2838}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8b1c9f5cbb54a7dc", "problem_id": "b4b5cd14b1cacc29", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $86. It is discounted by 40%. You buy 11 backpacks. What is the total cost?", "output": "Discounted price = 86 × (1 − 40/100) = 86 × 60/100 = 258/5.\nSubtotal = 11 × 258/5 = 2838/5.\nCheck: recomputing the formula yields 2838/5.\nFinal answer: $\\frac{2838}{5}$", "final_answer": "\\frac{2838}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6737d81185f7d450", "problem_id": "4dfc6435ee7b876b", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A store sells a backpack for $760. It is discounted by 40%. You buy 8 backpacks. What is the total cost?", "output": "1) Discounted price = 760 × (1 − 40/100) = 760 × 60/100 = 456.\n2) Subtotal = 8 × 456 = 3648.\nFinal answer: $3648$", "final_answer": "3648", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "52e5b6d4149a1a82", "problem_id": "06e4bc2eebfcb779", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} 2 x^{3} + x^{2} + 4 x + 1$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} + x^{2} + 3 x + 1.\n3) Then the limit is 7.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: 8.\nFinal answer: $8$", "final_answer": "8", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6438feeb268ee5dd", "problem_id": "fcd98b1fc86df073", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A disease has prevalence $\\frac{1}{50}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{23}{200}$.\n3) Compute P(D|+) = $\\frac{17}{115}$.\nFinal answer: $\\frac{17}{115}$", "final_answer": "\\frac{17}{115}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3522236eb3061661", "problem_id": "fcd98b1fc86df073", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A disease has prevalence $\\frac{1}{50}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{23}{200}$.\nCompute P(D|+) = $\\frac{17}{115}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{17}{115}$", "final_answer": "\\frac{17}{115}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b62adf4e3d43228", "problem_id": "6ee57c14c77c055e", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 14 times, what is the probability of exactly 10 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=14, k=10, p=$\\frac{2}{3}$ and simplify.\n4) Here C(14,10) = 1001.\nFinal answer: $\\frac{1025024}{4782969}$", "final_answer": "\\frac{1025024}{4782969}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f645041053b45861", "problem_id": "6ee57c14c77c055e", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 14 times, what is the probability of exactly 10 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=14, k=10, p=$\\frac{2}{3}$ and simplify.\nHere C(14,10) = 1001.\nCheck: expression simplifies to $\\frac{1025024}{4782969}$.\nFinal answer: $\\frac{1025024}{4782969}$", "final_answer": "\\frac{1025024}{4782969}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6cf88037c17133f0", "problem_id": "a083d73e3e7e3aa8", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡0 (mod 3), x≡6 (mod 8). Give the solution modulo 24.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 24.\n3) Compute the CRT solution to get x ≡ 6 (mod 24).\nFinal answer: $x\\equiv 6\\pmod{24}$", "final_answer": "x\\equiv 6\\pmod{24}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d7e377c586da120e", "problem_id": "a083d73e3e7e3aa8", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡0 (mod 3), x≡6 (mod 8). Give the solution modulo 24.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 24.\nCompute the CRT solution to get x ≡ 6 (mod 24).\nCheck: 6 mod 3=0 and mod 8=6.\nFinal answer: $x\\equiv 6\\pmod{24}$", "final_answer": "x\\equiv 6\\pmod{24}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fe522e92bab1906d", "problem_id": "18d23f44c9702b43", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{13} x - \\frac{13 x}{3} + \\frac{14}{3} + 7 \\sqrt{13}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- 3 \\sqrt{13} - \\frac{13}{3}$, c=$\\frac{14}{3} + 7 \\sqrt{13}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{1054}{9} - 2 \\sqrt{13}$, so √Δ = $\\sqrt{\\frac{1054}{9} - 2 \\sqrt{13}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{13}{6} + \\frac{\\sqrt{1054 - 18 \\sqrt{13}}}{6} + \\frac{3 \\sqrt{13}}{2}$.\n7) x2 = $- \\frac{\\sqrt{1054 - 18 \\sqrt{13}}}{6} + \\frac{13}{6} + \\frac{3 \\sqrt{13}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{13}{6} + \\frac{\\sqrt{1054 - 18\\cdot \\sqrt{13}}}{6} + \\frac{3\\cdot \\sqrt{13}}{2}, - \\frac{\\sqrt{1054 - 18\\cdot \\sqrt{13}}}{6} + \\frac{13}{6} + \\frac{3\\cdot \\sqrt{13}}{2}\\}$", "final_answer": "x \\in \\{\\frac{13}{6} + \\frac{\\sqrt{1054 - 18\\cdot \\sqrt{13}}}{6} + \\frac{3\\cdot \\sqrt{13}}{2}, - \\frac{\\sqrt{1054 - 18\\cdot \\sqrt{13}}}{6} + \\frac{13}{6} + \\frac{3\\cdot \\sqrt{13}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "81ea160de97f8e2a", "problem_id": "18d23f44c9702b43", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{13} x - \\frac{13 x}{3} + \\frac{14}{3} + 7 \\sqrt{13}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- 3 \\sqrt{13} - \\frac{13}{3}$, c=$\\frac{14}{3} + 7 \\sqrt{13}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{1054}{9} - 2 \\sqrt{13}$, so √Δ = $\\sqrt{\\frac{1054}{9} - 2 \\sqrt{13}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{13}{6} + \\frac{\\sqrt{1054 - 18 \\sqrt{13}}}{6} + \\frac{3 \\sqrt{13}}{2}$.\nx2 = $- \\frac{\\sqrt{1054 - 18 \\sqrt{13}}}{6} + \\frac{13}{6} + \\frac{3 \\sqrt{13}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{13}{6} + \\frac{\\sqrt{1054 - 18\\cdot \\sqrt{13}}}{6} + \\frac{3\\cdot \\sqrt{13}}{2}, - \\frac{\\sqrt{1054 - 18\\cdot \\sqrt{13}}}{6} + \\frac{13}{6} + \\frac{3\\cdot \\sqrt{13}}{2}\\}$", "final_answer": "x \\in \\{\\frac{13}{6} + \\frac{\\sqrt{1054 - 18\\cdot \\sqrt{13}}}{6} + \\frac{3\\cdot \\sqrt{13}}{2}, - \\frac{\\sqrt{1054 - 18\\cdot \\sqrt{13}}}{6} + \\frac{13}{6} + \\frac{3\\cdot \\sqrt{13}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d4c030c6ec5f1080", "problem_id": "1566d4dd306ca2b8", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} - 2 x^{4} - 4 x^{3} - 3 x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{4} - 4 x^{3} - 3 x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nEvaluate at the bounds: F(2 \\pi) and F(0).\nSubtract: F(2 \\pi) − F(0) = $\\pi^{2} \\left(- \\frac{64 \\pi^{3}}{5} - 16 \\pi^{2} - 8 \\pi - 2\\right)$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\pi^{2}\\cdot \\left(- \\frac{64\\cdot \\pi^{3}}{5} - 16\\cdot \\pi^{2} - 8\\cdot \\pi - 2\\right)$", "final_answer": "\\pi^{2}\\cdot \\left(- \\frac{64\\cdot \\pi^{3}}{5} - 16\\cdot \\pi^{2} - 8\\cdot \\pi - 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fd2b188d9eacfcf7", "problem_id": "1566d4dd306ca2b8", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} - 2 x^{4} - 4 x^{3} - 3 x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{4} - 4 x^{3} - 3 x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nEvaluate at the bounds: F(2 \\pi) and F(0).\nSubtract: F(2 \\pi) − F(0) = $\\pi^{2} \\left(- \\frac{64 \\pi^{3}}{5} - 16 \\pi^{2} - 8 \\pi - 2\\right)$.\nFinal answer: $\\pi^{2}\\cdot \\left(- \\frac{64\\cdot \\pi^{3}}{5} - 16\\cdot \\pi^{2} - 8\\cdot \\pi - 2\\right)$", "final_answer": "\\pi^{2}\\cdot \\left(- \\frac{64\\cdot \\pi^{3}}{5} - 16\\cdot \\pi^{2} - 8\\cdot \\pi - 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "17c505e4d28b6625", "problem_id": "422366d0410bc6ff", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A disease has prevalence $\\frac{1}{25}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{81}{1250}$.\n3) Compute P(D|+) = $\\frac{5}{9}$.\nFinal answer: $\\frac{5}{9}$", "final_answer": "\\frac{5}{9}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "add1fbfbaf1a708a", "problem_id": "422366d0410bc6ff", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A disease has prevalence $\\frac{1}{25}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{81}{1250}$.\nCompute P(D|+) = $\\frac{5}{9}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{5}{9}$", "final_answer": "\\frac{5}{9}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4f9946f59d7edc5a", "problem_id": "91a0322f11df7cd8", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 29 liters of a 25% salt solution. You have a 15% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 29−x liters is 60% solution.\nSalt equation: 15x + 60(29−x) = 25·29.\n(15-60)x = (25-60)·29 ⇒ x = 203/9.\nThen 29−x = 58/9.\nCheck: 15·203/9 + 60·58/9 = 25·29.\nFinal answer: $203/9 L of 15% and 58/9 L of 60%$", "final_answer": "203/9 L of 15% and 58/9 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a0a303ce935fda71", "problem_id": "91a0322f11df7cd8", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "You want 29 liters of a 25% salt solution. You have a 15% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 29−x liters is 60% solution.\n2) Salt equation: 15x + 60(29−x) = 25·29.\n3) (15-60)x = (25-60)·29 ⇒ x = 203/9.\n4) Then 29−x = 58/9.\nFinal answer: $203/9 L of 15% and 58/9 L of 60%$", "final_answer": "203/9 L of 15% and 58/9 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dc733074eb0bc443", "problem_id": "472fc953af585e42", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 29 red balls and 24 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(53,7).\n3) Favorable samples: choose k reds and n−k blues: C(29,2)·C(24,5).\n4) Compute totals: C(53,7) = 154143080, and favorable = 17256624.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{308154}{2752555}$", "final_answer": "\\frac{308154}{2752555}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5099365b32d7be53", "problem_id": "472fc953af585e42", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "An urn has 29 red balls and 24 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "$\\frac{308154}{2752555}$", "final_answer": "\\frac{308154}{2752555}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d3d30654634756dc", "problem_id": "192c30ef456c4825", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 x \\sin{\\left(x \\right)}) = 3 x \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 x \\sin{\\left(x \\right)}) = 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$", "final_answer": "- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "032a009460c1e168", "problem_id": "b3651c2e5fb8b985", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} - 2 x^{4} + 4 x^{3} + x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{4} + 4 x^{3} + x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $\\frac{3 \\pi \\left(- 22 \\pi^{4} - 20 - 5 \\pi + 5 \\pi^{2} + 25 \\pi^{3}\\right)}{5}$.\n4) Simplify the result.\nFinal answer: $\\frac{3\\cdot \\pi\\cdot \\left(- 22\\cdot \\pi^{4} - 20 - 5\\cdot \\pi + 5\\cdot \\pi^{2} + 25\\cdot \\pi^{3}\\right)}{5}$", "final_answer": "\\frac{3\\cdot \\pi\\cdot \\left(- 22\\cdot \\pi^{4} - 20 - 5\\cdot \\pi + 5\\cdot \\pi^{2} + 25\\cdot \\pi^{3}\\right)}{5}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "519157fb8e1a9eb7", "problem_id": "f064fab28afe76fd", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 10 times, what is the probability of exactly 7 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=7, p=$\\frac{1}{5}$ and simplify.\nFinal answer: $\\frac{1536}{1953125}$", "final_answer": "\\frac{1536}{1953125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "583f7a48fe583f53", "problem_id": "f064fab28afe76fd", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 10 times, what is the probability of exactly 7 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=7, p=$\\frac{1}{5}$ and simplify.\nHere C(10,7) = 120.\nCheck: expression simplifies to $\\frac{1536}{1953125}$.\nFinal answer: $\\frac{1536}{1953125}$", "final_answer": "\\frac{1536}{1953125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "18ceed059ead5e4d", "problem_id": "01912f0232eb4552", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Two friends split $1806 in the ratio 8:6. How much does each receive?", "output": "1) Total ratio parts = 8+6 = 14.\n2) Each part = 1806 ÷ 14 = 129.\n3) First share = 8 × 129 = 1032.\n4) Second share = 6 × 129 = 774.\nFinal answer: $774$", "final_answer": "774", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "18de64f60eba5306", "problem_id": "01912f0232eb4552", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Two friends split $1806 in the ratio 8:6. How much does each receive?", "output": "Total ratio parts = 8+6 = 14.\nEach part = 1806 ÷ 14 = 129.\nFirst share = 8 × 129 = 1032.\nSecond share = 6 × 129 = 774.\nCheck: 1032+774=1806.\nFinal answer: $774$", "final_answer": "774", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "89c2c52e83dbae1b", "problem_id": "036da99082bba48c", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 2 \\sqrt{7} x + 5 x - 4 \\sqrt{7} + 6$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$5 - 2 \\sqrt{7}$, c=$6 - 4 \\sqrt{7}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $29 - 4 \\sqrt{7}$, so √Δ = $\\sqrt{29 - 4 \\sqrt{7}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{5}{2} + \\frac{\\sqrt{29 - 4 \\sqrt{7}}}{2} + \\sqrt{7}$.\n7) x2 = $- \\frac{5}{2} - \\frac{\\sqrt{29 - 4 \\sqrt{7}}}{2} + \\sqrt{7}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{5}{2} + \\frac{\\sqrt{29 - 4\\cdot \\sqrt{7}}}{2} + \\sqrt{7}, - \\frac{5}{2} - \\frac{\\sqrt{29 - 4\\cdot \\sqrt{7}}}{2} + \\sqrt{7}\\}$", "final_answer": "x \\in \\{- \\frac{5}{2} + \\frac{\\sqrt{29 - 4\\cdot \\sqrt{7}}}{2} + \\sqrt{7}, - \\frac{5}{2} - \\frac{\\sqrt{29 - 4\\cdot \\sqrt{7}}}{2} + \\sqrt{7}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dc12392f703ffd0d", "problem_id": "036da99082bba48c", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 2 \\sqrt{7} x + 5 x - 4 \\sqrt{7} + 6$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$5 - 2 \\sqrt{7}$, c=$6 - 4 \\sqrt{7}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $29 - 4 \\sqrt{7}$, so √Δ = $\\sqrt{29 - 4 \\sqrt{7}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{5}{2} + \\frac{\\sqrt{29 - 4 \\sqrt{7}}}{2} + \\sqrt{7}$.\nx2 = $- \\frac{5}{2} - \\frac{\\sqrt{29 - 4 \\sqrt{7}}}{2} + \\sqrt{7}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{5}{2} + \\frac{\\sqrt{29 - 4\\cdot \\sqrt{7}}}{2} + \\sqrt{7}, - \\frac{5}{2} - \\frac{\\sqrt{29 - 4\\cdot \\sqrt{7}}}{2} + \\sqrt{7}\\}$", "final_answer": "x \\in \\{- \\frac{5}{2} + \\frac{\\sqrt{29 - 4\\cdot \\sqrt{7}}}{2} + \\sqrt{7}, - \\frac{5}{2} - \\frac{\\sqrt{29 - 4\\cdot \\sqrt{7}}}{2} + \\sqrt{7}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c3024c8340da4336", "problem_id": "1dcb4282d988f7af", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_5=37$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_5=27.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=27 and k=5.\n4) Compute C(31,4) = 31465.\nFinal answer: $31465$", "final_answer": "31465", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e420a57fa2ba2f5c", "problem_id": "1dcb4282d988f7af", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_5=37$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_5=27.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=27 and k=5.\nFinal answer: $31465$", "final_answer": "31465", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4e20a9003b56bae9", "problem_id": "3ed6935ae0b69247", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{5} x}{2} + \\frac{31 x}{10} - \\frac{3 \\sqrt{5}}{10} + \\frac{3}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{31}{10} - \\frac{\\sqrt{5}}{2}$, c=$\\frac{3}{2} - \\frac{3 \\sqrt{5}}{10}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{243}{50} - \\frac{19 \\sqrt{5}}{10}$, so √Δ = $\\sqrt{\\frac{243}{50} - \\frac{19 \\sqrt{5}}{10}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{31}{20} + \\frac{\\sqrt{486 - 190 \\sqrt{5}}}{20} + \\frac{\\sqrt{5}}{4}$.\n7) x2 = $- \\frac{31}{20} - \\frac{\\sqrt{486 - 190 \\sqrt{5}}}{20} + \\frac{\\sqrt{5}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{31}{20} + \\frac{\\sqrt{486 - 190\\cdot \\sqrt{5}}}{20} + \\frac{\\sqrt{5}}{4}, - \\frac{31}{20} - \\frac{\\sqrt{486 - 190\\cdot \\sqrt{5}}}{20} + \\frac{\\sqrt{5}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{31}{20} + \\frac{\\sqrt{486 - 190\\cdot \\sqrt{5}}}{20} + \\frac{\\sqrt{5}}{4}, - \\frac{31}{20} - \\frac{\\sqrt{486 - 190\\cdot \\sqrt{5}}}{20} + \\frac{\\sqrt{5}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6f799777e52b6f4f", "problem_id": "3ed6935ae0b69247", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{5} x}{2} + \\frac{31 x}{10} - \\frac{3 \\sqrt{5}}{10} + \\frac{3}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{31}{10} - \\frac{\\sqrt{5}}{2}$, c=$\\frac{3}{2} - \\frac{3 \\sqrt{5}}{10}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{243}{50} - \\frac{19 \\sqrt{5}}{10}$, so √Δ = $\\sqrt{\\frac{243}{50} - \\frac{19 \\sqrt{5}}{10}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{31}{20} + \\frac{\\sqrt{486 - 190 \\sqrt{5}}}{20} + \\frac{\\sqrt{5}}{4}$.\nx2 = $- \\frac{31}{20} - \\frac{\\sqrt{486 - 190 \\sqrt{5}}}{20} + \\frac{\\sqrt{5}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{31}{20} + \\frac{\\sqrt{486 - 190\\cdot \\sqrt{5}}}{20} + \\frac{\\sqrt{5}}{4}, - \\frac{31}{20} - \\frac{\\sqrt{486 - 190\\cdot \\sqrt{5}}}{20} + \\frac{\\sqrt{5}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{31}{20} + \\frac{\\sqrt{486 - 190\\cdot \\sqrt{5}}}{20} + \\frac{\\sqrt{5}}{4}, - \\frac{31}{20} - \\frac{\\sqrt{486 - 190\\cdot \\sqrt{5}}}{20} + \\frac{\\sqrt{5}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "700ecbc7f9d90e17", "problem_id": "6f3a478956ce613a", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} \\sin{\\left(x \\right)} + x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $x^{3} \\sin{\\left(x \\right)} + x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $x^{3}\\cdot \\cos{\\left(x \\right)} + 3\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} + x^{2}\\cdot \\cos{\\left(x \\right)} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - 3\\cdot \\cos{\\left(x \\right)}$", "final_answer": "x^{3}\\cdot \\cos{\\left(x \\right)} + 3\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} + x^{2}\\cdot \\cos{\\left(x \\right)} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - 3\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72138a9dd05b6e1c", "problem_id": "399ece830e72a8be", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - 2 x^{3} - 5 x^{2} - 21 x - 83$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{3} - 5 x^{2} - 22 x - 83.\n3) Then the limit is -379.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: -375.\nFinal answer: $-375$", "final_answer": "-375", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "db29c0bd17ce0c52", "problem_id": "1445ecd6c8255908", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "You want 21 liters of a 37% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 21−x liters is 50% solution.\n2) Salt equation: 25x + 50(21−x) = 37·21.\n3) (25-50)x = (37-50)·21 ⇒ x = 273/25.\n4) Then 21−x = 252/25.\nFinal answer: $273/25 L of 25% and 252/25 L of 50%$", "final_answer": "273/25 L of 25% and 252/25 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a97bfd773d98673b", "problem_id": "8ca33e415b9c53cf", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} + 3 x + 4$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x^{2}) = - 4 x^{2}$.\n3) So $f'(x) = 3 - 4 x^{2}$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x^{2}) = - 4 x$.\nTherefore the correct derivative is $f'(x) = 3 - 4 x$.\nFinal answer: $3 - 4 x$", "final_answer": "3 - 4 x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c2370fcae01eb344", "problem_id": "c4deb458d76be316", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Two friends split $370 in the ratio 8:2. How much does each receive?", "output": "Total ratio parts = 8+2 = 10.\nEach part = 370 ÷ 10 = 37.\nFirst share = 8 × 37 = 296.\nFinal answer: $74$", "final_answer": "74", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "77dff6febf3816d6", "problem_id": "c4deb458d76be316", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $370 in the ratio 8:2. How much does each receive?", "output": "Total ratio parts = 8+2 = 10.\nEach part = 370 ÷ 10 = 37.\nFirst share = 8 × 37 = 296.\nSecond share = 2 × 37 = 74.\nCheck: 296+74=370.\nFinal answer: $74$", "final_answer": "74", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9794a8768a1cc342", "problem_id": "920802bac6bb9d12", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 4 x - 126$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 4 x - 126$ = $2 \\left(x - 9\\right) \\left(x + 7\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $9$.\n5) From the second factor: x = $-7$.\nFinal answer: $x \\in \\{9, -7\\}$", "final_answer": "x \\in \\{9, -7\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "61b922f8cb18927d", "problem_id": "920802bac6bb9d12", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 4 x - 126$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} - 4 x - 126$ = $2 \\left(x - 9\\right) \\left(x + 7\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $9$.\nFrom the second factor: x = $-7$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{9, -7\\}$", "final_answer": "x \\in \\{9, -7\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8c1934cd91f868d6", "problem_id": "b15b4812339e667d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} x^{3} + 2 x^{2} + 2 x + 4$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to x^{3} + 2 x^{2} + x + 4.\n3) Then the limit is 8.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: 9.\nFinal answer: $9$", "final_answer": "9", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "694386dff76a61da", "problem_id": "644d2c037cda35c9", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{11} x}{2} + \\frac{17 x}{10} - \\frac{3 \\sqrt{11}}{10} + \\frac{3}{10}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{17}{10} - \\frac{3 \\sqrt{11}}{2}$, c=$\\frac{3}{10} - \\frac{3 \\sqrt{11}}{10}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{661}{25} - \\frac{39 \\sqrt{11}}{10}$, so √Δ = $\\sqrt{\\frac{661}{25} - \\frac{39 \\sqrt{11}}{10}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{17}{20} + \\frac{\\sqrt{2644 - 390 \\sqrt{11}}}{20} + \\frac{3 \\sqrt{11}}{4}$.\n7) x2 = $- \\frac{\\sqrt{2644 - 390 \\sqrt{11}}}{20} - \\frac{17}{20} + \\frac{3 \\sqrt{11}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{17}{20} + \\frac{\\sqrt{2644 - 390\\cdot \\sqrt{11}}}{20} + \\frac{3\\cdot \\sqrt{11}}{4}, - \\frac{\\sqrt{2644 - 390\\cdot \\sqrt{11}}}{20} - \\frac{17}{20} + \\frac{3\\cdot \\sqrt{11}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{17}{20} + \\frac{\\sqrt{2644 - 390\\cdot \\sqrt{11}}}{20} + \\frac{3\\cdot \\sqrt{11}}{4}, - \\frac{\\sqrt{2644 - 390\\cdot \\sqrt{11}}}{20} - \\frac{17}{20} + \\frac{3\\cdot \\sqrt{11}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9df8bfa47c7cf53a", "problem_id": "4ba58947bd9241ad", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Find the multiplicative inverse of 9 modulo 23.", "output": "We want x such that 9x ≡ 1 (mod 23).\nUse extended Euclid to find s,t with 9s + 23t = 1.\nOne Bezout identity is: 9·(-5) + 23·(2) = 1.\nFinal answer: $x\\equiv 18\\pmod{23}$", "final_answer": "x\\equiv 18\\pmod{23}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "49188aeef6d6e5a4", "problem_id": "4ba58947bd9241ad", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Find the multiplicative inverse of 9 modulo 23.", "output": "1) We want x such that 9x ≡ 1 (mod 23).\n2) Use extended Euclid to find s,t with 9s + 23t = 1.\n3) One Bezout identity is: 9·(-5) + 23·(2) = 1.\n4) Reduce s modulo 23: x ≡ 18 (mod 23).\nFinal answer: $x\\equiv 18\\pmod{23}$", "final_answer": "x\\equiv 18\\pmod{23}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ba869f69ff38d116", "problem_id": "4adab0f19983cad9", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 19 times, what is the probability of exactly 8 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=19, k=8, p=$\\frac{3}{8}$ and simplify.\n4) Here C(19,8) = 75582.\nFinal answer: $\\frac{12106774951171875}{72057594037927936}$", "final_answer": "\\frac{12106774951171875}{72057594037927936}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "375bf103ea109401", "problem_id": "4adab0f19983cad9", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 19 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=19, k=8, p=$\\frac{3}{8}$ and simplify.\nFinal answer: $\\frac{12106774951171875}{72057594037927936}$", "final_answer": "\\frac{12106774951171875}{72057594037927936}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b8a94de81f66e058", "problem_id": "5ef90e5b03edde0a", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A disease has prevalence $\\frac{1}{25}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{67}{500}$.\nCompute P(D|+) = $\\frac{19}{67}$.\nFinal answer: $\\frac{19}{67}$", "final_answer": "\\frac{19}{67}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "89d0d685dd664d67", "problem_id": "5ef90e5b03edde0a", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A disease has prevalence $\\frac{1}{25}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{67}{500}$.\n3) Compute P(D|+) = $\\frac{19}{67}$.\nFinal answer: $\\frac{19}{67}$", "final_answer": "\\frac{19}{67}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "40737603e997b2f5", "problem_id": "152d9bd489336606", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 11 x + 30$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} + 11 x + 30$ = $\\left(x + 5\\right) \\left(x + 6\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-5$.\n5) From the second factor: x = $-6$.\nFinal answer: $x \\in \\{-5, -6\\}$", "final_answer": "x \\in \\{-5, -6\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "286ac71dd3696f43", "problem_id": "152d9bd489336606", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 11 x + 30$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} + 11 x + 30$ = $\\left(x + 5\\right) \\left(x + 6\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $-5$.\nFrom the second factor: x = $-6$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-5, -6\\}$", "final_answer": "x \\in \\{-5, -6\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c56a74ce36ed94fd", "problem_id": "e69cea3dec3cb6a7", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "How many binary strings of length 15 contain exactly 7 ones with no two ones adjacent?", "output": "1) Reserve 6 mandatory zeros between the 7 ones.\n2) This leaves n−k+1 = 9 valid slots for the ones.\n3) Choose the slots: C(9,7) = 36.\nFinal answer: $36$", "final_answer": "36", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9c1381dcac5a1e19", "problem_id": "8a6217f9111d785d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 - 3 x$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 3 - 4 x.\n3) Then the limit is 11.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 9.\nFinal answer: $9$", "final_answer": "9", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "07fdce62aecfa4a9", "problem_id": "fa8a8eff55596e05", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Two friends split $1530 in the ratio 9:6. How much does each receive?", "output": "1) Total ratio parts = 9+6 = 15.\n2) Each part = 1530 ÷ 15 = 102.\n3) First share = 9 × 102 = 918.\n4) Second share = 6 × 102 = 612.\nFinal answer: $612$", "final_answer": "612", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ece929a8cce986c", "problem_id": "fa8a8eff55596e05", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $1530 in the ratio 9:6. How much does each receive?", "output": "Total ratio parts = 9+6 = 15.\nEach part = 1530 ÷ 15 = 102.\nFirst share = 9 × 102 = 918.\nSecond share = 6 × 102 = 612.\nCheck: 918+612=1530.\nFinal answer: $612$", "final_answer": "612", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "67cc171cf9107e24", "problem_id": "01a1e790dc8229e1", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{7} x + \\frac{9 x}{4} - 7 + \\frac{21 \\sqrt{7}}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{9}{4} - 3 \\sqrt{7}$, c=$-7 + \\frac{21 \\sqrt{7}}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{1537}{16} - \\frac{69 \\sqrt{7}}{2}$, so √Δ = $\\sqrt{\\frac{1537}{16} - \\frac{69 \\sqrt{7}}{2}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{9}{8} + \\frac{\\sqrt{1537 - 552 \\sqrt{7}}}{8} + \\frac{3 \\sqrt{7}}{2}$.\n7) x2 = $- \\frac{9}{8} - \\frac{\\sqrt{1537 - 552 \\sqrt{7}}}{8} + \\frac{3 \\sqrt{7}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{9}{8} + \\frac{\\sqrt{1537 - 552\\cdot \\sqrt{7}}}{8} + \\frac{3\\cdot \\sqrt{7}}{2}, - \\frac{9}{8} - \\frac{\\sqrt{1537 - 552\\cdot \\sqrt{7}}}{8} + \\frac{3\\cdot \\sqrt{7}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{9}{8} + \\frac{\\sqrt{1537 - 552\\cdot \\sqrt{7}}}{8} + \\frac{3\\cdot \\sqrt{7}}{2}, - \\frac{9}{8} - \\frac{\\sqrt{1537 - 552\\cdot \\sqrt{7}}}{8} + \\frac{3\\cdot \\sqrt{7}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f6948e9d621af7ec", "problem_id": "01a1e790dc8229e1", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{7} x + \\frac{9 x}{4} - 7 + \\frac{21 \\sqrt{7}}{4}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{9}{4} - 3 \\sqrt{7}$, c=$-7 + \\frac{21 \\sqrt{7}}{4}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{- \\frac{9}{8} + \\frac{\\sqrt{1537 - 552\\cdot \\sqrt{7}}}{8} + \\frac{3\\cdot \\sqrt{7}}{2}, - \\frac{9}{8} - \\frac{\\sqrt{1537 - 552\\cdot \\sqrt{7}}}{8} + \\frac{3\\cdot \\sqrt{7}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{9}{8} + \\frac{\\sqrt{1537 - 552\\cdot \\sqrt{7}}}{8} + \\frac{3\\cdot \\sqrt{7}}{2}, - \\frac{9}{8} - \\frac{\\sqrt{1537 - 552\\cdot \\sqrt{7}}}{8} + \\frac{3\\cdot \\sqrt{7}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c558cb11e3b1bdd8", "problem_id": "ef9eda0afc0bfb09", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $114 in the ratio 2:4. How much does each receive?", "output": "Total ratio parts = 2+4 = 6.\nEach part = 114 ÷ 6 = 19.\nFirst share = 2 × 19 = 38.\nSecond share = 4 × 19 = 76.\nCheck: 38+76=114.\nFinal answer: $76$", "final_answer": "76", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c97f5472fa193438", "problem_id": "ef9eda0afc0bfb09", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Two friends split $114 in the ratio 2:4. How much does each receive?", "output": "1) Total ratio parts = 2+4 = 6.\n2) Each part = 114 ÷ 6 = 19.\n3) First share = 2 × 19 = 38.\n4) Second share = 4 × 19 = 76.\nFinal answer: $76$", "final_answer": "76", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fff8e10ad0c6f58c", "problem_id": "9a696011efc887bd", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{3} + x^{2} - x + 3$.", "output": "1) Differentiate term-by-term:\n2) d/dx(3) = 0.\n3) d/dx(x^{2}) = 2 x.\n4) d/dx(- x) = -1.\n5) d/dx(3 x^{3}) = 9 x^{2}.\n6) Combine: f'(x) = 9 x^{2} + 2 x - 1.\nFinal answer: $9\\cdot x^{2} + 2\\cdot x - 1$", "final_answer": "9\\cdot x^{2} + 2\\cdot x - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4bfe8e37cc773e6d", "problem_id": "6ba99aa955fab2e0", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} + 4 x$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(4 x) = 4 x$.\n3) So $f'(x) = - 6 x^{2} + 4 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(4 x) = 4$.\nTherefore the correct derivative is $f'(x) = 2 \\left(2 - 3 x^{2}\\right)$.\nFinal answer: $2 \\left(2 - 3 x^{2}\\right)$", "final_answer": "2 \\left(2 - 3 x^{2}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "947ac2ec44c30514", "problem_id": "c26225878303ef66", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 57 x + 270$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} + 57 x + 270$ = $3 \\left(x + 9\\right) \\left(x + 10\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $-9$.\nFrom the second factor: x = $-10$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-9, -10\\}$", "final_answer": "x \\in \\{-9, -10\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e53d0b97206cb1dc", "problem_id": "c26225878303ef66", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 57 x + 270$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} + 57 x + 270$ = $3 \\left(x + 9\\right) \\left(x + 10\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-9$.\n5) From the second factor: x = $-10$.\nFinal answer: $x \\in \\{-9, -10\\}$", "final_answer": "x \\in \\{-9, -10\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7e4c077fbbbb7b59", "problem_id": "bf7d28dedf807dab", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 45 x + 150$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} - 45 x + 150$ = $3 \\left(x - 10\\right) \\left(x - 5\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $10$.\n5) From the second factor: x = $5$.\nFinal answer: $x \\in \\{10, 5\\}$", "final_answer": "x \\in \\{10, 5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "046cce55d4b6557f", "problem_id": "bf7d28dedf807dab", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 45 x + 150$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} - 45 x + 150$ = $3 \\left(x - 10\\right) \\left(x - 5\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFinal answer: $x \\in \\{10, 5\\}$", "final_answer": "x \\in \\{10, 5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eb1ee59b94ed2cd0", "problem_id": "d74f5ee1be45ddb9", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 7 red balls and 15 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(22,8).\n3) Favorable samples: choose k reds and n−k blues: C(7,1)·C(15,7).\n4) Compute totals: C(22,8) = 319770, and favorable = 45045.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{91}{646}$", "final_answer": "\\frac{91}{646}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c0f5b38edf95c78d", "problem_id": "d74f5ee1be45ddb9", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "An urn has 7 red balls and 15 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(22,8).\nFavorable samples: choose k reds and n−k blues: C(7,1)·C(15,7).\nFinal answer: $\\frac{91}{646}$", "final_answer": "\\frac{91}{646}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "56fd47c605cb3cd3", "problem_id": "ab3b4e7a6be46e58", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A disease has prevalence $\\frac{2}{25}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{259}{2500}$.\n3) Compute P(D|+) = $\\frac{190}{259}$.\nFinal answer: $\\frac{190}{259}$", "final_answer": "\\frac{190}{259}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "84c5919e13f701fb", "problem_id": "ab3b4e7a6be46e58", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A disease has prevalence $\\frac{2}{25}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{259}{2500}$.\nCompute P(D|+) = $\\frac{190}{259}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{190}{259}$", "final_answer": "\\frac{190}{259}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0314538231e44a0d", "problem_id": "de675bee631f4f73", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 26 liters of a 25% salt solution. You have a 10% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 26−x liters is 50% solution.\nSalt equation: 10x + 50(26−x) = 25·26.\n(10-50)x = (25-50)·26 ⇒ x = 65/4.\nThen 26−x = 39/4.\nCheck: 10·65/4 + 50·39/4 = 25·26.\nFinal answer: $65/4 L of 10% and 39/4 L of 50%$", "final_answer": "65/4 L of 10% and 39/4 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ba572fc15a3505f", "problem_id": "de675bee631f4f73", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "You want 26 liters of a 25% salt solution. You have a 10% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 26−x liters is 50% solution.\n2) Salt equation: 10x + 50(26−x) = 25·26.\n3) (10-50)x = (25-50)·26 ⇒ x = 65/4.\n4) Then 26−x = 39/4.\nFinal answer: $65/4 L of 10% and 39/4 L of 50%$", "final_answer": "65/4 L of 10% and 39/4 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cf2e7d04419784a7", "problem_id": "fc988c40e70e2e93", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the congruences: x≡7 (mod 9), x≡2 (mod 11). Give the solution modulo 99.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 99.\n3) Compute the CRT solution to get x ≡ 79 (mod 99).\nFinal answer: $x\\equiv 79\\pmod{99}$", "final_answer": "x\\equiv 79\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2c25c8196caa4b48", "problem_id": "fc988c40e70e2e93", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡7 (mod 9), x≡2 (mod 11). Give the solution modulo 99.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 99.\nCompute the CRT solution to get x ≡ 79 (mod 99).\nCheck: 79 mod 9=7 and mod 11=2.\nFinal answer: $x\\equiv 79\\pmod{99}$", "final_answer": "x\\equiv 79\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5531954e6219f059", "problem_id": "eef3dc92f49610e2", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many binary strings of length 10 contain exactly 5 ones with no two ones adjacent?", "output": "1) Reserve 4 mandatory zeros between the 5 ones.\n2) This leaves n−k+1 = 6 valid slots for the ones.\n3) Choose the slots: C(6,5) = 6.\nFinal answer: $6$", "final_answer": "6", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d8987b45574f6a1e", "problem_id": "eef3dc92f49610e2", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many binary strings of length 10 contain exactly 5 ones with no two ones adjacent?", "output": "Reserve 4 mandatory zeros between the 5 ones.\nThis leaves n−k+1 = 6 valid slots for the ones.\nChoose the slots: C(6,5) = 6.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $6$", "final_answer": "6", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0ba6235fe417d888", "problem_id": "4f060fc8a978793b", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(3 x^{2} + 2 x + 2\\right) \\cos{\\left(x - 3 \\right)}$.", "output": "1) Let f(x) = $\\left(x + 2\\right) e^{x} + \\left(3 x^{2} + 2 x + 2\\right) \\cos{\\left(x - 3 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\left(x + 2\\right) e^{x} + 2 \\left(3 x + 1\\right) \\cos{\\left(x - 3 \\right)} - \\left(3 x^{2} + 2 x + 2\\right) \\sin{\\left(x - 3 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 2\\right)\\cdot e^{x} + \\left(6\\cdot x + 2\\right)\\cdot \\cos{\\left(x - 3 \\right)} - \\left(3\\cdot x^{2} + 2\\cdot x + 2\\right)\\cdot \\sin{\\left(x - 3 \\right)} + e^{x}$", "final_answer": "\\left(x + 2\\right)\\cdot e^{x} + \\left(6\\cdot x + 2\\right)\\cdot \\cos{\\left(x - 3 \\right)} - \\left(3\\cdot x^{2} + 2\\cdot x + 2\\right)\\cdot \\sin{\\left(x - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fca971e63cd1b146", "problem_id": "4f060fc8a978793b", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(3 x^{2} + 2 x + 2\\right) \\cos{\\left(x - 3 \\right)}$.", "output": "Let f(x) = $\\left(x + 2\\right) e^{x} + \\left(3 x^{2} + 2 x + 2\\right) \\cos{\\left(x - 3 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\left(x + 2\\right) e^{x} + 2 \\left(3 x + 1\\right) \\cos{\\left(x - 3 \\right)} - \\left(3 x^{2} + 2 x + 2\\right) \\sin{\\left(x - 3 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 2\\right)\\cdot e^{x} + \\left(6\\cdot x + 2\\right)\\cdot \\cos{\\left(x - 3 \\right)} - \\left(3\\cdot x^{2} + 2\\cdot x + 2\\right)\\cdot \\sin{\\left(x - 3 \\right)} + e^{x}$", "final_answer": "\\left(x + 2\\right)\\cdot e^{x} + \\left(6\\cdot x + 2\\right)\\cdot \\cos{\\left(x - 3 \\right)} - \\left(3\\cdot x^{2} + 2\\cdot x + 2\\right)\\cdot \\sin{\\left(x - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1124a825fb451960", "problem_id": "b901e968fc6ed2f0", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} - 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 1$ at $x=-1$.", "output": "Differentiate: f'(x) = $6 x^{2} - 6 x + 2 \\cos{\\left(x \\right)} + 2$.\nSlope at x=-1: m = f'(-1) = $2 \\cos{\\left(1 \\right)} + 14$.\nPoint on curve: (-1, f(-1)) = (-1, $-8 - 2 \\sin{\\left(1 \\right)}$).\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 14\\cdot x - 2\\cdot \\sin{\\left(1 \\right)} + 2\\cdot \\cos{\\left(1 \\right)} + 6$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 14\\cdot x - 2\\cdot \\sin{\\left(1 \\right)} + 2\\cdot \\cos{\\left(1 \\right)} + 6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2da0e5265e1ef65f", "problem_id": "b901e968fc6ed2f0", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} - 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 1$ at $x=-1$.", "output": "Differentiate: f'(x) = $6 x^{2} - 6 x + 2 \\cos{\\left(x \\right)} + 2$.\nSlope at x=-1: m = f'(-1) = $2 \\cos{\\left(1 \\right)} + 14$.\nPoint on curve: (-1, f(-1)) = (-1, $-8 - 2 \\sin{\\left(1 \\right)}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -8 - 2 \\sin{\\left(1 \\right)} = 2 \\cos{\\left(1 \\right)} + 14(x − -1).\nOptionally expand: y = $2 x \\cos{\\left(1 \\right)} + 14 x - 2 \\sin{\\left(1 \\right)} + 2 \\cos{\\left(1 \\right)} + 6$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 14\\cdot x - 2\\cdot \\sin{\\left(1 \\right)} + 2\\cdot \\cos{\\left(1 \\right)} + 6$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 14\\cdot x - 2\\cdot \\sin{\\left(1 \\right)} + 2\\cdot \\cos{\\left(1 \\right)} + 6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fcc36a1dccb4f524", "problem_id": "dea526067e22028d", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 21 times, what is the probability of exactly 17 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=21, k=17, p=$\\frac{2}{7}$ and simplify.\n4) Here C(21,17) = 5985.\nFinal answer: $\\frac{70041600000}{79792266297612001}$", "final_answer": "\\frac{70041600000}{79792266297612001}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "64db28dcdffeaf7b", "problem_id": "dea526067e22028d", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 21 times, what is the probability of exactly 17 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=21, k=17, p=$\\frac{2}{7}$ and simplify.\nHere C(21,17) = 5985.\nCheck: expression simplifies to $\\frac{70041600000}{79792266297612001}$.\nFinal answer: $\\frac{70041600000}{79792266297612001}$", "final_answer": "\\frac{70041600000}{79792266297612001}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "647e2b22fdde6f9c", "problem_id": "7bb871fdd2bc4ebc", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 3 modulo 29.", "output": "We want x such that 3x ≡ 1 (mod 29).\nUse extended Euclid to find s,t with 3s + 29t = 1.\nOne Bezout identity is: 3·(10) + 29·(-1) = 1.\nReduce s modulo 29: x ≡ 10 (mod 29).\nCheck: 3·10 mod 29 = 1.\nFinal answer: $x\\equiv 10\\pmod{29}$", "final_answer": "x\\equiv 10\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "28f8a4cfe3add60f", "problem_id": "7bb871fdd2bc4ebc", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the multiplicative inverse of 3 modulo 29.", "output": "1) We want x such that 3x ≡ 1 (mod 29).\n2) Use extended Euclid to find s,t with 3s + 29t = 1.\n3) One Bezout identity is: 3·(10) + 29·(-1) = 1.\n4) Reduce s modulo 29: x ≡ 10 (mod 29).\nFinal answer: $x\\equiv 10\\pmod{29}$", "final_answer": "x\\equiv 10\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "af75e3ef5c2b88f6", "problem_id": "755a7e23efb2939a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} - x^{2} + 4 x - 16$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} + 3 x - 16.\n3) Then the limit is -44.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: -48.\nFinal answer: $-48$", "final_answer": "-48", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "46dae824e4031da7", "problem_id": "1793bf9b810143a1", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} - x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{3 \\pi^{3}}{8} - \\frac{9 \\pi^{2}}{8} - 2 + 3 \\pi$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{3\\cdot \\pi^{3}}{8} - \\frac{9\\cdot \\pi^{2}}{8} - 2 + 3\\cdot \\pi$", "final_answer": "- \\frac{3\\cdot \\pi^{3}}{8} - \\frac{9\\cdot \\pi^{2}}{8} - 2 + 3\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bf1230b38487be70", "problem_id": "109ba1d0b8cacfd8", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $380 in the ratio 5:5. How much does each receive?", "output": "Total ratio parts = 5+5 = 10.\nEach part = 380 ÷ 10 = 38.\nFirst share = 5 × 38 = 190.\nSecond share = 5 × 38 = 190.\nCheck: 190+190=380.\nFinal answer: $190$", "final_answer": "190", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bb0a7188bba1ffc9", "problem_id": "109ba1d0b8cacfd8", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Two friends split $380 in the ratio 5:5. How much does each receive?", "output": "Total ratio parts = 5+5 = 10.\nEach part = 380 ÷ 10 = 38.\nFirst share = 5 × 38 = 190.\nFinal answer: $190$", "final_answer": "190", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a0bee9b5cb57f3c6", "problem_id": "00b0d2573f0a59d4", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the multiplicative inverse of 31 modulo 59.", "output": "1) We want x such that 31x ≡ 1 (mod 59).\n2) Use extended Euclid to find s,t with 31s + 59t = 1.\n3) One Bezout identity is: 31·(-19) + 59·(10) = 1.\n4) Reduce s modulo 59: x ≡ 40 (mod 59).\nFinal answer: $x\\equiv 40\\pmod{59}$", "final_answer": "x\\equiv 40\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c0f077cbe521556e", "problem_id": "00b0d2573f0a59d4", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Find the multiplicative inverse of 31 modulo 59.", "output": "We want x such that 31x ≡ 1 (mod 59).\nUse extended Euclid to find s,t with 31s + 59t = 1.\nOne Bezout identity is: 31·(-19) + 59·(10) = 1.\nFinal answer: $x\\equiv 40\\pmod{59}$", "final_answer": "x\\equiv 40\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "982945a5291fd317", "problem_id": "65d41fd5a950adbf", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 66 liters of a 35% salt solution. You have a 15% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 66−x liters is 70% solution.\nSalt equation: 15x + 70(66−x) = 35·66.\n(15-70)x = (35-70)·66 ⇒ x = 42.\nThen 66−x = 24.\nCheck: 15·42 + 70·24 = 35·66.\nFinal answer: $42 L of 15% and 24 L of 70%$", "final_answer": "42 L of 15% and 24 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5abc3df78b16dd92", "problem_id": "65d41fd5a950adbf", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "You want 66 liters of a 35% salt solution. You have a 15% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 66−x liters is 70% solution.\n2) Salt equation: 15x + 70(66−x) = 35·66.\n3) (15-70)x = (35-70)·66 ⇒ x = 42.\n4) Then 66−x = 24.\nFinal answer: $42 L of 15% and 24 L of 70%$", "final_answer": "42 L of 15% and 24 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fbb6bb7e35b41e23", "problem_id": "b63083f4964345d5", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A cyclist rides at 59 miles per hour for 1/2 hours, then at 43 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 59 × 1/2 = 59/2 miles.\n2) Second leg distance = 43 × 5/2 = 215/2 miles.\n3) Total distance = 59/2 + 215/2 = 137 miles.\nFinal answer: $137$", "final_answer": "137", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c2d824c3d654507e", "problem_id": "f3c5ad0f91d4a7ee", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 \\cos{\\left(2 x \\right)}) = - 3 \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = - 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 3 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 \\cos{\\left(2 x \\right)}) = - 6 \\sin{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.\nFinal answer: $- 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$", "final_answer": "- 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "4526bf7a997ef8d2", "problem_id": "6cd110299462eec7", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{5}$. If it is flipped 8 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=8, k=4, p=$\\frac{3}{5}$ and simplify.\n4) Here C(8,4) = 70.\nFinal answer: $\\frac{18144}{78125}$", "final_answer": "\\frac{18144}{78125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "48f58e0136e6b9d6", "problem_id": "bc07347abcdf219c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 5 - x$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 5 - 2 x.\n3) Then the limit is 9.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 7.\nFinal answer: $7$", "final_answer": "7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c2f4fa01b75819b7", "problem_id": "628f312039578ae2", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "You want 25 liters of a 30% salt solution. You have a 10% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 25−x liters is 50% solution.\n2) Salt equation: 10x + 50(25−x) = 30·25.\n3) (10-50)x = (30-50)·25 ⇒ x = 25/2.\n4) Then 25−x = 25/2.\nFinal answer: $25/2 L of 10% and 25/2 L of 50%$", "final_answer": "25/2 L of 10% and 25/2 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "14a263273a1d0e2d", "problem_id": "628f312039578ae2", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "You want 25 liters of a 30% salt solution. You have a 10% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 25−x liters is 50% solution.\nSalt equation: 10x + 50(25−x) = 30·25.\n(10-50)x = (30-50)·25 ⇒ x = 25/2.\nFinal answer: $25/2 L of 10% and 25/2 L of 50%$", "final_answer": "25/2 L of 10% and 25/2 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "437269a580a7b675", "problem_id": "e3000c6b13e01f35", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(- x^{3} - 3 x^{2} + x - 2\\right) \\cos{\\left(3 x - 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x e^{x}) = 2 x e^{x}$.\n3) So $f'(x) = 3 x^{3} \\sin{\\left(3 x - 1 \\right)} + 9 x^{2} \\sin{\\left(3 x - 1 \\right)} - 3 x^{2} \\cos{\\left(3 x - 1 \\right)} + 2 x e^{x} - 3 x \\sin{\\left(3 x - 1 \\right)} - 6 x \\cos{\\left(3 x - 1 \\right)} + 4 e^{x} + 6 \\sin{\\left(3 x - 1 \\right)} + \\cos{\\left(3 x - 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x e^{x}) = 2 x e^{x} + 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x + 2\\right) e^{x} - 3 \\left(x \\left(x \\left(- x - 3\\right) + 1\\right) - 2\\right) \\sin{\\left(3 x - 1 \\right)} + \\left(- 3 x^{2} - 6 x + 1\\right) \\cos{\\left(3 x - 1 \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(x + 2\\right) e^{x} - 3 \\left(x \\left(x \\left(- x - 3\\right) + 1\\right) - 2\\right) \\sin{\\left(3 x - 1 \\right)} + \\left(- 3 x^{2} - 6 x + 1\\right) \\cos{\\left(3 x - 1 \\right)} + 2 e^{x}$", "final_answer": "2 \\left(x + 2\\right) e^{x} - 3 \\left(x \\left(x \\left(- x - 3\\right) + 1\\right) - 2\\right) \\sin{\\left(3 x - 1 \\right)} + \\left(- 3 x^{2} - 6 x + 1\\right) \\cos{\\left(3 x - 1 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e29f0dfa8996fa10", "problem_id": "956b8efea6f17046", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 16 times, what is the probability of exactly 10 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=16, k=10, p=$\\frac{2}{3}$ and simplify.\nFinal answer: $\\frac{8200192}{43046721}$", "final_answer": "\\frac{8200192}{43046721}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9c428706900ffece", "problem_id": "14f1e25b0575b2b6", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 55 miles per hour for 3/2 hours, then at 59 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 55 × 3/2 = 165/2 miles.\nSecond leg distance = 59 × 2 = 118 miles.\nTotal distance = 165/2 + 118 = 401/2 miles.\nCheck: adding both legs gives 401/2.\nFinal answer: $\\frac{401}{2}$", "final_answer": "\\frac{401}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fedba23cfc69cef4", "problem_id": "14f1e25b0575b2b6", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "A cyclist rides at 55 miles per hour for 3/2 hours, then at 59 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 55 × 3/2 = 165/2 miles.\nSecond leg distance = 59 × 2 = 118 miles.\nTotal distance = 165/2 + 118 = 401/2 miles.\nFinal answer: $\\frac{401}{2}$", "final_answer": "\\frac{401}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "538b7b0e3bcfd5ed", "problem_id": "539610d513a978fe", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_8=38$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_8=22.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=22 and k=8.\nCompute C(29,7) = 1560780.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $1560780$", "final_answer": "1560780", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c044df9ceb2ed433", "problem_id": "539610d513a978fe", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_8=38$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_8=22.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=22 and k=8.\n4) Compute C(29,7) = 1560780.\nFinal answer: $1560780$", "final_answer": "1560780", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "99bb432731ae9ed0", "problem_id": "3f848624a11456c5", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 5), x≡5 (mod 11). Give the solution modulo 55.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 55.\nCompute the CRT solution to get x ≡ 27 (mod 55).\nCheck: 27 mod 5=2 and mod 11=5.\nFinal answer: $x\\equiv 27\\pmod{55}$", "final_answer": "x\\equiv 27\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "644942e36ba64c54", "problem_id": "3f848624a11456c5", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡2 (mod 5), x≡5 (mod 11). Give the solution modulo 55.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 55.\n3) Compute the CRT solution to get x ≡ 27 (mod 55).\nFinal answer: $x\\equiv 27\\pmod{55}$", "final_answer": "x\\equiv 27\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2c74864c1ad22e9c", "problem_id": "26dcb7bec08ba440", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "You want 51 liters of a 45% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 51−x liters is 70% solution.\n2) Salt equation: 25x + 70(51−x) = 45·51.\n3) (25-70)x = (45-70)·51 ⇒ x = 85/3.\n4) Then 51−x = 68/3.\nFinal answer: $85/3 L of 25% and 68/3 L of 70%$", "final_answer": "85/3 L of 25% and 68/3 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0225c57e271f1884", "problem_id": "414fde83bfa8e3ed", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 2 x^{2} + x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}) = \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} + \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 6 x^{2} + 9} - \\frac{6 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} - \\frac{12 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} - \\frac{2 x^{2}}{2 x^{4} - 12 x^{2} + 18} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 6} - \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3} + \\frac{24 x}{2 x^{4} - 12 x^{2} + 18} - \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3} + \\frac{9 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} + \\frac{18 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} - \\frac{6}{2 x^{4} - 12 x^{2} + 18}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}) = - \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - 3\\right)^{2}} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x^{2} - 3\\right)} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3} + \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 4 x \\left(- 2 x^{2} + x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right) \\left(4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8 x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{2 \\left(x^{2} - 3\\right)^{2}}$.\nFinal answer: $\\frac{- 4 x \\left(- 2 x^{2} + x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right) \\left(4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8 x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{2 \\left(x^{2} - 3\\right)^{2}}$", "final_answer": "\\frac{- 4 x \\left(- 2 x^{2} + x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right) \\left(4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8 x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{2 \\left(x^{2} - 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "4eb9b4f8d178394f", "problem_id": "564166eb1691da16", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} x^{3} + 2 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $x^{3} + 2 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\n2) Evaluate at the bounds: F(2 \\pi) and F(0).\n3) Subtract: F(2 \\pi) − F(0) = $\\pi^{2} \\left(-6 + \\frac{16 \\pi}{3} + 4 \\pi^{2}\\right)$.\n4) Simplify the result.\nFinal answer: $\\pi^{2}\\cdot \\left(-6 + \\frac{16\\cdot \\pi}{3} + 4\\cdot \\pi^{2}\\right)$", "final_answer": "\\pi^{2}\\cdot \\left(-6 + \\frac{16\\cdot \\pi}{3} + 4\\cdot \\pi^{2}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bf4e556348e27b11", "problem_id": "564166eb1691da16", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} x^{3} + 2 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $x^{3} + 2 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\nEvaluate at the bounds: F(2 \\pi) and F(0).\nSubtract: F(2 \\pi) − F(0) = $\\pi^{2} \\left(-6 + \\frac{16 \\pi}{3} + 4 \\pi^{2}\\right)$.\nFinal answer: $\\pi^{2}\\cdot \\left(-6 + \\frac{16\\cdot \\pi}{3} + 4\\cdot \\pi^{2}\\right)$", "final_answer": "\\pi^{2}\\cdot \\left(-6 + \\frac{16\\cdot \\pi}{3} + 4\\cdot \\pi^{2}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "13a28219ca286ffd", "problem_id": "9dfa3b414e7fdc13", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A store sells a backpack for $435. It is discounted by 10%. You buy 14 backpacks. What is the total cost?", "output": "1) Discounted price = 435 × (1 − 10/100) = 435 × 90/100 = 783/2.\n2) Subtotal = 14 × 783/2 = 5481.\nFinal answer: $5481$", "final_answer": "5481", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b8a2c6d879d6aede", "problem_id": "9dfa3b414e7fdc13", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "A store sells a backpack for $435. It is discounted by 10%. You buy 14 backpacks. What is the total cost?", "output": "$5481$", "final_answer": "5481", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "75688ecbabf7735c", "problem_id": "645c0cf84c13278b", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $326. It is discounted by 10%. You buy 12 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 326 × (1 − 10/100) = 326 × 90/100 = 1467/5.\nSubtotal = 12 × 1467/5 = 17604/5.\nTotal with tax = 17604/5 × 105/100 = 92421/25.\nCheck: recomputing the formula yields 92421/25.\nFinal answer: $\\frac{92421}{25}$", "final_answer": "\\frac{92421}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8105b8c9f8bf8ed3", "problem_id": "645c0cf84c13278b", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A store sells a backpack for $326. It is discounted by 10%. You buy 12 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 326 × (1 − 10/100) = 326 × 90/100 = 1467/5.\n2) Subtotal = 12 × 1467/5 = 17604/5.\n3) Total with tax = 17604/5 × 105/100 = 92421/25.\nFinal answer: $\\frac{92421}{25}$", "final_answer": "\\frac{92421}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "23eb8e2094223f44", "problem_id": "3fb10ec6fd6a68ae", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 7 x - \\frac{8}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$7$, c=$- \\frac{8}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $81$, so √Δ = $9$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{1}{3}$.\n7) x2 = $- \\frac{8}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{1}{3}, - \\frac{8}{3}\\}$", "final_answer": "x \\in \\{\\frac{1}{3}, - \\frac{8}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3cb2f7bd508710a3", "problem_id": "ee8ca69cc5ede5ab", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} x + 1$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 1.\n3) Then the limit is 1.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -1.\nFinal answer: $-1$", "final_answer": "-1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8f1c9ebabdcabfe0", "problem_id": "7f39e7054e690e42", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the multiplicative inverse of 32 modulo 41.", "output": "1) We want x such that 32x ≡ 1 (mod 41).\n2) Use extended Euclid to find s,t with 32s + 41t = 1.\n3) One Bezout identity is: 32·(9) + 41·(-7) = 1.\n4) Reduce s modulo 41: x ≡ 9 (mod 41).\nFinal answer: $x\\equiv 9\\pmod{41}$", "final_answer": "x\\equiv 9\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c66df7e9cd136a83", "problem_id": "7f39e7054e690e42", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 32 modulo 41.", "output": "We want x such that 32x ≡ 1 (mod 41).\nUse extended Euclid to find s,t with 32s + 41t = 1.\nOne Bezout identity is: 32·(9) + 41·(-7) = 1.\nReduce s modulo 41: x ≡ 9 (mod 41).\nCheck: 32·9 mod 41 = 1.\nFinal answer: $x\\equiv 9\\pmod{41}$", "final_answer": "x\\equiv 9\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "05a2d83c869feb9e", "problem_id": "4397a6840bbcf678", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many binary strings of length 17 contain exactly 8 ones with no two ones adjacent?", "output": "1) Reserve 7 mandatory zeros between the 8 ones.\n2) This leaves n−k+1 = 10 valid slots for the ones.\n3) Choose the slots: C(10,8) = 45.\nFinal answer: $45$", "final_answer": "45", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9d726e8445db46a4", "problem_id": "4397a6840bbcf678", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many binary strings of length 17 contain exactly 8 ones with no two ones adjacent?", "output": "Reserve 7 mandatory zeros between the 8 ones.\nThis leaves n−k+1 = 10 valid slots for the ones.\nChoose the slots: C(10,8) = 45.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $45$", "final_answer": "45", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cdac773008a17403", "problem_id": "d5e0f1f771c20f50", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{15 x}{4} - \\frac{\\sqrt{11} x}{2} + \\frac{3 \\sqrt{11}}{8} + \\frac{9}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{15}{4} - \\frac{\\sqrt{11}}{2}$, c=$\\frac{3 \\sqrt{11}}{8} + \\frac{9}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{9 \\sqrt{11}}{4} + \\frac{125}{16}$, so √Δ = $\\sqrt{\\frac{9 \\sqrt{11}}{4} + \\frac{125}{16}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{11}}{4} + \\frac{15}{8} + \\frac{\\sqrt{36 \\sqrt{11} + 125}}{8}$.\n7) x2 = $- \\frac{\\sqrt{36 \\sqrt{11} + 125}}{8} + \\frac{\\sqrt{11}}{4} + \\frac{15}{8}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{11}}{4} + \\frac{15}{8} + \\frac{\\sqrt{36\\cdot \\sqrt{11} + 125}}{8}, - \\frac{\\sqrt{36\\cdot \\sqrt{11} + 125}}{8} + \\frac{\\sqrt{11}}{4} + \\frac{15}{8}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{11}}{4} + \\frac{15}{8} + \\frac{\\sqrt{36\\cdot \\sqrt{11} + 125}}{8}, - \\frac{\\sqrt{36\\cdot \\sqrt{11} + 125}}{8} + \\frac{\\sqrt{11}}{4} + \\frac{15}{8}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "27287cfcc49b6de6", "problem_id": "d5e0f1f771c20f50", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{15 x}{4} - \\frac{\\sqrt{11} x}{2} + \\frac{3 \\sqrt{11}}{8} + \\frac{9}{4}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{15}{4} - \\frac{\\sqrt{11}}{2}$, c=$\\frac{3 \\sqrt{11}}{8} + \\frac{9}{4}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{9 \\sqrt{11}}{4} + \\frac{125}{16}$, so √Δ = $\\sqrt{\\frac{9 \\sqrt{11}}{4} + \\frac{125}{16}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{11}}{4} + \\frac{15}{8} + \\frac{\\sqrt{36 \\sqrt{11} + 125}}{8}$.\nx2 = $- \\frac{\\sqrt{36 \\sqrt{11} + 125}}{8} + \\frac{\\sqrt{11}}{4} + \\frac{15}{8}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{11}}{4} + \\frac{15}{8} + \\frac{\\sqrt{36\\cdot \\sqrt{11} + 125}}{8}, - \\frac{\\sqrt{36\\cdot \\sqrt{11} + 125}}{8} + \\frac{\\sqrt{11}}{4} + \\frac{15}{8}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{11}}{4} + \\frac{15}{8} + \\frac{\\sqrt{36\\cdot \\sqrt{11} + 125}}{8}, - \\frac{\\sqrt{36\\cdot \\sqrt{11} + 125}}{8} + \\frac{\\sqrt{11}}{4} + \\frac{15}{8}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8b1837587d8dd366", "problem_id": "11b13bfcee5121d7", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 75$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} - 75$ = $3 \\left(x - 5\\right) \\left(x + 5\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $5$.\n5) From the second factor: x = $-5$.\nFinal answer: $x \\in \\{5, -5\\}$", "final_answer": "x \\in \\{5, -5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a3268742df0a461", "problem_id": "11b13bfcee5121d7", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 75$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} - 75$ = $3 \\left(x - 5\\right) \\left(x + 5\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $5$.\nFrom the second factor: x = $-5$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{5, -5\\}$", "final_answer": "x \\in \\{5, -5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "997baf86d45ac3ef", "problem_id": "eae1754caad67564", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the linear system:\n$- x + 4 y$ = $-13$\n$3 x + 4 y$ = $7$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $3$: $- 3 x + 12 y$ = $-39$.\n3) Multiply the second equation by $-1$: $- 3 x - 4 y$ = $-7$.\n4) Subtract to eliminate x: $16 y$ = $-32$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, -2\\right)$", "final_answer": "(x,y)=\\left(5, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "02e5913c4fc36684", "problem_id": "eae1754caad67564", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the linear system:\n$- x + 4 y$ = $-13$\n$3 x + 4 y$ = $7$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $- 3 x + 12 y$ = $-39$.\nMultiply the second equation by $-1$: $- 3 x - 4 y$ = $-7$.\nFinal answer: $(x,y)=\\left(5, -2\\right)$", "final_answer": "(x,y)=\\left(5, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5ebb34d5ebe691af", "problem_id": "4c1989da639ec4c7", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - x^{2} - 5 x - 17$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{3} - x^{2} + 3 x + 3$ and a = 4.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 3 x^{2} - 2 x + 3$.\nEvaluate at a: p'(4) = $-53$.\nCheck: the identity gives $-53$.\nFinal answer: $-53$", "final_answer": "-53", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "70b2a25efa187dd1", "problem_id": "4c1989da639ec4c7", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - x^{2} - 5 x - 17$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{3} - x^{2} + 3 x + 3$ and a = 4.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-53$", "final_answer": "-53", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "15af1b2f139d603c", "problem_id": "a263d09b3bfe0a96", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Worker A finishes a job in 18 hours and Worker B finishes it in 9 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/18 job/hour. B's rate = 1/9 job/hour.\nCombined rate = 1/18 + 1/9 = 1/6 job/hour.\nTime = 1 ÷ (combined rate) = 6 hours.\nFinal answer: $6$", "final_answer": "6", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "009f7e637f7ee1c2", "problem_id": "a263d09b3bfe0a96", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Worker A finishes a job in 18 hours and Worker B finishes it in 9 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "$6$", "final_answer": "6", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "179b6c7c57ccb039", "problem_id": "a9836c54ff5c5377", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{3} + 3 x^{2} + 2 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{2 x - 3}$.", "output": "Let f(x) = $\\frac{x^{3} + 3 x^{2} + 2 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{2 x - 3}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{- 4 x^{3} - 12 x^{2} - 8 x + \\left(2 x - 3\\right) \\left(6 x^{2} + 12 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 4 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8}{2 \\left(2 x - 3\\right)^{2}}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\frac{- 4\\cdot x^{3} - 12\\cdot x^{2} - 8\\cdot x + \\left(2\\cdot x - 3\\right)\\cdot \\left(6\\cdot x^{2} + 12\\cdot x + \\left(2\\cdot x - 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 4\\right) - \\left(8\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 8}{2\\cdot \\left(2\\cdot x - 3\\right)^{2}}$", "final_answer": "\\frac{- 4\\cdot x^{3} - 12\\cdot x^{2} - 8\\cdot x + \\left(2\\cdot x - 3\\right)\\cdot \\left(6\\cdot x^{2} + 12\\cdot x + \\left(2\\cdot x - 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 4\\right) - \\left(8\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 8}{2\\cdot \\left(2\\cdot x - 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b86eeb63295adff", "problem_id": "a9836c54ff5c5377", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{3} + 3 x^{2} + 2 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{2 x - 3}$.", "output": "1) Let f(x) = $\\frac{x^{3} + 3 x^{2} + 2 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{2 x - 3}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{- 4 x^{3} - 12 x^{2} - 8 x + \\left(2 x - 3\\right) \\left(6 x^{2} + 12 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 4 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8}{2 \\left(2 x - 3\\right)^{2}}$.\nFinal answer: $\\frac{- 4\\cdot x^{3} - 12\\cdot x^{2} - 8\\cdot x + \\left(2\\cdot x - 3\\right)\\cdot \\left(6\\cdot x^{2} + 12\\cdot x + \\left(2\\cdot x - 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 4\\right) - \\left(8\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 8}{2\\cdot \\left(2\\cdot x - 3\\right)^{2}}$", "final_answer": "\\frac{- 4\\cdot x^{3} - 12\\cdot x^{2} - 8\\cdot x + \\left(2\\cdot x - 3\\right)\\cdot \\left(6\\cdot x^{2} + 12\\cdot x + \\left(2\\cdot x - 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 4\\right) - \\left(8\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 8}{2\\cdot \\left(2\\cdot x - 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "df050a505e9281e4", "problem_id": "0f34d05a51e4df64", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $47. It is discounted by 15%. You buy 12 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 47 × (1 − 15/100) = 47 × 85/100 = 799/20.\nSubtotal = 12 × 799/20 = 2397/5.\nTotal with tax = 2397/5 × 109/100 = 261273/500.\nCheck: recomputing the formula yields 261273/500.\nFinal answer: $\\frac{261273}{500}$", "final_answer": "\\frac{261273}{500}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "16308ad7c47bda69", "problem_id": "0f34d05a51e4df64", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A store sells a backpack for $47. It is discounted by 15%. You buy 12 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 47 × (1 − 15/100) = 47 × 85/100 = 799/20.\n2) Subtotal = 12 × 799/20 = 2397/5.\n3) Total with tax = 2397/5 × 109/100 = 261273/500.\nFinal answer: $\\frac{261273}{500}$", "final_answer": "\\frac{261273}{500}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d67bd6ecb3cee8e4", "problem_id": "c616d98274e54d55", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Find the multiplicative inverse of 18 modulo 37.", "output": "1) We want x such that 18x ≡ 1 (mod 37).\n2) Use extended Euclid to find s,t with 18s + 37t = 1.\n3) One Bezout identity is: 18·(-2) + 37·(1) = 1.\n4) Reduce s modulo 37: x ≡ 35 (mod 37).\nFinal answer: $x\\equiv 35\\pmod{37}$", "final_answer": "x\\equiv 35\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bfb088c50592417f", "problem_id": "c616d98274e54d55", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the multiplicative inverse of 18 modulo 37.", "output": "We want x such that 18x ≡ 1 (mod 37).\nUse extended Euclid to find s,t with 18s + 37t = 1.\nOne Bezout identity is: 18·(-2) + 37·(1) = 1.\nReduce s modulo 37: x ≡ 35 (mod 37).\nCheck: 18·35 mod 37 = 1.\nFinal answer: $x\\equiv 35\\pmod{37}$", "final_answer": "x\\equiv 35\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f402530174cd016e", "problem_id": "dc9e8b6bffa8c192", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{3} + 3 x^{2} + x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 3 x^{3} + 3 x^{2} + x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2$.\nCompute an antiderivative: F(x) = $- \\frac{3 x^{4}}{4} + x^{3} + \\frac{x^{2}}{2} + 2 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - \\frac{3\\cdot x^{4}}{4} + x^{3} + \\frac{x^{2}}{2} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{3\\cdot x^{4}}{4} + x^{3} + \\frac{x^{2}}{2} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "134ffacb513d2db7", "problem_id": "dc9e8b6bffa8c192", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{3} + 3 x^{2} + x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 3 x^{3} + 3 x^{2} + x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2$.\nCompute an antiderivative: F(x) = $- \\frac{3 x^{4}}{4} + x^{3} + \\frac{x^{2}}{2} + 2 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $C - \\frac{3\\cdot x^{4}}{4} + x^{3} + \\frac{x^{2}}{2} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{3\\cdot x^{4}}{4} + x^{3} + \\frac{x^{2}}{2} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3a37d2538ca009a0", "problem_id": "8e2d725c84148cee", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} x^{4} - 2 x^{3} + 4 x^{2} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $x^{4} - 2 x^{3} + 4 x^{2} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\pi).\n3) Subtract: F(\\pi) − F(- \\pi) = $\\frac{2 \\pi \\left(15 + 20 \\pi^{2} + 3 \\pi^{4}\\right)}{15}$.\n4) Simplify the result.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(15 + 20\\cdot \\pi^{2} + 3\\cdot \\pi^{4}\\right)}{15}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(15 + 20\\cdot \\pi^{2} + 3\\cdot \\pi^{4}\\right)}{15}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5c028102965d6268", "problem_id": "8e2d725c84148cee", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} x^{4} - 2 x^{3} + 4 x^{2} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $x^{4} - 2 x^{3} + 4 x^{2} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\nEvaluate at the bounds: F(\\pi) and F(- \\pi).\nSubtract: F(\\pi) − F(- \\pi) = $\\frac{2 \\pi \\left(15 + 20 \\pi^{2} + 3 \\pi^{4}\\right)}{15}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(15 + 20\\cdot \\pi^{2} + 3\\cdot \\pi^{4}\\right)}{15}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(15 + 20\\cdot \\pi^{2} + 3\\cdot \\pi^{4}\\right)}{15}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "421efaaac1f72cf6", "problem_id": "0a975c4906b5dedb", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "You want 23 liters of a 47% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 23−x liters is 70% solution.\n2) Salt equation: 25x + 70(23−x) = 47·23.\n3) (25-70)x = (47-70)·23 ⇒ x = 529/45.\n4) Then 23−x = 506/45.\nFinal answer: $529/45 L of 25% and 506/45 L of 70%$", "final_answer": "529/45 L of 25% and 506/45 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7fc66851c40449b6", "problem_id": "0a975c4906b5dedb", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "You want 23 liters of a 47% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 23−x liters is 70% solution.\nSalt equation: 25x + 70(23−x) = 47·23.\n(25-70)x = (47-70)·23 ⇒ x = 529/45.\nThen 23−x = 506/45.\nCheck: 25·529/45 + 70·506/45 = 47·23.\nFinal answer: $529/45 L of 25% and 506/45 L of 70%$", "final_answer": "529/45 L of 25% and 506/45 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8079c71f0163d8f1", "problem_id": "d863e9a5a2f53248", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡2 (mod 4), x≡5 (mod 11). Give the solution modulo 44.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 44.\n3) Compute the CRT solution to get x ≡ 38 (mod 44).\nFinal answer: $x\\equiv 38\\pmod{44}$", "final_answer": "x\\equiv 38\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e3d71cf77db2a68d", "problem_id": "d863e9a5a2f53248", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 4), x≡5 (mod 11). Give the solution modulo 44.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 44.\nCompute the CRT solution to get x ≡ 38 (mod 44).\nCheck: 38 mod 4=2 and mod 11=5.\nFinal answer: $x\\equiv 38\\pmod{44}$", "final_answer": "x\\equiv 38\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "31fca32ef3de845a", "problem_id": "83e03b53a015a482", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{33 x}{5} - \\sqrt{11} x + \\frac{8 \\sqrt{11}}{5} + 8$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{33}{5} - \\sqrt{11}$, c=$\\frac{8 \\sqrt{11}}{5} + 8$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{34 \\sqrt{11}}{5} + \\frac{564}{25}$, so √Δ = $\\sqrt{\\frac{34 \\sqrt{11}}{5} + \\frac{564}{25}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{11}}{2} + \\frac{33}{10} + \\frac{\\sqrt{170 \\sqrt{11} + 564}}{10}$.\n7) x2 = $- \\frac{\\sqrt{170 \\sqrt{11} + 564}}{10} + \\frac{\\sqrt{11}}{2} + \\frac{33}{10}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{11}}{2} + \\frac{33}{10} + \\frac{\\sqrt{170\\cdot \\sqrt{11} + 564}}{10}, - \\frac{\\sqrt{170\\cdot \\sqrt{11} + 564}}{10} + \\frac{\\sqrt{11}}{2} + \\frac{33}{10}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{11}}{2} + \\frac{33}{10} + \\frac{\\sqrt{170\\cdot \\sqrt{11} + 564}}{10}, - \\frac{\\sqrt{170\\cdot \\sqrt{11} + 564}}{10} + \\frac{\\sqrt{11}}{2} + \\frac{33}{10}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "28f0af80719a609e", "problem_id": "83e03b53a015a482", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{33 x}{5} - \\sqrt{11} x + \\frac{8 \\sqrt{11}}{5} + 8$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{33}{5} - \\sqrt{11}$, c=$\\frac{8 \\sqrt{11}}{5} + 8$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{\\sqrt{11}}{2} + \\frac{33}{10} + \\frac{\\sqrt{170\\cdot \\sqrt{11} + 564}}{10}, - \\frac{\\sqrt{170\\cdot \\sqrt{11} + 564}}{10} + \\frac{\\sqrt{11}}{2} + \\frac{33}{10}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{11}}{2} + \\frac{33}{10} + \\frac{\\sqrt{170\\cdot \\sqrt{11} + 564}}{10}, - \\frac{\\sqrt{170\\cdot \\sqrt{11} + 564}}{10} + \\frac{\\sqrt{11}}{2} + \\frac{33}{10}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f7b96416f5c735e3", "problem_id": "b7d06cbe2689090e", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$- 5 x - y$ = $41$\n$- 2 y$ = $-8$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $-5$: $10 y$ = $40$.\nSubtract to eliminate x: $- 10 y$ = $-40$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $-9$.\nCheck: plug (x,y)=(-9,4) into both equations.\nFinal answer: $(x,y)=\\left(-9, 4\\right)$", "final_answer": "(x,y)=\\left(-9, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f6ba6987b225f00c", "problem_id": "b7d06cbe2689090e", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$- 5 x - y$ = $41$\n$- 2 y$ = $-8$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $-5$: $10 y$ = $40$.\n4) Subtract to eliminate x: $- 10 y$ = $-40$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $-9$.\nFinal answer: $(x,y)=\\left(-9, 4\\right)$", "final_answer": "(x,y)=\\left(-9, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "45819dccf61d8acd", "problem_id": "8946817ec8c5f214", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} - x^{4} - x^{3} - 3 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{4} - x^{3} - 3 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\nEvaluate at the bounds: F(\\pi) and F(- \\pi).\nSubtract: F(\\pi) − F(- \\pi) = $\\frac{2 \\pi \\left(- \\pi^{4} - 5 \\pi^{2} - 15\\right)}{5}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(- \\pi^{4} - 5\\cdot \\pi^{2} - 15\\right)}{5}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(- \\pi^{4} - 5\\cdot \\pi^{2} - 15\\right)}{5}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f2f3a0b79436bd90", "problem_id": "8946817ec8c5f214", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} - x^{4} - x^{3} - 3 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{4} - x^{3} - 3 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\pi).\n3) Subtract: F(\\pi) − F(- \\pi) = $\\frac{2 \\pi \\left(- \\pi^{4} - 5 \\pi^{2} - 15\\right)}{5}$.\n4) Simplify the result.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(- \\pi^{4} - 5\\cdot \\pi^{2} - 15\\right)}{5}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(- \\pi^{4} - 5\\cdot \\pi^{2} - 15\\right)}{5}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8aebd918e197f66f", "problem_id": "7dd28420924931e1", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{3} + \\frac{2 x^{2}}{3} + \\frac{x}{3} + \\left(x^{2} + x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - \\frac{1}{3}}{x^{2} + x + 1}$.", "output": "1) Let f(x) = $\\frac{x^{3} + \\frac{2 x^{2}}{3} + \\frac{x}{3} + \\left(x^{2} + x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - \\frac{1}{3}}{x^{2} + x + 1}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{2 \\left(- 2 x - 1\\right) \\left(3 x^{3} + 2 x^{2} + x + 3 \\left(x^{2} + x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) + \\left(x^{2} + x + 1\\right) \\left(18 x^{2} + 8 x + 6 \\left(2 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3 \\left(x^{2} + x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x^{2} + x + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{6 \\left(x^{2} + x + 1\\right)^{2}}$.\nFinal answer: $\\frac{\\left(- 4\\cdot x - 2\\right)\\cdot \\left(3\\cdot x^{3} + 2\\cdot x^{2} + x + \\left(3\\cdot x^{2} + 3\\cdot x + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 1\\right) + \\left(x^{2} + x + 1\\right)\\cdot \\left(18\\cdot x^{2} + 8\\cdot x + \\left(12\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(3\\cdot x^{2} + 3\\cdot x + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x^{2} + 6\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\right)}{6\\cdot \\left(x^{2} + x + 1\\right)^{2}}$", "final_answer": "\\frac{\\left(- 4\\cdot x - 2\\right)\\cdot \\left(3\\cdot x^{3} + 2\\cdot x^{2} + x + \\left(3\\cdot x^{2} + 3\\cdot x + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 1\\right) + \\left(x^{2} + x + 1\\right)\\cdot \\left(18\\cdot x^{2} + 8\\cdot x + \\left(12\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(3\\cdot x^{2} + 3\\cdot x + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x^{2} + 6\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\right)}{6\\cdot \\left(x^{2} + x + 1\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db4651b5759df45c", "problem_id": "7dd28420924931e1", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{3} + \\frac{2 x^{2}}{3} + \\frac{x}{3} + \\left(x^{2} + x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - \\frac{1}{3}}{x^{2} + x + 1}$.", "output": "Let f(x) = $\\frac{x^{3} + \\frac{2 x^{2}}{3} + \\frac{x}{3} + \\left(x^{2} + x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - \\frac{1}{3}}{x^{2} + x + 1}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{2 \\left(- 2 x - 1\\right) \\left(3 x^{3} + 2 x^{2} + x + 3 \\left(x^{2} + x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) + \\left(x^{2} + x + 1\\right) \\left(18 x^{2} + 8 x + 6 \\left(2 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3 \\left(x^{2} + x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x^{2} + x + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{6 \\left(x^{2} + x + 1\\right)^{2}}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\frac{\\left(- 4\\cdot x - 2\\right)\\cdot \\left(3\\cdot x^{3} + 2\\cdot x^{2} + x + \\left(3\\cdot x^{2} + 3\\cdot x + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 1\\right) + \\left(x^{2} + x + 1\\right)\\cdot \\left(18\\cdot x^{2} + 8\\cdot x + \\left(12\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(3\\cdot x^{2} + 3\\cdot x + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x^{2} + 6\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\right)}{6\\cdot \\left(x^{2} + x + 1\\right)^{2}}$", "final_answer": "\\frac{\\left(- 4\\cdot x - 2\\right)\\cdot \\left(3\\cdot x^{3} + 2\\cdot x^{2} + x + \\left(3\\cdot x^{2} + 3\\cdot x + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 1\\right) + \\left(x^{2} + x + 1\\right)\\cdot \\left(18\\cdot x^{2} + 8\\cdot x + \\left(12\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(3\\cdot x^{2} + 3\\cdot x + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x^{2} + 6\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\right)}{6\\cdot \\left(x^{2} + x + 1\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "06cd6ad706365795", "problem_id": "62f3aab58bf8282a", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 34 liters of a 40% salt solution. You have a 15% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 34−x liters is 70% solution.\nSalt equation: 15x + 70(34−x) = 40·34.\n(15-70)x = (40-70)·34 ⇒ x = 204/11.\nThen 34−x = 170/11.\nCheck: 15·204/11 + 70·170/11 = 40·34.\nFinal answer: $204/11 L of 15% and 170/11 L of 70%$", "final_answer": "204/11 L of 15% and 170/11 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "916b5141a9a0db52", "problem_id": "62f3aab58bf8282a", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "You want 34 liters of a 40% salt solution. You have a 15% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 34−x liters is 70% solution.\n2) Salt equation: 15x + 70(34−x) = 40·34.\n3) (15-70)x = (40-70)·34 ⇒ x = 204/11.\n4) Then 34−x = 170/11.\nFinal answer: $204/11 L of 15% and 170/11 L of 70%$", "final_answer": "204/11 L of 15% and 170/11 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1696177236c38d9b", "problem_id": "08249263f91a30f3", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Two friends split $888 in the ratio 3:5. How much does each receive?", "output": "1) Total ratio parts = 3+5 = 8.\n2) Each part = 888 ÷ 8 = 111.\n3) First share = 3 × 111 = 333.\n4) Second share = 5 × 111 = 555.\nFinal answer: $555$", "final_answer": "555", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "48c068b874b2e8e5", "problem_id": "08249263f91a30f3", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $888 in the ratio 3:5. How much does each receive?", "output": "Total ratio parts = 3+5 = 8.\nEach part = 888 ÷ 8 = 111.\nFirst share = 3 × 111 = 333.\nSecond share = 5 × 111 = 555.\nCheck: 333+555=888.\nFinal answer: $555$", "final_answer": "555", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "64f363ec5fef87a6", "problem_id": "4fbe19566b24c506", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Two friends split $2048 in the ratio 7:9. How much does each receive?", "output": "1) Total ratio parts = 7+9 = 16.\n2) Each part = 2048 ÷ 16 = 128.\n3) First share = 7 × 128 = 896.\n4) Second share = 9 × 128 = 1152.\nFinal answer: $1152$", "final_answer": "1152", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4e11ce6b522eb119", "problem_id": "4fbe19566b24c506", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Two friends split $2048 in the ratio 7:9. How much does each receive?", "output": "Total ratio parts = 7+9 = 16.\nEach part = 2048 ÷ 16 = 128.\nFirst share = 7 × 128 = 896.\nSecond share = 9 × 128 = 1152.\nCheck: 896+1152=2048.\nFinal answer: $1152$", "final_answer": "1152", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8aa60495e3a3491b", "problem_id": "99b06a6bb87dc2b6", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{2} - 2 x$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $2 x - 2$.\n5) Evaluate at a: p'(2) = $2$.\nFinal answer: $2$", "final_answer": "2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1db8a1db15170edd", "problem_id": "80cd2d9d70c1bc48", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{3} \\sin{\\left(x \\right)} - x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x^{3} \\sin{\\left(x \\right)}) = 2 x^{3} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = 2 x^{3} \\cos{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x^{3} \\sin{\\left(x \\right)}) = 2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)}$.\nFinal answer: $2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)}$", "final_answer": "2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b9d78f0cc4b1afba", "problem_id": "c287ad3f17e3f5d3", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Worker A finishes a job in 15 hours and Worker B finishes it in 19 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/15 job/hour. B's rate = 1/19 job/hour.\n2) Combined rate = 1/15 + 1/19 = 34/285 job/hour.\n3) Time = 1 ÷ (combined rate) = 285/34 hours.\nFinal answer: $\\frac{285}{34}$", "final_answer": "\\frac{285}{34}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "42fd0580eaaf1ad9", "problem_id": "c287ad3f17e3f5d3", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Worker A finishes a job in 15 hours and Worker B finishes it in 19 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/15 job/hour. B's rate = 1/19 job/hour.\nCombined rate = 1/15 + 1/19 = 34/285 job/hour.\nTime = 1 ÷ (combined rate) = 285/34 hours.\nCheck: (combined rate)×(time) = 34/285×285/34 = 1.\nFinal answer: $\\frac{285}{34}$", "final_answer": "\\frac{285}{34}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "57ea544f4cec082e", "problem_id": "dde54addd5bba6dc", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the linear system:\n$5 x - 7 y$ = $-20$\n$- x - 3 y$ = $-18$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $- 5 x + 7 y$ = $20$.\n3) Multiply the second equation by $5$: $- 5 x - 15 y$ = $-90$.\n4) Subtract to eliminate x: $22 y$ = $110$.\n5) Solve for y: y = $5$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, 5\\right)$", "final_answer": "(x,y)=\\left(3, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e462be170412a08f", "problem_id": "90ce192073de0a89", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{4} - 2 x^{3} + x^{2} - 2 x$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x^{2}) = 2 x^{2}$.\n3) So $f'(x) = 8 x^{3} - 4 x^{2} - 2$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x^{2}) = 2 x$.\nTherefore the correct derivative is $f'(x) = 2 \\left(4 x^{3} - 3 x^{2} + x - 1\\right)$.\nFinal answer: $2 \\left(4 x^{3} - 3 x^{2} + x - 1\\right)$", "final_answer": "2 \\left(4 x^{3} - 3 x^{2} + x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6df460a5e98f3f73", "problem_id": "e304530bcd6e4f76", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A disease has prevalence $\\frac{3}{200}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{14}{125}$.\n3) Compute P(D|+) = $\\frac{27}{224}$.\nFinal answer: $\\frac{27}{224}$", "final_answer": "\\frac{27}{224}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ae7a24e7032e147b", "problem_id": "e304530bcd6e4f76", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A disease has prevalence $\\frac{3}{200}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{14}{125}$.\nCompute P(D|+) = $\\frac{27}{224}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{27}{224}$", "final_answer": "\\frac{27}{224}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "231f5041bcab5fc9", "problem_id": "78506250070eab8e", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "You want 25 liters of a 45% salt solution. You have a 15% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 25−x liters is 50% solution.\nSalt equation: 15x + 50(25−x) = 45·25.\n(15-50)x = (45-50)·25 ⇒ x = 25/7.\nThen 25−x = 150/7.\nCheck: 15·25/7 + 50·150/7 = 45·25.\nFinal answer: $25/7 L of 15% and 150/7 L of 50%$", "final_answer": "25/7 L of 15% and 150/7 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1853d048ec58a346", "problem_id": "78506250070eab8e", "style": "answer_only", "instruction": "Final answer only.", "input": "You want 25 liters of a 45% salt solution. You have a 15% solution and a 50% solution. How many liters of each should you mix?", "output": "25/7 L of 15% and 150/7 L of 50%", "final_answer": "25/7 L of 15% and 150/7 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "74921cb2948db476", "problem_id": "683f51bf6c7a9c8e", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{4} - x^{3} - x^{2} - 2 x - 3$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-3) = 0.\n3) d/dx(x^{4}) = 4 x^{3}.\n4) d/dx(- x^{2}) = - 2 x.\n5) d/dx(- x^{3}) = - 3 x^{2}.\n6) d/dx(- 2 x) = -2.\n7) Combine: f'(x) = 4 x^{3} - 3 x^{2} - 2 x - 2.\nFinal answer: $4\\cdot x^{3} - 3\\cdot x^{2} - 2\\cdot x - 2$", "final_answer": "4\\cdot x^{3} - 3\\cdot x^{2} - 2\\cdot x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a341153cdc689b3f", "problem_id": "683f51bf6c7a9c8e", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{4} - x^{3} - x^{2} - 2 x - 3$.", "output": "Differentiate term-by-term:\nd/dx(-3) = 0.\nd/dx(x^{4}) = 4 x^{3}.\nd/dx(- x^{2}) = - 2 x.\nd/dx(- x^{3}) = - 3 x^{2}.\nd/dx(- 2 x) = -2.\nCombine: f'(x) = 4 x^{3} - 3 x^{2} - 2 x - 2.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $4\\cdot x^{3} - 3\\cdot x^{2} - 2\\cdot x - 2$", "final_answer": "4\\cdot x^{3} - 3\\cdot x^{2} - 2\\cdot x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "407e761dfc184201", "problem_id": "8ace5bf22ab7eae3", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{3} - x^{2} - x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1}{x^{2} + 2}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 2}) = \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 2}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 8 x^{2} + 8} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 8 x^{2} + 8} - \\frac{6 x^{4}}{2 x^{4} + 8 x^{2} + 8} + \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 4 x^{2} + 4} + \\frac{4 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 8 x^{2} + 8} + \\frac{8 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 8 x^{2} + 8} - \\frac{34 x^{2}}{2 x^{4} + 8 x^{2} + 8} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 4} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 2} - \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 2} - \\frac{4 x}{2 x^{4} + 8 x^{2} + 8} - \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 2} + \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 8 x^{2} + 8} + \\frac{8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 8 x^{2} + 8} - \\frac{4}{2 x^{4} + 8 x^{2} + 8}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 2}) = - \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 2\\right)^{2}} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x^{2} + 2\\right)} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 2} + \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 2}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 4 x \\left(- 3 x^{3} - x^{2} - x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) + \\left(x^{2} + 2\\right) \\left(- 18 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4 x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x^{2} + 2\\right)^{2}}$.\nFinal answer: $\\frac{- 4 x \\left(- 3 x^{3} - x^{2} - x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) + \\left(x^{2} + 2\\right) \\left(- 18 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4 x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x^{2} + 2\\right)^{2}}$", "final_answer": "\\frac{- 4 x \\left(- 3 x^{3} - x^{2} - x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) + \\left(x^{2} + 2\\right) \\left(- 18 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4 x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x^{2} + 2\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2804ee4b81242833", "problem_id": "3abee4f70a3997b5", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{3} + 3 x^{2} - x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 3 x^{3} + 3 x^{2} - x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3$.\n3) Compute an antiderivative: F(x) = $- \\frac{3 x^{4}}{4} + x^{3} - \\frac{x^{2}}{2} - 3 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{3\\cdot x^{4}}{4} + x^{3} - \\frac{x^{2}}{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{3\\cdot x^{4}}{4} + x^{3} - \\frac{x^{2}}{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6abec8db5d521f1c", "problem_id": "8becc4ad099fc928", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 3 x^{2} + 12 x - 35$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 3 x^{3} + 3 x^{2} + x - 1$ and a = -3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 9 x^{2} + 6 x + 1$.\nEvaluate at a: p'(-3) = $-98$.\nCheck: the identity gives $-98$.\nFinal answer: $-98$", "final_answer": "-98", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d765bc1fa9ccd959", "problem_id": "8becc4ad099fc928", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 3 x^{2} + 12 x - 35$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 3 x^{3} + 3 x^{2} + x - 1$ and a = -3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 9 x^{2} + 6 x + 1$.\n5) Evaluate at a: p'(-3) = $-98$.\nFinal answer: $-98$", "final_answer": "-98", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "602431b12292cedf", "problem_id": "9395b4e943f77422", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 12 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=6, p=$\\frac{2}{7}$ and simplify.\n4) Here C(12,6) = 924.\nFinal answer: $\\frac{132000000}{1977326743}$", "final_answer": "\\frac{132000000}{1977326743}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "69c9ee3d1760d514", "problem_id": "9395b4e943f77422", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 12 times, what is the probability of exactly 6 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=12, k=6, p=$\\frac{2}{7}$ and simplify.\nHere C(12,6) = 924.\nCheck: expression simplifies to $\\frac{132000000}{1977326743}$.\nFinal answer: $\\frac{132000000}{1977326743}$", "final_answer": "\\frac{132000000}{1977326743}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "92834c0656229a37", "problem_id": "84277557b220502a", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the linear system:\n$2 x + 4 y$ = $-2$\n$- 4 x + y$ = $4$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $- 8 x - 16 y$ = $8$.\nMultiply the second equation by $2$: $- 8 x + 2 y$ = $8$.\nFinal answer: $(x,y)=\\left(-1, 0\\right)$", "final_answer": "(x,y)=\\left(-1, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "50b7523bedda8c3a", "problem_id": "84277557b220502a", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$2 x + 4 y$ = $-2$\n$- 4 x + y$ = $4$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $- 8 x - 16 y$ = $8$.\n3) Multiply the second equation by $2$: $- 8 x + 2 y$ = $8$.\n4) Subtract to eliminate x: $- 18 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, 0\\right)$", "final_answer": "(x,y)=\\left(-1, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f4620ff3a68ca3c1", "problem_id": "1dd74dc926005ace", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "How many binary strings of length 11 contain exactly 5 ones with no two ones adjacent?", "output": "1) Reserve 4 mandatory zeros between the 5 ones.\n2) This leaves n−k+1 = 7 valid slots for the ones.\n3) Choose the slots: C(7,5) = 21.\nFinal answer: $21$", "final_answer": "21", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1524716c30f44796", "problem_id": "1dd74dc926005ace", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many binary strings of length 11 contain exactly 5 ones with no two ones adjacent?", "output": "Reserve 4 mandatory zeros between the 5 ones.\nThis leaves n−k+1 = 7 valid slots for the ones.\nChoose the slots: C(7,5) = 21.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $21$", "final_answer": "21", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5e488188c4d0edd3", "problem_id": "beeb6a5043b46324", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 13 x + 42$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} + 13 x + 42$ = $\\left(x + 6\\right) \\left(x + 7\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-6$.\n5) From the second factor: x = $-7$.\nFinal answer: $x \\in \\{-6, -7\\}$", "final_answer": "x \\in \\{-6, -7\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "97bfab6920ad6c84", "problem_id": "beeb6a5043b46324", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 13 x + 42$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} + 13 x + 42$ = $\\left(x + 6\\right) \\left(x + 7\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $-6$.\nFrom the second factor: x = $-7$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-6, -7\\}$", "final_answer": "x \\in \\{-6, -7\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b8df0eb5b6c16639", "problem_id": "7257931eb0860ae1", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $520. It is discounted by 10%. You buy 7 backpacks. What is the total cost?", "output": "Discounted price = 520 × (1 − 10/100) = 520 × 90/100 = 468.\nSubtotal = 7 × 468 = 3276.\nCheck: recomputing the formula yields 3276.\nFinal answer: $3276$", "final_answer": "3276", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b6695836d42bb4a2", "problem_id": "7257931eb0860ae1", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A store sells a backpack for $520. It is discounted by 10%. You buy 7 backpacks. What is the total cost?", "output": "1) Discounted price = 520 × (1 − 10/100) = 520 × 90/100 = 468.\n2) Subtotal = 7 × 468 = 3276.\nFinal answer: $3276$", "final_answer": "3276", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6e1ac3842f6e5784", "problem_id": "edc2071dd59bbdd8", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_7=22$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_7=15.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=15 and k=7.\n4) Compute C(21,6) = 54264.\nFinal answer: $54264$", "final_answer": "54264", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5ebdb0d76175856d", "problem_id": "edc2071dd59bbdd8", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_7=22$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_7=15.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=15 and k=7.\nFinal answer: $54264$", "final_answer": "54264", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "710dc7d635a37522", "problem_id": "9e84074b80b7df79", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(- x^{2} - 2 x - 1\\right) \\cos{\\left(x^{2} + x - 1 \\right)}$.", "output": "1) Let f(x) = $\\left(x + 2\\right) e^{x} + \\left(- x^{2} - 2 x - 1\\right) \\cos{\\left(x^{2} + x - 1 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $2 \\left(- x - 1\\right) \\cos{\\left(x^{2} + x - 1 \\right)} + \\left(x + 2\\right) e^{x} - \\left(2 x + 1\\right) \\left(- x^{2} - 2 x - 1\\right) \\sin{\\left(x^{2} + x - 1 \\right)} + e^{x}$.\nFinal answer: $\\left(- 2\\cdot x - 2\\right)\\cdot \\cos{\\left(x^{2} + x - 1 \\right)} + \\left(x + 2\\right)\\cdot e^{x} - \\left(2\\cdot x + 1\\right)\\cdot \\left(- x^{2} - 2\\cdot x - 1\\right)\\cdot \\sin{\\left(x^{2} + x - 1 \\right)} + e^{x}$", "final_answer": "\\left(- 2\\cdot x - 2\\right)\\cdot \\cos{\\left(x^{2} + x - 1 \\right)} + \\left(x + 2\\right)\\cdot e^{x} - \\left(2\\cdot x + 1\\right)\\cdot \\left(- x^{2} - 2\\cdot x - 1\\right)\\cdot \\sin{\\left(x^{2} + x - 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2ef7d9d8e54490da", "problem_id": "9e84074b80b7df79", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(- x^{2} - 2 x - 1\\right) \\cos{\\left(x^{2} + x - 1 \\right)}$.", "output": "Let f(x) = $\\left(x + 2\\right) e^{x} + \\left(- x^{2} - 2 x - 1\\right) \\cos{\\left(x^{2} + x - 1 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $2 \\left(- x - 1\\right) \\cos{\\left(x^{2} + x - 1 \\right)} + \\left(x + 2\\right) e^{x} - \\left(2 x + 1\\right) \\left(- x^{2} - 2 x - 1\\right) \\sin{\\left(x^{2} + x - 1 \\right)} + e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\left(- 2\\cdot x - 2\\right)\\cdot \\cos{\\left(x^{2} + x - 1 \\right)} + \\left(x + 2\\right)\\cdot e^{x} - \\left(2\\cdot x + 1\\right)\\cdot \\left(- x^{2} - 2\\cdot x - 1\\right)\\cdot \\sin{\\left(x^{2} + x - 1 \\right)} + e^{x}$", "final_answer": "\\left(- 2\\cdot x - 2\\right)\\cdot \\cos{\\left(x^{2} + x - 1 \\right)} + \\left(x + 2\\right)\\cdot e^{x} - \\left(2\\cdot x + 1\\right)\\cdot \\left(- x^{2} - 2\\cdot x - 1\\right)\\cdot \\sin{\\left(x^{2} + x - 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "03392f2f812e9cf1", "problem_id": "e17bf3cc4d28163f", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} + 3 x^{2} - 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=1$.", "output": "1) Differentiate: f'(x) = $- 9 x^{2} + 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 3$.\n2) Slope at x=1: m = f'(1) = $-6 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}$.\n3) Point on curve: (1, f(1)) = (1, $-5 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -5 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} = -6 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}(x − 1).\n6) Optionally expand: y = $- 6 x - 2 x \\sin{\\left(2 \\right)} + \\frac{x e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 1 + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = - 6\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 1 + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 6\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 1 + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a887aa258a605c43", "problem_id": "e17bf3cc4d28163f", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} + 3 x^{2} - 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=1$.", "output": "Differentiate: f'(x) = $- 9 x^{2} + 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 3$.\nSlope at x=1: m = f'(1) = $-6 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}$.\nPoint on curve: (1, f(1)) = (1, $-5 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}}$).\nFinal answer: $y = - 6\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 1 + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 6\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 1 + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bc168b2a7abab11b", "problem_id": "a307689524598b01", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + \\frac{69 x}{5} + \\frac{36}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$\\frac{69}{5}$, c=$\\frac{36}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{2601}{25}$, so √Δ = $\\frac{51}{5}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{3}{5}$.\nx2 = $-4$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{3}{5}, -4\\}$", "final_answer": "x \\in \\{- \\frac{3}{5}, -4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a7e316c2416ad48b", "problem_id": "a307689524598b01", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + \\frac{69 x}{5} + \\frac{36}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$\\frac{69}{5}$, c=$\\frac{36}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{2601}{25}$, so √Δ = $\\frac{51}{5}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{3}{5}$.\n7) x2 = $-4$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{3}{5}, -4\\}$", "final_answer": "x \\in \\{- \\frac{3}{5}, -4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b519db59776de60f", "problem_id": "36dd0cfa55a1b7f4", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 21 x + 18$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} - 21 x + 18$ = $3 \\left(x - 6\\right) \\left(x - 1\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $6$.\n5) From the second factor: x = $1$.\nFinal answer: $x \\in \\{6, 1\\}$", "final_answer": "x \\in \\{6, 1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e956409842177fd7", "problem_id": "36dd0cfa55a1b7f4", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 21 x + 18$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} - 21 x + 18$ = $3 \\left(x - 6\\right) \\left(x - 1\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $6$.\nFrom the second factor: x = $1$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{6, 1\\}$", "final_answer": "x \\in \\{6, 1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "229210aec5baf02c", "problem_id": "d965a2634383bde8", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A cyclist rides at 79 miles per hour for 3/2 hours, then at 53 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 79 × 3/2 = 237/2 miles.\n2) Second leg distance = 53 × 2 = 106 miles.\n3) Total distance = 237/2 + 106 = 449/2 miles.\nFinal answer: $\\frac{449}{2}$", "final_answer": "\\frac{449}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2dfa70eb31f37bae", "problem_id": "d965a2634383bde8", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A cyclist rides at 79 miles per hour for 3/2 hours, then at 53 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 79 × 3/2 = 237/2 miles.\nSecond leg distance = 53 × 2 = 106 miles.\nTotal distance = 237/2 + 106 = 449/2 miles.\nCheck: adding both legs gives 449/2.\nFinal answer: $\\frac{449}{2}$", "final_answer": "\\frac{449}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "49430e038827d763", "problem_id": "48dbd6030f3b25ea", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $3 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} + 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)}$", "final_answer": "3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} + 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "59b4a0327aeaf7cd", "problem_id": "48dbd6030f3b25ea", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $3 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)}$.\nFinal answer: $3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} + 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)}$", "final_answer": "3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} + 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "15291491ed730cd8", "problem_id": "3c539f2219eb9e70", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_7=37$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_7=30.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=30 and k=7.\nCompute C(36,6) = 1947792.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $1947792$", "final_answer": "1947792", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a2ea256aad03492", "problem_id": "3c539f2219eb9e70", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_7=37$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_7=30.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=30 and k=7.\nFinal answer: $1947792$", "final_answer": "1947792", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "85bd44ac58fa3b39", "problem_id": "aaad1354f7760d33", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Two friends split $288 in the ratio 4:8. How much does each receive?", "output": "1) Total ratio parts = 4+8 = 12.\n2) Each part = 288 ÷ 12 = 24.\n3) First share = 4 × 24 = 96.\n4) Second share = 8 × 24 = 192.\nFinal answer: $192$", "final_answer": "192", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fb950b06adb0e672", "problem_id": "5657bd67d9d2a4a2", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} \\sin{\\left(x \\right)} - 3 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 \\cos{\\left(2 x \\right)}) = - 2 \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 \\cos{\\left(2 x \\right)}) = - 4 \\sin{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$", "final_answer": "3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "340e0d41871c3b73", "problem_id": "f988366da0f25484", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{3} - x^{2} - 3 x + \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{3 x - 2}$.", "output": "1) Let f(x) = $\\frac{- 3 x^{3} - x^{2} - 3 x + \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{3 x - 2}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{18 x^{3} + 6 x^{2} + 18 x + \\left(3 x - 2\\right) \\left(- 18 x^{2} - 4 x + \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6\\right) - 6 \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 12}{2 \\left(3 x - 2\\right)^{2}}$.\nFinal answer: $\\frac{18\\cdot x^{3} + 6\\cdot x^{2} + 18\\cdot x + \\left(3\\cdot x - 2\\right)\\cdot \\left(- 18\\cdot x^{2} - 4\\cdot x + \\left(3\\cdot x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x - 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 6\\right) - \\left(18\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 12}{2\\cdot \\left(3\\cdot x - 2\\right)^{2}}$", "final_answer": "\\frac{18\\cdot x^{3} + 6\\cdot x^{2} + 18\\cdot x + \\left(3\\cdot x - 2\\right)\\cdot \\left(- 18\\cdot x^{2} - 4\\cdot x + \\left(3\\cdot x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x - 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 6\\right) - \\left(18\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 12}{2\\cdot \\left(3\\cdot x - 2\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "10c51397248337ba", "problem_id": "f988366da0f25484", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{3} - x^{2} - 3 x + \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{3 x - 2}$.", "output": "Let f(x) = $\\frac{- 3 x^{3} - x^{2} - 3 x + \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{3 x - 2}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{18 x^{3} + 6 x^{2} + 18 x + \\left(3 x - 2\\right) \\left(- 18 x^{2} - 4 x + \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6\\right) - 6 \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 12}{2 \\left(3 x - 2\\right)^{2}}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\frac{18\\cdot x^{3} + 6\\cdot x^{2} + 18\\cdot x + \\left(3\\cdot x - 2\\right)\\cdot \\left(- 18\\cdot x^{2} - 4\\cdot x + \\left(3\\cdot x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x - 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 6\\right) - \\left(18\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 12}{2\\cdot \\left(3\\cdot x - 2\\right)^{2}}$", "final_answer": "\\frac{18\\cdot x^{3} + 6\\cdot x^{2} + 18\\cdot x + \\left(3\\cdot x - 2\\right)\\cdot \\left(- 18\\cdot x^{2} - 4\\cdot x + \\left(3\\cdot x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x - 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 6\\right) - \\left(18\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 12}{2\\cdot \\left(3\\cdot x - 2\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "13b31bb789d06015", "problem_id": "645a879382bec921", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} - 3 x^{4} + 3 x^{3} - x^{2} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{4} + 3 x^{3} - x^{2} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{99 \\pi^{5}}{160} - \\frac{45 \\pi^{4}}{64} - \\frac{3 \\pi^{2}}{2} - \\frac{3 \\pi^{3}}{8} - 3 \\pi + 5$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{99\\cdot \\pi^{5}}{160} - \\frac{45\\cdot \\pi^{4}}{64} - \\frac{3\\cdot \\pi^{2}}{2} - \\frac{3\\cdot \\pi^{3}}{8} - 3\\cdot \\pi + 5$", "final_answer": "- \\frac{99\\cdot \\pi^{5}}{160} - \\frac{45\\cdot \\pi^{4}}{64} - \\frac{3\\cdot \\pi^{2}}{2} - \\frac{3\\cdot \\pi^{3}}{8} - 3\\cdot \\pi + 5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e5e5a2892f52b59c", "problem_id": "645a879382bec921", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} - 3 x^{4} + 3 x^{3} - x^{2} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{4} + 3 x^{3} - x^{2} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{99 \\pi^{5}}{160} - \\frac{45 \\pi^{4}}{64} - \\frac{3 \\pi^{2}}{2} - \\frac{3 \\pi^{3}}{8} - 3 \\pi + 5$.\n4) Simplify the result.\nFinal answer: $- \\frac{99\\cdot \\pi^{5}}{160} - \\frac{45\\cdot \\pi^{4}}{64} - \\frac{3\\cdot \\pi^{2}}{2} - \\frac{3\\cdot \\pi^{3}}{8} - 3\\cdot \\pi + 5$", "final_answer": "- \\frac{99\\cdot \\pi^{5}}{160} - \\frac{45\\cdot \\pi^{4}}{64} - \\frac{3\\cdot \\pi^{2}}{2} - \\frac{3\\cdot \\pi^{3}}{8} - 3\\cdot \\pi + 5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cb7d983854536542", "problem_id": "994b635cba8097f4", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Worker A finishes a job in 4 hours and Worker B finishes it in 10 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/4 job/hour. B's rate = 1/10 job/hour.\n2) Combined rate = 1/4 + 1/10 = 7/20 job/hour.\n3) Time = 1 ÷ (combined rate) = 20/7 hours.\nFinal answer: $\\frac{20}{7}$", "final_answer": "\\frac{20}{7}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "04ee0958d728475b", "problem_id": "90b5c9913946f531", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- x^{2} - x - 2\\right) \\cos{\\left(2 x^{2} + 2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x \\cos{\\left(2 x^{2} + 2 x \\right)}) = x \\sin{\\left(2 x \\left(x + 1\\right) \\right)}$.\n3) So $f'(x) = 4 x^{3} \\sin{\\left(2 x^{2} + 2 x \\right)} + 2 x^{2} \\sin{\\left(2 x^{2} + 2 x \\right)} + 2 x e^{x} + x \\sin{\\left(2 x^{2} + 2 x \\right)} + 8 x \\sin{\\left(2 x^{2} + 2 x \\right)} - 2 x \\cos{\\left(2 x^{2} + 2 x \\right)} + 4 e^{x} + 4 \\sin{\\left(2 x^{2} + 2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x \\cos{\\left(2 x^{2} + 2 x \\right)}) = x \\left(4 x + 2\\right) \\sin{\\left(2 x^{2} + 2 x \\right)} - \\cos{\\left(2 x^{2} + 2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(- 2 x - 1\\right) \\cos{\\left(2 x^{2} + 2 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 2 \\left(2 x + 1\\right) \\left(x \\left(- x - 1\\right) - 2\\right) \\sin{\\left(2 x^{2} + 2 x \\right)} + 2 e^{x}$.\nFinal answer: $\\left(- 2 x - 1\\right) \\cos{\\left(2 x^{2} + 2 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 2 \\left(2 x + 1\\right) \\left(x \\left(- x - 1\\right) - 2\\right) \\sin{\\left(2 x^{2} + 2 x \\right)} + 2 e^{x}$", "final_answer": "\\left(- 2 x - 1\\right) \\cos{\\left(2 x^{2} + 2 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 2 \\left(2 x + 1\\right) \\left(x \\left(- x - 1\\right) - 2\\right) \\sin{\\left(2 x^{2} + 2 x \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6997754a37075f4d", "problem_id": "00632b5c552212fc", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{11} x + \\frac{14 x}{5} - \\frac{4 \\sqrt{11}}{5} + \\frac{8}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{14}{5} - \\sqrt{11}$, c=$\\frac{8}{5} - \\frac{4 \\sqrt{11}}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{311}{25} - \\frac{12 \\sqrt{11}}{5}$, so √Δ = $\\sqrt{\\frac{311}{25} - \\frac{12 \\sqrt{11}}{5}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{7}{5} + \\frac{\\sqrt{311 - 60 \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}$.\nx2 = $- \\frac{7}{5} - \\frac{\\sqrt{311 - 60 \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{7}{5} + \\frac{\\sqrt{311 - 60\\cdot \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}, - \\frac{7}{5} - \\frac{\\sqrt{311 - 60\\cdot \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{7}{5} + \\frac{\\sqrt{311 - 60\\cdot \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}, - \\frac{7}{5} - \\frac{\\sqrt{311 - 60\\cdot \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d8c082603bae58a6", "problem_id": "00632b5c552212fc", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{11} x + \\frac{14 x}{5} - \\frac{4 \\sqrt{11}}{5} + \\frac{8}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{14}{5} - \\sqrt{11}$, c=$\\frac{8}{5} - \\frac{4 \\sqrt{11}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{311}{25} - \\frac{12 \\sqrt{11}}{5}$, so √Δ = $\\sqrt{\\frac{311}{25} - \\frac{12 \\sqrt{11}}{5}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{7}{5} + \\frac{\\sqrt{311 - 60 \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}$.\n7) x2 = $- \\frac{7}{5} - \\frac{\\sqrt{311 - 60 \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{7}{5} + \\frac{\\sqrt{311 - 60\\cdot \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}, - \\frac{7}{5} - \\frac{\\sqrt{311 - 60\\cdot \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{7}{5} + \\frac{\\sqrt{311 - 60\\cdot \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}, - \\frac{7}{5} - \\frac{\\sqrt{311 - 60\\cdot \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "82a0a933ab3f46e2", "problem_id": "52c2d5dc48680aa9", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=- x^{2} + x - 3$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-3) = 0.\n3) d/dx(x) = 1.\n4) d/dx(- x^{2}) = - 2 x.\n5) Combine: f'(x) = 1 - 2 x.\nFinal answer: $1 - 2\\cdot x$", "final_answer": "1 - 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b8117928aad8cc7a", "problem_id": "52c2d5dc48680aa9", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{2} + x - 3$.", "output": "Differentiate term-by-term:\nd/dx(-3) = 0.\nd/dx(x) = 1.\nFinal answer: $1 - 2\\cdot x$", "final_answer": "1 - 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ed2736a766eea411", "problem_id": "1d522edd8fd061e5", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "You want 35 liters of a 40% salt solution. You have a 10% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 35−x liters is 60% solution.\n2) Salt equation: 10x + 60(35−x) = 40·35.\n3) (10-60)x = (40-60)·35 ⇒ x = 14.\n4) Then 35−x = 21.\nFinal answer: $14 L of 10% and 21 L of 60%$", "final_answer": "14 L of 10% and 21 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5826e6a2a61ca90e", "problem_id": "1d522edd8fd061e5", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "You want 35 liters of a 40% salt solution. You have a 10% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 35−x liters is 60% solution.\nSalt equation: 10x + 60(35−x) = 40·35.\n(10-60)x = (40-60)·35 ⇒ x = 14.\nThen 35−x = 21.\nCheck: 10·14 + 60·21 = 40·35.\nFinal answer: $14 L of 10% and 21 L of 60%$", "final_answer": "14 L of 10% and 21 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "26c5b0582764b0d8", "problem_id": "782a2880d9f340b4", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} + x^{2} - 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 3 x^{2} + 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 3$.\n2) Slope at x=2: m = f'(2) = $-11 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\n3) Point on curve: (2, f(2)) = (2, $-9 + \\cos{\\left(4 \\right)} + e$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -9 + \\cos{\\left(4 \\right)} + e = -11 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\n6) Optionally expand: y = $- 11 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 13$.\nFinal answer: $y = - 11\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 13$", "final_answer": "y = - 11\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 13", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8724562be92bc9db", "problem_id": "782a2880d9f340b4", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} + x^{2} - 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=2$.", "output": "Differentiate: f'(x) = $- 3 x^{2} + 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 3$.\nSlope at x=2: m = f'(2) = $-11 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\nPoint on curve: (2, f(2)) = (2, $-9 + \\cos{\\left(4 \\right)} + e$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -9 + \\cos{\\left(4 \\right)} + e = -11 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\nOptionally expand: y = $- 11 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 13$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 11\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 13$", "final_answer": "y = - 11\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 13", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a983f2cfbc03e7c4", "problem_id": "08dad0ffebbc8c3d", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x - 4$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{2} - 2 x$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 2 x - 2$.\n5) Evaluate at a: p'(2) = $-6$.\nFinal answer: $-6$", "final_answer": "-6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8134c82cade1e12d", "problem_id": "08dad0ffebbc8c3d", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x - 4$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{2} - 2 x$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 2 x - 2$.\nEvaluate at a: p'(2) = $-6$.\nCheck: the identity gives $-6$.\nFinal answer: $-6$", "final_answer": "-6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cd524cd4056dc8ae", "problem_id": "9b5e9e5e01c0a7ae", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$x$ = $2$\n$11 x - y$ = $17$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $11$: $11 x$ = $22$.\n3) Multiply the second equation by $1$: $11 x - y$ = $17$.\n4) Subtract to eliminate x: $y$ = $5$.\n5) Solve for y: y = $5$.\n6) Substitute back to find x: x = $2$.\nFinal answer: $(x,y)=\\left(2, 5\\right)$", "final_answer": "(x,y)=\\left(2, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "74c79574e038a8b2", "problem_id": "9b5e9e5e01c0a7ae", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$x$ = $2$\n$11 x - y$ = $17$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $11$: $11 x$ = $22$.\nMultiply the second equation by $1$: $11 x - y$ = $17$.\nSubtract to eliminate x: $y$ = $5$.\nSolve for y: y = $5$.\nSubstitute back to find x: x = $2$.\nCheck: plug (x,y)=(2,5) into both equations.\nFinal answer: $(x,y)=\\left(2, 5\\right)$", "final_answer": "(x,y)=\\left(2, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0a5435156f74be0b", "problem_id": "acdd413b5fdca8a9", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x - 2$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 2 x^{2} + 2 x$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-6$", "final_answer": "-6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "24f2c3e0e49f306f", "problem_id": "f3c5ad0f91d4a7ee", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\sin{\\left(x \\right)}) = - \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\sin{\\left(x \\right)}) = - \\cos{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.\nFinal answer: $- 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$", "final_answer": "- 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5a7bc79357415924", "problem_id": "9a345ef92e12ac4d", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $1890 in the ratio 8:6. How much does each receive?", "output": "1) Total ratio parts = 8+6 = 14.\n2) Each part = 1890 ÷ 14 = 135.\n3) First share = 8 × 135 = 1080.\n4) Second share = 6 × 135 = 810.\nFinal answer: $810$", "final_answer": "810", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ad8ab4215ded3143", "problem_id": "9a345ef92e12ac4d", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Two friends split $1890 in the ratio 8:6. How much does each receive?", "output": "Total ratio parts = 8+6 = 14.\nEach part = 1890 ÷ 14 = 135.\nFirst share = 8 × 135 = 1080.\nFinal answer: $810$", "final_answer": "810", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c010be79d2f12505", "problem_id": "9eb0936006f1b647", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$3 y$ = $12$\n$x - y$ = $-6$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $3 y$ = $12$.\n3) Multiply the second equation by $0$: $0$ = $0$.\n4) Subtract to eliminate x: $3 y$ = $12$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, 4\\right)$", "final_answer": "(x,y)=\\left(-2, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "df5cd85466bfd703", "problem_id": "9eb0936006f1b647", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$3 y$ = $12$\n$x - y$ = $-6$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $3 y$ = $12$.\nMultiply the second equation by $0$: $0$ = $0$.\nSubtract to eliminate x: $3 y$ = $12$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $-2$.\nCheck: plug (x,y)=(-2,4) into both equations.\nFinal answer: $(x,y)=\\left(-2, 4\\right)$", "final_answer": "(x,y)=\\left(-2, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f7f87e33219c3ae5", "problem_id": "cc5bf0a30ffd57f9", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x^{3} - 2 x - 2$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{4} - 2 x^{3} - 2 x^{2} + 2 x + 3$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $4 x^{3} - 6 x^{2} - 4 x + 2$.\n5) Evaluate at a: p'(2) = $2$.\nFinal answer: $2$", "final_answer": "2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1ee38e9a4cb1c872", "problem_id": "cc5bf0a30ffd57f9", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x^{3} - 2 x - 2$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $x^{4} - 2 x^{3} - 2 x^{2} + 2 x + 3$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $4 x^{3} - 6 x^{2} - 4 x + 2$.\nEvaluate at a: p'(2) = $2$.\nCheck: the identity gives $2$.\nFinal answer: $2$", "final_answer": "2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "05f5f27a56e2bb34", "problem_id": "5610b2360a289d8e", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$11 x - 3 y$ = $-56$\n$11 x + 8 y$ = $-12$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $11$: $121 x - 33 y$ = $-616$.\n3) Multiply the second equation by $11$: $121 x + 88 y$ = $-132$.\n4) Subtract to eliminate x: $- 121 y$ = $-484$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $-4$.\nFinal answer: $(x,y)=\\left(-4, 4\\right)$", "final_answer": "(x,y)=\\left(-4, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "486d1b3019f31197", "problem_id": "5610b2360a289d8e", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$11 x - 3 y$ = $-56$\n$11 x + 8 y$ = $-12$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $11$: $121 x - 33 y$ = $-616$.\nMultiply the second equation by $11$: $121 x + 88 y$ = $-132$.\nSubtract to eliminate x: $- 121 y$ = $-484$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $-4$.\nCheck: plug (x,y)=(-4,4) into both equations.\nFinal answer: $(x,y)=\\left(-4, 4\\right)$", "final_answer": "(x,y)=\\left(-4, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2be874d3ad4a3c2c", "problem_id": "9c0e6a5d83cac615", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{3} - 2 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $3 x^{3} - 2 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1$.\n3) Compute an antiderivative: F(x) = $\\frac{3 x^{4}}{4} - \\frac{2 x^{3}}{3} - x^{2} - x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{3\\cdot x^{4}}{4} - \\frac{2\\cdot x^{3}}{3} - x^{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{3\\cdot x^{4}}{4} - \\frac{2\\cdot x^{3}}{3} - x^{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "46cd14adc09ae621", "problem_id": "9c0e6a5d83cac615", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{3} - 2 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $3 x^{3} - 2 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1$.\nCompute an antiderivative: F(x) = $\\frac{3 x^{4}}{4} - \\frac{2 x^{3}}{3} - x^{2} - x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{3\\cdot x^{4}}{4} - \\frac{2\\cdot x^{3}}{3} - x^{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{3\\cdot x^{4}}{4} - \\frac{2\\cdot x^{3}}{3} - x^{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a8c0aee94f75c1c", "problem_id": "91be852a4be0da32", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} 3 x^{3} + 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{3} + 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $-2 + 5 \\pi + \\frac{15 \\pi^{2}}{4} + \\frac{65 \\pi^{3}}{8} + \\frac{765 \\pi^{4}}{64}$.\n4) Simplify the result.\nFinal answer: $-2 + 5\\cdot \\pi + \\frac{15\\cdot \\pi^{2}}{4} + \\frac{65\\cdot \\pi^{3}}{8} + \\frac{765\\cdot \\pi^{4}}{64}$", "final_answer": "-2 + 5\\cdot \\pi + \\frac{15\\cdot \\pi^{2}}{4} + \\frac{65\\cdot \\pi^{3}}{8} + \\frac{765\\cdot \\pi^{4}}{64}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2c7c2d900ce138ce", "problem_id": "91be852a4be0da32", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} 3 x^{3} + 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{3} + 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $-2 + 5 \\pi + \\frac{15 \\pi^{2}}{4} + \\frac{65 \\pi^{3}}{8} + \\frac{765 \\pi^{4}}{64}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $-2 + 5\\cdot \\pi + \\frac{15\\cdot \\pi^{2}}{4} + \\frac{65\\cdot \\pi^{3}}{8} + \\frac{765\\cdot \\pi^{4}}{64}$", "final_answer": "-2 + 5\\cdot \\pi + \\frac{15\\cdot \\pi^{2}}{4} + \\frac{65\\cdot \\pi^{3}}{8} + \\frac{765\\cdot \\pi^{4}}{64}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c6379e585883e1e1", "problem_id": "4f2433c1712cc523", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$- 2 x - 8 y$ = $-42$\n$11 x + 8 y$ = $123$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $11$: $- 22 x - 88 y$ = $-462$.\nMultiply the second equation by $-2$: $- 22 x - 16 y$ = $-246$.\nSubtract to eliminate x: $- 72 y$ = $-216$.\nSolve for y: y = $3$.\nSubstitute back to find x: x = $9$.\nCheck: plug (x,y)=(9,3) into both equations.\nFinal answer: $(x,y)=\\left(9, 3\\right)$", "final_answer": "(x,y)=\\left(9, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e4f7665601d589d3", "problem_id": "4f2433c1712cc523", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$- 2 x - 8 y$ = $-42$\n$11 x + 8 y$ = $123$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $11$: $- 22 x - 88 y$ = $-462$.\n3) Multiply the second equation by $-2$: $- 22 x - 16 y$ = $-246$.\n4) Subtract to eliminate x: $- 72 y$ = $-216$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $9$.\nFinal answer: $(x,y)=\\left(9, 3\\right)$", "final_answer": "(x,y)=\\left(9, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "83c23d031b7a95a4", "problem_id": "ff64fd6a8ff032fc", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Two friends split $1392 in the ratio 8:8. How much does each receive?", "output": "1) Total ratio parts = 8+8 = 16.\n2) Each part = 1392 ÷ 16 = 87.\n3) First share = 8 × 87 = 696.\n4) Second share = 8 × 87 = 696.\nFinal answer: $696$", "final_answer": "696", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "594fa5cdf9877a4b", "problem_id": "ff64fd6a8ff032fc", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Two friends split $1392 in the ratio 8:8. How much does each receive?", "output": "Total ratio parts = 8+8 = 16.\nEach part = 1392 ÷ 16 = 87.\nFirst share = 8 × 87 = 696.\nSecond share = 8 × 87 = 696.\nCheck: 696+696=1392.\nFinal answer: $696$", "final_answer": "696", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "91c32972901e9885", "problem_id": "b869ad7040807f94", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{22 x}{3} + \\frac{20}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$\\frac{22}{3}$, c=$\\frac{20}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{4}{9}$, so √Δ = $\\frac{2}{3}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{5}{3}$.\n7) x2 = $-2$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{5}{3}, -2\\}$", "final_answer": "x \\in \\{- \\frac{5}{3}, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "186ba84d35685ad2", "problem_id": "b869ad7040807f94", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{22 x}{3} + \\frac{20}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$\\frac{22}{3}$, c=$\\frac{20}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{4}{9}$, so √Δ = $\\frac{2}{3}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{5}{3}$.\nx2 = $-2$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{5}{3}, -2\\}$", "final_answer": "x \\in \\{- \\frac{5}{3}, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "98a2f94593c71498", "problem_id": "c323dc62c89936b1", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the multiplicative inverse of 34 modulo 37.", "output": "1) We want x such that 34x ≡ 1 (mod 37).\n2) Use extended Euclid to find s,t with 34s + 37t = 1.\n3) One Bezout identity is: 34·(12) + 37·(-11) = 1.\n4) Reduce s modulo 37: x ≡ 12 (mod 37).\nFinal answer: $x\\equiv 12\\pmod{37}$", "final_answer": "x\\equiv 12\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "297b4ff0f328619d", "problem_id": "c323dc62c89936b1", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 34 modulo 37.", "output": "We want x such that 34x ≡ 1 (mod 37).\nUse extended Euclid to find s,t with 34s + 37t = 1.\nOne Bezout identity is: 34·(12) + 37·(-11) = 1.\nReduce s modulo 37: x ≡ 12 (mod 37).\nCheck: 34·12 mod 37 = 1.\nFinal answer: $x\\equiv 12\\pmod{37}$", "final_answer": "x\\equiv 12\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "20e64b8db9e6fcd4", "problem_id": "c2b5f5c54c174436", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Worker A finishes a job in 10 hours and Worker B finishes it in 8 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/10 job/hour. B's rate = 1/8 job/hour.\nCombined rate = 1/10 + 1/8 = 9/40 job/hour.\nTime = 1 ÷ (combined rate) = 40/9 hours.\nFinal answer: $\\frac{40}{9}$", "final_answer": "\\frac{40}{9}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7b65158fc0d5ee94", "problem_id": "c2b5f5c54c174436", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Worker A finishes a job in 10 hours and Worker B finishes it in 8 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/10 job/hour. B's rate = 1/8 job/hour.\n2) Combined rate = 1/10 + 1/8 = 9/40 job/hour.\n3) Time = 1 ÷ (combined rate) = 40/9 hours.\nFinal answer: $\\frac{40}{9}$", "final_answer": "\\frac{40}{9}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "93240809f7fcb4cf", "problem_id": "ffc833387e88e11e", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_4=27$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=27 and k=4.\n4) Compute C(30,3) = 4060.\nFinal answer: $4060$", "final_answer": "4060", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f01dbc654a5d327", "problem_id": "ffc833387e88e11e", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_4=27$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=27 and k=4.\nFinal answer: $4060$", "final_answer": "4060", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "257b01e62be5e537", "problem_id": "1c79bedd36efa53e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 3 x^{3} - 3 x^{2} + 2 x - 5$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} - 3 x^{2} + x - 5.\n3) Then the limit is -12.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: -13.\nFinal answer: $-13$", "final_answer": "-13", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "fe85ffdfb7420c8b", "problem_id": "60b0132d2714ed61", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} x + 1$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{2}$ and a = 1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $2 x$.\n5) Evaluate at a: p'(1) = $2$.\nFinal answer: $2$", "final_answer": "2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bb2b0f3735bf8dc7", "problem_id": "2a5bf4a3eaf8fe24", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} 3 x^{3} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{3} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\pi).\n3) Subtract: F(\\pi) − F(- \\pi) = $4 \\pi$.\n4) Simplify the result.\nFinal answer: $4\\cdot \\pi$", "final_answer": "4\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c1a83439c3491dc0", "problem_id": "2a5bf4a3eaf8fe24", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} 3 x^{3} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{3} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2$.\nEvaluate at the bounds: F(\\pi) and F(- \\pi).\nSubtract: F(\\pi) − F(- \\pi) = $4 \\pi$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $4\\cdot \\pi$", "final_answer": "4\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d79eff14da331265", "problem_id": "e719ef7795f5d8c0", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$2 x + 9 y$ = $-97$\n$11 x - 4 y$ = $-52$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $11$: $22 x + 99 y$ = $-1067$.\n3) Multiply the second equation by $2$: $22 x - 8 y$ = $-104$.\n4) Subtract to eliminate x: $107 y$ = $-963$.\n5) Solve for y: y = $-9$.\n6) Substitute back to find x: x = $-8$.\nFinal answer: $(x,y)=\\left(-8, -9\\right)$", "final_answer": "(x,y)=\\left(-8, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1b8e0eb2131b134f", "problem_id": "e719ef7795f5d8c0", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$2 x + 9 y$ = $-97$\n$11 x - 4 y$ = $-52$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $11$: $22 x + 99 y$ = $-1067$.\nMultiply the second equation by $2$: $22 x - 8 y$ = $-104$.\nSubtract to eliminate x: $107 y$ = $-963$.\nSolve for y: y = $-9$.\nSubstitute back to find x: x = $-8$.\nCheck: plug (x,y)=(-8,-9) into both equations.\nFinal answer: $(x,y)=\\left(-8, -9\\right)$", "final_answer": "(x,y)=\\left(-8, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b1ff95e753decacf", "problem_id": "ada46df922c050bc", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Worker A finishes a job in 21 hours and Worker B finishes it in 8 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/21 job/hour. B's rate = 1/8 job/hour.\nCombined rate = 1/21 + 1/8 = 29/168 job/hour.\nTime = 1 ÷ (combined rate) = 168/29 hours.\nCheck: (combined rate)×(time) = 29/168×168/29 = 1.\nFinal answer: $\\frac{168}{29}$", "final_answer": "\\frac{168}{29}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bea6176ce9730516", "problem_id": "ada46df922c050bc", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Worker A finishes a job in 21 hours and Worker B finishes it in 8 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/21 job/hour. B's rate = 1/8 job/hour.\n2) Combined rate = 1/21 + 1/8 = 29/168 job/hour.\n3) Time = 1 ÷ (combined rate) = 168/29 hours.\nFinal answer: $\\frac{168}{29}$", "final_answer": "\\frac{168}{29}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e2305b9e91a4106c", "problem_id": "e18150a3c6b67b40", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$- 2 x + 3 y$ = $12$\n$- 8 x - 8 y$ = $48$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-8$: $16 x - 24 y$ = $-96$.\n3) Multiply the second equation by $-2$: $16 x + 16 y$ = $-96$.\n4) Subtract to eliminate x: $- 40 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $-6$.\nFinal answer: $(x,y)=\\left(-6, 0\\right)$", "final_answer": "(x,y)=\\left(-6, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "20d75c8e0a9ca414", "problem_id": "e18150a3c6b67b40", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$- 2 x + 3 y$ = $12$\n$- 8 x - 8 y$ = $48$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-8$: $16 x - 24 y$ = $-96$.\nMultiply the second equation by $-2$: $16 x + 16 y$ = $-96$.\nSubtract to eliminate x: $- 40 y$ = $0$.\nSolve for y: y = $0$.\nSubstitute back to find x: x = $-6$.\nCheck: plug (x,y)=(-6,0) into both equations.\nFinal answer: $(x,y)=\\left(-6, 0\\right)$", "final_answer": "(x,y)=\\left(-6, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e50a391174ca7d0c", "problem_id": "b41c9fbf976de78a", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$2 x - 3 y$ = $2$\n$- 4 y$ = $8$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $2$: $- 8 y$ = $16$.\n4) Subtract to eliminate x: $8 y$ = $-16$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, -2\\right)$", "final_answer": "(x,y)=\\left(-2, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "47a8aa24479d62f7", "problem_id": "b41c9fbf976de78a", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$2 x - 3 y$ = $2$\n$- 4 y$ = $8$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $2$: $- 8 y$ = $16$.\nSubtract to eliminate x: $8 y$ = $-16$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $-2$.\nCheck: plug (x,y)=(-2,-2) into both equations.\nFinal answer: $(x,y)=\\left(-2, -2\\right)$", "final_answer": "(x,y)=\\left(-2, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cd5ea6bfff08ed6e", "problem_id": "770e2975b990c857", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{2} + 2 \\sin{\\left(x \\right)} - 3$ at $x=0$.", "output": "1) Differentiate: f'(x) = $6 x + 2 \\cos{\\left(x \\right)}$.\n2) Slope at x=0: m = f'(0) = $2$.\n3) Point on curve: (0, f(0)) = (0, $-3$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -3 = 2(x − 0).\n6) Optionally expand: y = $2 x - 3$.\nFinal answer: $y = 2\\cdot x - 3$", "final_answer": "y = 2\\cdot x - 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "221780b93e6dbd47", "problem_id": "770e2975b990c857", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{2} + 2 \\sin{\\left(x \\right)} - 3$ at $x=0$.", "output": "Differentiate: f'(x) = $6 x + 2 \\cos{\\left(x \\right)}$.\nSlope at x=0: m = f'(0) = $2$.\nPoint on curve: (0, f(0)) = (0, $-3$).\nFinal answer: $y = 2\\cdot x - 3$", "final_answer": "y = 2\\cdot x - 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b81176e18fdbfd59", "problem_id": "c71792387e0c8cf7", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$- 2 x - 3 y$ = $-7$\n$2 x$ = $-8$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $- 4 x - 6 y$ = $-14$.\n3) Multiply the second equation by $-2$: $- 4 x$ = $16$.\n4) Subtract to eliminate x: $- 6 y$ = $-30$.\n5) Solve for y: y = $5$.\n6) Substitute back to find x: x = $-4$.\nFinal answer: $(x,y)=\\left(-4, 5\\right)$", "final_answer": "(x,y)=\\left(-4, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e9090cddb986feba", "problem_id": "32d8d20afd531b76", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{3} + x^{2} - \\frac{x}{2} + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - \\frac{1}{2}}{x - 4}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}) = \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4}$.\n3) So $f'(x) = \\frac{4 x^{3}}{2 x^{2} - 16 x + 32} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{22 x^{2}}{2 x^{2} - 16 x + 32} - \\frac{8 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{16 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{16 x}{2 x^{2} - 16 x + 32} + \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 8 x + 16} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 8} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} - \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} + \\frac{16 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{32 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{5}{2 x^{2} - 16 x + 32} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}) = \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x - 4\\right)} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x - 4\\right)^{2}} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 2 x^{3} - 2 x^{2} + x + \\left(x - 4\\right) \\left(6 x^{2} + 4 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{2 \\left(x - 4\\right)^{2}}$.\nFinal answer: $\\frac{- 2 x^{3} - 2 x^{2} + x + \\left(x - 4\\right) \\left(6 x^{2} + 4 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{2 \\left(x - 4\\right)^{2}}$", "final_answer": "\\frac{- 2 x^{3} - 2 x^{2} + x + \\left(x - 4\\right) \\left(6 x^{2} + 4 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{2 \\left(x - 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6803ea1fc1ef578d", "problem_id": "6bb5d86b915a15a8", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A disease has prevalence $\\frac{1}{25}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{21}{250}$.\n3) Compute P(D|+) = $\\frac{3}{7}$.\nFinal answer: $\\frac{3}{7}$", "final_answer": "\\frac{3}{7}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3690cd18fbd7dd58", "problem_id": "6bb5d86b915a15a8", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A disease has prevalence $\\frac{1}{25}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{21}{250}$.\nCompute P(D|+) = $\\frac{3}{7}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{3}{7}$", "final_answer": "\\frac{3}{7}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "849efd090e7952d4", "problem_id": "871e00af79571721", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A disease has prevalence $\\frac{3}{50}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{29}{200}$.\n3) Compute P(D|+) = $\\frac{51}{145}$.\nFinal answer: $\\frac{51}{145}$", "final_answer": "\\frac{51}{145}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e7a6135960852e1c", "problem_id": "adfe202ac964bfdc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{4} + 3 x^{3} + 3 x^{2} - 3 x + 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(3 x^{3}) = 9 x^{3}$.\n3) So $f'(x) = 25 x^{3} + 6 x - 3$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(3 x^{3}) = 9 x^{2}$.\nTherefore the correct derivative is $f'(x) = 16 x^{3} + 9 x^{2} + 6 x - 3$.\nFinal answer: $16 x^{3} + 9 x^{2} + 6 x - 3$", "final_answer": "16 x^{3} + 9 x^{2} + 6 x - 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "232fa6562fb8377c", "problem_id": "f2b253315096701c", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "You want 69 liters of a 47% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 69−x liters is 70% solution.\n2) Salt equation: 25x + 70(69−x) = 47·69.\n3) (25-70)x = (47-70)·69 ⇒ x = 529/15.\n4) Then 69−x = 506/15.\nFinal answer: $529/15 L of 25% and 506/15 L of 70%$", "final_answer": "529/15 L of 25% and 506/15 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6b49c98e2dd783e4", "problem_id": "f2b253315096701c", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "You want 69 liters of a 47% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 69−x liters is 70% solution.\nSalt equation: 25x + 70(69−x) = 47·69.\n(25-70)x = (47-70)·69 ⇒ x = 529/15.\nThen 69−x = 506/15.\nCheck: 25·529/15 + 70·506/15 = 47·69.\nFinal answer: $529/15 L of 25% and 506/15 L of 70%$", "final_answer": "529/15 L of 25% and 506/15 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c03ed23331062070", "problem_id": "dbe34a8b224711d0", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 x^{3} - 4 x^{2} + 11 x - 19$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{4} + 2 x^{3} + 3 x^{2} + 3 x - 1$ and a = -2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $12 x^{3} + 6 x^{2} + 6 x + 3$.\n5) Evaluate at a: p'(-2) = $-81$.\nFinal answer: $-81$", "final_answer": "-81", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b05d842c190fe64f", "problem_id": "dbe34a8b224711d0", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 x^{3} - 4 x^{2} + 11 x - 19$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $3 x^{4} + 2 x^{3} + 3 x^{2} + 3 x - 1$ and a = -2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $12 x^{3} + 6 x^{2} + 6 x + 3$.\nEvaluate at a: p'(-2) = $-81$.\nCheck: the identity gives $-81$.\nFinal answer: $-81$", "final_answer": "-81", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7e09e5a245815ae5", "problem_id": "ee4275a7c99e600b", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$5 x - 5 y$ = $-40$\n$7 x + 3 y$ = $34$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $7$: $35 x - 35 y$ = $-280$.\n3) Multiply the second equation by $5$: $35 x + 15 y$ = $170$.\n4) Subtract to eliminate x: $- 50 y$ = $-450$.\n5) Solve for y: y = $9$.\n6) Substitute back to find x: x = $1$.\nFinal answer: $(x,y)=\\left(1, 9\\right)$", "final_answer": "(x,y)=\\left(1, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "56daa55989b73451", "problem_id": "ee4275a7c99e600b", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$5 x - 5 y$ = $-40$\n$7 x + 3 y$ = $34$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $7$: $35 x - 35 y$ = $-280$.\nMultiply the second equation by $5$: $35 x + 15 y$ = $170$.\nSubtract to eliminate x: $- 50 y$ = $-450$.\nSolve for y: y = $9$.\nSubstitute back to find x: x = $1$.\nCheck: plug (x,y)=(1,9) into both equations.\nFinal answer: $(x,y)=\\left(1, 9\\right)$", "final_answer": "(x,y)=\\left(1, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "995b83817a79d7b9", "problem_id": "09f37b2d4477ce8b", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 10 times, what is the probability of exactly 8 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=8, p=$\\frac{1}{6}$ and simplify.\n4) Here C(10,8) = 45.\nFinal answer: $\\frac{125}{6718464}$", "final_answer": "\\frac{125}{6718464}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e1d8267360542d6", "problem_id": "127845561a322a32", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{3} + 4 x^{2} + 7 x + 11$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} + 4 x^{2} + 6 x + 11.\n3) Then the limit is 55.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 57.\nFinal answer: $57$", "final_answer": "57", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8b546aaa1ef3a6be", "problem_id": "960630399ae4b845", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(- 3 x^{2} - x + 2\\right) \\cos{\\left(3 x - 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x \\cos{\\left(3 x - 1 \\right)}) = x \\sin{\\left(3 x - 1 \\right)}$.\n3) So $f'(x) = 9 x^{2} \\sin{\\left(3 x - 1 \\right)} + x e^{x} + x \\sin{\\left(3 x - 1 \\right)} - 6 x \\cos{\\left(3 x - 1 \\right)} + 2 e^{x} - 6 \\sin{\\left(3 x - 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x \\cos{\\left(3 x - 1 \\right)}) = 3 x \\sin{\\left(3 x - 1 \\right)} - \\cos{\\left(3 x - 1 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(- 6 x - 1\\right) \\cos{\\left(3 x - 1 \\right)} + \\left(x + 1\\right) e^{x} - 3 \\left(x \\left(- 3 x - 1\\right) + 2\\right) \\sin{\\left(3 x - 1 \\right)} + e^{x}$.\nFinal answer: $\\left(- 6 x - 1\\right) \\cos{\\left(3 x - 1 \\right)} + \\left(x + 1\\right) e^{x} - 3 \\left(x \\left(- 3 x - 1\\right) + 2\\right) \\sin{\\left(3 x - 1 \\right)} + e^{x}$", "final_answer": "\\left(- 6 x - 1\\right) \\cos{\\left(3 x - 1 \\right)} + \\left(x + 1\\right) e^{x} - 3 \\left(x \\left(- 3 x - 1\\right) + 2\\right) \\sin{\\left(3 x - 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "75ed43111b0203d9", "problem_id": "499bbb0910db0390", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} - 3 x^{2} + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $- 6 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)}$.\n2) Slope at x=-1: m = f'(-1) = $\\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\n3) Point on curve: (-1, f(-1)) = (-1, $\\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} = \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}(x − -1).\n6) Optionally expand: y = $\\frac{x}{2 e^{\\frac{1}{2}}} + 2 x \\sin{\\left(2 \\right)} + \\cos{\\left(2 \\right)} + \\frac{3}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dea87fb4609600aa", "problem_id": "b9e1c0e12a27da3d", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "An urn has 7 red balls and 11 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(18,6).\n3) Favorable samples: choose k reds and n−k blues: C(7,2)·C(11,4).\n4) Compute totals: C(18,6) = 18564, and favorable = 6930.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{165}{442}$", "final_answer": "\\frac{165}{442}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1a29299dd8173d8e", "problem_id": "b9e1c0e12a27da3d", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 7 red balls and 11 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(18,6).\nFavorable samples: choose k reds and n−k blues: C(7,2)·C(11,4).\nCompute totals: C(18,6) = 18564, and favorable = 6930.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{165}{442}$.\nFinal answer: $\\frac{165}{442}$", "final_answer": "\\frac{165}{442}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e7ad6f894e242473", "problem_id": "c8810b2eb4f78db6", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + x - 4$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$1$, c=$-4$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $49$, so √Δ = $7$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $1$.\n7) x2 = $- \\frac{4}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{1, - \\frac{4}{3}\\}$", "final_answer": "x \\in \\{1, - \\frac{4}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a52270d46543e29b", "problem_id": "c8810b2eb4f78db6", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + x - 4$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$1$, c=$-4$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{1, - \\frac{4}{3}\\}$", "final_answer": "x \\in \\{1, - \\frac{4}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "182a48128a580d69", "problem_id": "2fa50dffd89f8e84", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 12 x + 36$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} - 12 x + 36$ = $\\left(x - 6\\right)^{2}$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $6$.\nFrom the second factor: x = $6$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{6, 6\\}$", "final_answer": "x \\in \\{6, 6\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9978c608197226bc", "problem_id": "2fa50dffd89f8e84", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 12 x + 36$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} - 12 x + 36$ = $\\left(x - 6\\right)^{2}$.\nSet each factor equal to 0 and solve each linear equation.\nFinal answer: $x \\in \\{6, 6\\}$", "final_answer": "x \\in \\{6, 6\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ce091de2932e116c", "problem_id": "2f3fe191e32dfe01", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 7 red balls and 12 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(19,6).\nFavorable samples: choose k reds and n−k blues: C(7,3)·C(12,3).\nCompute totals: C(19,6) = 27132, and favorable = 7700.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{275}{969}$.\nFinal answer: $\\frac{275}{969}$", "final_answer": "\\frac{275}{969}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dc2faff7fa3a41b5", "problem_id": "2f3fe191e32dfe01", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 7 red balls and 12 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(19,6).\n3) Favorable samples: choose k reds and n−k blues: C(7,3)·C(12,3).\n4) Compute totals: C(19,6) = 27132, and favorable = 7700.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{275}{969}$", "final_answer": "\\frac{275}{969}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d5d92d36e3f278c9", "problem_id": "f9125b49ac9ca289", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Two friends split $1209 in the ratio 8:5. How much does each receive?", "output": "1) Total ratio parts = 8+5 = 13.\n2) Each part = 1209 ÷ 13 = 93.\n3) First share = 8 × 93 = 744.\n4) Second share = 5 × 93 = 465.\nFinal answer: $465$", "final_answer": "465", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3ee59a15a9128a79", "problem_id": "f9125b49ac9ca289", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $1209 in the ratio 8:5. How much does each receive?", "output": "Total ratio parts = 8+5 = 13.\nEach part = 1209 ÷ 13 = 93.\nFirst share = 8 × 93 = 744.\nSecond share = 5 × 93 = 465.\nCheck: 744+465=1209.\nFinal answer: $465$", "final_answer": "465", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "462cec9d2be7626b", "problem_id": "bfda8b2283764b53", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$x + 11 y$ = $41$\n$4 x + 9 y$ = $59$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $4 x + 44 y$ = $164$.\nMultiply the second equation by $1$: $4 x + 9 y$ = $59$.\nSubtract to eliminate x: $35 y$ = $105$.\nSolve for y: y = $3$.\nSubstitute back to find x: x = $8$.\nCheck: plug (x,y)=(8,3) into both equations.\nFinal answer: $(x,y)=\\left(8, 3\\right)$", "final_answer": "(x,y)=\\left(8, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aa74d58d087d1040", "problem_id": "bfda8b2283764b53", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$x + 11 y$ = $41$\n$4 x + 9 y$ = $59$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $4 x + 44 y$ = $164$.\n3) Multiply the second equation by $1$: $4 x + 9 y$ = $59$.\n4) Subtract to eliminate x: $35 y$ = $105$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $8$.\nFinal answer: $(x,y)=\\left(8, 3\\right)$", "final_answer": "(x,y)=\\left(8, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "99d70172f87b889e", "problem_id": "f2180e82c1812c9a", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 1$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} - 1$ = $\\left(x - 1\\right) \\left(x + 1\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $1$.\nFrom the second factor: x = $-1$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{1, -1\\}$", "final_answer": "x \\in \\{1, -1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "844c72b84b7ca2d3", "problem_id": "f2180e82c1812c9a", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 1$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} - 1$ = $\\left(x - 1\\right) \\left(x + 1\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFinal answer: $x \\in \\{1, -1\\}$", "final_answer": "x \\in \\{1, -1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5e21cb0ee1e9d50f", "problem_id": "c157bd1f2df41e16", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 2 x - \\frac{20}{9}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$-2$, c=$- \\frac{20}{9}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{196}{9}$, so √Δ = $\\frac{14}{3}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{5}{3}$.\n7) x2 = $- \\frac{2}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{5}{3}, - \\frac{2}{3}\\}$", "final_answer": "x \\in \\{\\frac{5}{3}, - \\frac{2}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "05582209bc1b2479", "problem_id": "0afa354efc24e7f3", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - 3 x^{2} - 11 x - 43$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} - 12 x - 43.\n3) Then the limit is -139.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: -135.\nFinal answer: $-135$", "final_answer": "-135", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "4f36434b2705bb4f", "problem_id": "848a7bb7abc58db0", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "You want 50 liters of a 50% salt solution. You have a 30% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 50−x liters is 70% solution.\n2) Salt equation: 30x + 70(50−x) = 50·50.\n3) (30-70)x = (50-70)·50 ⇒ x = 25.\n4) Then 50−x = 25.\nFinal answer: $25 L of 30% and 25 L of 70%$", "final_answer": "25 L of 30% and 25 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a70568a4c086eba", "problem_id": "aae1fb5eaa1aecc6", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} - 3 x^{2} + 11 x - 41$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} + 10 x - 41.\n3) Then the limit is -129.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: -133.\nFinal answer: $-133$", "final_answer": "-133", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f2855716dc6c6267", "problem_id": "a229db38963f5070", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Two friends split $1575 in the ratio 1:8. How much does each receive?", "output": "1) Total ratio parts = 1+8 = 9.\n2) Each part = 1575 ÷ 9 = 175.\n3) First share = 1 × 175 = 175.\n4) Second share = 8 × 175 = 1400.\nFinal answer: $1400$", "final_answer": "1400", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a3a20aa8f740ae51", "problem_id": "d6bb2ee279f828cb", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} + x + 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(3 x^{2}) = 6 x^{2}$.\n3) So $f'(x) = 6 x^{2} + 1$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(3 x^{2}) = 6 x$.\nTherefore the correct derivative is $f'(x) = 6 x + 1$.\nFinal answer: $6 x + 1$", "final_answer": "6 x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "4dced1616e54d934", "problem_id": "05546e2d620452c5", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A cyclist rides at 33 miles per hour for 1/2 hours, then at 48 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 33 × 1/2 = 33/2 miles.\nSecond leg distance = 48 × 3 = 144 miles.\nTotal distance = 33/2 + 144 = 321/2 miles.\nCheck: adding both legs gives 321/2.\nFinal answer: $\\frac{321}{2}$", "final_answer": "\\frac{321}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "954325e6234dbc53", "problem_id": "05546e2d620452c5", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A cyclist rides at 33 miles per hour for 1/2 hours, then at 48 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 33 × 1/2 = 33/2 miles.\n2) Second leg distance = 48 × 3 = 144 miles.\n3) Total distance = 33/2 + 144 = 321/2 miles.\nFinal answer: $\\frac{321}{2}$", "final_answer": "\\frac{321}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c7044acad74bb05d", "problem_id": "2207868cead6cefd", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the congruences: x≡2 (mod 4), x≡7 (mod 9). Give the solution modulo 36.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 36.\nCompute the CRT solution to get x ≡ 34 (mod 36).\nFinal answer: $x\\equiv 34\\pmod{36}$", "final_answer": "x\\equiv 34\\pmod{36}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fa53c2e43f7db90a", "problem_id": "2207868cead6cefd", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡2 (mod 4), x≡7 (mod 9). Give the solution modulo 36.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 36.\n3) Compute the CRT solution to get x ≡ 34 (mod 36).\nFinal answer: $x\\equiv 34\\pmod{36}$", "final_answer": "x\\equiv 34\\pmod{36}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "de86289dc34d05fa", "problem_id": "02f5ba068679179f", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(- x^{3} + 2 x^{2} - 2 x - 2\\right) \\cos{\\left(2 x - 2 \\right)}$.", "output": "1) Let f(x) = $\\left(x - 3\\right) e^{x} + \\left(- x^{3} + 2 x^{2} - 2 x - 2\\right) \\cos{\\left(2 x - 2 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\left(x - 3\\right) e^{x} + \\left(- 3 x^{2} + 4 x - 2\\right) \\cos{\\left(2 x - 2 \\right)} - 2 \\left(- x^{3} + 2 x^{2} - 2 x - 2\\right) \\sin{\\left(2 x - 2 \\right)} + e^{x}$.\nFinal answer: $\\left(x - 3\\right)\\cdot e^{x} + \\left(- 3\\cdot x^{2} + 4\\cdot x - 2\\right)\\cdot \\cos{\\left(2\\cdot x - 2 \\right)} - \\left(- 2\\cdot x^{3} + 4\\cdot x^{2} - 4\\cdot x - 4\\right)\\cdot \\sin{\\left(2\\cdot x - 2 \\right)} + e^{x}$", "final_answer": "\\left(x - 3\\right)\\cdot e^{x} + \\left(- 3\\cdot x^{2} + 4\\cdot x - 2\\right)\\cdot \\cos{\\left(2\\cdot x - 2 \\right)} - \\left(- 2\\cdot x^{3} + 4\\cdot x^{2} - 4\\cdot x - 4\\right)\\cdot \\sin{\\left(2\\cdot x - 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "492d612c6ff53dda", "problem_id": "02f5ba068679179f", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(- x^{3} + 2 x^{2} - 2 x - 2\\right) \\cos{\\left(2 x - 2 \\right)}$.", "output": "Let f(x) = $\\left(x - 3\\right) e^{x} + \\left(- x^{3} + 2 x^{2} - 2 x - 2\\right) \\cos{\\left(2 x - 2 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\left(x - 3\\right) e^{x} + \\left(- 3 x^{2} + 4 x - 2\\right) \\cos{\\left(2 x - 2 \\right)} - 2 \\left(- x^{3} + 2 x^{2} - 2 x - 2\\right) \\sin{\\left(2 x - 2 \\right)} + e^{x}$.\nFinal answer: $\\left(x - 3\\right)\\cdot e^{x} + \\left(- 3\\cdot x^{2} + 4\\cdot x - 2\\right)\\cdot \\cos{\\left(2\\cdot x - 2 \\right)} - \\left(- 2\\cdot x^{3} + 4\\cdot x^{2} - 4\\cdot x - 4\\right)\\cdot \\sin{\\left(2\\cdot x - 2 \\right)} + e^{x}$", "final_answer": "\\left(x - 3\\right)\\cdot e^{x} + \\left(- 3\\cdot x^{2} + 4\\cdot x - 2\\right)\\cdot \\cos{\\left(2\\cdot x - 2 \\right)} - \\left(- 2\\cdot x^{3} + 4\\cdot x^{2} - 4\\cdot x - 4\\right)\\cdot \\sin{\\left(2\\cdot x - 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bc58a4dc257cde0c", "problem_id": "a534b0892681a1e7", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$- 7 x$ = $-7$\n$2 x + 4 y$ = $-34$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $- 14 x$ = $-14$.\n3) Multiply the second equation by $-7$: $- 14 x - 28 y$ = $238$.\n4) Subtract to eliminate x: $28 y$ = $-252$.\n5) Solve for y: y = $-9$.\n6) Substitute back to find x: x = $1$.\nFinal answer: $(x,y)=\\left(1, -9\\right)$", "final_answer": "(x,y)=\\left(1, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ffa01e8fa4242271", "problem_id": "a534b0892681a1e7", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- 7 x$ = $-7$\n$2 x + 4 y$ = $-34$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $2$: $- 14 x$ = $-14$.\nMultiply the second equation by $-7$: $- 14 x - 28 y$ = $238$.\nSubtract to eliminate x: $28 y$ = $-252$.\nSolve for y: y = $-9$.\nSubstitute back to find x: x = $1$.\nCheck: plug (x,y)=(1,-9) into both equations.\nFinal answer: $(x,y)=\\left(1, -9\\right)$", "final_answer": "(x,y)=\\left(1, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "173bfa6a646ceee5", "problem_id": "e3000c6b13e01f35", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(- x^{3} - 3 x^{2} + x - 2\\right) \\cos{\\left(3 x - 1 \\right)}$.", "output": "1) Let f(x) = $2 \\left(x + 2\\right) e^{x} + \\left(- x^{3} - 3 x^{2} + x - 2\\right) \\cos{\\left(3 x - 1 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $2 \\left(x + 2\\right) e^{x} + \\left(- 3 x^{2} - 6 x + 1\\right) \\cos{\\left(3 x - 1 \\right)} - 3 \\left(- x^{3} - 3 x^{2} + x - 2\\right) \\sin{\\left(3 x - 1 \\right)} + 2 e^{x}$.\nFinal answer: $\\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(- 3\\cdot x^{2} - 6\\cdot x + 1\\right)\\cdot \\cos{\\left(3\\cdot x - 1 \\right)} - \\left(- 3\\cdot x^{3} - 9\\cdot x^{2} + 3\\cdot x - 6\\right)\\cdot \\sin{\\left(3\\cdot x - 1 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(- 3\\cdot x^{2} - 6\\cdot x + 1\\right)\\cdot \\cos{\\left(3\\cdot x - 1 \\right)} - \\left(- 3\\cdot x^{3} - 9\\cdot x^{2} + 3\\cdot x - 6\\right)\\cdot \\sin{\\left(3\\cdot x - 1 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "14c960214f3607c2", "problem_id": "e3000c6b13e01f35", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(- x^{3} - 3 x^{2} + x - 2\\right) \\cos{\\left(3 x - 1 \\right)}$.", "output": "Let f(x) = $2 \\left(x + 2\\right) e^{x} + \\left(- x^{3} - 3 x^{2} + x - 2\\right) \\cos{\\left(3 x - 1 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $2 \\left(x + 2\\right) e^{x} + \\left(- 3 x^{2} - 6 x + 1\\right) \\cos{\\left(3 x - 1 \\right)} - 3 \\left(- x^{3} - 3 x^{2} + x - 2\\right) \\sin{\\left(3 x - 1 \\right)} + 2 e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(- 3\\cdot x^{2} - 6\\cdot x + 1\\right)\\cdot \\cos{\\left(3\\cdot x - 1 \\right)} - \\left(- 3\\cdot x^{3} - 9\\cdot x^{2} + 3\\cdot x - 6\\right)\\cdot \\sin{\\left(3\\cdot x - 1 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(- 3\\cdot x^{2} - 6\\cdot x + 1\\right)\\cdot \\cos{\\left(3\\cdot x - 1 \\right)} - \\left(- 3\\cdot x^{3} - 9\\cdot x^{2} + 3\\cdot x - 6\\right)\\cdot \\sin{\\left(3\\cdot x - 1 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "89cbef88c8c98c4a", "problem_id": "d77a3f5cbecd3116", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} - 3 x^{2} + x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $9 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 1$.\n2) Slope at x=-2: m = f'(-2) = $2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e} + 49$.\n3) Point on curve: (-2, f(-2)) = (-2, $-40 + \\cos{\\left(4 \\right)} + e^{-1}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -40 + \\cos{\\left(4 \\right)} + e^{-1} = 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e} + 49(x − -2).\n6) Optionally expand: y = $2 x \\sin{\\left(4 \\right)} + \\frac{x}{2 e} + 49 x + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 58$.\nFinal answer: $y = 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 49\\cdot x + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 58$", "final_answer": "y = 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 49\\cdot x + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 58", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "66720b5a024a7d41", "problem_id": "d77a3f5cbecd3116", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} - 3 x^{2} + x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=-2$.", "output": "Differentiate: f'(x) = $9 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 1$.\nSlope at x=-2: m = f'(-2) = $2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e} + 49$.\nPoint on curve: (-2, f(-2)) = (-2, $-40 + \\cos{\\left(4 \\right)} + e^{-1}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -40 + \\cos{\\left(4 \\right)} + e^{-1} = 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e} + 49(x − -2).\nOptionally expand: y = $2 x \\sin{\\left(4 \\right)} + \\frac{x}{2 e} + 49 x + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 58$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 49\\cdot x + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 58$", "final_answer": "y = 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 49\\cdot x + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 58", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "be2be0f177627d81", "problem_id": "50962f02415211b5", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 x}{2} - \\sqrt{2} x - \\frac{9}{2} - \\frac{3 \\sqrt{2}}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{3}{2} - \\sqrt{2}$, c=$- \\frac{9}{2} - \\frac{3 \\sqrt{2}}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $9 \\sqrt{2} + \\frac{89}{4}$, so √Δ = $\\sqrt{9 \\sqrt{2} + \\frac{89}{4}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{2}}{2} + \\frac{3}{4} + \\frac{\\sqrt{36 \\sqrt{2} + 89}}{4}$.\n7) x2 = $- \\frac{\\sqrt{36 \\sqrt{2} + 89}}{4} + \\frac{\\sqrt{2}}{2} + \\frac{3}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{2}}{2} + \\frac{3}{4} + \\frac{\\sqrt{36\\cdot \\sqrt{2} + 89}}{4}, - \\frac{\\sqrt{36\\cdot \\sqrt{2} + 89}}{4} + \\frac{\\sqrt{2}}{2} + \\frac{3}{4}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{2}}{2} + \\frac{3}{4} + \\frac{\\sqrt{36\\cdot \\sqrt{2} + 89}}{4}, - \\frac{\\sqrt{36\\cdot \\sqrt{2} + 89}}{4} + \\frac{\\sqrt{2}}{2} + \\frac{3}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8a5b3ab087617ba6", "problem_id": "50962f02415211b5", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 x}{2} - \\sqrt{2} x - \\frac{9}{2} - \\frac{3 \\sqrt{2}}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{3}{2} - \\sqrt{2}$, c=$- \\frac{9}{2} - \\frac{3 \\sqrt{2}}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $9 \\sqrt{2} + \\frac{89}{4}$, so √Δ = $\\sqrt{9 \\sqrt{2} + \\frac{89}{4}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{2}}{2} + \\frac{3}{4} + \\frac{\\sqrt{36 \\sqrt{2} + 89}}{4}$.\nx2 = $- \\frac{\\sqrt{36 \\sqrt{2} + 89}}{4} + \\frac{\\sqrt{2}}{2} + \\frac{3}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{2}}{2} + \\frac{3}{4} + \\frac{\\sqrt{36\\cdot \\sqrt{2} + 89}}{4}, - \\frac{\\sqrt{36\\cdot \\sqrt{2} + 89}}{4} + \\frac{\\sqrt{2}}{2} + \\frac{3}{4}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{2}}{2} + \\frac{3}{4} + \\frac{\\sqrt{36\\cdot \\sqrt{2} + 89}}{4}, - \\frac{\\sqrt{36\\cdot \\sqrt{2} + 89}}{4} + \\frac{\\sqrt{2}}{2} + \\frac{3}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e69dd55bc6137681", "problem_id": "6dd3183d7ce389ae", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A disease has prevalence $\\frac{3}{50}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{13}{125}$.\n3) Compute P(D|+) = $\\frac{57}{104}$.\nFinal answer: $\\frac{57}{104}$", "final_answer": "\\frac{57}{104}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a70568a4c086eba", "problem_id": "aae1fb5eaa1aecc6", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} - 3 x^{2} + 11 x - 41$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} + 10 x - 41.\n3) Then the limit is -129.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: -133.\nFinal answer: $-133$", "final_answer": "-133", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6410aae79c02c1aa", "problem_id": "49db87a164778292", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A cyclist rides at 63 miles per hour for 2 hours, then at 39 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 63 × 2 = 126 miles.\n2) Second leg distance = 39 × 2 = 78 miles.\n3) Total distance = 126 + 78 = 204 miles.\nFinal answer: $204$", "final_answer": "204", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bf4c927c9db4ca49", "problem_id": "49db87a164778292", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 63 miles per hour for 2 hours, then at 39 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 63 × 2 = 126 miles.\nSecond leg distance = 39 × 2 = 78 miles.\nTotal distance = 126 + 78 = 204 miles.\nCheck: adding both legs gives 204.\nFinal answer: $204$", "final_answer": "204", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e365506e18401ab5", "problem_id": "e89bceed71deb113", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many binary strings of length 14 contain exactly 5 ones with no two ones adjacent?", "output": "Reserve 4 mandatory zeros between the 5 ones.\nThis leaves n−k+1 = 10 valid slots for the ones.\nChoose the slots: C(10,5) = 252.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $252$", "final_answer": "252", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "46c4fbda702ba77c", "problem_id": "e89bceed71deb113", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "How many binary strings of length 14 contain exactly 5 ones with no two ones adjacent?", "output": "Reserve 4 mandatory zeros between the 5 ones.\nThis leaves n−k+1 = 10 valid slots for the ones.\nChoose the slots: C(10,5) = 252.\nFinal answer: $252$", "final_answer": "252", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5679762c2a2da403", "problem_id": "95be78d1be32240a", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_9=32$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_9=23.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=23 and k=9.\n4) Compute C(31,8) = 7888725.\nFinal answer: $7888725$", "final_answer": "7888725", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4fab4d0b2f4390aa", "problem_id": "95be78d1be32240a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_9=32$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_9=23.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=23 and k=9.\nCompute C(31,8) = 7888725.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $7888725$", "final_answer": "7888725", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9164553d2fd66985", "problem_id": "effc55a3494b5316", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 14 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=14, k=5, p=$\\frac{2}{7}$ and simplify.\n4) Here C(14,5) = 2002.\nFinal answer: $\\frac{17875000000}{96889010407}$", "final_answer": "\\frac{17875000000}{96889010407}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "23c7e13f1540f0c8", "problem_id": "effc55a3494b5316", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 14 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=14, k=5, p=$\\frac{2}{7}$ and simplify.\nHere C(14,5) = 2002.\nCheck: expression simplifies to $\\frac{17875000000}{96889010407}$.\nFinal answer: $\\frac{17875000000}{96889010407}$", "final_answer": "\\frac{17875000000}{96889010407}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aadf970ba45a865f", "problem_id": "9bbeb39f202378b5", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡10 (mod 11), x≡7 (mod 8). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 87 (mod 88).\nFinal answer: $x\\equiv 87\\pmod{88}$", "final_answer": "x\\equiv 87\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "342ee61552060b9b", "problem_id": "9bbeb39f202378b5", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡10 (mod 11), x≡7 (mod 8). Give the solution modulo 88.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 88.\nCompute the CRT solution to get x ≡ 87 (mod 88).\nCheck: 87 mod 11=10 and mod 8=7.\nFinal answer: $x\\equiv 87\\pmod{88}$", "final_answer": "x\\equiv 87\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "85fd2e1905d4563d", "problem_id": "afddbee639b720a3", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "You want 53 liters of a 40% salt solution. You have a 20% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 53−x liters is 50% solution.\n2) Salt equation: 20x + 50(53−x) = 40·53.\n3) (20-50)x = (40-50)·53 ⇒ x = 53/3.\n4) Then 53−x = 106/3.\nFinal answer: $53/3 L of 20% and 106/3 L of 50%$", "final_answer": "53/3 L of 20% and 106/3 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7200b6e528879cbd", "problem_id": "da68ca0158e4bdcc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{4} - 2 x^{2} - 2 x + 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- x^{4}) = - 4 x^{4}$.\n3) So $f'(x) = - 4 x^{4} - 4 x - 2$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- x^{4}) = - 4 x^{3}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- 2 x^{3} - 2 x - 1\\right)$.\nFinal answer: $2 \\left(- 2 x^{3} - 2 x - 1\\right)$", "final_answer": "2 \\left(- 2 x^{3} - 2 x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e9724fdbd171733d", "problem_id": "aba2666775358f91", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$5 x^{2} - 5 \\sqrt{7} x - 7 x + 7 \\sqrt{7}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$5$, b=$- 5 \\sqrt{7} - 7$, c=$7 \\sqrt{7}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $224 - 70 \\sqrt{7}$, so √Δ = $\\sqrt{224 - 70 \\sqrt{7}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{224 - 70 \\sqrt{7}}}{10} + \\frac{7}{10} + \\frac{\\sqrt{7}}{2}$.\n7) x2 = $- \\frac{\\sqrt{224 - 70 \\sqrt{7}}}{10} + \\frac{7}{10} + \\frac{\\sqrt{7}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{224 - 70\\cdot \\sqrt{7}}}{10} + \\frac{7}{10} + \\frac{\\sqrt{7}}{2}, - \\frac{\\sqrt{224 - 70\\cdot \\sqrt{7}}}{10} + \\frac{7}{10} + \\frac{\\sqrt{7}}{2}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{224 - 70\\cdot \\sqrt{7}}}{10} + \\frac{7}{10} + \\frac{\\sqrt{7}}{2}, - \\frac{\\sqrt{224 - 70\\cdot \\sqrt{7}}}{10} + \\frac{7}{10} + \\frac{\\sqrt{7}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f497576d6ab67b91", "problem_id": "aba2666775358f91", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$5 x^{2} - 5 \\sqrt{7} x - 7 x + 7 \\sqrt{7}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$5$, b=$- 5 \\sqrt{7} - 7$, c=$7 \\sqrt{7}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $224 - 70 \\sqrt{7}$, so √Δ = $\\sqrt{224 - 70 \\sqrt{7}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{224 - 70 \\sqrt{7}}}{10} + \\frac{7}{10} + \\frac{\\sqrt{7}}{2}$.\nx2 = $- \\frac{\\sqrt{224 - 70 \\sqrt{7}}}{10} + \\frac{7}{10} + \\frac{\\sqrt{7}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{224 - 70\\cdot \\sqrt{7}}}{10} + \\frac{7}{10} + \\frac{\\sqrt{7}}{2}, - \\frac{\\sqrt{224 - 70\\cdot \\sqrt{7}}}{10} + \\frac{7}{10} + \\frac{\\sqrt{7}}{2}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{224 - 70\\cdot \\sqrt{7}}}{10} + \\frac{7}{10} + \\frac{\\sqrt{7}}{2}, - \\frac{\\sqrt{224 - 70\\cdot \\sqrt{7}}}{10} + \\frac{7}{10} + \\frac{\\sqrt{7}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "15457e8fa5e5ca2f", "problem_id": "80224c89fe71d9d5", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Worker A finishes a job in 9 hours and Worker B finishes it in 5 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/9 job/hour. B's rate = 1/5 job/hour.\n2) Combined rate = 1/9 + 1/5 = 14/45 job/hour.\n3) Time = 1 ÷ (combined rate) = 45/14 hours.\nFinal answer: $\\frac{45}{14}$", "final_answer": "\\frac{45}{14}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "52a3e7e351110e67", "problem_id": "80224c89fe71d9d5", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Worker A finishes a job in 9 hours and Worker B finishes it in 5 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/9 job/hour. B's rate = 1/5 job/hour.\nCombined rate = 1/9 + 1/5 = 14/45 job/hour.\nTime = 1 ÷ (combined rate) = 45/14 hours.\nCheck: (combined rate)×(time) = 14/45×45/14 = 1.\nFinal answer: $\\frac{45}{14}$", "final_answer": "\\frac{45}{14}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db7dd9424b7ec4ff", "problem_id": "cdf049942ef3f132", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Worker A finishes a job in 18 hours and Worker B finishes it in 14 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/18 job/hour. B's rate = 1/14 job/hour.\n2) Combined rate = 1/18 + 1/14 = 8/63 job/hour.\n3) Time = 1 ÷ (combined rate) = 63/8 hours.\nFinal answer: $\\frac{63}{8}$", "final_answer": "\\frac{63}{8}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "209974d22567e1a3", "problem_id": "cdf049942ef3f132", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Worker A finishes a job in 18 hours and Worker B finishes it in 14 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/18 job/hour. B's rate = 1/14 job/hour.\nCombined rate = 1/18 + 1/14 = 8/63 job/hour.\nTime = 1 ÷ (combined rate) = 63/8 hours.\nCheck: (combined rate)×(time) = 8/63×63/8 = 1.\nFinal answer: $\\frac{63}{8}$", "final_answer": "\\frac{63}{8}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "305626fd82da47e8", "problem_id": "a3343dda6c927aaf", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} - x + \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{x + 6}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 6}) = \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 6}$.\n3) So $f'(x) = \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 24 x + 72} + \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + 24 x + 72} - \\frac{2 x^{2}}{2 x^{2} + 24 x + 72} + \\frac{12 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 24 x + 72} + \\frac{24 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + 24 x + 72} - \\frac{24 x}{2 x^{2} + 24 x + 72} + \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 12 x + 36} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x + 12} - \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 6} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 6} + \\frac{36 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 24 x + 72} + \\frac{72 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + 24 x + 72} - \\frac{6}{2 x^{2} + 24 x + 72} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 6}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 6}) = \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x + 6\\right)} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 6} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x + 6\\right)^{2}} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 6}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 x^{2} + 2 x + \\left(x + 6\\right) \\left(- 4 x + \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 2 \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6}{2 \\left(x + 6\\right)^{2}}$.\nFinal answer: $\\frac{2 x^{2} + 2 x + \\left(x + 6\\right) \\left(- 4 x + \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 2 \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6}{2 \\left(x + 6\\right)^{2}}$", "final_answer": "\\frac{2 x^{2} + 2 x + \\left(x + 6\\right) \\left(- 4 x + \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 2 \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6}{2 \\left(x + 6\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c1f97aa1af77ea6c", "problem_id": "f69d5ac15b747e68", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡2 (mod 4), x≡4 (mod 7). Give the solution modulo 28.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 28.\n3) Compute the CRT solution to get x ≡ 18 (mod 28).\nFinal answer: $x\\equiv 18\\pmod{28}$", "final_answer": "x\\equiv 18\\pmod{28}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "87417c219b9d9e72", "problem_id": "4688a6df9c433add", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{2} x - \\frac{x}{2} - \\frac{3}{2} + \\frac{9 \\sqrt{2}}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- 3 \\sqrt{2} - \\frac{1}{2}$, c=$- \\frac{3}{2} + \\frac{9 \\sqrt{2}}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{97}{4} - 15 \\sqrt{2}$, so √Δ = $\\sqrt{\\frac{97}{4} - 15 \\sqrt{2}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{1}{4} + \\frac{\\sqrt{97 - 60 \\sqrt{2}}}{4} + \\frac{3 \\sqrt{2}}{2}$.\n7) x2 = $- \\frac{\\sqrt{97 - 60 \\sqrt{2}}}{4} + \\frac{1}{4} + \\frac{3 \\sqrt{2}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{1}{4} + \\frac{\\sqrt{97 - 60\\cdot \\sqrt{2}}}{4} + \\frac{3\\cdot \\sqrt{2}}{2}, - \\frac{\\sqrt{97 - 60\\cdot \\sqrt{2}}}{4} + \\frac{1}{4} + \\frac{3\\cdot \\sqrt{2}}{2}\\}$", "final_answer": "x \\in \\{\\frac{1}{4} + \\frac{\\sqrt{97 - 60\\cdot \\sqrt{2}}}{4} + \\frac{3\\cdot \\sqrt{2}}{2}, - \\frac{\\sqrt{97 - 60\\cdot \\sqrt{2}}}{4} + \\frac{1}{4} + \\frac{3\\cdot \\sqrt{2}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eef37a9c883514bd", "problem_id": "4688a6df9c433add", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{2} x - \\frac{x}{2} - \\frac{3}{2} + \\frac{9 \\sqrt{2}}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- 3 \\sqrt{2} - \\frac{1}{2}$, c=$- \\frac{3}{2} + \\frac{9 \\sqrt{2}}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{97}{4} - 15 \\sqrt{2}$, so √Δ = $\\sqrt{\\frac{97}{4} - 15 \\sqrt{2}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{1}{4} + \\frac{\\sqrt{97 - 60 \\sqrt{2}}}{4} + \\frac{3 \\sqrt{2}}{2}$.\nx2 = $- \\frac{\\sqrt{97 - 60 \\sqrt{2}}}{4} + \\frac{1}{4} + \\frac{3 \\sqrt{2}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{1}{4} + \\frac{\\sqrt{97 - 60\\cdot \\sqrt{2}}}{4} + \\frac{3\\cdot \\sqrt{2}}{2}, - \\frac{\\sqrt{97 - 60\\cdot \\sqrt{2}}}{4} + \\frac{1}{4} + \\frac{3\\cdot \\sqrt{2}}{2}\\}$", "final_answer": "x \\in \\{\\frac{1}{4} + \\frac{\\sqrt{97 - 60\\cdot \\sqrt{2}}}{4} + \\frac{3\\cdot \\sqrt{2}}{2}, - \\frac{\\sqrt{97 - 60\\cdot \\sqrt{2}}}{4} + \\frac{1}{4} + \\frac{3\\cdot \\sqrt{2}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bbc879cacda3d3a8", "problem_id": "a73883a2247c745a", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡11 (mod 13). Give the solution modulo 39.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 39.\n3) Compute the CRT solution to get x ≡ 37 (mod 39).\nFinal answer: $x\\equiv 37\\pmod{39}$", "final_answer": "x\\equiv 37\\pmod{39}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f3fa69672bc8db16", "problem_id": "a73883a2247c745a", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡11 (mod 13). Give the solution modulo 39.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 39.\nCompute the CRT solution to get x ≡ 37 (mod 39).\nFinal answer: $x\\equiv 37\\pmod{39}$", "final_answer": "x\\equiv 37\\pmod{39}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f28fc6d4c7c9ac68", "problem_id": "8958c4f95b42765e", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} - 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)}$ at $x=0$.", "output": "Differentiate: f'(x) = $9 x^{2} - 6 x + 2 \\cos{\\left(x \\right)} + 3$.\nSlope at x=0: m = f'(0) = $5$.\nPoint on curve: (0, f(0)) = (0, $0$).\nFinal answer: $y = 5\\cdot x$", "final_answer": "y = 5\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a7db42635ef029a5", "problem_id": "8958c4f95b42765e", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} - 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)}$ at $x=0$.", "output": "1) Differentiate: f'(x) = $9 x^{2} - 6 x + 2 \\cos{\\left(x \\right)} + 3$.\n2) Slope at x=0: m = f'(0) = $5$.\n3) Point on curve: (0, f(0)) = (0, $0$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 0 = 5(x − 0).\n6) Optionally expand: y = $5 x$.\nFinal answer: $y = 5\\cdot x$", "final_answer": "y = 5\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c9773858c7f3bfcd", "problem_id": "f4ad4f4d5ae03998", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} 3 x^{3} - 3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{3} - 3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{45 \\pi^{4}}{64} - \\frac{9 \\pi^{3}}{8} - 2 + \\frac{3 \\pi}{2} + \\frac{9 \\pi^{2}}{8}$.\nFinal answer: $- \\frac{45\\cdot \\pi^{4}}{64} - \\frac{9\\cdot \\pi^{3}}{8} - 2 + \\frac{3\\cdot \\pi}{2} + \\frac{9\\cdot \\pi^{2}}{8}$", "final_answer": "- \\frac{45\\cdot \\pi^{4}}{64} - \\frac{9\\cdot \\pi^{3}}{8} - 2 + \\frac{3\\cdot \\pi}{2} + \\frac{9\\cdot \\pi^{2}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "98ab2497abbe189d", "problem_id": "f4ad4f4d5ae03998", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} 3 x^{3} - 3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{3} - 3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{45 \\pi^{4}}{64} - \\frac{9 \\pi^{3}}{8} - 2 + \\frac{3 \\pi}{2} + \\frac{9 \\pi^{2}}{8}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{45\\cdot \\pi^{4}}{64} - \\frac{9\\cdot \\pi^{3}}{8} - 2 + \\frac{3\\cdot \\pi}{2} + \\frac{9\\cdot \\pi^{2}}{8}$", "final_answer": "- \\frac{45\\cdot \\pi^{4}}{64} - \\frac{9\\cdot \\pi^{3}}{8} - 2 + \\frac{3\\cdot \\pi}{2} + \\frac{9\\cdot \\pi^{2}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "198d215eb2aa120d", "problem_id": "1a66912c31abd50b", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Two friends split $2282 in the ratio 9:5. How much does each receive?", "output": "Total ratio parts = 9+5 = 14.\nEach part = 2282 ÷ 14 = 163.\nFirst share = 9 × 163 = 1467.\nFinal answer: $815$", "final_answer": "815", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e4372fc73c7691af", "problem_id": "1a66912c31abd50b", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $2282 in the ratio 9:5. How much does each receive?", "output": "1) Total ratio parts = 9+5 = 14.\n2) Each part = 2282 ÷ 14 = 163.\n3) First share = 9 × 163 = 1467.\n4) Second share = 5 × 163 = 815.\nFinal answer: $815$", "final_answer": "815", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9aac373b1eeefcd4", "problem_id": "e5229ea51ac48d4a", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$x + 2 y$ = $6$\n$12 x + 5 y$ = $-4$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $12$: $12 x + 24 y$ = $72$.\n3) Multiply the second equation by $1$: $12 x + 5 y$ = $-4$.\n4) Subtract to eliminate x: $19 y$ = $76$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, 4\\right)$", "final_answer": "(x,y)=\\left(-2, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f061f50cf136e70", "problem_id": "e5229ea51ac48d4a", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$x + 2 y$ = $6$\n$12 x + 5 y$ = $-4$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $12$: $12 x + 24 y$ = $72$.\nMultiply the second equation by $1$: $12 x + 5 y$ = $-4$.\nSubtract to eliminate x: $19 y$ = $76$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $-2$.\nCheck: plug (x,y)=(-2,4) into both equations.\nFinal answer: $(x,y)=\\left(-2, 4\\right)$", "final_answer": "(x,y)=\\left(-2, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d5f4a97fd6fc1f41", "problem_id": "367fddcabc069d25", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Two friends split $180 in the ratio 2:1. How much does each receive?", "output": "1) Total ratio parts = 2+1 = 3.\n2) Each part = 180 ÷ 3 = 60.\n3) First share = 2 × 60 = 120.\n4) Second share = 1 × 60 = 60.\nFinal answer: $60$", "final_answer": "60", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a371ca9ffa4462b4", "problem_id": "fd2bcac6548591c0", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\cos{\\left(2 x \\right)}) = - \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\cos{\\left(2 x \\right)}) = - 2 \\sin{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$", "final_answer": "x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "9b1fe93e279e978e", "problem_id": "e2a00a8325a17d07", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 14 times, what is the probability of exactly 12 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=14, k=12, p=$\\frac{1}{6}$ and simplify.\nHere C(14,12) = 91.\nCheck: expression simplifies to $\\frac{2275}{78364164096}$.\nFinal answer: $\\frac{2275}{78364164096}$", "final_answer": "\\frac{2275}{78364164096}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "10be1d28c8034000", "problem_id": "e2a00a8325a17d07", "style": "answer_only", "instruction": "Final answer only.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 14 times, what is the probability of exactly 12 heads?", "output": "$\\frac{2275}{78364164096}$", "final_answer": "\\frac{2275}{78364164096}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "41747c4904e297cb", "problem_id": "928d29637e6ee390", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the linear system:\n$5 x - 7 y$ = $47$\n$x - 5 y$ = $13$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $5 x - 7 y$ = $47$.\nMultiply the second equation by $5$: $5 x - 25 y$ = $65$.\nFinal answer: $(x,y)=\\left(8, -1\\right)$", "final_answer": "(x,y)=\\left(8, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0b40febab44dc218", "problem_id": "928d29637e6ee390", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$5 x - 7 y$ = $47$\n$x - 5 y$ = $13$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $5 x - 7 y$ = $47$.\nMultiply the second equation by $5$: $5 x - 25 y$ = $65$.\nSubtract to eliminate x: $18 y$ = $-18$.\nSolve for y: y = $-1$.\nSubstitute back to find x: x = $8$.\nCheck: plug (x,y)=(8,-1) into both equations.\nFinal answer: $(x,y)=\\left(8, -1\\right)$", "final_answer": "(x,y)=\\left(8, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "646f876014ac1e16", "problem_id": "49b6dcbb85dd582a", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the equation of the tangent line to $y=4 x^{2} + 3 x + 1$ at $x=2$.", "output": "1) Differentiate: f'(x) = $8 x + 3$.\n2) Slope at x=2: m = f'(2) = $19$.\n3) Point on curve: (2, f(2)) = (2, $23$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 23 = 19(x − 2).\n6) Optionally expand: y = $19 x - 15$.\nFinal answer: $y = 19\\cdot x - 15$", "final_answer": "y = 19\\cdot x - 15", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ecc3ec921f9a1a5", "problem_id": "49b6dcbb85dd582a", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the equation of the tangent line to $y=4 x^{2} + 3 x + 1$ at $x=2$.", "output": "Differentiate: f'(x) = $8 x + 3$.\nSlope at x=2: m = f'(2) = $19$.\nPoint on curve: (2, f(2)) = (2, $23$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 23 = 19(x − 2).\nOptionally expand: y = $19 x - 15$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 19\\cdot x - 15$", "final_answer": "y = 19\\cdot x - 15", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d3cdf3c31756c42c", "problem_id": "f63f7a44d91b8951", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 2 x^{2} + 3 x + \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 6}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 6}) = - \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 6}$.\n3) So $f'(x) = \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 24 x + 72} + \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 24 x + 72} - \\frac{4 x^{2}}{2 x^{2} - 24 x + 72} - \\frac{12 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 24 x + 72} - \\frac{24 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 24 x + 72} + \\frac{48 x}{2 x^{2} - 24 x + 72} + \\frac{36 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 24 x + 72} + \\frac{72 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 24 x + 72} - \\frac{36}{2 x^{2} - 24 x + 72} - \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 12 x + 36} - \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 6} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 6} + \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 6}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 6}) = - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 6} - \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 6} + \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x - 6\\right)^{2}}$.\nTherefore the correct derivative is $f'(x) = \\frac{4 x^{2} - 6 x + \\left(x - 6\\right) \\left(- 8 x + \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 2 \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x - 6\\right)^{2}}$.\nFinal answer: $\\frac{4 x^{2} - 6 x + \\left(x - 6\\right) \\left(- 8 x + \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 2 \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x - 6\\right)^{2}}$", "final_answer": "\\frac{4 x^{2} - 6 x + \\left(x - 6\\right) \\left(- 8 x + \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 2 \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x - 6\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ef34989a96dbb1a7", "problem_id": "26c01a5ed1a774f1", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} - 4 x^{3} - 2 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 4 x^{3} - 2 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\pi).\n3) Subtract: F(\\pi) − F(- \\pi) = $\\frac{4 \\pi \\left(3 - \\pi^{2}\\right)}{3}$.\n4) Simplify the result.\nFinal answer: $\\frac{4\\cdot \\pi\\cdot \\left(3 - \\pi^{2}\\right)}{3}$", "final_answer": "\\frac{4\\cdot \\pi\\cdot \\left(3 - \\pi^{2}\\right)}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1728c65b4517999c", "problem_id": "4d76c83e123f21f8", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$- 3 x + y$ = $15$\n$2 x - y$ = $-11$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $- 6 x + 2 y$ = $30$.\n3) Multiply the second equation by $-3$: $- 6 x + 3 y$ = $33$.\n4) Subtract to eliminate x: $- y$ = $-3$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $-4$.\nFinal answer: $(x,y)=\\left(-4, 3\\right)$", "final_answer": "(x,y)=\\left(-4, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3cb2f7bd508710a3", "problem_id": "ee8ca69cc5ede5ab", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} x + 1$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 1.\n3) Then the limit is 1.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -1.\nFinal answer: $-1$", "final_answer": "-1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6cb66b6e4c03c053", "problem_id": "a299a942722775b8", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 11 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=6, p=$\\frac{4}{5}$ and simplify.\n4) Here C(11,6) = 462.\nFinal answer: $\\frac{1892352}{48828125}$", "final_answer": "\\frac{1892352}{48828125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "289144e81afcd858", "problem_id": "a299a942722775b8", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 11 times, what is the probability of exactly 6 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=11, k=6, p=$\\frac{4}{5}$ and simplify.\nHere C(11,6) = 462.\nCheck: expression simplifies to $\\frac{1892352}{48828125}$.\nFinal answer: $\\frac{1892352}{48828125}$", "final_answer": "\\frac{1892352}{48828125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b4ec8450c19a937a", "problem_id": "f784b6187e2b5632", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 21 times, what is the probability of exactly 9 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=21, k=9, p=$\\frac{1}{6}$ and simplify.\n4) Here C(21,9) = 293930.\nFinal answer: $\\frac{35880126953125}{10968475320188928}$", "final_answer": "\\frac{35880126953125}{10968475320188928}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ef7a2bec3fed0920", "problem_id": "f784b6187e2b5632", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 21 times, what is the probability of exactly 9 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=21, k=9, p=$\\frac{1}{6}$ and simplify.\nHere C(21,9) = 293930.\nCheck: expression simplifies to $\\frac{35880126953125}{10968475320188928}$.\nFinal answer: $\\frac{35880126953125}{10968475320188928}$", "final_answer": "\\frac{35880126953125}{10968475320188928}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ae7255b056ff8cd8", "problem_id": "f0cf4e381fc2b52b", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A disease has prevalence $\\frac{3}{100}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{36}{625}$.\nCompute P(D|+) = $\\frac{95}{192}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{95}{192}$", "final_answer": "\\frac{95}{192}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7b32df4ebac3e4c2", "problem_id": "f0cf4e381fc2b52b", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A disease has prevalence $\\frac{3}{100}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{36}{625}$.\n3) Compute P(D|+) = $\\frac{95}{192}$.\nFinal answer: $\\frac{95}{192}$", "final_answer": "\\frac{95}{192}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5092722c052f1467", "problem_id": "59178cd3534cf68e", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 17 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=17, k=3, p=$\\frac{1}{8}$ and simplify.\n4) Here C(17,3) = 680.\nFinal answer: $\\frac{57648961192165}{281474976710656}$", "final_answer": "\\frac{57648961192165}{281474976710656}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f3861f4ad0c3dd2f", "problem_id": "59178cd3534cf68e", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 17 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=17, k=3, p=$\\frac{1}{8}$ and simplify.\nHere C(17,3) = 680.\nCheck: expression simplifies to $\\frac{57648961192165}{281474976710656}$.\nFinal answer: $\\frac{57648961192165}{281474976710656}$", "final_answer": "\\frac{57648961192165}{281474976710656}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2e6a09e4684437ab", "problem_id": "7d230d0b8696bc73", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "You want 25 liters of a 47% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 25−x liters is 70% solution.\n2) Salt equation: 25x + 70(25−x) = 47·25.\n3) (25-70)x = (47-70)·25 ⇒ x = 115/9.\n4) Then 25−x = 110/9.\nFinal answer: $115/9 L of 25% and 110/9 L of 70%$", "final_answer": "115/9 L of 25% and 110/9 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "529e6796eff7909e", "problem_id": "7d230d0b8696bc73", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 25 liters of a 47% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 25−x liters is 70% solution.\nSalt equation: 25x + 70(25−x) = 47·25.\n(25-70)x = (47-70)·25 ⇒ x = 115/9.\nThen 25−x = 110/9.\nCheck: 25·115/9 + 70·110/9 = 47·25.\nFinal answer: $115/9 L of 25% and 110/9 L of 70%$", "final_answer": "115/9 L of 25% and 110/9 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e036d83fb8f5ad02", "problem_id": "1adca454b7863c15", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A disease has prevalence $\\frac{1}{50}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{117}{1000}$.\n3) Compute P(D|+) = $\\frac{19}{117}$.\nFinal answer: $\\frac{19}{117}$", "final_answer": "\\frac{19}{117}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6dfa209b4bbf3f02", "problem_id": "d8d35b6d9871079c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- \\frac{3 x^{2}}{2} - x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1}{x - 4}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}) = \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4}$.\n3) So $f'(x) = \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{3 x^{2}}{2 x^{2} - 16 x + 32} - \\frac{8 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{16 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{24 x}{2 x^{2} - 16 x + 32} + \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 8 x + 16} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 8} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} - \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} + \\frac{16 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{32 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{10}{2 x^{2} - 16 x + 32} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}) = \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x - 4\\right)} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x - 4\\right)^{2}} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}$.\nTherefore the correct derivative is $f'(x) = \\frac{3 x^{2} + 2 x + \\left(x - 4\\right) \\left(- 6 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{2 \\left(x - 4\\right)^{2}}$.\nFinal answer: $\\frac{3 x^{2} + 2 x + \\left(x - 4\\right) \\left(- 6 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{2 \\left(x - 4\\right)^{2}}$", "final_answer": "\\frac{3 x^{2} + 2 x + \\left(x - 4\\right) \\left(- 6 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{2 \\left(x - 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7e0d12db2a51eead", "problem_id": "7ee57131ee381922", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A disease has prevalence $\\frac{1}{100}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{49}{1250}$.\nCompute P(D|+) = $\\frac{95}{392}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{95}{392}$", "final_answer": "\\frac{95}{392}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "60588b03b3be929f", "problem_id": "5baffeeac5a4af65", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{2} - 4$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- x^{2}) = - 2 x^{2}$.\n3) So $f'(x) = - 2 x^{2}$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- x^{2}) = - 2 x$.\nTherefore the correct derivative is $f'(x) = - 2 x$.\nFinal answer: $- 2 x$", "final_answer": "- 2 x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6bff298d0444d67c", "problem_id": "560cd721d5266f11", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1$.\n3) Compute an antiderivative: F(x) = $x^{3} - \\frac{3 x^{2}}{2} - x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + x^{3} - \\frac{3\\cdot x^{2}}{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + x^{3} - \\frac{3\\cdot x^{2}}{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c463ff8672ff9a2c", "problem_id": "560cd721d5266f11", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1$.\nCompute an antiderivative: F(x) = $x^{3} - \\frac{3 x^{2}}{2} - x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $C + x^{3} - \\frac{3\\cdot x^{2}}{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + x^{3} - \\frac{3\\cdot x^{2}}{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5fe830a57c0f4f19", "problem_id": "3a201a11ae9c1414", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 10 times, what is the probability of exactly 7 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=7, p=$\\frac{1}{2}$ and simplify.\n4) Here C(10,7) = 120.\nFinal answer: $\\frac{15}{128}$", "final_answer": "\\frac{15}{128}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a6834f25de97abd", "problem_id": "30b1fab2e37dc270", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{3} + 3 x^{2} - 3 x - 2$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(3 x^{2}) = 6 x^{2}$.\n3) So $f'(x) = 15 x^{2} - 3$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(3 x^{2}) = 6 x$.\nTherefore the correct derivative is $f'(x) = 3 \\left(3 x^{2} + 2 x - 1\\right)$.\nFinal answer: $3 \\left(3 x^{2} + 2 x - 1\\right)$", "final_answer": "3 \\left(3 x^{2} + 2 x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "1f2beebae5723fb7", "problem_id": "683f14afc0e03db0", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A store sells a backpack for $175. It is discounted by 20%. You buy 5 backpacks. What is the total cost?", "output": "1) Discounted price = 175 × (1 − 20/100) = 175 × 80/100 = 140.\n2) Subtotal = 5 × 140 = 700.\nFinal answer: $700$", "final_answer": "700", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "331fc29345dbfd00", "problem_id": "683f14afc0e03db0", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A store sells a backpack for $175. It is discounted by 20%. You buy 5 backpacks. What is the total cost?", "output": "Discounted price = 175 × (1 − 20/100) = 175 × 80/100 = 140.\nSubtotal = 5 × 140 = 700.\nCheck: recomputing the formula yields 700.\nFinal answer: $700$", "final_answer": "700", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2dcc21aedb6c2996", "problem_id": "2fe2ec883bedd622", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A store sells a backpack for $262. It is discounted by 30%. You buy 13 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 262 × (1 − 30/100) = 262 × 70/100 = 917/5.\n2) Subtotal = 13 × 917/5 = 11921/5.\n3) Total with tax = 11921/5 × 105/100 = 250341/100.\nFinal answer: $\\frac{250341}{100}$", "final_answer": "\\frac{250341}{100}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff32d43022c3aeac", "problem_id": "2fe2ec883bedd622", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A store sells a backpack for $262. It is discounted by 30%. You buy 13 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 262 × (1 − 30/100) = 262 × 70/100 = 917/5.\nSubtotal = 13 × 917/5 = 11921/5.\nTotal with tax = 11921/5 × 105/100 = 250341/100.\nCheck: recomputing the formula yields 250341/100.\nFinal answer: $\\frac{250341}{100}$", "final_answer": "\\frac{250341}{100}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "89a93fd7f870dc39", "problem_id": "5e6db33c3dbe76a3", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} - 3 x^{2} + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)}$ at $x=0$.", "output": "1) Differentiate: f'(x) = $6 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)}$.\n2) Slope at x=0: m = f'(0) = $\\frac{1}{2}$.\n3) Point on curve: (0, f(0)) = (0, $2$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 2 = \\frac{1}{2}(x − 0).\n6) Optionally expand: y = $\\frac{x}{2} + 2$.\nFinal answer: $y = \\frac{x}{2} + 2$", "final_answer": "y = \\frac{x}{2} + 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "213eba75985948b1", "problem_id": "ca37e305e6636670", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{3 x^{2} - 3 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{2 x - 3}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 3}) = - \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 3}$.\n3) So $f'(x) = \\frac{4 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{2} - 24 x + 18} + \\frac{8 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{2} - 24 x + 18} + \\frac{12 x^{2}}{8 x^{2} - 24 x + 18} - \\frac{12 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{2} - 24 x + 18} - \\frac{24 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{2} - 24 x + 18} - \\frac{36 x}{8 x^{2} - 24 x + 18} + \\frac{9 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{2} - 24 x + 18} + \\frac{18 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{2} - 24 x + 18} + \\frac{26}{8 x^{2} - 24 x + 18} - \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{4 x^{2} - 12 x + 9} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{4 x - 6} - \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 3} + \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 3}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 3}) = - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(2 x - 3\\right)} - \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 3} + \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x - 3\\right)^{2}}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 12 x^{2} + 12 x + \\left(2 x - 3\\right) \\left(12 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6\\right) - 4 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8}{2 \\left(2 x - 3\\right)^{2}}$.\nFinal answer: $\\frac{- 12 x^{2} + 12 x + \\left(2 x - 3\\right) \\left(12 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6\\right) - 4 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8}{2 \\left(2 x - 3\\right)^{2}}$", "final_answer": "\\frac{- 12 x^{2} + 12 x + \\left(2 x - 3\\right) \\left(12 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6\\right) - 4 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8}{2 \\left(2 x - 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a6f9577584a15ece", "problem_id": "173c98000e772a37", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$- 8 x + 4 y$ = $0$\n$- 3 x - 7 y$ = $17$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $24 x - 12 y$ = $0$.\n3) Multiply the second equation by $-8$: $24 x + 56 y$ = $-136$.\n4) Subtract to eliminate x: $- 68 y$ = $136$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, -2\\right)$", "final_answer": "(x,y)=\\left(-1, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fdcf986552e0042e", "problem_id": "173c98000e772a37", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$- 8 x + 4 y$ = $0$\n$- 3 x - 7 y$ = $17$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $24 x - 12 y$ = $0$.\nMultiply the second equation by $-8$: $24 x + 56 y$ = $-136$.\nSubtract to eliminate x: $- 68 y$ = $136$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $-1$.\nCheck: plug (x,y)=(-1,-2) into both equations.\nFinal answer: $(x,y)=\\left(-1, -2\\right)$", "final_answer": "(x,y)=\\left(-1, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b2619682c9d40b3d", "problem_id": "c1979fe929408e9d", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡7 (mod 9), x≡3 (mod 5). Give the solution modulo 45.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 45.\n3) Compute the CRT solution to get x ≡ 43 (mod 45).\nFinal answer: $x\\equiv 43\\pmod{45}$", "final_answer": "x\\equiv 43\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9a309b381e886af1", "problem_id": "c1979fe929408e9d", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡7 (mod 9), x≡3 (mod 5). Give the solution modulo 45.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 45.\nCompute the CRT solution to get x ≡ 43 (mod 45).\nCheck: 43 mod 9=7 and mod 5=3.\nFinal answer: $x\\equiv 43\\pmod{45}$", "final_answer": "x\\equiv 43\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0b21a00e1ceac971", "problem_id": "4041b7f26b03d432", "style": "answer_only", "instruction": "Final answer only.", "input": "A coin lands heads with probability p=$\\frac{1}{7}$. If it is flipped 10 times, what is the probability of exactly 7 heads?", "output": "$\\frac{25920}{282475249}$", "final_answer": "\\frac{25920}{282475249}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "76c92eea6d152ae6", "problem_id": "4041b7f26b03d432", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{7}$. If it is flipped 10 times, what is the probability of exactly 7 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=7, p=$\\frac{1}{7}$ and simplify.\n4) Here C(10,7) = 120.\nFinal answer: $\\frac{25920}{282475249}$", "final_answer": "\\frac{25920}{282475249}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "99c9da64fcf9bfee", "problem_id": "8567a75a884d0ed7", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$- 8 x + 7 y$ = $17$\n$- 6 x - 8 y$ = $26$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-6$: $48 x - 42 y$ = $-102$.\n3) Multiply the second equation by $-8$: $48 x + 64 y$ = $-208$.\n4) Subtract to eliminate x: $- 106 y$ = $106$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, -1\\right)$", "final_answer": "(x,y)=\\left(-3, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "65a52449bd5bd4da", "problem_id": "8567a75a884d0ed7", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$- 8 x + 7 y$ = $17$\n$- 6 x - 8 y$ = $26$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-6$: $48 x - 42 y$ = $-102$.\nMultiply the second equation by $-8$: $48 x + 64 y$ = $-208$.\nSubtract to eliminate x: $- 106 y$ = $106$.\nSolve for y: y = $-1$.\nSubstitute back to find x: x = $-3$.\nCheck: plug (x,y)=(-3,-1) into both equations.\nFinal answer: $(x,y)=\\left(-3, -1\\right)$", "final_answer": "(x,y)=\\left(-3, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b70ee7d731184aff", "problem_id": "69ba216b333fcafe", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{2} - 2 x - 4$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $6 x - 2$.\n2) Slope at x=-2: m = f'(-2) = $-14$.\n3) Point on curve: (-2, f(-2)) = (-2, $12$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 12 = -14(x − -2).\n6) Optionally expand: y = $- 14 x - 16$.\nFinal answer: $y = - 14\\cdot x - 16$", "final_answer": "y = - 14\\cdot x - 16", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8546288ecc113f9a", "problem_id": "69ba216b333fcafe", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{2} - 2 x - 4$ at $x=-2$.", "output": "Differentiate: f'(x) = $6 x - 2$.\nSlope at x=-2: m = f'(-2) = $-14$.\nPoint on curve: (-2, f(-2)) = (-2, $12$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 12 = -14(x − -2).\nOptionally expand: y = $- 14 x - 16$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 14\\cdot x - 16$", "final_answer": "y = - 14\\cdot x - 16", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8309fa4845cd0e11", "problem_id": "a22093ef784ceab2", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 10 times, what is the probability of exactly 6 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=6, p=$\\frac{1}{10}$ and simplify.\nHere C(10,6) = 210.\nCheck: expression simplifies to $\\frac{137781}{1000000000}$.\nFinal answer: $\\frac{137781}{1000000000}$", "final_answer": "\\frac{137781}{1000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5727362d8a9f7789", "problem_id": "a22093ef784ceab2", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 10 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=6, p=$\\frac{1}{10}$ and simplify.\n4) Here C(10,6) = 210.\nFinal answer: $\\frac{137781}{1000000000}$", "final_answer": "\\frac{137781}{1000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f66369f05356ae49", "problem_id": "7e96155b1fac8c28", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the linear system:\n$7 x$ = $-14$\n$- 2 x - 8 y$ = $-20$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $- 14 x$ = $28$.\nMultiply the second equation by $7$: $- 14 x - 56 y$ = $-140$.\nFinal answer: $(x,y)=\\left(-2, 3\\right)$", "final_answer": "(x,y)=\\left(-2, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0148075723983ec4", "problem_id": "7e96155b1fac8c28", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$7 x$ = $-14$\n$- 2 x - 8 y$ = $-20$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $- 14 x$ = $28$.\n3) Multiply the second equation by $7$: $- 14 x - 56 y$ = $-140$.\n4) Subtract to eliminate x: $56 y$ = $168$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, 3\\right)$", "final_answer": "(x,y)=\\left(-2, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1db0525bd3e1b7b8", "problem_id": "749845bcb27e1e36", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A cyclist rides at 55 miles per hour for 2 hours, then at 63 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 55 × 2 = 110 miles.\n2) Second leg distance = 63 × 5/2 = 315/2 miles.\n3) Total distance = 110 + 315/2 = 535/2 miles.\nFinal answer: $\\frac{535}{2}$", "final_answer": "\\frac{535}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "832a1936094c8074", "problem_id": "749845bcb27e1e36", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "A cyclist rides at 55 miles per hour for 2 hours, then at 63 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 55 × 2 = 110 miles.\nSecond leg distance = 63 × 5/2 = 315/2 miles.\nTotal distance = 110 + 315/2 = 535/2 miles.\nFinal answer: $\\frac{535}{2}$", "final_answer": "\\frac{535}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0b221ca43cfab71c", "problem_id": "d534291c7de5045a", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} + 2 x^{2} + 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=0$.", "output": "1) Differentiate: f'(x) = $6 x^{2} + 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 3$.\n2) Slope at x=0: m = f'(0) = $\\frac{7}{2}$.\n3) Point on curve: (0, f(0)) = (0, $1$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 1 = \\frac{7}{2}(x − 0).\n6) Optionally expand: y = $\\frac{7 x}{2} + 1$.\nFinal answer: $y = \\frac{7\\cdot x}{2} + 1$", "final_answer": "y = \\frac{7\\cdot x}{2} + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7046cf39aff8af4d", "problem_id": "d534291c7de5045a", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} + 2 x^{2} + 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=0$.", "output": "Differentiate: f'(x) = $6 x^{2} + 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 3$.\nSlope at x=0: m = f'(0) = $\\frac{7}{2}$.\nPoint on curve: (0, f(0)) = (0, $1$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 1 = \\frac{7}{2}(x − 0).\nOptionally expand: y = $\\frac{7 x}{2} + 1$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = \\frac{7\\cdot x}{2} + 1$", "final_answer": "y = \\frac{7\\cdot x}{2} + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "36e50a1db5744515", "problem_id": "2aff119ccb94f44f", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 \\cos{\\left(2 x \\right)}) = - 2 \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 \\cos{\\left(2 x \\right)}) = - 4 \\sin{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$", "final_answer": "x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6d7281323b346b67", "problem_id": "8d22424c34a58b87", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "You want 62 liters of a 45% salt solution. You have a 15% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 62−x liters is 60% solution.\nSalt equation: 15x + 60(62−x) = 45·62.\n(15-60)x = (45-60)·62 ⇒ x = 62/3.\nThen 62−x = 124/3.\nCheck: 15·62/3 + 60·124/3 = 45·62.\nFinal answer: $62/3 L of 15% and 124/3 L of 60%$", "final_answer": "62/3 L of 15% and 124/3 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a48e47b14c278ca0", "problem_id": "beaf7b6780904fb7", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - x^{2} + 5 x - 8$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{3} + 3 x^{2} + 2 x$ and a = -2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 3 x^{2} + 6 x + 2$.\n5) Evaluate at a: p'(-2) = $-22$.\nFinal answer: $-22$", "final_answer": "-22", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "251805455cd269d3", "problem_id": "beaf7b6780904fb7", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - x^{2} + 5 x - 8$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{3} + 3 x^{2} + 2 x$ and a = -2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 3 x^{2} + 6 x + 2$.\nEvaluate at a: p'(-2) = $-22$.\nCheck: the identity gives $-22$.\nFinal answer: $-22$", "final_answer": "-22", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "753550730e0fd5cd", "problem_id": "24e932094023ca7c", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "You want 17 liters of a 35% salt solution. You have a 30% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 17−x liters is 40% solution.\n2) Salt equation: 30x + 40(17−x) = 35·17.\n3) (30-40)x = (35-40)·17 ⇒ x = 17/2.\n4) Then 17−x = 17/2.\nFinal answer: $17/2 L of 30% and 17/2 L of 40%$", "final_answer": "17/2 L of 30% and 17/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a501d0724de5a563", "problem_id": "24e932094023ca7c", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "You want 17 liters of a 35% salt solution. You have a 30% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 17−x liters is 40% solution.\nSalt equation: 30x + 40(17−x) = 35·17.\n(30-40)x = (35-40)·17 ⇒ x = 17/2.\nFinal answer: $17/2 L of 30% and 17/2 L of 40%$", "final_answer": "17/2 L of 30% and 17/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c22fc5cfcdcf813b", "problem_id": "59c32a17a05252ed", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 3 x^{3} + 11 x^{2} - 31 x + 92$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 3 x^{4} + 2 x^{3} + 2 x^{2} - x - 3$ and a = -3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 12 x^{3} + 6 x^{2} + 4 x - 1$.\n5) Evaluate at a: p'(-3) = $365$.\nFinal answer: $365$", "final_answer": "365", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1b305e2bfff9dd14", "problem_id": "59c32a17a05252ed", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 3 x^{3} + 11 x^{2} - 31 x + 92$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 3 x^{4} + 2 x^{3} + 2 x^{2} - x - 3$ and a = -3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 12 x^{3} + 6 x^{2} + 4 x - 1$.\nEvaluate at a: p'(-3) = $365$.\nCheck: the identity gives $365$.\nFinal answer: $365$", "final_answer": "365", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f5e0f07350f8c76d", "problem_id": "287a60ccac5652ad", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Worker A finishes a job in 7 hours and Worker B finishes it in 4 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/7 job/hour. B's rate = 1/4 job/hour.\n2) Combined rate = 1/7 + 1/4 = 11/28 job/hour.\n3) Time = 1 ÷ (combined rate) = 28/11 hours.\nFinal answer: $\\frac{28}{11}$", "final_answer": "\\frac{28}{11}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4f02c91c07ab5590", "problem_id": "287a60ccac5652ad", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Worker A finishes a job in 7 hours and Worker B finishes it in 4 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/7 job/hour. B's rate = 1/4 job/hour.\nCombined rate = 1/7 + 1/4 = 11/28 job/hour.\nTime = 1 ÷ (combined rate) = 28/11 hours.\nCheck: (combined rate)×(time) = 11/28×28/11 = 1.\nFinal answer: $\\frac{28}{11}$", "final_answer": "\\frac{28}{11}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9b8be6f3aa6c27a9", "problem_id": "7e633854e80c9eff", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{6} x}{2} - \\frac{17 x}{10} + \\frac{3}{5} + \\frac{9 \\sqrt{6}}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{3 \\sqrt{6}}{2} - \\frac{17}{10}$, c=$\\frac{3}{5} + \\frac{9 \\sqrt{6}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{1399}{100} - \\frac{21 \\sqrt{6}}{10}$, so √Δ = $\\sqrt{\\frac{1399}{100} - \\frac{21 \\sqrt{6}}{10}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{17}{20} + \\frac{\\sqrt{1399 - 210 \\sqrt{6}}}{20} + \\frac{3 \\sqrt{6}}{4}$.\n7) x2 = $- \\frac{\\sqrt{1399 - 210 \\sqrt{6}}}{20} + \\frac{17}{20} + \\frac{3 \\sqrt{6}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{17}{20} + \\frac{\\sqrt{1399 - 210\\cdot \\sqrt{6}}}{20} + \\frac{3\\cdot \\sqrt{6}}{4}, - \\frac{\\sqrt{1399 - 210\\cdot \\sqrt{6}}}{20} + \\frac{17}{20} + \\frac{3\\cdot \\sqrt{6}}{4}\\}$", "final_answer": "x \\in \\{\\frac{17}{20} + \\frac{\\sqrt{1399 - 210\\cdot \\sqrt{6}}}{20} + \\frac{3\\cdot \\sqrt{6}}{4}, - \\frac{\\sqrt{1399 - 210\\cdot \\sqrt{6}}}{20} + \\frac{17}{20} + \\frac{3\\cdot \\sqrt{6}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "170bd7a51daee98b", "problem_id": "7e633854e80c9eff", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{6} x}{2} - \\frac{17 x}{10} + \\frac{3}{5} + \\frac{9 \\sqrt{6}}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{3 \\sqrt{6}}{2} - \\frac{17}{10}$, c=$\\frac{3}{5} + \\frac{9 \\sqrt{6}}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{1399}{100} - \\frac{21 \\sqrt{6}}{10}$, so √Δ = $\\sqrt{\\frac{1399}{100} - \\frac{21 \\sqrt{6}}{10}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{17}{20} + \\frac{\\sqrt{1399 - 210 \\sqrt{6}}}{20} + \\frac{3 \\sqrt{6}}{4}$.\nx2 = $- \\frac{\\sqrt{1399 - 210 \\sqrt{6}}}{20} + \\frac{17}{20} + \\frac{3 \\sqrt{6}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{17}{20} + \\frac{\\sqrt{1399 - 210\\cdot \\sqrt{6}}}{20} + \\frac{3\\cdot \\sqrt{6}}{4}, - \\frac{\\sqrt{1399 - 210\\cdot \\sqrt{6}}}{20} + \\frac{17}{20} + \\frac{3\\cdot \\sqrt{6}}{4}\\}$", "final_answer": "x \\in \\{\\frac{17}{20} + \\frac{\\sqrt{1399 - 210\\cdot \\sqrt{6}}}{20} + \\frac{3\\cdot \\sqrt{6}}{4}, - \\frac{\\sqrt{1399 - 210\\cdot \\sqrt{6}}}{20} + \\frac{17}{20} + \\frac{3\\cdot \\sqrt{6}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1285ef2b10e3e341", "problem_id": "8f7819df658f0ab2", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$- x + 2 y$ = $-1$\n$- 2 x$ = $6$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $2 x - 4 y$ = $2$.\n3) Multiply the second equation by $-1$: $2 x$ = $-6$.\n4) Subtract to eliminate x: $- 4 y$ = $8$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, -2\\right)$", "final_answer": "(x,y)=\\left(-3, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "823eff923d95e217", "problem_id": "8f7819df658f0ab2", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$- x + 2 y$ = $-1$\n$- 2 x$ = $6$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $2 x - 4 y$ = $2$.\nMultiply the second equation by $-1$: $2 x$ = $-6$.\nSubtract to eliminate x: $- 4 y$ = $8$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $-3$.\nCheck: plug (x,y)=(-3,-2) into both equations.\nFinal answer: $(x,y)=\\left(-3, -2\\right)$", "final_answer": "(x,y)=\\left(-3, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5cb9f2495751bdd4", "problem_id": "e54a0591e9874891", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 6 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=6, k=4, p=$\\frac{1}{8}$ and simplify.\n4) Here C(6,4) = 15.\nFinal answer: $\\frac{735}{262144}$", "final_answer": "\\frac{735}{262144}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6181c039d4848d7e", "problem_id": "e54a0591e9874891", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 6 times, what is the probability of exactly 4 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=6, k=4, p=$\\frac{1}{8}$ and simplify.\nFinal answer: $\\frac{735}{262144}$", "final_answer": "\\frac{735}{262144}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3c693f7a166f3dcc", "problem_id": "ae534fbe21217fdf", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$- 8 x$ = $0$\n$- 2 x + 4 y$ = $36$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $16 x$ = $0$.\n3) Multiply the second equation by $-8$: $16 x - 32 y$ = $-288$.\n4) Subtract to eliminate x: $32 y$ = $288$.\n5) Solve for y: y = $9$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, 9\\right)$", "final_answer": "(x,y)=\\left(0, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3c64ab4763692149", "problem_id": "ae534fbe21217fdf", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$- 8 x$ = $0$\n$- 2 x + 4 y$ = $36$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $16 x$ = $0$.\nMultiply the second equation by $-8$: $16 x - 32 y$ = $-288$.\nSubtract to eliminate x: $32 y$ = $288$.\nSolve for y: y = $9$.\nSubstitute back to find x: x = $0$.\nCheck: plug (x,y)=(0,9) into both equations.\nFinal answer: $(x,y)=\\left(0, 9\\right)$", "final_answer": "(x,y)=\\left(0, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "881d7bb6704ab058", "problem_id": "05200c7ac8673838", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - 3 x^{2} + 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 2$ at $x=-1$.", "output": "Differentiate: f'(x) = $- 9 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 2$.\nSlope at x=-1: m = f'(-1) = $-1 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nPoint on curve: (-1, f(-1)) = (-1, $\\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}}$).\nFinal answer: $y = - x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 1 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 1 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4395ad91fad540b1", "problem_id": "05200c7ac8673838", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - 3 x^{2} + 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 2$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $- 9 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 2$.\n2) Slope at x=-1: m = f'(-1) = $-1 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\n3) Point on curve: (-1, f(-1)) = (-1, $\\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} = -1 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}(x − -1).\n6) Optionally expand: y = $- x + \\frac{x}{2 e^{\\frac{1}{2}}} + 2 x \\sin{\\left(2 \\right)} - 1 + \\cos{\\left(2 \\right)} + \\frac{3}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = - x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 1 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 1 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1fb6ca466d159760", "problem_id": "1dae8a7f043f90e8", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "How many binary strings of length 16 contain exactly 2 ones with no two ones adjacent?", "output": "1) Reserve 1 mandatory zeros between the 2 ones.\n2) This leaves n−k+1 = 15 valid slots for the ones.\n3) Choose the slots: C(15,2) = 105.\nFinal answer: $105$", "final_answer": "105", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f00bae0e7677c809", "problem_id": "1dae8a7f043f90e8", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many binary strings of length 16 contain exactly 2 ones with no two ones adjacent?", "output": "Reserve 1 mandatory zeros between the 2 ones.\nThis leaves n−k+1 = 15 valid slots for the ones.\nChoose the slots: C(15,2) = 105.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $105$", "final_answer": "105", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "afb703a93f4a34dc", "problem_id": "fbb6c0e4245372b2", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Worker A finishes a job in 9 hours and Worker B finishes it in 20 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/9 job/hour. B's rate = 1/20 job/hour.\n2) Combined rate = 1/9 + 1/20 = 29/180 job/hour.\n3) Time = 1 ÷ (combined rate) = 180/29 hours.\nFinal answer: $\\frac{180}{29}$", "final_answer": "\\frac{180}{29}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c4e0990420135c86", "problem_id": "fbb6c0e4245372b2", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Worker A finishes a job in 9 hours and Worker B finishes it in 20 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/9 job/hour. B's rate = 1/20 job/hour.\nCombined rate = 1/9 + 1/20 = 29/180 job/hour.\nTime = 1 ÷ (combined rate) = 180/29 hours.\nCheck: (combined rate)×(time) = 29/180×180/29 = 1.\nFinal answer: $\\frac{180}{29}$", "final_answer": "\\frac{180}{29}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "49c579976c36cd5e", "problem_id": "e6dcef8eea04ac05", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A store sells a backpack for $84. It is discounted by 15%. You buy 13 backpacks. What is the total cost?", "output": "1) Discounted price = 84 × (1 − 15/100) = 84 × 85/100 = 357/5.\n2) Subtotal = 13 × 357/5 = 4641/5.\nFinal answer: $\\frac{4641}{5}$", "final_answer": "\\frac{4641}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e3fd6984095a8b82", "problem_id": "e6dcef8eea04ac05", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A store sells a backpack for $84. It is discounted by 15%. You buy 13 backpacks. What is the total cost?", "output": "Discounted price = 84 × (1 − 15/100) = 84 × 85/100 = 357/5.\nSubtotal = 13 × 357/5 = 4641/5.\nCheck: recomputing the formula yields 4641/5.\nFinal answer: $\\frac{4641}{5}$", "final_answer": "\\frac{4641}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8d2804bc921bad88", "problem_id": "2fc0ebf4951960f9", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Worker A finishes a job in 19 hours and Worker B finishes it in 17 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/19 job/hour. B's rate = 1/17 job/hour.\nCombined rate = 1/19 + 1/17 = 36/323 job/hour.\nTime = 1 ÷ (combined rate) = 323/36 hours.\nCheck: (combined rate)×(time) = 36/323×323/36 = 1.\nFinal answer: $\\frac{323}{36}$", "final_answer": "\\frac{323}{36}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9e02a87ebd37ec92", "problem_id": "2fc0ebf4951960f9", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Worker A finishes a job in 19 hours and Worker B finishes it in 17 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/19 job/hour. B's rate = 1/17 job/hour.\n2) Combined rate = 1/19 + 1/17 = 36/323 job/hour.\n3) Time = 1 ÷ (combined rate) = 323/36 hours.\nFinal answer: $\\frac{323}{36}$", "final_answer": "\\frac{323}{36}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "864f532fd33c6af1", "problem_id": "94b5f2cba0c3399f", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{13} x + 2 x - 3 \\sqrt{13} - 3$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$2 - \\sqrt{13}$, c=$- 3 \\sqrt{13} - 3$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $8 \\sqrt{13} + 29$, so √Δ = $\\sqrt{8 \\sqrt{13} + 29}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $-1 + \\frac{\\sqrt{13}}{2} + \\frac{\\sqrt{8 \\sqrt{13} + 29}}{2}$.\n7) x2 = $- \\frac{\\sqrt{8 \\sqrt{13} + 29}}{2} - 1 + \\frac{\\sqrt{13}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{-1 + \\frac{\\sqrt{13}}{2} + \\frac{\\sqrt{8\\cdot \\sqrt{13} + 29}}{2}, - \\frac{\\sqrt{8\\cdot \\sqrt{13} + 29}}{2} - 1 + \\frac{\\sqrt{13}}{2}\\}$", "final_answer": "x \\in \\{-1 + \\frac{\\sqrt{13}}{2} + \\frac{\\sqrt{8\\cdot \\sqrt{13} + 29}}{2}, - \\frac{\\sqrt{8\\cdot \\sqrt{13} + 29}}{2} - 1 + \\frac{\\sqrt{13}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "36d4367aeacbc963", "problem_id": "c4aa75c753ae9dfe", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x^{2} - 5 x - 13$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{2} - 6 x - 13.\n3) Then the limit is -33.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -31.\nFinal answer: $-31$", "final_answer": "-31", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "eba749e131ad29bb", "problem_id": "62aa740f20ddec42", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A cyclist rides at 53 miles per hour for 2 hours, then at 49 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 53 × 2 = 106 miles.\n2) Second leg distance = 49 × 1 = 49 miles.\n3) Total distance = 106 + 49 = 155 miles.\nFinal answer: $155$", "final_answer": "155", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4ba29b0332872d5a", "problem_id": "62aa740f20ddec42", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 53 miles per hour for 2 hours, then at 49 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 53 × 2 = 106 miles.\nSecond leg distance = 49 × 1 = 49 miles.\nTotal distance = 106 + 49 = 155 miles.\nCheck: adding both legs gives 155.\nFinal answer: $155$", "final_answer": "155", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8bdc46d49e3b394a", "problem_id": "232140c6d24e0324", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} x^{2} - 6 x + 22$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to x^{2} - 7 x + 22.\n3) Then the limit is 66.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: 62.\nFinal answer: $62$", "final_answer": "62", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0c1d2c8c7e1e7d31", "problem_id": "93a14e280055a489", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} - 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(0).\n3) Subtract: F(2 \\pi) − F(0) = $4 \\pi \\left(- 2 \\pi^{2} - 1 + \\pi\\right)$.\n4) Simplify the result.\nFinal answer: $4\\cdot \\pi\\cdot \\left(- 2\\cdot \\pi^{2} - 1 + \\pi\\right)$", "final_answer": "4\\cdot \\pi\\cdot \\left(- 2\\cdot \\pi^{2} - 1 + \\pi\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "74e225b1e49f8bfa", "problem_id": "bdfc7ed1e6d5f604", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(x^{3} + x^{2} + 3\\right) \\cos{\\left(x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 \\cos{\\left(x \\right)}) = - 3 \\sin{\\left(x \\right)}$.\n3) So $f'(x) = - x^{3} \\sin{\\left(x \\right)} - x^{2} \\sin{\\left(x \\right)} + 3 x^{2} \\cos{\\left(x \\right)} + x e^{x} + 2 x \\cos{\\left(x \\right)} - 2 e^{x} - 3 \\sin{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 \\cos{\\left(x \\right)}) = - 3 \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = x \\left(3 x + 2\\right) \\cos{\\left(x \\right)} + \\left(x - 3\\right) e^{x} - \\left(x^{2} \\left(x + 1\\right) + 3\\right) \\sin{\\left(x \\right)} + e^{x}$.\nFinal answer: $x \\left(3 x + 2\\right) \\cos{\\left(x \\right)} + \\left(x - 3\\right) e^{x} - \\left(x^{2} \\left(x + 1\\right) + 3\\right) \\sin{\\left(x \\right)} + e^{x}$", "final_answer": "x \\left(3 x + 2\\right) \\cos{\\left(x \\right)} + \\left(x - 3\\right) e^{x} - \\left(x^{2} \\left(x + 1\\right) + 3\\right) \\sin{\\left(x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "fd19ebd2be5e4438", "problem_id": "f62025dc29616b6b", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A cyclist rides at 34 miles per hour for 3/2 hours, then at 54 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 34 × 3/2 = 51 miles.\n2) Second leg distance = 54 × 1 = 54 miles.\n3) Total distance = 51 + 54 = 105 miles.\nFinal answer: $105$", "final_answer": "105", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eaa5325ac3279501", "problem_id": "eb595ba9d4c7ea3f", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A disease has prevalence $\\frac{1}{50}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{29}{250}$.\n3) Compute P(D|+) = $\\frac{9}{58}$.\nFinal answer: $\\frac{9}{58}$", "final_answer": "\\frac{9}{58}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ab52e67a85241944", "problem_id": "c2516c611d74dc4f", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{3} + x^{2}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(4 x^{3}) = 12 x^{3}$.\n3) So $f'(x) = 12 x^{3} + 2 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(4 x^{3}) = 12 x^{2}$.\nTherefore the correct derivative is $f'(x) = 2 x \\left(6 x + 1\\right)$.\nFinal answer: $2 x \\left(6 x + 1\\right)$", "final_answer": "2 x \\left(6 x + 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "38072ba4f0dd4048", "problem_id": "26e9853dcb83fb74", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{4} - 2 x^{3} - x^{2} + x - 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(3 x^{4}) = 12 x^{4}$.\n3) So $f'(x) = 12 x^{4} - 6 x^{2} - 2 x + 1$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(3 x^{4}) = 12 x^{3}$.\nTherefore the correct derivative is $f'(x) = 12 x^{3} - 6 x^{2} - 2 x + 1$.\nFinal answer: $12 x^{3} - 6 x^{2} - 2 x + 1$", "final_answer": "12 x^{3} - 6 x^{2} - 2 x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b79d887001cfe703", "problem_id": "9526358e9333e54f", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "You want 42 liters of a 35% salt solution. You have a 30% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 42−x liters is 40% solution.\n2) Salt equation: 30x + 40(42−x) = 35·42.\n3) (30-40)x = (35-40)·42 ⇒ x = 21.\n4) Then 42−x = 21.\nFinal answer: $21 L of 30% and 21 L of 40%$", "final_answer": "21 L of 30% and 21 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f5ab66a9b870102b", "problem_id": "580831d785aa4964", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$3 x - 8 y$ = $-56$\n$7 x + 3 y$ = $-44$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $7$: $21 x - 56 y$ = $-392$.\n3) Multiply the second equation by $3$: $21 x + 9 y$ = $-132$.\n4) Subtract to eliminate x: $- 65 y$ = $-260$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $-8$.\nFinal answer: $(x,y)=\\left(-8, 4\\right)$", "final_answer": "(x,y)=\\left(-8, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e2182d35f66a49c9", "problem_id": "580831d785aa4964", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the linear system:\n$3 x - 8 y$ = $-56$\n$7 x + 3 y$ = $-44$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $7$: $21 x - 56 y$ = $-392$.\nMultiply the second equation by $3$: $21 x + 9 y$ = $-132$.\nFinal answer: $(x,y)=\\left(-8, 4\\right)$", "final_answer": "(x,y)=\\left(-8, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d78b0daf8dad49d7", "problem_id": "4204b9ccb13be038", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 10 red balls and 9 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(19,4).\n3) Favorable samples: choose k reds and n−k blues: C(10,1)·C(9,3).\n4) Compute totals: C(19,4) = 3876, and favorable = 840.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{70}{323}$", "final_answer": "\\frac{70}{323}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e7291d5ff0b234cc", "problem_id": "4204b9ccb13be038", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 10 red balls and 9 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(19,4).\nFavorable samples: choose k reds and n−k blues: C(10,1)·C(9,3).\nCompute totals: C(19,4) = 3876, and favorable = 840.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{70}{323}$.\nFinal answer: $\\frac{70}{323}$", "final_answer": "\\frac{70}{323}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "84ca01cc583d9b5d", "problem_id": "fab1df36d58d0ec1", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡5 (mod 9), x≡1 (mod 8). Give the solution modulo 72.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 72.\n3) Compute the CRT solution to get x ≡ 41 (mod 72).\nFinal answer: $x\\equiv 41\\pmod{72}$", "final_answer": "x\\equiv 41\\pmod{72}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c5582b6782a8ec00", "problem_id": "22cbb7d6762ab061", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(- x^{2} - x - 2\\right) \\cos{\\left(x - 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x e^{x}) = 2 x e^{x}$.\n3) So $f'(x) = x^{2} \\sin{\\left(x - 3 \\right)} + 2 x e^{x} + x \\sin{\\left(x - 3 \\right)} - 2 x \\cos{\\left(x - 3 \\right)} + 4 e^{x} + 2 \\sin{\\left(x - 3 \\right)} - \\cos{\\left(x - 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x e^{x}) = 2 x e^{x} + 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = \\left(- 2 x - 1\\right) \\cos{\\left(x - 3 \\right)} + 2 \\left(x + 2\\right) e^{x} - \\left(x \\left(- x - 1\\right) - 2\\right) \\sin{\\left(x - 3 \\right)} + 2 e^{x}$.\nFinal answer: $\\left(- 2 x - 1\\right) \\cos{\\left(x - 3 \\right)} + 2 \\left(x + 2\\right) e^{x} - \\left(x \\left(- x - 1\\right) - 2\\right) \\sin{\\left(x - 3 \\right)} + 2 e^{x}$", "final_answer": "\\left(- 2 x - 1\\right) \\cos{\\left(x - 3 \\right)} + 2 \\left(x + 2\\right) e^{x} - \\left(x \\left(- x - 1\\right) - 2\\right) \\sin{\\left(x - 3 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f8b1b150d946558b", "problem_id": "9a462b9c58f12542", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Worker A finishes a job in 19 hours and Worker B finishes it in 4 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/19 job/hour. B's rate = 1/4 job/hour.\nCombined rate = 1/19 + 1/4 = 23/76 job/hour.\nTime = 1 ÷ (combined rate) = 76/23 hours.\nFinal answer: $\\frac{76}{23}$", "final_answer": "\\frac{76}{23}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ed3f4ed9243d65b7", "problem_id": "9a462b9c58f12542", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Worker A finishes a job in 19 hours and Worker B finishes it in 4 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/19 job/hour. B's rate = 1/4 job/hour.\nCombined rate = 1/19 + 1/4 = 23/76 job/hour.\nTime = 1 ÷ (combined rate) = 76/23 hours.\nCheck: (combined rate)×(time) = 23/76×76/23 = 1.\nFinal answer: $\\frac{76}{23}$", "final_answer": "\\frac{76}{23}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0f57138f45a78f1b", "problem_id": "fa44bc997a777d50", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 3 x^{3} - x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{3} - x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $- 3 \\pi^{3} - \\frac{9 \\pi^{2}}{2} - 4 + 6 \\pi + \\frac{45 \\pi^{4}}{4}$.\n4) Simplify the result.\nFinal answer: $- 3\\cdot \\pi^{3} - \\frac{9\\cdot \\pi^{2}}{2} - 4 + 6\\cdot \\pi + \\frac{45\\cdot \\pi^{4}}{4}$", "final_answer": "- 3\\cdot \\pi^{3} - \\frac{9\\cdot \\pi^{2}}{2} - 4 + 6\\cdot \\pi + \\frac{45\\cdot \\pi^{4}}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "17f4f92edaf5d737", "problem_id": "32d8d20afd531b76", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{3} + x^{2} - \\frac{x}{2} + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - \\frac{1}{2}}{x - 4}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}) = - \\frac{4 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4}$.\n3) So $f'(x) = \\frac{4 x^{3}}{2 x^{2} - 16 x + 32} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{22 x^{2}}{2 x^{2} - 16 x + 32} - \\frac{8 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{16 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{16 x}{2 x^{2} - 16 x + 32} + \\frac{16 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{32 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{5}{2 x^{2} - 16 x + 32} - \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 8 x + 16} - \\frac{4 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} + \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4} + \\frac{4 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}) = - \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4} - \\frac{4 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} + \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x - 4\\right)^{2}}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 2 x^{3} - 2 x^{2} + x + \\left(x - 4\\right) \\left(6 x^{2} + 4 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{2 \\left(x - 4\\right)^{2}}$.\nFinal answer: $\\frac{- 2 x^{3} - 2 x^{2} + x + \\left(x - 4\\right) \\left(6 x^{2} + 4 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{2 \\left(x - 4\\right)^{2}}$", "final_answer": "\\frac{- 2 x^{3} - 2 x^{2} + x + \\left(x - 4\\right) \\left(6 x^{2} + 4 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{2 \\left(x - 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0ade0b077fe45a91", "problem_id": "686cda7250ab98c5", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Two friends split $644 in the ratio 6:8. How much does each receive?", "output": "Total ratio parts = 6+8 = 14.\nEach part = 644 ÷ 14 = 46.\nFirst share = 6 × 46 = 276.\nFinal answer: $368$", "final_answer": "368", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "203d73b94106a111", "problem_id": "686cda7250ab98c5", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Two friends split $644 in the ratio 6:8. How much does each receive?", "output": "1) Total ratio parts = 6+8 = 14.\n2) Each part = 644 ÷ 14 = 46.\n3) First share = 6 × 46 = 276.\n4) Second share = 8 × 46 = 368.\nFinal answer: $368$", "final_answer": "368", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5db6a38d3b2190e6", "problem_id": "fa441c067b9c0bfe", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} - x^{2} + 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)}$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $- 3 x^{2} - 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 2$.\n2) Slope at x=-2: m = f'(-2) = $-6 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}$.\n3) Point on curve: (-2, f(-2)) = (-2, $\\cos{\\left(4 \\right)} + e^{-1}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(4 \\right)} + e^{-1} = -6 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}(x − -2).\n6) Optionally expand: y = $- 6 x + 2 x \\sin{\\left(4 \\right)} + \\frac{x}{2 e} - 12 + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$.\nFinal answer: $y = - 6\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 12 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$", "final_answer": "y = - 6\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 12 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "43e2883aa48f4a83", "problem_id": "d62e0e0ca7029ce5", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} - x - 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x^{2}) = 2 x^{2}$.\n3) So $f'(x) = 2 x^{2} - 1$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x^{2}) = 2 x$.\nTherefore the correct derivative is $f'(x) = 2 x - 1$.\nFinal answer: $2 x - 1$", "final_answer": "2 x - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7cb26da07d81680f", "problem_id": "442acb0d8f272748", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Worker A finishes a job in 20 hours and Worker B finishes it in 6 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/20 job/hour. B's rate = 1/6 job/hour.\nCombined rate = 1/20 + 1/6 = 13/60 job/hour.\nTime = 1 ÷ (combined rate) = 60/13 hours.\nFinal answer: $\\frac{60}{13}$", "final_answer": "\\frac{60}{13}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "70a3e92a851eb13e", "problem_id": "442acb0d8f272748", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Worker A finishes a job in 20 hours and Worker B finishes it in 6 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/20 job/hour. B's rate = 1/6 job/hour.\nCombined rate = 1/20 + 1/6 = 13/60 job/hour.\nTime = 1 ÷ (combined rate) = 60/13 hours.\nCheck: (combined rate)×(time) = 13/60×60/13 = 1.\nFinal answer: $\\frac{60}{13}$", "final_answer": "\\frac{60}{13}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e01c647b3faefa7d", "problem_id": "0abd6e8bd5c0def6", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$6 x$ = $-36$\n$4 y$ = $-12$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $6$: $24 y$ = $-72$.\n4) Subtract to eliminate x: $- 24 y$ = $72$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $-6$.\nFinal answer: $(x,y)=\\left(-6, -3\\right)$", "final_answer": "(x,y)=\\left(-6, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fa82c8a88919d046", "problem_id": "0abd6e8bd5c0def6", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$6 x$ = $-36$\n$4 y$ = $-12$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $6$: $24 y$ = $-72$.\nSubtract to eliminate x: $- 24 y$ = $72$.\nSolve for y: y = $-3$.\nSubstitute back to find x: x = $-6$.\nCheck: plug (x,y)=(-6,-3) into both equations.\nFinal answer: $(x,y)=\\left(-6, -3\\right)$", "final_answer": "(x,y)=\\left(-6, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7a8cccec29c30080", "problem_id": "ee99da3a8f13cf9c", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A store sells a backpack for $285. It is discounted by 35%. You buy 11 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 285 × (1 − 35/100) = 285 × 65/100 = 741/4.\n2) Subtotal = 11 × 741/4 = 8151/4.\n3) Total with tax = 8151/4 × 105/100 = 171171/80.\nFinal answer: $\\frac{171171}{80}$", "final_answer": "\\frac{171171}{80}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ab00f7e828e4ec92", "problem_id": "ee99da3a8f13cf9c", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "A store sells a backpack for $285. It is discounted by 35%. You buy 11 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 285 × (1 − 35/100) = 285 × 65/100 = 741/4.\nSubtotal = 11 × 741/4 = 8151/4.\nTotal with tax = 8151/4 × 105/100 = 171171/80.\nFinal answer: $\\frac{171171}{80}$", "final_answer": "\\frac{171171}{80}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b21223dd1d1c47f1", "problem_id": "1eaf2e005c2256e0", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 10 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=2, p=$\\frac{1}{5}$ and simplify.\n4) Here C(10,2) = 45.\nFinal answer: $\\frac{589824}{1953125}$", "final_answer": "\\frac{589824}{1953125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f09312f37d97c27c", "problem_id": "1eaf2e005c2256e0", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 10 times, what is the probability of exactly 2 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=2, p=$\\frac{1}{5}$ and simplify.\nFinal answer: $\\frac{589824}{1953125}$", "final_answer": "\\frac{589824}{1953125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f82dd359c3f133e9", "problem_id": "b8c4b35b3e29c5da", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A store sells a backpack for $326. It is discounted by 25%. You buy 9 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 326 × (1 − 25/100) = 326 × 75/100 = 489/2.\n2) Subtotal = 9 × 489/2 = 4401/2.\n3) Total with tax = 4401/2 × 105/100 = 92421/40.\nFinal answer: $\\frac{92421}{40}$", "final_answer": "\\frac{92421}{40}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7f31e937e274eceb", "problem_id": "b8c4b35b3e29c5da", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A store sells a backpack for $326. It is discounted by 25%. You buy 9 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 326 × (1 − 25/100) = 326 × 75/100 = 489/2.\nSubtotal = 9 × 489/2 = 4401/2.\nTotal with tax = 4401/2 × 105/100 = 92421/40.\nFinal answer: $\\frac{92421}{40}$", "final_answer": "\\frac{92421}{40}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4c0d05f380421519", "problem_id": "f130d5c2a95b1078", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Worker A finishes a job in 7 hours and Worker B finishes it in 13 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/7 job/hour. B's rate = 1/13 job/hour.\nCombined rate = 1/7 + 1/13 = 20/91 job/hour.\nTime = 1 ÷ (combined rate) = 91/20 hours.\nCheck: (combined rate)×(time) = 20/91×91/20 = 1.\nFinal answer: $\\frac{91}{20}$", "final_answer": "\\frac{91}{20}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e8d91e911b85e69", "problem_id": "d8ba5ecd6ddbe0bf", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 3 x^{3} - 6 x^{2} - 9 x - 15$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{3} - 6 x^{2} - 10 x - 15.\n3) Then the limit is -83.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -81.\nFinal answer: $-81$", "final_answer": "-81", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "54a4dd6604d37e3f", "problem_id": "72867f4568cf93a8", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 8 x + 15$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} - 8 x + 15$ = $\\left(x - 5\\right) \\left(x - 3\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $5$.\n5) From the second factor: x = $3$.\nFinal answer: $x \\in \\{5, 3\\}$", "final_answer": "x \\in \\{5, 3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d4f0eeab205a58c4", "problem_id": "72867f4568cf93a8", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 8 x + 15$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} - 8 x + 15$ = $\\left(x - 5\\right) \\left(x - 3\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $5$.\nFrom the second factor: x = $3$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{5, 3\\}$", "final_answer": "x \\in \\{5, 3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "137df9dbe15561f1", "problem_id": "07887a7afe71ed64", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{2} x + \\frac{26 x}{5} - \\frac{\\sqrt{2}}{5} + 1$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{26}{5} - \\sqrt{2}$, c=$1 - \\frac{\\sqrt{2}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{626}{25} - \\frac{48 \\sqrt{2}}{5}$, so √Δ = $\\sqrt{\\frac{626}{25} - \\frac{48 \\sqrt{2}}{5}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{13}{5} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{626 - 240 \\sqrt{2}}}{10}$.\n7) x2 = $- \\frac{13}{5} - \\frac{\\sqrt{626 - 240 \\sqrt{2}}}{10} + \\frac{\\sqrt{2}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{13}{5} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{626 - 240\\cdot \\sqrt{2}}}{10}, - \\frac{13}{5} - \\frac{\\sqrt{626 - 240\\cdot \\sqrt{2}}}{10} + \\frac{\\sqrt{2}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{13}{5} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{626 - 240\\cdot \\sqrt{2}}}{10}, - \\frac{13}{5} - \\frac{\\sqrt{626 - 240\\cdot \\sqrt{2}}}{10} + \\frac{\\sqrt{2}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6c9d278c8a43940e", "problem_id": "07887a7afe71ed64", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{2} x + \\frac{26 x}{5} - \\frac{\\sqrt{2}}{5} + 1$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{26}{5} - \\sqrt{2}$, c=$1 - \\frac{\\sqrt{2}}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{626}{25} - \\frac{48 \\sqrt{2}}{5}$, so √Δ = $\\sqrt{\\frac{626}{25} - \\frac{48 \\sqrt{2}}{5}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{13}{5} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{626 - 240 \\sqrt{2}}}{10}$.\nx2 = $- \\frac{13}{5} - \\frac{\\sqrt{626 - 240 \\sqrt{2}}}{10} + \\frac{\\sqrt{2}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{13}{5} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{626 - 240\\cdot \\sqrt{2}}}{10}, - \\frac{13}{5} - \\frac{\\sqrt{626 - 240\\cdot \\sqrt{2}}}{10} + \\frac{\\sqrt{2}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{13}{5} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{626 - 240\\cdot \\sqrt{2}}}{10}, - \\frac{13}{5} - \\frac{\\sqrt{626 - 240\\cdot \\sqrt{2}}}{10} + \\frac{\\sqrt{2}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dab86a7057d51e89", "problem_id": "221cb3121addc7db", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A store sells a backpack for $635. It is discounted by 20%. You buy 6 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 635 × (1 − 20/100) = 635 × 80/100 = 508.\n2) Subtotal = 6 × 508 = 3048.\n3) Total with tax = 3048 × 105/100 = 16002/5.\nFinal answer: $\\frac{16002}{5}$", "final_answer": "\\frac{16002}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fbfd04839919215f", "problem_id": "9ef4bedea90b0247", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(2 x^{2} - 3\\right) \\cos{\\left(x^{2} + x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x^{2} \\cos{\\left(\\left(x^{2} + x\\right) + 3 \\right)}) = - 2 x^{2} \\sin{\\left(x^{2} + x + 3 \\right)}$.\n3) So $f'(x) = - 2 x^{2} \\sin{\\left(x^{2} + x + 3 \\right)} + 2 x e^{x} + 6 x \\sin{\\left(\\left(x^{2} + x\\right) + 3 \\right)} - 4 e^{x} + 3 \\sin{\\left(\\left(x^{2} + x\\right) + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x^{2} \\cos{\\left(\\left(x^{2} + x\\right) + 3 \\right)}) = - 2 x^{2} \\left(2 x + 1\\right) \\sin{\\left(\\left(x^{2} + x\\right) + 3 \\right)} + 4 x \\cos{\\left(\\left(x^{2} + x\\right) + 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = 4 x \\cos{\\left(\\left(x^{2} + x\\right) + 3 \\right)} + 2 \\left(x - 3\\right) e^{x} - \\left(2 x + 1\\right) \\left(2 x^{2} - 3\\right) \\sin{\\left(\\left(x^{2} + x\\right) + 3 \\right)} + 2 e^{x}$.\nFinal answer: $4 x \\cos{\\left(\\left(x^{2} + x\\right) + 3 \\right)} + 2 \\left(x - 3\\right) e^{x} - \\left(2 x + 1\\right) \\left(2 x^{2} - 3\\right) \\sin{\\left(\\left(x^{2} + x\\right) + 3 \\right)} + 2 e^{x}$", "final_answer": "4 x \\cos{\\left(\\left(x^{2} + x\\right) + 3 \\right)} + 2 \\left(x - 3\\right) e^{x} - \\left(2 x + 1\\right) \\left(2 x^{2} - 3\\right) \\sin{\\left(\\left(x^{2} + x\\right) + 3 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "344c8e43968dfe17", "problem_id": "38ba58e5425a9a35", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$4 x + 4 y$ = $48$\n$6 x - 5 y$ = $17$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $6$: $24 x + 24 y$ = $288$.\n3) Multiply the second equation by $4$: $24 x - 20 y$ = $68$.\n4) Subtract to eliminate x: $44 y$ = $220$.\n5) Solve for y: y = $5$.\n6) Substitute back to find x: x = $7$.\nFinal answer: $(x,y)=\\left(7, 5\\right)$", "final_answer": "(x,y)=\\left(7, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fbfd9802c6106eb3", "problem_id": "38ba58e5425a9a35", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the linear system:\n$4 x + 4 y$ = $48$\n$6 x - 5 y$ = $17$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $6$: $24 x + 24 y$ = $288$.\nMultiply the second equation by $4$: $24 x - 20 y$ = $68$.\nFinal answer: $(x,y)=\\left(7, 5\\right)$", "final_answer": "(x,y)=\\left(7, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f814098aef920b03", "problem_id": "b6e139def8ec2ff0", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=30$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=30 and k=5.\n4) Compute C(34,4) = 46376.\nFinal answer: $46376$", "final_answer": "46376", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "578ada488fa7f470", "problem_id": "b6e139def8ec2ff0", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=30$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=30 and k=5.\nCompute C(34,4) = 46376.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $46376$", "final_answer": "46376", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aa3a13aa643db2b1", "problem_id": "f6e38ff1fe3a3242", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} - x^{4} + 3 x^{3} - 4 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{4} + 3 x^{3} - 4 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\pi).\n3) Subtract: F(\\pi) − F(- \\pi) = $\\frac{2 \\pi \\left(- 3 \\pi^{4} - 20 \\pi^{2} - 45\\right)}{15}$.\n4) Simplify the result.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(- 3\\cdot \\pi^{4} - 20\\cdot \\pi^{2} - 45\\right)}{15}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(- 3\\cdot \\pi^{4} - 20\\cdot \\pi^{2} - 45\\right)}{15}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5430a3f40dc1ce7d", "problem_id": "f6e38ff1fe3a3242", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} - x^{4} + 3 x^{3} - 4 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{4} + 3 x^{3} - 4 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\nEvaluate at the bounds: F(\\pi) and F(- \\pi).\nSubtract: F(\\pi) − F(- \\pi) = $\\frac{2 \\pi \\left(- 3 \\pi^{4} - 20 \\pi^{2} - 45\\right)}{15}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(- 3\\cdot \\pi^{4} - 20\\cdot \\pi^{2} - 45\\right)}{15}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(- 3\\cdot \\pi^{4} - 20\\cdot \\pi^{2} - 45\\right)}{15}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "640a81e7de032453", "problem_id": "ac6e562eec8aade8", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡7 (mod 9), x≡4 (mod 10). Give the solution modulo 90.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 90.\n3) Compute the CRT solution to get x ≡ 34 (mod 90).\nFinal answer: $x\\equiv 34\\pmod{90}$", "final_answer": "x\\equiv 34\\pmod{90}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c6f38e90a32271ec", "problem_id": "ac6e562eec8aade8", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡7 (mod 9), x≡4 (mod 10). Give the solution modulo 90.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 90.\nCompute the CRT solution to get x ≡ 34 (mod 90).\nCheck: 34 mod 9=7 and mod 10=4.\nFinal answer: $x\\equiv 34\\pmod{90}$", "final_answer": "x\\equiv 34\\pmod{90}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8c4c2abb88aa7fd0", "problem_id": "e569bc076aa0ab82", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 26 x + 84$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} - 26 x + 84$ = $2 \\left(x - 7\\right) \\left(x - 6\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $7$.\nFrom the second factor: x = $6$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{7, 6\\}$", "final_answer": "x \\in \\{7, 6\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1e593b04df969ba1", "problem_id": "e569bc076aa0ab82", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 26 x + 84$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 26 x + 84$ = $2 \\left(x - 7\\right) \\left(x - 6\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $7$.\n5) From the second factor: x = $6$.\nFinal answer: $x \\in \\{7, 6\\}$", "final_answer": "x \\in \\{7, 6\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "05dcad944fb44ed3", "problem_id": "a16b7cae570566f2", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$- 10 x + 4 y$ = $-58$\n$- 5 x + 6 y$ = $-17$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-5$: $50 x - 20 y$ = $290$.\nMultiply the second equation by $-10$: $50 x - 60 y$ = $170$.\nSubtract to eliminate x: $40 y$ = $120$.\nSolve for y: y = $3$.\nSubstitute back to find x: x = $7$.\nCheck: plug (x,y)=(7,3) into both equations.\nFinal answer: $(x,y)=\\left(7, 3\\right)$", "final_answer": "(x,y)=\\left(7, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5c69821a788c2d0f", "problem_id": "888ac461b1afacb4", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} 2 x^{2} + 4 x + 1$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{2} + 3 x + 1.\n3) Then the limit is 6.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: 7.\nFinal answer: $7$", "final_answer": "7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f4792c1b39d28225", "problem_id": "501eb9a491da244f", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} + 1$ at $x=-2$.", "output": "Differentiate: f'(x) = $6 x + 2 \\cos{\\left(x \\right)} + 3$.\nSlope at x=-2: m = f'(-2) = $-9 + 2 \\cos{\\left(2 \\right)}$.\nPoint on curve: (-2, f(-2)) = (-2, $7 - 2 \\sin{\\left(2 \\right)}$).\nFinal answer: $y = - 9\\cdot x + 2\\cdot x\\cdot \\cos{\\left(2 \\right)} - 11 - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)}$", "final_answer": "y = - 9\\cdot x + 2\\cdot x\\cdot \\cos{\\left(2 \\right)} - 11 - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a9144bbe2776e416", "problem_id": "501eb9a491da244f", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} + 1$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $6 x + 2 \\cos{\\left(x \\right)} + 3$.\n2) Slope at x=-2: m = f'(-2) = $-9 + 2 \\cos{\\left(2 \\right)}$.\n3) Point on curve: (-2, f(-2)) = (-2, $7 - 2 \\sin{\\left(2 \\right)}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 7 - 2 \\sin{\\left(2 \\right)} = -9 + 2 \\cos{\\left(2 \\right)}(x − -2).\n6) Optionally expand: y = $- 9 x + 2 x \\cos{\\left(2 \\right)} - 11 - 2 \\sin{\\left(2 \\right)} + 4 \\cos{\\left(2 \\right)}$.\nFinal answer: $y = - 9\\cdot x + 2\\cdot x\\cdot \\cos{\\left(2 \\right)} - 11 - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)}$", "final_answer": "y = - 9\\cdot x + 2\\cdot x\\cdot \\cos{\\left(2 \\right)} - 11 - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff5604fcf6811b00", "problem_id": "54b0c7c081ef3152", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$- 9 x + y$ = $67$\n$- 8 y$ = $-32$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $-9$: $72 y$ = $288$.\nSubtract to eliminate x: $- 72 y$ = $-288$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $-7$.\nCheck: plug (x,y)=(-7,4) into both equations.\nFinal answer: $(x,y)=\\left(-7, 4\\right)$", "final_answer": "(x,y)=\\left(-7, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "714e0001407d862d", "problem_id": "54b0c7c081ef3152", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$- 9 x + y$ = $67$\n$- 8 y$ = $-32$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $-9$: $72 y$ = $288$.\n4) Subtract to eliminate x: $- 72 y$ = $-288$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $-7$.\nFinal answer: $(x,y)=\\left(-7, 4\\right)$", "final_answer": "(x,y)=\\left(-7, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c4841718fd10202d", "problem_id": "8beb79840d86b9ae", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} - 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $- 6 \\pi^{3} - 9 \\pi - 4 + \\frac{9 \\pi^{2}}{2}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- 6\\cdot \\pi^{3} - 9\\cdot \\pi - 4 + \\frac{9\\cdot \\pi^{2}}{2}$", "final_answer": "- 6\\cdot \\pi^{3} - 9\\cdot \\pi - 4 + \\frac{9\\cdot \\pi^{2}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "57e15b68fdca1dd0", "problem_id": "8beb79840d86b9ae", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} - 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $- 6 \\pi^{3} - 9 \\pi - 4 + \\frac{9 \\pi^{2}}{2}$.\n4) Simplify the result.\nFinal answer: $- 6\\cdot \\pi^{3} - 9\\cdot \\pi - 4 + \\frac{9\\cdot \\pi^{2}}{2}$", "final_answer": "- 6\\cdot \\pi^{3} - 9\\cdot \\pi - 4 + \\frac{9\\cdot \\pi^{2}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a579d2c44ecafa28", "problem_id": "73b38415fde16c5f", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A store sells a backpack for $495. It is discounted by 25%. You buy 6 backpacks. What is the total cost?", "output": "Discounted price = 495 × (1 − 25/100) = 495 × 75/100 = 1485/4.\nSubtotal = 6 × 1485/4 = 4455/2.\nCheck: recomputing the formula yields 4455/2.\nFinal answer: $\\frac{4455}{2}$", "final_answer": "\\frac{4455}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "164c2922f501f39b", "problem_id": "73b38415fde16c5f", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A store sells a backpack for $495. It is discounted by 25%. You buy 6 backpacks. What is the total cost?", "output": "Discounted price = 495 × (1 − 25/100) = 495 × 75/100 = 1485/4.\nSubtotal = 6 × 1485/4 = 4455/2.\nFinal answer: $\\frac{4455}{2}$", "final_answer": "\\frac{4455}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8ffea7565f57dfac", "problem_id": "f432591b68e54d06", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A cyclist rides at 79 miles per hour for 1 hours, then at 60 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 79 × 1 = 79 miles.\n2) Second leg distance = 60 × 5/2 = 150 miles.\n3) Total distance = 79 + 150 = 229 miles.\nFinal answer: $229$", "final_answer": "229", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4746de7965cbf404", "problem_id": "c9caa646a1f664d6", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} + 4 x - 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(2 x^{2}) = 4 x^{2}$.\n3) So $f'(x) = 4 x^{2} + 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(2 x^{2}) = 4 x$.\nTherefore the correct derivative is $f'(x) = 4 \\left(x + 1\\right)$.\nFinal answer: $4 \\left(x + 1\\right)$", "final_answer": "4 \\left(x + 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e102426cf9bd1127", "problem_id": "d585c7523a386ba3", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$4 x + 6 y$ = $-4$\n$- x - 8 y$ = $40$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $- 4 x - 6 y$ = $4$.\n3) Multiply the second equation by $4$: $- 4 x - 32 y$ = $160$.\n4) Subtract to eliminate x: $26 y$ = $-156$.\n5) Solve for y: y = $-6$.\n6) Substitute back to find x: x = $8$.\nFinal answer: $(x,y)=\\left(8, -6\\right)$", "final_answer": "(x,y)=\\left(8, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "832e617653c13696", "problem_id": "d585c7523a386ba3", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$4 x + 6 y$ = $-4$\n$- x - 8 y$ = $40$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $- 4 x - 6 y$ = $4$.\nMultiply the second equation by $4$: $- 4 x - 32 y$ = $160$.\nSubtract to eliminate x: $26 y$ = $-156$.\nSolve for y: y = $-6$.\nSubstitute back to find x: x = $8$.\nCheck: plug (x,y)=(8,-6) into both equations.\nFinal answer: $(x,y)=\\left(8, -6\\right)$", "final_answer": "(x,y)=\\left(8, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4efadd47a6853c4d", "problem_id": "6eabf884c2a02142", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A disease has prevalence $\\frac{1}{100}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{43}{400}$.\n3) Compute P(D|+) = $\\frac{17}{215}$.\nFinal answer: $\\frac{17}{215}$", "final_answer": "\\frac{17}{215}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "383caca12e55a326", "problem_id": "6eabf884c2a02142", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A disease has prevalence $\\frac{1}{100}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{43}{400}$.\nCompute P(D|+) = $\\frac{17}{215}$.\nFinal answer: $\\frac{17}{215}$", "final_answer": "\\frac{17}{215}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "12288daeac57ac7c", "problem_id": "4afdb3a31bcd075c", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{3} - x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 3 x^{3} - x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 2$.\nCompute an antiderivative: F(x) = $- \\frac{3 x^{4}}{4} - \\frac{x^{3}}{3} - x^{2} - 2 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - \\frac{3\\cdot x^{4}}{4} - \\frac{x^{3}}{3} - x^{2} - 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{3\\cdot x^{4}}{4} - \\frac{x^{3}}{3} - x^{2} - 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c5eb92da2b0998b3", "problem_id": "4afdb3a31bcd075c", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{3} - x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 3 x^{3} - x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 2$.\n3) Compute an antiderivative: F(x) = $- \\frac{3 x^{4}}{4} - \\frac{x^{3}}{3} - x^{2} - 2 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{3\\cdot x^{4}}{4} - \\frac{x^{3}}{3} - x^{2} - 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{3\\cdot x^{4}}{4} - \\frac{x^{3}}{3} - x^{2} - 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3ae586b3aa9ebb94", "problem_id": "ffd8b696d7112f6b", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Find the equation of the tangent line to $y=4 x^{3} - x^{2} - 4$ at $x=0$.", "output": "1) Differentiate: f'(x) = $12 x^{2} - 2 x$.\n2) Slope at x=0: m = f'(0) = $0$.\n3) Point on curve: (0, f(0)) = (0, $-4$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -4 = 0(x − 0).\n6) Optionally expand: y = $-4$.\nFinal answer: $y = -4$", "final_answer": "y = -4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8c1934cd91f868d6", "problem_id": "b15b4812339e667d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} x^{3} + 2 x^{2} + 2 x + 4$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to x^{3} + 2 x^{2} + x + 4.\n3) Then the limit is 8.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: 9.\nFinal answer: $9$", "final_answer": "9", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "972b4a16f449304b", "problem_id": "c9caa646a1f664d6", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} + 4 x - 1$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-1) = 0.\n3) d/dx(2 x^{2}) = 4 x.\n4) d/dx(4 x) = 4.\n5) Combine: f'(x) = 4 \\left(x + 1\\right).\nFinal answer: $4\\cdot x + 4$", "final_answer": "4\\cdot x + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ab2a01c1cb23f1f5", "problem_id": "c9caa646a1f664d6", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} + 4 x - 1$.", "output": "Differentiate term-by-term:\nd/dx(-1) = 0.\nd/dx(2 x^{2}) = 4 x.\nd/dx(4 x) = 4.\nCombine: f'(x) = 4 \\left(x + 1\\right).\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $4\\cdot x + 4$", "final_answer": "4\\cdot x + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "99c7cc1f0b6c3228", "problem_id": "86efa5cec7f7d1de", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 7 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=7, k=3, p=$\\frac{1}{8}$ and simplify.\n4) Here C(7,3) = 35.\nFinal answer: $\\frac{84035}{2097152}$", "final_answer": "\\frac{84035}{2097152}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5454c0677c41ca7d", "problem_id": "86efa5cec7f7d1de", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 7 times, what is the probability of exactly 3 heads?", "output": "$\\frac{84035}{2097152}$", "final_answer": "\\frac{84035}{2097152}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fdce5fbb83aa5301", "problem_id": "61a5079b9e49d7ba", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Worker A finishes a job in 22 hours and Worker B finishes it in 21 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/22 job/hour. B's rate = 1/21 job/hour.\n2) Combined rate = 1/22 + 1/21 = 43/462 job/hour.\n3) Time = 1 ÷ (combined rate) = 462/43 hours.\nFinal answer: $\\frac{462}{43}$", "final_answer": "\\frac{462}{43}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c23bd98d84af1b2e", "problem_id": "61a5079b9e49d7ba", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Worker A finishes a job in 22 hours and Worker B finishes it in 21 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/22 job/hour. B's rate = 1/21 job/hour.\nCombined rate = 1/22 + 1/21 = 43/462 job/hour.\nTime = 1 ÷ (combined rate) = 462/43 hours.\nCheck: (combined rate)×(time) = 43/462×462/43 = 1.\nFinal answer: $\\frac{462}{43}$", "final_answer": "\\frac{462}{43}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7bdf8530c49f3c3a", "problem_id": "b382b66f05c25a17", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A cyclist rides at 51 miles per hour for 1 hours, then at 38 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 51 × 1 = 51 miles.\n2) Second leg distance = 38 × 2 = 76 miles.\n3) Total distance = 51 + 76 = 127 miles.\nFinal answer: $127$", "final_answer": "127", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "75397c9b7f2db045", "problem_id": "b382b66f05c25a17", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 51 miles per hour for 1 hours, then at 38 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 51 × 1 = 51 miles.\nSecond leg distance = 38 × 2 = 76 miles.\nTotal distance = 51 + 76 = 127 miles.\nCheck: adding both legs gives 127.\nFinal answer: $127$", "final_answer": "127", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "18e24a65a0eaac7d", "problem_id": "81bdf5f170e55a9a", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{3} + x^{2} - x + \\left(2 x^{2} + x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{2 x^{2} + x - 2}$.", "output": "1) Let f(x) = $\\frac{- x^{3} + x^{2} - x + \\left(2 x^{2} + x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{2 x^{2} + x - 2}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{2 \\left(- 4 x - 1\\right) \\left(- x^{3} + x^{2} - x + \\left(2 x^{2} + x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(2 x^{2} + x - 2\\right) \\left(- 6 x^{2} + 4 x + 2 \\left(4 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(2 x^{2} + x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x^{2} + x - 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(2 x^{2} + x - 2\\right)^{2}}$.\nFinal answer: $\\frac{\\left(- 8\\cdot x - 2\\right)\\cdot \\left(- x^{3} + x^{2} - x + \\left(2\\cdot x^{2} + x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 3\\right) + \\left(2\\cdot x^{2} + x - 2\\right)\\cdot \\left(- 6\\cdot x^{2} + 4\\cdot x + \\left(8\\cdot x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x^{2} + 2\\cdot x - 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 2\\right)}{2\\cdot \\left(2\\cdot x^{2} + x - 2\\right)^{2}}$", "final_answer": "\\frac{\\left(- 8\\cdot x - 2\\right)\\cdot \\left(- x^{3} + x^{2} - x + \\left(2\\cdot x^{2} + x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 3\\right) + \\left(2\\cdot x^{2} + x - 2\\right)\\cdot \\left(- 6\\cdot x^{2} + 4\\cdot x + \\left(8\\cdot x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x^{2} + 2\\cdot x - 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 2\\right)}{2\\cdot \\left(2\\cdot x^{2} + x - 2\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b5a3ed7ec3d82b74", "problem_id": "81bdf5f170e55a9a", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{3} + x^{2} - x + \\left(2 x^{2} + x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{2 x^{2} + x - 2}$.", "output": "Let f(x) = $\\frac{- x^{3} + x^{2} - x + \\left(2 x^{2} + x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{2 x^{2} + x - 2}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{2 \\left(- 4 x - 1\\right) \\left(- x^{3} + x^{2} - x + \\left(2 x^{2} + x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(2 x^{2} + x - 2\\right) \\left(- 6 x^{2} + 4 x + 2 \\left(4 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(2 x^{2} + x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x^{2} + x - 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(2 x^{2} + x - 2\\right)^{2}}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\frac{\\left(- 8\\cdot x - 2\\right)\\cdot \\left(- x^{3} + x^{2} - x + \\left(2\\cdot x^{2} + x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 3\\right) + \\left(2\\cdot x^{2} + x - 2\\right)\\cdot \\left(- 6\\cdot x^{2} + 4\\cdot x + \\left(8\\cdot x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x^{2} + 2\\cdot x - 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 2\\right)}{2\\cdot \\left(2\\cdot x^{2} + x - 2\\right)^{2}}$", "final_answer": "\\frac{\\left(- 8\\cdot x - 2\\right)\\cdot \\left(- x^{3} + x^{2} - x + \\left(2\\cdot x^{2} + x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 3\\right) + \\left(2\\cdot x^{2} + x - 2\\right)\\cdot \\left(- 6\\cdot x^{2} + 4\\cdot x + \\left(8\\cdot x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x^{2} + 2\\cdot x - 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 2\\right)}{2\\cdot \\left(2\\cdot x^{2} + x - 2\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "654b3eddc815e3b6", "problem_id": "668d640b4416785d", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the linear system:\n$- x$ = $9$\n$- 5 x - 6 y$ = $87$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-5$: $5 x$ = $-45$.\n3) Multiply the second equation by $-1$: $5 x + 6 y$ = $-87$.\n4) Subtract to eliminate x: $- 6 y$ = $42$.\n5) Solve for y: y = $-7$.\n6) Substitute back to find x: x = $-9$.\nFinal answer: $(x,y)=\\left(-9, -7\\right)$", "final_answer": "(x,y)=\\left(-9, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "982be1302411c848", "problem_id": "668d640b4416785d", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the linear system:\n$- x$ = $9$\n$- 5 x - 6 y$ = $87$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-5$: $5 x$ = $-45$.\nMultiply the second equation by $-1$: $5 x + 6 y$ = $-87$.\nFinal answer: $(x,y)=\\left(-9, -7\\right)$", "final_answer": "(x,y)=\\left(-9, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "663a153640a65e3d", "problem_id": "49882216d0f286e6", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - x^{3} - 7 x^{2} - 25 x - 99$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{4} - 3 x^{3} + 3 x^{2} + x + 1$ and a = 4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 4 x^{3} - 9 x^{2} + 6 x + 1$.\n5) Evaluate at a: p'(4) = $-375$.\nFinal answer: $-375$", "final_answer": "-375", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e51da8a02cc3f462", "problem_id": "49882216d0f286e6", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - x^{3} - 7 x^{2} - 25 x - 99$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{4} - 3 x^{3} + 3 x^{2} + x + 1$ and a = 4.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 4 x^{3} - 9 x^{2} + 6 x + 1$.\nEvaluate at a: p'(4) = $-375$.\nCheck: the identity gives $-375$.\nFinal answer: $-375$", "final_answer": "-375", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "489922b7c58f8953", "problem_id": "d428d313166c0ad4", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_3=28$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=28 and k=3.\n4) Compute C(30,2) = 435.\nFinal answer: $435$", "final_answer": "435", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d4add95da870c26b", "problem_id": "e3962e367ed3f02a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{2} + x - 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(4 x^{2}) = 8 x^{2}$.\n3) So $f'(x) = 8 x^{2} + 1$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(4 x^{2}) = 8 x$.\nTherefore the correct derivative is $f'(x) = 8 x + 1$.\nFinal answer: $8 x + 1$", "final_answer": "8 x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8663e3e3c5956a88", "problem_id": "8f99f8f80a1a2692", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{5} x}{2} + \\frac{7 x}{10} - \\frac{3 \\sqrt{5}}{5} - \\frac{3}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{7}{10} - \\frac{\\sqrt{5}}{2}$, c=$- \\frac{3 \\sqrt{5}}{5} - \\frac{3}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{17 \\sqrt{5}}{10} + \\frac{207}{50}$, so √Δ = $\\sqrt{\\frac{17 \\sqrt{5}}{10} + \\frac{207}{50}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{7}{20} + \\frac{\\sqrt{5}}{4} + \\frac{\\sqrt{170 \\sqrt{5} + 414}}{20}$.\n7) x2 = $- \\frac{\\sqrt{170 \\sqrt{5} + 414}}{20} - \\frac{7}{20} + \\frac{\\sqrt{5}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{7}{20} + \\frac{\\sqrt{5}}{4} + \\frac{\\sqrt{170\\cdot \\sqrt{5} + 414}}{20}, - \\frac{\\sqrt{170\\cdot \\sqrt{5} + 414}}{20} - \\frac{7}{20} + \\frac{\\sqrt{5}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{7}{20} + \\frac{\\sqrt{5}}{4} + \\frac{\\sqrt{170\\cdot \\sqrt{5} + 414}}{20}, - \\frac{\\sqrt{170\\cdot \\sqrt{5} + 414}}{20} - \\frac{7}{20} + \\frac{\\sqrt{5}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "135d800dc446584b", "problem_id": "8f99f8f80a1a2692", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{5} x}{2} + \\frac{7 x}{10} - \\frac{3 \\sqrt{5}}{5} - \\frac{3}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{7}{10} - \\frac{\\sqrt{5}}{2}$, c=$- \\frac{3 \\sqrt{5}}{5} - \\frac{3}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{17 \\sqrt{5}}{10} + \\frac{207}{50}$, so √Δ = $\\sqrt{\\frac{17 \\sqrt{5}}{10} + \\frac{207}{50}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{7}{20} + \\frac{\\sqrt{5}}{4} + \\frac{\\sqrt{170 \\sqrt{5} + 414}}{20}$.\nx2 = $- \\frac{\\sqrt{170 \\sqrt{5} + 414}}{20} - \\frac{7}{20} + \\frac{\\sqrt{5}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{7}{20} + \\frac{\\sqrt{5}}{4} + \\frac{\\sqrt{170\\cdot \\sqrt{5} + 414}}{20}, - \\frac{\\sqrt{170\\cdot \\sqrt{5} + 414}}{20} - \\frac{7}{20} + \\frac{\\sqrt{5}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{7}{20} + \\frac{\\sqrt{5}}{4} + \\frac{\\sqrt{170\\cdot \\sqrt{5} + 414}}{20}, - \\frac{\\sqrt{170\\cdot \\sqrt{5} + 414}}{20} - \\frac{7}{20} + \\frac{\\sqrt{5}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "22affe723ce43714", "problem_id": "fdd595970dbd052a", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the multiplicative inverse of 13 modulo 43.", "output": "We want x such that 13x ≡ 1 (mod 43).\nUse extended Euclid to find s,t with 13s + 43t = 1.\nOne Bezout identity is: 13·(10) + 43·(-3) = 1.\nReduce s modulo 43: x ≡ 10 (mod 43).\nCheck: 13·10 mod 43 = 1.\nFinal answer: $x\\equiv 10\\pmod{43}$", "final_answer": "x\\equiv 10\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f6e2c4dc43ac78b", "problem_id": "fdd595970dbd052a", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Find the multiplicative inverse of 13 modulo 43.", "output": "1) We want x such that 13x ≡ 1 (mod 43).\n2) Use extended Euclid to find s,t with 13s + 43t = 1.\n3) One Bezout identity is: 13·(10) + 43·(-3) = 1.\n4) Reduce s modulo 43: x ≡ 10 (mod 43).\nFinal answer: $x\\equiv 10\\pmod{43}$", "final_answer": "x\\equiv 10\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cfb1588a712ee241", "problem_id": "8f1f1f5fa9ed5fb0", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int \\left(2 x + 1\\right) \\cos{\\left(x \\right)} + \\left(3 x + 1\\right) e^{3 x}\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $\\left(2 x + 1\\right) \\cos{\\left(x \\right)} + \\left(3 x + 1\\right) e^{3 x}$.\nCompute an antiderivative: F(x) = $x e^{3 x} + 2 x \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + x\\cdot e^{3\\cdot x} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + x\\cdot e^{3\\cdot x} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0768e2fcc39e3856", "problem_id": "8f1f1f5fa9ed5fb0", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int \\left(2 x + 1\\right) \\cos{\\left(x \\right)} + \\left(3 x + 1\\right) e^{3 x}\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $\\left(2 x + 1\\right) \\cos{\\left(x \\right)} + \\left(3 x + 1\\right) e^{3 x}$.\n3) Compute an antiderivative: F(x) = $x e^{3 x} + 2 x \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + x\\cdot e^{3\\cdot x} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + x\\cdot e^{3\\cdot x} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "793b30c7877d80dd", "problem_id": "c47267e3d72d040c", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "You want 68 liters of a 32% salt solution. You have a 25% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 68−x liters is 40% solution.\n2) Salt equation: 25x + 40(68−x) = 32·68.\n3) (25-40)x = (32-40)·68 ⇒ x = 544/15.\n4) Then 68−x = 476/15.\nFinal answer: $544/15 L of 25% and 476/15 L of 40%$", "final_answer": "544/15 L of 25% and 476/15 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "732718e86ff2b7dc", "problem_id": "c47267e3d72d040c", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "You want 68 liters of a 32% salt solution. You have a 25% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 68−x liters is 40% solution.\nSalt equation: 25x + 40(68−x) = 32·68.\n(25-40)x = (32-40)·68 ⇒ x = 544/15.\nThen 68−x = 476/15.\nCheck: 25·544/15 + 40·476/15 = 32·68.\nFinal answer: $544/15 L of 25% and 476/15 L of 40%$", "final_answer": "544/15 L of 25% and 476/15 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ec272899cd3933b", "problem_id": "b1aa584f3095e0c5", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{13} x + 3 x - 4 \\sqrt{13} - 4$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$3 - \\sqrt{13}$, c=$- 4 \\sqrt{13} - 4$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $10 \\sqrt{13} + 38$, so √Δ = $\\sqrt{10 \\sqrt{13} + 38}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{3}{2} + \\frac{\\sqrt{13}}{2} + \\frac{\\sqrt{10 \\sqrt{13} + 38}}{2}$.\n7) x2 = $- \\frac{\\sqrt{10 \\sqrt{13} + 38}}{2} - \\frac{3}{2} + \\frac{\\sqrt{13}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{3}{2} + \\frac{\\sqrt{13}}{2} + \\frac{\\sqrt{10\\cdot \\sqrt{13} + 38}}{2}, - \\frac{\\sqrt{10\\cdot \\sqrt{13} + 38}}{2} - \\frac{3}{2} + \\frac{\\sqrt{13}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{3}{2} + \\frac{\\sqrt{13}}{2} + \\frac{\\sqrt{10\\cdot \\sqrt{13} + 38}}{2}, - \\frac{\\sqrt{10\\cdot \\sqrt{13} + 38}}{2} - \\frac{3}{2} + \\frac{\\sqrt{13}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bb1af19d466368ad", "problem_id": "b1aa584f3095e0c5", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{13} x + 3 x - 4 \\sqrt{13} - 4$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$3 - \\sqrt{13}$, c=$- 4 \\sqrt{13} - 4$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $10 \\sqrt{13} + 38$, so √Δ = $\\sqrt{10 \\sqrt{13} + 38}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{3}{2} + \\frac{\\sqrt{13}}{2} + \\frac{\\sqrt{10 \\sqrt{13} + 38}}{2}$.\nx2 = $- \\frac{\\sqrt{10 \\sqrt{13} + 38}}{2} - \\frac{3}{2} + \\frac{\\sqrt{13}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{3}{2} + \\frac{\\sqrt{13}}{2} + \\frac{\\sqrt{10\\cdot \\sqrt{13} + 38}}{2}, - \\frac{\\sqrt{10\\cdot \\sqrt{13} + 38}}{2} - \\frac{3}{2} + \\frac{\\sqrt{13}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{3}{2} + \\frac{\\sqrt{13}}{2} + \\frac{\\sqrt{10\\cdot \\sqrt{13} + 38}}{2}, - \\frac{\\sqrt{10\\cdot \\sqrt{13} + 38}}{2} - \\frac{3}{2} + \\frac{\\sqrt{13}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d7f942ad6a86350e", "problem_id": "ded43c2601cb8dc8", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "An urn has 15 red balls and 14 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(29,12).\n3) Favorable samples: choose k reds and n−k blues: C(15,2)·C(14,10).\n4) Compute totals: C(29,12) = 51895935, and favorable = 105105.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{77}{38019}$", "final_answer": "\\frac{77}{38019}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "31082b5a22c3c36c", "problem_id": "ded43c2601cb8dc8", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "An urn has 15 red balls and 14 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(29,12).\nFavorable samples: choose k reds and n−k blues: C(15,2)·C(14,10).\nFinal answer: $\\frac{77}{38019}$", "final_answer": "\\frac{77}{38019}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "549be35171267134", "problem_id": "45b220dbbaf7b89a", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} - x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{3 \\pi^{3}}{8} - 3 \\pi - \\frac{3 \\pi^{2}}{8} - 2$.\n4) Simplify the result.\nFinal answer: $- \\frac{3\\cdot \\pi^{3}}{8} - 3\\cdot \\pi - \\frac{3\\cdot \\pi^{2}}{8} - 2$", "final_answer": "- \\frac{3\\cdot \\pi^{3}}{8} - 3\\cdot \\pi - \\frac{3\\cdot \\pi^{2}}{8} - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72c7808671550e1b", "problem_id": "45b220dbbaf7b89a", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} - x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{3 \\pi^{3}}{8} - 3 \\pi - \\frac{3 \\pi^{2}}{8} - 2$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{3\\cdot \\pi^{3}}{8} - 3\\cdot \\pi - \\frac{3\\cdot \\pi^{2}}{8} - 2$", "final_answer": "- \\frac{3\\cdot \\pi^{3}}{8} - 3\\cdot \\pi - \\frac{3\\cdot \\pi^{2}}{8} - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b96cadd5c53849c5", "problem_id": "caf2905fd85b5b4e", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 23 miles per hour for 1 hours, then at 31 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 23 × 1 = 23 miles.\nSecond leg distance = 31 × 2 = 62 miles.\nTotal distance = 23 + 62 = 85 miles.\nCheck: adding both legs gives 85.\nFinal answer: $85$", "final_answer": "85", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e721ee7c7607ab0", "problem_id": "caf2905fd85b5b4e", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A cyclist rides at 23 miles per hour for 1 hours, then at 31 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 23 × 1 = 23 miles.\n2) Second leg distance = 31 × 2 = 62 miles.\n3) Total distance = 23 + 62 = 85 miles.\nFinal answer: $85$", "final_answer": "85", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a72893cf6132cf77", "problem_id": "698b47db7b9a12d2", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$4 x + 4 y$ = $24$\n$x - 6 y$ = $-57$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $4 x + 4 y$ = $24$.\nMultiply the second equation by $4$: $4 x - 24 y$ = $-228$.\nSubtract to eliminate x: $28 y$ = $252$.\nSolve for y: y = $9$.\nSubstitute back to find x: x = $-3$.\nCheck: plug (x,y)=(-3,9) into both equations.\nFinal answer: $(x,y)=\\left(-3, 9\\right)$", "final_answer": "(x,y)=\\left(-3, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1c70fdbd49db548c", "problem_id": "0d1ed7df721c833d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(3 x^{2} + x + 1\\right) \\cos{\\left(x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x e^{x}) = 2 x e^{x}$.\n3) So $f'(x) = - 3 x^{2} \\sin{\\left(x + 3 \\right)} + 2 x e^{x} - x \\sin{\\left(x + 3 \\right)} + 6 x \\cos{\\left(x + 3 \\right)} - 6 e^{x} - \\sin{\\left(x + 3 \\right)} + \\cos{\\left(x + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x e^{x}) = 2 x e^{x} + 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x - 3\\right) e^{x} + \\left(6 x + 1\\right) \\cos{\\left(x + 3 \\right)} - \\left(x \\left(3 x + 1\\right) + 1\\right) \\sin{\\left(x + 3 \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(x - 3\\right) e^{x} + \\left(6 x + 1\\right) \\cos{\\left(x + 3 \\right)} - \\left(x \\left(3 x + 1\\right) + 1\\right) \\sin{\\left(x + 3 \\right)} + 2 e^{x}$", "final_answer": "2 \\left(x - 3\\right) e^{x} + \\left(6 x + 1\\right) \\cos{\\left(x + 3 \\right)} - \\left(x \\left(3 x + 1\\right) + 1\\right) \\sin{\\left(x + 3 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "cf0dc13e5bee6451", "problem_id": "03e6159ba1e9fb23", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many binary strings of length 21 contain exactly 5 ones with no two ones adjacent?", "output": "Reserve 4 mandatory zeros between the 5 ones.\nThis leaves n−k+1 = 17 valid slots for the ones.\nChoose the slots: C(17,5) = 6188.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $6188$", "final_answer": "6188", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e06dc4c598a4cf1e", "problem_id": "03e6159ba1e9fb23", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "How many binary strings of length 21 contain exactly 5 ones with no two ones adjacent?", "output": "Reserve 4 mandatory zeros between the 5 ones.\nThis leaves n−k+1 = 17 valid slots for the ones.\nChoose the slots: C(17,5) = 6188.\nFinal answer: $6188$", "final_answer": "6188", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bae36170a010bda3", "problem_id": "dd71e6911fc0698b", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$9 x + 9 y$ = $54$\n$5 x - 10 y$ = $-30$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $5$: $45 x + 45 y$ = $270$.\n3) Multiply the second equation by $9$: $45 x - 90 y$ = $-270$.\n4) Subtract to eliminate x: $135 y$ = $540$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $2$.\nFinal answer: $(x,y)=\\left(2, 4\\right)$", "final_answer": "(x,y)=\\left(2, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "972cff76fb9c188d", "problem_id": "dd71e6911fc0698b", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Solve the linear system:\n$9 x + 9 y$ = $54$\n$5 x - 10 y$ = $-30$", "output": "$(x,y)=\\left(2, 4\\right)$", "final_answer": "(x,y)=\\left(2, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e35662aed3501c5b", "problem_id": "b82e99176332d679", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} x \\left(- x - 2\\right)$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} - 3 x.\n3) Then the limit is 2.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: 1.\nFinal answer: $1$", "final_answer": "1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "017fd4cb6aef11f0", "problem_id": "1e650323c65e3d6b", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int \\left(2 x + 1\\right) \\cos{\\left(x \\right)} + \\left(3 x + 3\\right) e^{3 x}\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $\\left(2 x + 1\\right) \\cos{\\left(x \\right)} + \\left(3 x + 3\\right) e^{3 x}$.\nCompute an antiderivative: F(x) = $2 x \\sin{\\left(x \\right)} + \\frac{\\left(3 x + 2\\right) e^{3 x}}{3} + \\sin{\\left(x \\right)} + 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\frac{\\left(3\\cdot x + 2\\right)\\cdot e^{3\\cdot x}}{3} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\frac{\\left(3\\cdot x + 2\\right)\\cdot e^{3\\cdot x}}{3} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6c99d1bcaa895200", "problem_id": "d0b4412203167893", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A cyclist rides at 90 miles per hour for 3/2 hours, then at 41 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 90 × 3/2 = 135 miles.\nSecond leg distance = 41 × 2 = 82 miles.\nTotal distance = 135 + 82 = 217 miles.\nCheck: adding both legs gives 217.\nFinal answer: $217$", "final_answer": "217", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "21341f1c3367d430", "problem_id": "d0b4412203167893", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A cyclist rides at 90 miles per hour for 3/2 hours, then at 41 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 90 × 3/2 = 135 miles.\n2) Second leg distance = 41 × 2 = 82 miles.\n3) Total distance = 135 + 82 = 217 miles.\nFinal answer: $217$", "final_answer": "217", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b81ba43e0c169710", "problem_id": "8b0c60d038cb1cb5", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 29 miles per hour for 1 hours, then at 25 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 29 × 1 = 29 miles.\nSecond leg distance = 25 × 1 = 25 miles.\nTotal distance = 29 + 25 = 54 miles.\nCheck: adding both legs gives 54.\nFinal answer: $54$", "final_answer": "54", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c51b0d06b998ad8f", "problem_id": "8b0c60d038cb1cb5", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A cyclist rides at 29 miles per hour for 1 hours, then at 25 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 29 × 1 = 29 miles.\n2) Second leg distance = 25 × 1 = 25 miles.\n3) Total distance = 29 + 25 = 54 miles.\nFinal answer: $54$", "final_answer": "54", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "821b2f25f0c0a2ec", "problem_id": "6b545b5a431dc809", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 26 red balls and 18 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(44,12).\nFavorable samples: choose k reds and n−k blues: C(26,2)·C(18,10).\nCompute totals: C(44,12) = 21090682613, and favorable = 14221350.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{5850}{8675723}$.\nFinal answer: $\\frac{5850}{8675723}$", "final_answer": "\\frac{5850}{8675723}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "58bdf2fe2389e715", "problem_id": "6b545b5a431dc809", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 26 red balls and 18 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(44,12).\n3) Favorable samples: choose k reds and n−k blues: C(26,2)·C(18,10).\n4) Compute totals: C(44,12) = 21090682613, and favorable = 14221350.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{5850}{8675723}$", "final_answer": "\\frac{5850}{8675723}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b502a3c5e089185a", "problem_id": "cef2704c92e478da", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A store sells a backpack for $76. It is discounted by 40%. You buy 12 backpacks. What is the total cost?", "output": "1) Discounted price = 76 × (1 − 40/100) = 76 × 60/100 = 228/5.\n2) Subtotal = 12 × 228/5 = 2736/5.\nFinal answer: $\\frac{2736}{5}$", "final_answer": "\\frac{2736}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8a7ba5f969e17176", "problem_id": "cef2704c92e478da", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A store sells a backpack for $76. It is discounted by 40%. You buy 12 backpacks. What is the total cost?", "output": "Discounted price = 76 × (1 − 40/100) = 76 × 60/100 = 228/5.\nSubtotal = 12 × 228/5 = 2736/5.\nCheck: recomputing the formula yields 2736/5.\nFinal answer: $\\frac{2736}{5}$", "final_answer": "\\frac{2736}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a2ff8f57689d8c1b", "problem_id": "12717b20a9259407", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 2 x^{2} \\sin{\\left(x \\right)}) = - 2 x^{2} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = 2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 2 x^{2} \\sin{\\left(x \\right)}) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$", "final_answer": "2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a0e8c428f3034257", "problem_id": "0a8518c7436336af", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A cyclist rides at 62 miles per hour for 3/2 hours, then at 79 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 62 × 3/2 = 93 miles.\n2) Second leg distance = 79 × 5/2 = 395/2 miles.\n3) Total distance = 93 + 395/2 = 581/2 miles.\nFinal answer: $\\frac{581}{2}$", "final_answer": "\\frac{581}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1c737808f2320167", "problem_id": "1add520283d5a549", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "You want 21 liters of a 35% salt solution. You have a 30% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 21−x liters is 40% solution.\n2) Salt equation: 30x + 40(21−x) = 35·21.\n3) (30-40)x = (35-40)·21 ⇒ x = 21/2.\n4) Then 21−x = 21/2.\nFinal answer: $21/2 L of 30% and 21/2 L of 40%$", "final_answer": "21/2 L of 30% and 21/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3441cc9315908b1c", "problem_id": "1add520283d5a549", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "You want 21 liters of a 35% salt solution. You have a 30% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 21−x liters is 40% solution.\nSalt equation: 30x + 40(21−x) = 35·21.\n(30-40)x = (35-40)·21 ⇒ x = 21/2.\nThen 21−x = 21/2.\nCheck: 30·21/2 + 40·21/2 = 35·21.\nFinal answer: $21/2 L of 30% and 21/2 L of 40%$", "final_answer": "21/2 L of 30% and 21/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "56eea90e6dfdb175", "problem_id": "6885ad579e41f46f", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$2 x + 4 y$ = $32$\n$- 7 x + 7 y$ = $35$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-7$: $- 14 x - 28 y$ = $-224$.\n3) Multiply the second equation by $2$: $- 14 x + 14 y$ = $70$.\n4) Subtract to eliminate x: $- 42 y$ = $-294$.\n5) Solve for y: y = $7$.\n6) Substitute back to find x: x = $2$.\nFinal answer: $(x,y)=\\left(2, 7\\right)$", "final_answer": "(x,y)=\\left(2, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eaced28760dc43de", "problem_id": "6885ad579e41f46f", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$2 x + 4 y$ = $32$\n$- 7 x + 7 y$ = $35$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-7$: $- 14 x - 28 y$ = $-224$.\nMultiply the second equation by $2$: $- 14 x + 14 y$ = $70$.\nSubtract to eliminate x: $- 42 y$ = $-294$.\nSolve for y: y = $7$.\nSubstitute back to find x: x = $2$.\nCheck: plug (x,y)=(2,7) into both equations.\nFinal answer: $(x,y)=\\left(2, 7\\right)$", "final_answer": "(x,y)=\\left(2, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c37bdb81c09a5d3a", "problem_id": "c70b1fa0bce0719e", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} 2 - x$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{2} - 2 x - 1$ and a = -4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 2 x - 2$.\n5) Evaluate at a: p'(-4) = $6$.\nFinal answer: $6$", "final_answer": "6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "97b327ab6b77894b", "problem_id": "c70b1fa0bce0719e", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} 2 - x$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{2} - 2 x - 1$ and a = -4.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 2 x - 2$.\nEvaluate at a: p'(-4) = $6$.\nCheck: the identity gives $6$.\nFinal answer: $6$", "final_answer": "6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2dbf9c7404dd1fa7", "problem_id": "a32add838778aaeb", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A cyclist rides at 58 miles per hour for 1/2 hours, then at 44 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 58 × 1/2 = 29 miles.\n2) Second leg distance = 44 × 3/2 = 66 miles.\n3) Total distance = 29 + 66 = 95 miles.\nFinal answer: $95$", "final_answer": "95", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bbd0db3d8ff1e6fb", "problem_id": "a32add838778aaeb", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A cyclist rides at 58 miles per hour for 1/2 hours, then at 44 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 58 × 1/2 = 29 miles.\nSecond leg distance = 44 × 3/2 = 66 miles.\nTotal distance = 29 + 66 = 95 miles.\nCheck: adding both legs gives 95.\nFinal answer: $95$", "final_answer": "95", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f5c085553c1ff9dd", "problem_id": "d8ba5ecd6ddbe0bf", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 3 x^{3} - 6 x^{2} - 9 x - 15$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 3 x^{4} + 3 x^{2} + 3 x$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 12 x^{3} + 6 x + 3$.\n5) Evaluate at a: p'(2) = $-81$.\nFinal answer: $-81$", "final_answer": "-81", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "38f8ff616e8bd5be", "problem_id": "9673fa0311086037", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x^{3} + 5 x^{2} + 11 x + 20$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to x^{3} + 5 x^{2} + 10 x + 20.\n3) Then the limit is 68.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 70.\nFinal answer: $70$", "final_answer": "70", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "034c6a86b4d7e49f", "problem_id": "791c104576047744", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the multiplicative inverse of 8 modulo 29.", "output": "1) We want x such that 8x ≡ 1 (mod 29).\n2) Use extended Euclid to find s,t with 8s + 29t = 1.\n3) One Bezout identity is: 8·(11) + 29·(-3) = 1.\n4) Reduce s modulo 29: x ≡ 11 (mod 29).\nFinal answer: $x\\equiv 11\\pmod{29}$", "final_answer": "x\\equiv 11\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "581d494836c2b660", "problem_id": "40056030801488e7", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{4} - 2 x^{3} + 4 x^{2} + 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x^{3}) = - 6 x^{3}$.\n3) So $f'(x) = - 2 x^{3} + 8 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x^{3}) = - 6 x^{2}$.\nTherefore the correct derivative is $f'(x) = 2 x \\left(2 x^{2} - 3 x + 4\\right)$.\nFinal answer: $2 x \\left(2 x^{2} - 3 x + 4\\right)$", "final_answer": "2 x \\left(2 x^{2} - 3 x + 4\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "84f996abd19ee94b", "problem_id": "2d618bdb4c634354", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "How many binary strings of length 25 contain exactly 5 ones with no two ones adjacent?", "output": "1) Reserve 4 mandatory zeros between the 5 ones.\n2) This leaves n−k+1 = 21 valid slots for the ones.\n3) Choose the slots: C(21,5) = 20349.\nFinal answer: $20349$", "final_answer": "20349", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b78434a1ce3b8f80", "problem_id": "2d618bdb4c634354", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many binary strings of length 25 contain exactly 5 ones with no two ones adjacent?", "output": "Reserve 4 mandatory zeros between the 5 ones.\nThis leaves n−k+1 = 21 valid slots for the ones.\nChoose the slots: C(21,5) = 20349.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $20349$", "final_answer": "20349", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "947273fcfa7679da", "problem_id": "ca8e18f5ad87a9d6", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Worker A finishes a job in 5 hours and Worker B finishes it in 21 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/5 job/hour. B's rate = 1/21 job/hour.\nCombined rate = 1/5 + 1/21 = 26/105 job/hour.\nTime = 1 ÷ (combined rate) = 105/26 hours.\nFinal answer: $\\frac{105}{26}$", "final_answer": "\\frac{105}{26}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2f7d271a69b090ff", "problem_id": "ca8e18f5ad87a9d6", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Worker A finishes a job in 5 hours and Worker B finishes it in 21 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/5 job/hour. B's rate = 1/21 job/hour.\n2) Combined rate = 1/5 + 1/21 = 26/105 job/hour.\n3) Time = 1 ÷ (combined rate) = 105/26 hours.\nFinal answer: $\\frac{105}{26}$", "final_answer": "\\frac{105}{26}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0393cc186885b687", "problem_id": "b50fecd697df74de", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_9=18$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=18 and k=9.\n4) Compute C(26,8) = 1562275.\nFinal answer: $1562275$", "final_answer": "1562275", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "317a753997562695", "problem_id": "b50fecd697df74de", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_9=18$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=18 and k=9.\nFinal answer: $1562275$", "final_answer": "1562275", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "df2cbdd193eae959", "problem_id": "6f207e9759e1c11d", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{3} - \\frac{3 x}{2} + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}$.", "output": "1) Let f(x) = $\\frac{- x^{3} - \\frac{3 x}{2} + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{- 2 x \\left(- 2 x^{3} - 3 x + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 3\\right)}{2 \\left(x^{2} - 3\\right)^{2}}$.\nFinal answer: $\\frac{- 2\\cdot x\\cdot \\left(- 2\\cdot x^{3} - 3\\cdot x + \\left(2\\cdot x^{2} - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right)\\cdot \\left(- 6\\cdot x^{2} + 4\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(x^{2} - 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 3\\right)}{2\\cdot \\left(x^{2} - 3\\right)^{2}}$", "final_answer": "\\frac{- 2\\cdot x\\cdot \\left(- 2\\cdot x^{3} - 3\\cdot x + \\left(2\\cdot x^{2} - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right)\\cdot \\left(- 6\\cdot x^{2} + 4\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(x^{2} - 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 3\\right)}{2\\cdot \\left(x^{2} - 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "de7f8d48daebe9d6", "problem_id": "e3c1868197425aeb", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{2} - x + \\left(2 x^{2} + 2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{2 x^{2} + 2 x - 5}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x^{2} + 2 x\\right) - 5}) = - \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + 2 x - 5}$.\n3) So $f'(x) = \\frac{4 x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} + 16 x^{3} - 32 x^{2} - 40 x + 50} + \\frac{8 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} + 16 x^{3} - 32 x^{2} - 40 x + 50} + \\frac{8 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} + 16 x^{3} - 32 x^{2} - 40 x + 50} + \\frac{16 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} + 16 x^{3} - 32 x^{2} - 40 x + 50} - \\frac{16 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} + 16 x^{3} - 32 x^{2} - 40 x + 50} - \\frac{32 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} + 16 x^{3} - 32 x^{2} - 40 x + 50} + \\frac{8 x^{2}}{8 x^{4} + 16 x^{3} - 32 x^{2} - 40 x + 50} - \\frac{20 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} + 16 x^{3} - 32 x^{2} - 40 x + 50} - \\frac{40 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} + 16 x^{3} - 32 x^{2} - 40 x + 50} - \\frac{44 x}{8 x^{4} + 16 x^{3} - 32 x^{2} - 40 x + 50} - \\frac{20 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{4 x^{4} + 8 x^{3} - 16 x^{2} - 20 x + 25} + \\frac{25 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} + 16 x^{3} - 32 x^{2} - 40 x + 50} + \\frac{50 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} + 16 x^{3} - 32 x^{2} - 40 x + 50} - \\frac{2}{8 x^{4} + 16 x^{3} - 32 x^{2} - 40 x + 50} - \\frac{10 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{4 x^{4} + 8 x^{3} - 16 x^{2} - 20 x + 25} + \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{4 x^{2} + 4 x - 10} - \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + 2 x - 5} + \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(2 x^{2} + 2 x\\right) - 5}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x^{2} + 2 x\\right) - 5}) = - \\frac{5 \\left(- 4 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(2 x^{2} + 2 x\\right) - 5\\right)^{2}} - \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(2 x^{2} + 2 x\\right) - 5\\right)} - \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(2 x^{2} + 2 x\\right) - 5}$.\nTherefore the correct derivative is $f'(x) = \\frac{4 \\left(- 2 x - 1\\right) \\left(x^{2} - x + \\left(2 x \\left(x + 1\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(2 x \\left(x + 1\\right) - 5\\right) \\left(4 x + 4 \\left(2 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(2 x \\left(x + 1\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x \\left(x + 1\\right) - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(2 x \\left(x + 1\\right) - 5\\right)^{2}}$.\nFinal answer: $\\frac{4 \\left(- 2 x - 1\\right) \\left(x^{2} - x + \\left(2 x \\left(x + 1\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(2 x \\left(x + 1\\right) - 5\\right) \\left(4 x + 4 \\left(2 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(2 x \\left(x + 1\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x \\left(x + 1\\right) - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(2 x \\left(x + 1\\right) - 5\\right)^{2}}$", "final_answer": "\\frac{4 \\left(- 2 x - 1\\right) \\left(x^{2} - x + \\left(2 x \\left(x + 1\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(2 x \\left(x + 1\\right) - 5\\right) \\left(4 x + 4 \\left(2 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(2 x \\left(x + 1\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x \\left(x + 1\\right) - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(2 x \\left(x + 1\\right) - 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "30e4b43606988174", "problem_id": "9c96ecad31ed736d", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many binary strings of length 25 contain exactly 9 ones with no two ones adjacent?", "output": "1) Reserve 8 mandatory zeros between the 9 ones.\n2) This leaves n−k+1 = 17 valid slots for the ones.\n3) Choose the slots: C(17,9) = 24310.\nFinal answer: $24310$", "final_answer": "24310", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8e1afec00a6bd60d", "problem_id": "9c96ecad31ed736d", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many binary strings of length 25 contain exactly 9 ones with no two ones adjacent?", "output": "Reserve 8 mandatory zeros between the 9 ones.\nThis leaves n−k+1 = 17 valid slots for the ones.\nChoose the slots: C(17,9) = 24310.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $24310$", "final_answer": "24310", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "45b009894bb3229a", "problem_id": "e28752807fddb085", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "You want 34 liters of a 30% salt solution. You have a 20% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 34−x liters is 40% solution.\n2) Salt equation: 20x + 40(34−x) = 30·34.\n3) (20-40)x = (30-40)·34 ⇒ x = 17.\n4) Then 34−x = 17.\nFinal answer: $17 L of 20% and 17 L of 40%$", "final_answer": "17 L of 20% and 17 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b39ea70d00b55407", "problem_id": "e28752807fddb085", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 34 liters of a 30% salt solution. You have a 20% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 34−x liters is 40% solution.\nSalt equation: 20x + 40(34−x) = 30·34.\n(20-40)x = (30-40)·34 ⇒ x = 17.\nThen 34−x = 17.\nCheck: 20·17 + 40·17 = 30·34.\nFinal answer: $17 L of 20% and 17 L of 40%$", "final_answer": "17 L of 20% and 17 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "83c601c430d982d2", "problem_id": "06446885ad096c3b", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡3 (mod 11), x≡6 (mod 7). Give the solution modulo 77.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 77.\nCompute the CRT solution to get x ≡ 69 (mod 77).\nCheck: 69 mod 11=3 and mod 7=6.\nFinal answer: $x\\equiv 69\\pmod{77}$", "final_answer": "x\\equiv 69\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4b2509e89b90d105", "problem_id": "06446885ad096c3b", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡3 (mod 11), x≡6 (mod 7). Give the solution modulo 77.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 77.\n3) Compute the CRT solution to get x ≡ 69 (mod 77).\nFinal answer: $x\\equiv 69\\pmod{77}$", "final_answer": "x\\equiv 69\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4921c06be36c7267", "problem_id": "ed5ef68c36f9b2d0", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{4} - 3 x^{3} - 3 x^{2} + 2 x + 3\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 3 x^{4} - 3 x^{3} - 3 x^{2} + 2 x + 3$.\n3) Compute an antiderivative: F(x) = $- \\frac{3 x^{5}}{5} - \\frac{3 x^{4}}{4} - x^{3} + x^{2} + 3 x$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{3\\cdot x^{5}}{5} - \\frac{3\\cdot x^{4}}{4} - x^{3} + x^{2} + 3\\cdot x$", "final_answer": "C - \\frac{3\\cdot x^{5}}{5} - \\frac{3\\cdot x^{4}}{4} - x^{3} + x^{2} + 3\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bd9e270efb68209a", "problem_id": "ed5ef68c36f9b2d0", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{4} - 3 x^{3} - 3 x^{2} + 2 x + 3\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 3 x^{4} - 3 x^{3} - 3 x^{2} + 2 x + 3$.\nCompute an antiderivative: F(x) = $- \\frac{3 x^{5}}{5} - \\frac{3 x^{4}}{4} - x^{3} + x^{2} + 3 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - \\frac{3\\cdot x^{5}}{5} - \\frac{3\\cdot x^{4}}{4} - x^{3} + x^{2} + 3\\cdot x$", "final_answer": "C - \\frac{3\\cdot x^{5}}{5} - \\frac{3\\cdot x^{4}}{4} - x^{3} + x^{2} + 3\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dd1a39764efe46de", "problem_id": "d8eaa743f245daad", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "You want 60 liters of a 32% salt solution. You have a 25% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 60−x liters is 40% solution.\n2) Salt equation: 25x + 40(60−x) = 32·60.\n3) (25-40)x = (32-40)·60 ⇒ x = 32.\n4) Then 60−x = 28.\nFinal answer: $32 L of 25% and 28 L of 40%$", "final_answer": "32 L of 25% and 28 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fe01016a65728bf0", "problem_id": "ca1db2df9672f21b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 \\cos{\\left(2 x \\right)}) = - 2 \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 \\cos{\\left(2 x \\right)}) = - 4 \\sin{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$.\nFinal answer: $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$", "final_answer": "- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "1290ce4727de76d8", "problem_id": "8fb772d6574dd87b", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A store sells a backpack for $129. It is discounted by 10%. You buy 3 backpacks. What is the total cost?", "output": "1) Discounted price = 129 × (1 − 10/100) = 129 × 90/100 = 1161/10.\n2) Subtotal = 3 × 1161/10 = 3483/10.\nFinal answer: $\\frac{3483}{10}$", "final_answer": "\\frac{3483}{10}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "95710b6f49eac5d3", "problem_id": "8fb772d6574dd87b", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $129. It is discounted by 10%. You buy 3 backpacks. What is the total cost?", "output": "Discounted price = 129 × (1 − 10/100) = 129 × 90/100 = 1161/10.\nSubtotal = 3 × 1161/10 = 3483/10.\nCheck: recomputing the formula yields 3483/10.\nFinal answer: $\\frac{3483}{10}$", "final_answer": "\\frac{3483}{10}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e026477f74193066", "problem_id": "5b61773b320bfd1c", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 15 red balls and 23 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(38,11).\n3) Favorable samples: choose k reds and n−k blues: C(15,2)·C(23,9).\n4) Compute totals: C(38,11) = 1203322288, and favorable = 85804950.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{18975}{266104}$", "final_answer": "\\frac{18975}{266104}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec2dc98678406ee7", "problem_id": "5b61773b320bfd1c", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 15 red balls and 23 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(38,11).\nFavorable samples: choose k reds and n−k blues: C(15,2)·C(23,9).\nCompute totals: C(38,11) = 1203322288, and favorable = 85804950.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{18975}{266104}$.\nFinal answer: $\\frac{18975}{266104}$", "final_answer": "\\frac{18975}{266104}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "03c3794eb897fadc", "problem_id": "51c07a52a4b502bd", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 11 x + 24$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} + 11 x + 24$ = $\\left(x + 3\\right) \\left(x + 8\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $-3$.\nFrom the second factor: x = $-8$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-3, -8\\}$", "final_answer": "x \\in \\{-3, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4766ed9af4b67151", "problem_id": "51c07a52a4b502bd", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 11 x + 24$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} + 11 x + 24$ = $\\left(x + 3\\right) \\left(x + 8\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-3$.\n5) From the second factor: x = $-8$.\nFinal answer: $x \\in \\{-3, -8\\}$", "final_answer": "x \\in \\{-3, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "80be49e8b3668b19", "problem_id": "7dcfb30d437bd4ea", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the multiplicative inverse of 33 modulo 43.", "output": "1) We want x such that 33x ≡ 1 (mod 43).\n2) Use extended Euclid to find s,t with 33s + 43t = 1.\n3) One Bezout identity is: 33·(-13) + 43·(10) = 1.\n4) Reduce s modulo 43: x ≡ 30 (mod 43).\nFinal answer: $x\\equiv 30\\pmod{43}$", "final_answer": "x\\equiv 30\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b52d3cf06f12bf50", "problem_id": "7dcfb30d437bd4ea", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the multiplicative inverse of 33 modulo 43.", "output": "We want x such that 33x ≡ 1 (mod 43).\nUse extended Euclid to find s,t with 33s + 43t = 1.\nOne Bezout identity is: 33·(-13) + 43·(10) = 1.\nReduce s modulo 43: x ≡ 30 (mod 43).\nCheck: 33·30 mod 43 = 1.\nFinal answer: $x\\equiv 30\\pmod{43}$", "final_answer": "x\\equiv 30\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a6670c400658ebfc", "problem_id": "2cc3a66de6b6b558", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡0 (mod 3), x≡3 (mod 7). Give the solution modulo 21.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 21.\nCompute the CRT solution to get x ≡ 3 (mod 21).\nCheck: 3 mod 3=0 and mod 7=3.\nFinal answer: $x\\equiv 3\\pmod{21}$", "final_answer": "x\\equiv 3\\pmod{21}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b2529ab3429b812d", "problem_id": "2cc3a66de6b6b558", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡0 (mod 3), x≡3 (mod 7). Give the solution modulo 21.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 21.\n3) Compute the CRT solution to get x ≡ 3 (mod 21).\nFinal answer: $x\\equiv 3\\pmod{21}$", "final_answer": "x\\equiv 3\\pmod{21}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1497b76a22ed3036", "problem_id": "d98b3930e6b9983b", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 77 miles per hour for 1/2 hours, then at 85 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 77 × 1/2 = 77/2 miles.\nSecond leg distance = 85 × 2 = 170 miles.\nTotal distance = 77/2 + 170 = 417/2 miles.\nCheck: adding both legs gives 417/2.\nFinal answer: $\\frac{417}{2}$", "final_answer": "\\frac{417}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "35e1c5dd78401924", "problem_id": "d98b3930e6b9983b", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A cyclist rides at 77 miles per hour for 1/2 hours, then at 85 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 77 × 1/2 = 77/2 miles.\n2) Second leg distance = 85 × 2 = 170 miles.\n3) Total distance = 77/2 + 170 = 417/2 miles.\nFinal answer: $\\frac{417}{2}$", "final_answer": "\\frac{417}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3821480aa35c20d0", "problem_id": "08fb46d50fda96b4", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡1 (mod 5), x≡6 (mod 11). Give the solution modulo 55.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 55.\nCompute the CRT solution to get x ≡ 6 (mod 55).\nCheck: 6 mod 5=1 and mod 11=6.\nFinal answer: $x\\equiv 6\\pmod{55}$", "final_answer": "x\\equiv 6\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "499793e8c87de773", "problem_id": "08fb46d50fda96b4", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡1 (mod 5), x≡6 (mod 11). Give the solution modulo 55.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 55.\n3) Compute the CRT solution to get x ≡ 6 (mod 55).\nFinal answer: $x\\equiv 6\\pmod{55}$", "final_answer": "x\\equiv 6\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5cd51d71ad1037ad", "problem_id": "c668871b1d785241", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "How many binary strings of length 23 contain exactly 7 ones with no two ones adjacent?", "output": "1) Reserve 6 mandatory zeros between the 7 ones.\n2) This leaves n−k+1 = 17 valid slots for the ones.\n3) Choose the slots: C(17,7) = 19448.\nFinal answer: $19448$", "final_answer": "19448", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b9ca84a51d172140", "problem_id": "c668871b1d785241", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many binary strings of length 23 contain exactly 7 ones with no two ones adjacent?", "output": "Reserve 6 mandatory zeros between the 7 ones.\nThis leaves n−k+1 = 17 valid slots for the ones.\nChoose the slots: C(17,7) = 19448.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $19448$", "final_answer": "19448", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "83ae05018db27180", "problem_id": "7cb98644bfb03507", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A cyclist rides at 25 miles per hour for 3/2 hours, then at 39 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 25 × 3/2 = 75/2 miles.\n2) Second leg distance = 39 × 2 = 78 miles.\n3) Total distance = 75/2 + 78 = 231/2 miles.\nFinal answer: $\\frac{231}{2}$", "final_answer": "\\frac{231}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9597b38064ab0934", "problem_id": "8c2b46771a27cbde", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} \\sin{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\sin{\\left(x \\right)}) = - \\cos{\\left(x \\right)}$.\n3) So $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\sin{\\left(x \\right)}) = - \\cos{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.\nFinal answer: $x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$", "final_answer": "x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "da9a8bba79494135", "problem_id": "a94396dbd7c9859d", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Worker A finishes a job in 14 hours and Worker B finishes it in 9 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/14 job/hour. B's rate = 1/9 job/hour.\n2) Combined rate = 1/14 + 1/9 = 23/126 job/hour.\n3) Time = 1 ÷ (combined rate) = 126/23 hours.\nFinal answer: $\\frac{126}{23}$", "final_answer": "\\frac{126}{23}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2798ae7465b20271", "problem_id": "a94396dbd7c9859d", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Worker A finishes a job in 14 hours and Worker B finishes it in 9 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/14 job/hour. B's rate = 1/9 job/hour.\nCombined rate = 1/14 + 1/9 = 23/126 job/hour.\nTime = 1 ÷ (combined rate) = 126/23 hours.\nCheck: (combined rate)×(time) = 23/126×126/23 = 1.\nFinal answer: $\\frac{126}{23}$", "final_answer": "\\frac{126}{23}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9b164ebc54febda5", "problem_id": "585e60885307f1c2", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Find the multiplicative inverse of 17 modulo 23.", "output": "1) We want x such that 17x ≡ 1 (mod 23).\n2) Use extended Euclid to find s,t with 17s + 23t = 1.\n3) One Bezout identity is: 17·(-4) + 23·(3) = 1.\n4) Reduce s modulo 23: x ≡ 19 (mod 23).\nFinal answer: $x\\equiv 19\\pmod{23}$", "final_answer": "x\\equiv 19\\pmod{23}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "10e489656d32c741", "problem_id": "22cbb7d6762ab061", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(- x^{2} - x - 2\\right) \\cos{\\left(x - 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x^{2} \\cos{\\left(x - 3 \\right)}) = x^{2} \\sin{\\left(x - 3 \\right)}$.\n3) So $f'(x) = x^{2} \\sin{\\left(x - 3 \\right)} + 2 x e^{x} + x \\sin{\\left(x - 3 \\right)} + 6 e^{x} + 2 \\sin{\\left(x - 3 \\right)} - \\cos{\\left(x - 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x^{2} \\cos{\\left(x - 3 \\right)}) = x^{2} \\sin{\\left(x - 3 \\right)} - 2 x \\cos{\\left(x - 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(- 2 x - 1\\right) \\cos{\\left(x - 3 \\right)} + 2 \\left(x + 2\\right) e^{x} - \\left(x \\left(- x - 1\\right) - 2\\right) \\sin{\\left(x - 3 \\right)} + 2 e^{x}$.\nFinal answer: $\\left(- 2 x - 1\\right) \\cos{\\left(x - 3 \\right)} + 2 \\left(x + 2\\right) e^{x} - \\left(x \\left(- x - 1\\right) - 2\\right) \\sin{\\left(x - 3 \\right)} + 2 e^{x}$", "final_answer": "\\left(- 2 x - 1\\right) \\cos{\\left(x - 3 \\right)} + 2 \\left(x + 2\\right) e^{x} - \\left(x \\left(- x - 1\\right) - 2\\right) \\sin{\\left(x - 3 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a75e56d5d3bae330", "problem_id": "d8bc1bc4b0ec4523", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Find the equation of the tangent line to $y=- x^{2} - 3 x + 2 \\sin{\\left(x \\right)}$ at $x=-2$.", "output": "Differentiate: f'(x) = $- 2 x + 2 \\cos{\\left(x \\right)} - 3$.\nSlope at x=-2: m = f'(-2) = $2 \\cos{\\left(2 \\right)} + 1$.\nPoint on curve: (-2, f(-2)) = (-2, $2 - 2 \\sin{\\left(2 \\right)}$).\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + x - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)} + 4$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + x - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)} + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1386478c22cf6066", "problem_id": "d8bc1bc4b0ec4523", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the equation of the tangent line to $y=- x^{2} - 3 x + 2 \\sin{\\left(x \\right)}$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $- 2 x + 2 \\cos{\\left(x \\right)} - 3$.\n2) Slope at x=-2: m = f'(-2) = $2 \\cos{\\left(2 \\right)} + 1$.\n3) Point on curve: (-2, f(-2)) = (-2, $2 - 2 \\sin{\\left(2 \\right)}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 2 - 2 \\sin{\\left(2 \\right)} = 2 \\cos{\\left(2 \\right)} + 1(x − -2).\n6) Optionally expand: y = $2 x \\cos{\\left(2 \\right)} + x - 2 \\sin{\\left(2 \\right)} + 4 \\cos{\\left(2 \\right)} + 4$.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + x - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)} + 4$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + x - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)} + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2fcb4d984c32af4c", "problem_id": "0832ee9cadf333e9", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 4 - 2 x^{2}\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $4 - 2 x^{2}$.\n3) Compute an antiderivative: F(x) = $- \\frac{2 x^{3}}{3} + 4 x$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{2\\cdot x^{3}}{3} + 4\\cdot x$", "final_answer": "C - \\frac{2\\cdot x^{3}}{3} + 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "634339470f7ec386", "problem_id": "0832ee9cadf333e9", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 4 - 2 x^{2}\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $4 - 2 x^{2}$.\nCompute an antiderivative: F(x) = $- \\frac{2 x^{3}}{3} + 4 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - \\frac{2\\cdot x^{3}}{3} + 4\\cdot x$", "final_answer": "C - \\frac{2\\cdot x^{3}}{3} + 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "28bb20960fde9893", "problem_id": "7fc11cd6001f3011", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$- 7 x$ = $21$\n$- 2 x - 8 y$ = $14$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $14 x$ = $-42$.\n3) Multiply the second equation by $-7$: $14 x + 56 y$ = $-98$.\n4) Subtract to eliminate x: $- 56 y$ = $56$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, -1\\right)$", "final_answer": "(x,y)=\\left(-3, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d7bebd0eb5e0f77", "problem_id": "5377178775cdd578", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{2} - 4 x$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- x^{2}) = - 2 x^{2}$.\n3) So $f'(x) = - 2 x^{2} - 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- x^{2}) = - 2 x$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- x - 2\\right)$.\nFinal answer: $2 \\left(- x - 2\\right)$", "final_answer": "2 \\left(- x - 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "07ba0e541db7a3c7", "problem_id": "5c107b6a4bb1aea3", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - 2 x^{2} - 10 x - 42$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 2 x^{3} - 2 x^{2} - 2 x - 2$ and a = 4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 6 x^{2} - 4 x - 2$.\n5) Evaluate at a: p'(4) = $-114$.\nFinal answer: $-114$", "final_answer": "-114", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "56f67ca8ca911e98", "problem_id": "276e32001b053792", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - 2 x^{2} - 11 x - 41$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{2} - 12 x - 41.\n3) Then the limit is -121.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: -117.\nFinal answer: $-117$", "final_answer": "-117", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0b65eae79c52cb82", "problem_id": "fc5587cf012bf340", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the linear system:\n$8 x + 7 y$ = $-11$\n$- 6 x - 8 y$ = $22$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-6$: $- 48 x - 42 y$ = $66$.\n3) Multiply the second equation by $8$: $- 48 x - 64 y$ = $176$.\n4) Subtract to eliminate x: $22 y$ = $-110$.\n5) Solve for y: y = $-5$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, -5\\right)$", "final_answer": "(x,y)=\\left(3, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "75e682223f7284ec", "problem_id": "fc5587cf012bf340", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$8 x + 7 y$ = $-11$\n$- 6 x - 8 y$ = $22$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-6$: $- 48 x - 42 y$ = $66$.\nMultiply the second equation by $8$: $- 48 x - 64 y$ = $176$.\nSubtract to eliminate x: $22 y$ = $-110$.\nSolve for y: y = $-5$.\nSubstitute back to find x: x = $3$.\nCheck: plug (x,y)=(3,-5) into both equations.\nFinal answer: $(x,y)=\\left(3, -5\\right)$", "final_answer": "(x,y)=\\left(3, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6c13f461bd23d3e3", "problem_id": "3df3f8a611be338b", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 8 x + \\frac{64}{9}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$-8$, c=$\\frac{64}{9}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{64}{9}$, so √Δ = $\\frac{8}{3}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{8}{3}$.\nx2 = $\\frac{4}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{8}{3}, \\frac{4}{3}\\}$", "final_answer": "x \\in \\{\\frac{8}{3}, \\frac{4}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5034934a6446b104", "problem_id": "3df3f8a611be338b", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 8 x + \\frac{64}{9}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$-8$, c=$\\frac{64}{9}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{64}{9}$, so √Δ = $\\frac{8}{3}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{8}{3}$.\n7) x2 = $\\frac{4}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{8}{3}, \\frac{4}{3}\\}$", "final_answer": "x \\in \\{\\frac{8}{3}, \\frac{4}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3567188f9f039549", "problem_id": "3e52b4d472b1c18d", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=- x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $- x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $- x^{2}\\cdot \\cos{\\left(x \\right)} - 2\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot x\\cdot \\cos{\\left(x \\right)} - 2\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- x^{2}\\cdot \\cos{\\left(x \\right)} - 2\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot x\\cdot \\cos{\\left(x \\right)} - 2\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1055f141d48d64fe", "problem_id": "3e52b4d472b1c18d", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $- x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- x^{2}\\cdot \\cos{\\left(x \\right)} - 2\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot x\\cdot \\cos{\\left(x \\right)} - 2\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- x^{2}\\cdot \\cos{\\left(x \\right)} - 2\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot x\\cdot \\cos{\\left(x \\right)} - 2\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "53a4f2732adddce6", "problem_id": "bf9a188c8d9bc031", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the multiplicative inverse of 10 modulo 59.", "output": "We want x such that 10x ≡ 1 (mod 59).\nUse extended Euclid to find s,t with 10s + 59t = 1.\nOne Bezout identity is: 10·(6) + 59·(-1) = 1.\nReduce s modulo 59: x ≡ 6 (mod 59).\nCheck: 10·6 mod 59 = 1.\nFinal answer: $x\\equiv 6\\pmod{59}$", "final_answer": "x\\equiv 6\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "53bc5b9ff78f61bf", "problem_id": "bf9a188c8d9bc031", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Find the multiplicative inverse of 10 modulo 59.", "output": "1) We want x such that 10x ≡ 1 (mod 59).\n2) Use extended Euclid to find s,t with 10s + 59t = 1.\n3) One Bezout identity is: 10·(6) + 59·(-1) = 1.\n4) Reduce s modulo 59: x ≡ 6 (mod 59).\nFinal answer: $x\\equiv 6\\pmod{59}$", "final_answer": "x\\equiv 6\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "239b892f483b805d", "problem_id": "ec8e01b149f3afaa", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{3} x - \\frac{11 x}{15} - \\frac{14}{15} + \\frac{7 \\sqrt{3}}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\sqrt{3} - \\frac{11}{15}$, c=$- \\frac{14}{15} + \\frac{7 \\sqrt{3}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{1636}{225} - \\frac{62 \\sqrt{3}}{15}$, so √Δ = $\\sqrt{\\frac{1636}{225} - \\frac{62 \\sqrt{3}}{15}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{1636 - 930 \\sqrt{3}}}{30} + \\frac{11}{30} + \\frac{\\sqrt{3}}{2}$.\n7) x2 = $- \\frac{\\sqrt{1636 - 930 \\sqrt{3}}}{30} + \\frac{11}{30} + \\frac{\\sqrt{3}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{1636 - 930\\cdot \\sqrt{3}}}{30} + \\frac{11}{30} + \\frac{\\sqrt{3}}{2}, - \\frac{\\sqrt{1636 - 930\\cdot \\sqrt{3}}}{30} + \\frac{11}{30} + \\frac{\\sqrt{3}}{2}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{1636 - 930\\cdot \\sqrt{3}}}{30} + \\frac{11}{30} + \\frac{\\sqrt{3}}{2}, - \\frac{\\sqrt{1636 - 930\\cdot \\sqrt{3}}}{30} + \\frac{11}{30} + \\frac{\\sqrt{3}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "133c538a4dc34203", "problem_id": "ec8e01b149f3afaa", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{3} x - \\frac{11 x}{15} - \\frac{14}{15} + \\frac{7 \\sqrt{3}}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\sqrt{3} - \\frac{11}{15}$, c=$- \\frac{14}{15} + \\frac{7 \\sqrt{3}}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{1636}{225} - \\frac{62 \\sqrt{3}}{15}$, so √Δ = $\\sqrt{\\frac{1636}{225} - \\frac{62 \\sqrt{3}}{15}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{1636 - 930 \\sqrt{3}}}{30} + \\frac{11}{30} + \\frac{\\sqrt{3}}{2}$.\nx2 = $- \\frac{\\sqrt{1636 - 930 \\sqrt{3}}}{30} + \\frac{11}{30} + \\frac{\\sqrt{3}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{1636 - 930\\cdot \\sqrt{3}}}{30} + \\frac{11}{30} + \\frac{\\sqrt{3}}{2}, - \\frac{\\sqrt{1636 - 930\\cdot \\sqrt{3}}}{30} + \\frac{11}{30} + \\frac{\\sqrt{3}}{2}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{1636 - 930\\cdot \\sqrt{3}}}{30} + \\frac{11}{30} + \\frac{\\sqrt{3}}{2}, - \\frac{\\sqrt{1636 - 930\\cdot \\sqrt{3}}}{30} + \\frac{11}{30} + \\frac{\\sqrt{3}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fb69615dfa39a577", "problem_id": "be71ce2db2bf4fe9", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A store sells a backpack for $106. It is discounted by 20%. You buy 2 backpacks. What is the total cost?", "output": "1) Discounted price = 106 × (1 − 20/100) = 106 × 80/100 = 424/5.\n2) Subtotal = 2 × 424/5 = 848/5.\nFinal answer: $\\frac{848}{5}$", "final_answer": "\\frac{848}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "278e878c4879c93c", "problem_id": "be71ce2db2bf4fe9", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $106. It is discounted by 20%. You buy 2 backpacks. What is the total cost?", "output": "Discounted price = 106 × (1 − 20/100) = 106 × 80/100 = 424/5.\nSubtotal = 2 × 424/5 = 848/5.\nCheck: recomputing the formula yields 848/5.\nFinal answer: $\\frac{848}{5}$", "final_answer": "\\frac{848}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f5383fb3439b294b", "problem_id": "8e480e5c6edb8e24", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$- 2 x - 2 y$ = $2$\n$4 x + 2 y$ = $-12$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $- 8 x - 8 y$ = $8$.\n3) Multiply the second equation by $-2$: $- 8 x - 4 y$ = $24$.\n4) Subtract to eliminate x: $- 4 y$ = $-16$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, 4\\right)$", "final_answer": "(x,y)=\\left(-5, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d5c277603b7f9d9f", "problem_id": "8e480e5c6edb8e24", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- 2 x - 2 y$ = $2$\n$4 x + 2 y$ = $-12$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $- 8 x - 8 y$ = $8$.\nMultiply the second equation by $-2$: $- 8 x - 4 y$ = $24$.\nSubtract to eliminate x: $- 4 y$ = $-16$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $-5$.\nCheck: plug (x,y)=(-5,4) into both equations.\nFinal answer: $(x,y)=\\left(-5, 4\\right)$", "final_answer": "(x,y)=\\left(-5, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3aef3dc454b30e66", "problem_id": "7c049fdcdbd72ce0", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + \\frac{13 x}{4} + \\frac{3}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{13}{4}$, c=$\\frac{3}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{121}{16}$, so √Δ = $\\frac{11}{4}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{1}{4}$.\n7) x2 = $-3$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{1}{4}, -3\\}$", "final_answer": "x \\in \\{- \\frac{1}{4}, -3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f2444f22fd6fb5d1", "problem_id": "2eae6a3d1f410b8c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} \\sin{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 \\cos{\\left(2 x \\right)}) = - 2 \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = - 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} + 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 \\cos{\\left(2 x \\right)}) = - 4 \\sin{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} + 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} + 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$", "final_answer": "- 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} + 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6a71642ac0b7edcb", "problem_id": "5dc2a20451b6e497", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡1 (mod 5). Give the solution modulo 15.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 15.\n3) Compute the CRT solution to get x ≡ 1 (mod 15).\nFinal answer: $x\\equiv 1\\pmod{15}$", "final_answer": "x\\equiv 1\\pmod{15}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "44bd2a93c7f1df1f", "problem_id": "5dc2a20451b6e497", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡1 (mod 5). Give the solution modulo 15.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 15.\nCompute the CRT solution to get x ≡ 1 (mod 15).\nCheck: 1 mod 3=1 and mod 5=1.\nFinal answer: $x\\equiv 1\\pmod{15}$", "final_answer": "x\\equiv 1\\pmod{15}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "65e4d9f3b2d79333", "problem_id": "2e3cfcfa59f9ea29", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{17 x}{4} - \\sqrt{2} x + \\frac{\\sqrt{2}}{4} + 1$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{17}{4} - \\sqrt{2}$, c=$\\frac{\\sqrt{2}}{4} + 1$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{15 \\sqrt{2}}{2} + \\frac{257}{16}$, so √Δ = $\\sqrt{\\frac{15 \\sqrt{2}}{2} + \\frac{257}{16}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{2}}{2} + \\frac{17}{8} + \\frac{\\sqrt{120 \\sqrt{2} + 257}}{8}$.\n7) x2 = $- \\frac{\\sqrt{120 \\sqrt{2} + 257}}{8} + \\frac{\\sqrt{2}}{2} + \\frac{17}{8}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{2}}{2} + \\frac{17}{8} + \\frac{\\sqrt{120\\cdot \\sqrt{2} + 257}}{8}, - \\frac{\\sqrt{120\\cdot \\sqrt{2} + 257}}{8} + \\frac{\\sqrt{2}}{2} + \\frac{17}{8}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{2}}{2} + \\frac{17}{8} + \\frac{\\sqrt{120\\cdot \\sqrt{2} + 257}}{8}, - \\frac{\\sqrt{120\\cdot \\sqrt{2} + 257}}{8} + \\frac{\\sqrt{2}}{2} + \\frac{17}{8}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "075fccad97319c34", "problem_id": "2e3cfcfa59f9ea29", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{17 x}{4} - \\sqrt{2} x + \\frac{\\sqrt{2}}{4} + 1$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{17}{4} - \\sqrt{2}$, c=$\\frac{\\sqrt{2}}{4} + 1$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{15 \\sqrt{2}}{2} + \\frac{257}{16}$, so √Δ = $\\sqrt{\\frac{15 \\sqrt{2}}{2} + \\frac{257}{16}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{2}}{2} + \\frac{17}{8} + \\frac{\\sqrt{120 \\sqrt{2} + 257}}{8}$.\nx2 = $- \\frac{\\sqrt{120 \\sqrt{2} + 257}}{8} + \\frac{\\sqrt{2}}{2} + \\frac{17}{8}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{2}}{2} + \\frac{17}{8} + \\frac{\\sqrt{120\\cdot \\sqrt{2} + 257}}{8}, - \\frac{\\sqrt{120\\cdot \\sqrt{2} + 257}}{8} + \\frac{\\sqrt{2}}{2} + \\frac{17}{8}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{2}}{2} + \\frac{17}{8} + \\frac{\\sqrt{120\\cdot \\sqrt{2} + 257}}{8}, - \\frac{\\sqrt{120\\cdot \\sqrt{2} + 257}}{8} + \\frac{\\sqrt{2}}{2} + \\frac{17}{8}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "18d999df337b3ad7", "problem_id": "e8d75bd826445784", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "You want 68 liters of a 35% salt solution. You have a 10% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 68−x liters is 70% solution.\n2) Salt equation: 10x + 70(68−x) = 35·68.\n3) (10-70)x = (35-70)·68 ⇒ x = 119/3.\n4) Then 68−x = 85/3.\nFinal answer: $119/3 L of 10% and 85/3 L of 70%$", "final_answer": "119/3 L of 10% and 85/3 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cb677ec1d439ece3", "problem_id": "e8d75bd826445784", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "You want 68 liters of a 35% salt solution. You have a 10% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 68−x liters is 70% solution.\nSalt equation: 10x + 70(68−x) = 35·68.\n(10-70)x = (35-70)·68 ⇒ x = 119/3.\nFinal answer: $119/3 L of 10% and 85/3 L of 70%$", "final_answer": "119/3 L of 10% and 85/3 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e765f489b86c8e5d", "problem_id": "929050f10f545aaa", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(3 x^{3} + 2 x^{2} + 3 x\\right) \\cos{\\left(2 x + 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x^{2} \\cos{\\left(2 x + 2 \\right)}) = - 2 x^{2} \\sin{\\left(2 x + 2 \\right)}$.\n3) So $f'(x) = - 6 x^{3} \\sin{\\left(2 x + 2 \\right)} - 2 x^{2} \\sin{\\left(2 x + 2 \\right)} + 9 x^{2} \\cos{\\left(2 x + 2 \\right)} + 2 x e^{x} - 6 x \\sin{\\left(2 x + 2 \\right)} + 6 e^{x} + 3 \\cos{\\left(2 x + 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x^{2} \\cos{\\left(2 x + 2 \\right)}) = - 4 x^{2} \\sin{\\left(2 x + 2 \\right)} + 4 x \\cos{\\left(2 x + 2 \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x \\left(x \\left(3 x + 2\\right) + 3\\right) \\sin{\\left(2 x + 2 \\right)} + 2 \\left(x + 2\\right) e^{x} + \\left(9 x^{2} + 4 x + 3\\right) \\cos{\\left(2 x + 2 \\right)} + 2 e^{x}$.\nFinal answer: $- 2 x \\left(x \\left(3 x + 2\\right) + 3\\right) \\sin{\\left(2 x + 2 \\right)} + 2 \\left(x + 2\\right) e^{x} + \\left(9 x^{2} + 4 x + 3\\right) \\cos{\\left(2 x + 2 \\right)} + 2 e^{x}$", "final_answer": "- 2 x \\left(x \\left(3 x + 2\\right) + 3\\right) \\sin{\\left(2 x + 2 \\right)} + 2 \\left(x + 2\\right) e^{x} + \\left(9 x^{2} + 4 x + 3\\right) \\cos{\\left(2 x + 2 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e0dd76e9d14b91ac", "problem_id": "9c3f5f5d5ea8a492", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - 3 x + 3$ at $x=0$.", "output": "1) Differentiate: f'(x) = $- 9 x^{2} - 3$.\n2) Slope at x=0: m = f'(0) = $-3$.\n3) Point on curve: (0, f(0)) = (0, $3$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 3 = -3(x − 0).\n6) Optionally expand: y = $3 - 3 x$.\nFinal answer: $y = 3 - 3\\cdot x$", "final_answer": "y = 3 - 3\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fc4d3e445727d698", "problem_id": "20a1dce88b1561ae", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the equation of the tangent line to $y=x^{3} + 3 x^{2} - x - 1$ at $x=-2$.", "output": "Differentiate: f'(x) = $3 x^{2} + 6 x - 1$.\nSlope at x=-2: m = f'(-2) = $-1$.\nPoint on curve: (-2, f(-2)) = (-2, $5$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 5 = -1(x − -2).\nOptionally expand: y = $3 - x$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 3 - x$", "final_answer": "y = 3 - x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bcc7fc2eec186404", "problem_id": "20a1dce88b1561ae", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the equation of the tangent line to $y=x^{3} + 3 x^{2} - x - 1$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $3 x^{2} + 6 x - 1$.\n2) Slope at x=-2: m = f'(-2) = $-1$.\n3) Point on curve: (-2, f(-2)) = (-2, $5$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 5 = -1(x − -2).\n6) Optionally expand: y = $3 - x$.\nFinal answer: $y = 3 - x$", "final_answer": "y = 3 - x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c0f2a1072625bca7", "problem_id": "147d9f386677f9c6", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the equation of the tangent line to $y=- 4 x^{2} + x + 1$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $1 - 8 x$.\n2) Slope at x=-2: m = f'(-2) = $17$.\n3) Point on curve: (-2, f(-2)) = (-2, $-17$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -17 = 17(x − -2).\n6) Optionally expand: y = $17 x + 17$.\nFinal answer: $y = 17\\cdot x + 17$", "final_answer": "y = 17\\cdot x + 17", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "26475e32f52b4e64", "problem_id": "8539dc75b22b44f4", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 3 x^{3} - 6 x^{2} + 21 x - 63$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} - 6 x^{2} + 20 x - 63.\n3) Then the limit is -258.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -261.\nFinal answer: $-261$", "final_answer": "-261", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5bfc942c30928795", "problem_id": "8bb5264bb4e149cf", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{5 x}{2} - \\frac{\\sqrt{6} x}{2} - \\frac{3}{2} - \\frac{\\sqrt{6}}{4}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{5}{2} - \\frac{\\sqrt{6}}{2}$, c=$- \\frac{3}{2} - \\frac{\\sqrt{6}}{4}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{\\sqrt{6}}{4} + \\frac{5}{4} + \\frac{\\sqrt{14\\cdot \\sqrt{6} + 55}}{4}, - \\frac{\\sqrt{14\\cdot \\sqrt{6} + 55}}{4} + \\frac{\\sqrt{6}}{4} + \\frac{5}{4}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{6}}{4} + \\frac{5}{4} + \\frac{\\sqrt{14\\cdot \\sqrt{6} + 55}}{4}, - \\frac{\\sqrt{14\\cdot \\sqrt{6} + 55}}{4} + \\frac{\\sqrt{6}}{4} + \\frac{5}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "24fdc49c007fd91a", "problem_id": "8bb5264bb4e149cf", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{5 x}{2} - \\frac{\\sqrt{6} x}{2} - \\frac{3}{2} - \\frac{\\sqrt{6}}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{5}{2} - \\frac{\\sqrt{6}}{2}$, c=$- \\frac{3}{2} - \\frac{\\sqrt{6}}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{7 \\sqrt{6}}{2} + \\frac{55}{4}$, so √Δ = $\\sqrt{\\frac{7 \\sqrt{6}}{2} + \\frac{55}{4}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{6}}{4} + \\frac{5}{4} + \\frac{\\sqrt{14 \\sqrt{6} + 55}}{4}$.\n7) x2 = $- \\frac{\\sqrt{14 \\sqrt{6} + 55}}{4} + \\frac{\\sqrt{6}}{4} + \\frac{5}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{6}}{4} + \\frac{5}{4} + \\frac{\\sqrt{14\\cdot \\sqrt{6} + 55}}{4}, - \\frac{\\sqrt{14\\cdot \\sqrt{6} + 55}}{4} + \\frac{\\sqrt{6}}{4} + \\frac{5}{4}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{6}}{4} + \\frac{5}{4} + \\frac{\\sqrt{14\\cdot \\sqrt{6} + 55}}{4}, - \\frac{\\sqrt{14\\cdot \\sqrt{6} + 55}}{4} + \\frac{\\sqrt{6}}{4} + \\frac{5}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8e6d75681ce4b934", "problem_id": "1540c67344431d7e", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A disease has prevalence $\\frac{3}{100}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{37}{500}$.\n3) Compute P(D|+) = $\\frac{51}{148}$.\nFinal answer: $\\frac{51}{148}$", "final_answer": "\\frac{51}{148}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3109720d9da3434f", "problem_id": "1540c67344431d7e", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A disease has prevalence $\\frac{3}{100}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{37}{500}$.\nCompute P(D|+) = $\\frac{51}{148}$.\nFinal answer: $\\frac{51}{148}$", "final_answer": "\\frac{51}{148}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8411ff310231cdd2", "problem_id": "798fa5b4e4c4f411", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} \\frac{- 2 x^{4} + 3 x^{2} + 20}{x + 2}$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to - \\frac{2 x^{4}}{x + 2} + \\frac{3 x^{2}}{x + 2} - x + \\frac{20}{x + 2}.\n3) Then the limit is \\text{NaN}.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 52.\nFinal answer: $52$", "final_answer": "52", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "490462c4ee1cb42b", "problem_id": "230e4faf47537280", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many binary strings of length 22 contain exactly 4 ones with no two ones adjacent?", "output": "1) Reserve 3 mandatory zeros between the 4 ones.\n2) This leaves n−k+1 = 19 valid slots for the ones.\n3) Choose the slots: C(19,4) = 3876.\nFinal answer: $3876$", "final_answer": "3876", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9787f493941cce38", "problem_id": "db0abdde49ff0774", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the congruences: x≡0 (mod 8), x≡0 (mod 11). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 0 (mod 88).\nFinal answer: $x\\equiv 0\\pmod{88}$", "final_answer": "x\\equiv 0\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8e7ed95c41c48558", "problem_id": "c94653b55cacc193", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} \\sin{\\left(x \\right)} - x^{2} \\sin{\\left(x \\right)} - 3 x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 \\sin{\\left(x \\right)}) = 3 \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 \\sin{\\left(x \\right)}) = 3 \\cos{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$", "final_answer": "- 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "4804beb97d665a3e", "problem_id": "5d1d9aee6cf3e8b1", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 2 x^{2} + 8 x - 21$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 2 x^{3} + 2 x^{2} + 3 x$ and a = -3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 6 x^{2} + 4 x + 3$.\nEvaluate at a: p'(-3) = $-63$.\nCheck: the identity gives $-63$.\nFinal answer: $-63$", "final_answer": "-63", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1ac5f441c2cccedf", "problem_id": "5d1d9aee6cf3e8b1", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 2 x^{2} + 8 x - 21$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 2 x^{3} + 2 x^{2} + 3 x$ and a = -3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 6 x^{2} + 4 x + 3$.\n5) Evaluate at a: p'(-3) = $-63$.\nFinal answer: $-63$", "final_answer": "-63", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "baa23c177f9a4f59", "problem_id": "d49791f6cb1fde5c", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 8 x - 90$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 8 x - 90$ = $2 \\left(x - 9\\right) \\left(x + 5\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $9$.\n5) From the second factor: x = $-5$.\nFinal answer: $x \\in \\{9, -5\\}$", "final_answer": "x \\in \\{9, -5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d0138b5cc3383812", "problem_id": "d49791f6cb1fde5c", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 8 x - 90$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} - 8 x - 90$ = $2 \\left(x - 9\\right) \\left(x + 5\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $9$.\nFrom the second factor: x = $-5$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{9, -5\\}$", "final_answer": "x \\in \\{9, -5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dd8d17c8831f3fc1", "problem_id": "47dfb0db42a96afd", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_10=39$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_10=9.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=9 and k=10.\n4) Compute C(18,9) = 48620.\nFinal answer: $48620$", "final_answer": "48620", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d5f080f4beb1d0e7", "problem_id": "7253d4677acf777a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} + 3 x + \\left(3 x - 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{3 x - 7}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{7 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 7}) = - \\frac{7 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 7}$.\n3) So $f'(x) = \\frac{9 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} - 84 x + 98} + \\frac{18 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} - 84 x + 98} - \\frac{6 x^{2}}{18 x^{2} - 84 x + 98} - \\frac{42 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} - 84 x + 98} - \\frac{84 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} - 84 x + 98} + \\frac{28 x}{18 x^{2} - 84 x + 98} + \\frac{49 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} - 84 x + 98} + \\frac{98 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} - 84 x + 98} - \\frac{48}{18 x^{2} - 84 x + 98} - \\frac{21 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{2} - 42 x + 49} + \\frac{7 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x - 14} - \\frac{7 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 7} + \\frac{7 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 7}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{7 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 7}) = - \\frac{7 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(3 x - 7\\right)} - \\frac{7 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 7} + \\frac{21 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(3 x - 7\\right)^{2}}$.\nTherefore the correct derivative is $f'(x) = \\frac{6 x^{2} - 18 x + \\left(3 x - 7\\right) \\left(- 4 x + \\left(3 x - 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x - 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 6 \\left(3 x - 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6}{2 \\left(3 x - 7\\right)^{2}}$.\nFinal answer: $\\frac{6 x^{2} - 18 x + \\left(3 x - 7\\right) \\left(- 4 x + \\left(3 x - 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x - 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 6 \\left(3 x - 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6}{2 \\left(3 x - 7\\right)^{2}}$", "final_answer": "\\frac{6 x^{2} - 18 x + \\left(3 x - 7\\right) \\left(- 4 x + \\left(3 x - 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x - 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 6 \\left(3 x - 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6}{2 \\left(3 x - 7\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b89b33a8dc1e0a1a", "problem_id": "34524cfb09183a83", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=31$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=31 and k=6.\n4) Compute C(36,5) = 376992.\nFinal answer: $376992$", "final_answer": "376992", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b816e6ec7988029f", "problem_id": "34524cfb09183a83", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=31$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=31 and k=6.\nCompute C(36,5) = 376992.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $376992$", "final_answer": "376992", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "85f9d3e29d2c5184", "problem_id": "1e19ebf396e031d3", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A store sells a backpack for $360. It is discounted by 25%. You buy 3 backpacks. What is the total cost?", "output": "1) Discounted price = 360 × (1 − 25/100) = 360 × 75/100 = 270.\n2) Subtotal = 3 × 270 = 810.\nFinal answer: $810$", "final_answer": "810", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ac8a4ec782b97a78", "problem_id": "e725b9b3a9f46c53", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(2 x^{2} + 2 x + 1\\right) \\cos{\\left(2 x^{2} + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x \\cos{\\left(2 x^{2} + 3 \\right)}) = - 2 x \\sin{\\left(2 x^{2} + 3 \\right)}$.\n3) So $f'(x) = - 8 x^{3} \\sin{\\left(2 x^{2} + 3 \\right)} + x e^{x} - 2 x \\sin{\\left(2 x^{2} + 3 \\right)} - 4 x \\sin{\\left(2 x^{2} + 3 \\right)} + 4 x \\cos{\\left(2 x^{2} + 3 \\right)} + 3 e^{x}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x \\cos{\\left(2 x^{2} + 3 \\right)}) = - 8 x^{2} \\sin{\\left(2 x^{2} + 3 \\right)} + 2 \\cos{\\left(2 x^{2} + 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = - 4 x \\left(2 x \\left(x + 1\\right) + 1\\right) \\sin{\\left(2 x^{2} + 3 \\right)} + \\left(x + 2\\right) e^{x} + 2 \\left(2 x + 1\\right) \\cos{\\left(2 x^{2} + 3 \\right)} + e^{x}$.\nFinal answer: $- 4 x \\left(2 x \\left(x + 1\\right) + 1\\right) \\sin{\\left(2 x^{2} + 3 \\right)} + \\left(x + 2\\right) e^{x} + 2 \\left(2 x + 1\\right) \\cos{\\left(2 x^{2} + 3 \\right)} + e^{x}$", "final_answer": "- 4 x \\left(2 x \\left(x + 1\\right) + 1\\right) \\sin{\\left(2 x^{2} + 3 \\right)} + \\left(x + 2\\right) e^{x} + 2 \\left(2 x + 1\\right) \\cos{\\left(2 x^{2} + 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b9344b0a8b54c0d8", "problem_id": "c9fb8b84088b56bd", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} - 3 x + \\left(x^{2} - x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{x^{2} - x + 6}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 6}) = \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 6}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 26 x^{2} - 24 x + 72} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 26 x^{2} - 24 x + 72} - \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 26 x^{2} - 24 x + 72} - \\frac{4 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 26 x^{2} - 24 x + 72} + \\frac{13 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 26 x^{2} - 24 x + 72} + \\frac{26 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 26 x^{2} - 24 x + 72} + \\frac{8 x^{2}}{2 x^{4} - 4 x^{3} + 26 x^{2} - 24 x + 72} - \\frac{12 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 26 x^{2} - 24 x + 72} - \\frac{24 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 26 x^{2} - 24 x + 72} - \\frac{32 x}{2 x^{4} - 4 x^{3} + 26 x^{2} - 24 x + 72} + \\frac{12 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 13 x^{2} - 12 x + 36} + \\frac{36 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 26 x^{2} - 24 x + 72} + \\frac{72 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 26 x^{2} - 24 x + 72} - \\frac{32}{2 x^{4} - 4 x^{3} + 26 x^{2} - 24 x + 72} - \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 13 x^{2} - 12 x + 36} + \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 6} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 6} - \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 6}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 6}) = \\frac{6 \\left(1 - 2 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(x^{2} - x\\right) + 6\\right)^{2}} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 6} + \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 6}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(1 - 2 x\\right) \\left(- x^{2} - 3 x + \\left(x \\left(x - 1\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2\\right) + \\left(x \\left(x - 1\\right) + 6\\right) \\left(- 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x \\left(x - 1\\right) + 6\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(1 - 2 x\\right) \\left(- x^{2} - 3 x + \\left(x \\left(x - 1\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2\\right) + \\left(x \\left(x - 1\\right) + 6\\right) \\left(- 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x \\left(x - 1\\right) + 6\\right)^{2}}$", "final_answer": "\\frac{2 \\left(1 - 2 x\\right) \\left(- x^{2} - 3 x + \\left(x \\left(x - 1\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2\\right) + \\left(x \\left(x - 1\\right) + 6\\right) \\left(- 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x \\left(x - 1\\right) + 6\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "bd4e374f62676efc", "problem_id": "a496190bdc6d874d", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the multiplicative inverse of 5 modulo 17.", "output": "1) We want x such that 5x ≡ 1 (mod 17).\n2) Use extended Euclid to find s,t with 5s + 17t = 1.\n3) One Bezout identity is: 5·(7) + 17·(-2) = 1.\n4) Reduce s modulo 17: x ≡ 7 (mod 17).\nFinal answer: $x\\equiv 7\\pmod{17}$", "final_answer": "x\\equiv 7\\pmod{17}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8d5e74b2ac36c643", "problem_id": "ce79677541f02a21", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 10 x + \\frac{25}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$-10$, c=$\\frac{25}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $0$, so √Δ = $0$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{5}{3}$.\n7) x2 = $\\frac{5}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{5}{3}, \\frac{5}{3}\\}$", "final_answer": "x \\in \\{\\frac{5}{3}, \\frac{5}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "99caf46137830c7a", "problem_id": "ce79677541f02a21", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 10 x + \\frac{25}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$-10$, c=$\\frac{25}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $0$, so √Δ = $0$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{5}{3}$.\nx2 = $\\frac{5}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{5}{3}, \\frac{5}{3}\\}$", "final_answer": "x \\in \\{\\frac{5}{3}, \\frac{5}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "652a31574ef61c7d", "problem_id": "1c0485c5d80ee598", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Find the multiplicative inverse of 22 modulo 37.", "output": "$x\\equiv 32\\pmod{37}$", "final_answer": "x\\equiv 32\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "503677a5c41f3f30", "problem_id": "1c0485c5d80ee598", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the multiplicative inverse of 22 modulo 37.", "output": "1) We want x such that 22x ≡ 1 (mod 37).\n2) Use extended Euclid to find s,t with 22s + 37t = 1.\n3) One Bezout identity is: 22·(-5) + 37·(3) = 1.\n4) Reduce s modulo 37: x ≡ 32 (mod 37).\nFinal answer: $x\\equiv 32\\pmod{37}$", "final_answer": "x\\equiv 32\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "89063c3cb0f9b4f1", "problem_id": "67d23280557fcb33", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $2190 in the ratio 8:7. How much does each receive?", "output": "Total ratio parts = 8+7 = 15.\nEach part = 2190 ÷ 15 = 146.\nFirst share = 8 × 146 = 1168.\nSecond share = 7 × 146 = 1022.\nCheck: 1168+1022=2190.\nFinal answer: $1022$", "final_answer": "1022", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cd3e8c378a0c8ebe", "problem_id": "67d23280557fcb33", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Two friends split $2190 in the ratio 8:7. How much does each receive?", "output": "1) Total ratio parts = 8+7 = 15.\n2) Each part = 2190 ÷ 15 = 146.\n3) First share = 8 × 146 = 1168.\n4) Second share = 7 × 146 = 1022.\nFinal answer: $1022$", "final_answer": "1022", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "517255adb86e50a7", "problem_id": "045199744d756260", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Find the equation of the tangent line to $y=- 4 x^{2} - 3 x + 1$ at $x=2$.", "output": "Differentiate: f'(x) = $- 8 x - 3$.\nSlope at x=2: m = f'(2) = $-19$.\nPoint on curve: (2, f(2)) = (2, $-21$).\nFinal answer: $y = 17 - 19\\cdot x$", "final_answer": "y = 17 - 19\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9db384f16f07253e", "problem_id": "045199744d756260", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the equation of the tangent line to $y=- 4 x^{2} - 3 x + 1$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 8 x - 3$.\n2) Slope at x=2: m = f'(2) = $-19$.\n3) Point on curve: (2, f(2)) = (2, $-21$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -21 = -19(x − 2).\n6) Optionally expand: y = $17 - 19 x$.\nFinal answer: $y = 17 - 19\\cdot x$", "final_answer": "y = 17 - 19\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "610e14a360031cab", "problem_id": "d62e0e0ca7029ce5", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} - x - 1$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-1) = 0.\n3) d/dx(x^{2}) = 2 x.\n4) d/dx(- x) = -1.\n5) Combine: f'(x) = 2 x - 1.\nFinal answer: $2\\cdot x - 1$", "final_answer": "2\\cdot x - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8f91c7a32104b4e8", "problem_id": "d62e0e0ca7029ce5", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} - x - 1$.", "output": "Differentiate term-by-term:\nd/dx(-1) = 0.\nd/dx(x^{2}) = 2 x.\nd/dx(- x) = -1.\nCombine: f'(x) = 2 x - 1.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $2\\cdot x - 1$", "final_answer": "2\\cdot x - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "73016664ab153afe", "problem_id": "f3d3b4e037485f8c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(x^{2} - 3 x - 2\\right) \\cos{\\left(- 3 x^{2} + 3 x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\n3) So $f'(x) = 6 x^{3} \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} - 21 x^{2} \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} + x e^{x} - 3 x \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} + 2 x \\cos{\\left(\\left(- 3 x^{2} + 3 x\\right) + 3 \\right)} + 3 e^{x} + 6 \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} - 3 \\cos{\\left(\\left(- 3 x^{2} + 3 x\\right) + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = - 3 \\left(1 - 2 x\\right) \\left(x \\left(x - 3\\right) - 2\\right) \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} + \\left(x + 2\\right) e^{x} + \\left(2 x - 3\\right) \\cos{\\left(\\left(- 3 x^{2} + 3 x\\right) + 3 \\right)} + e^{x}$.\nFinal answer: $- 3 \\left(1 - 2 x\\right) \\left(x \\left(x - 3\\right) - 2\\right) \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} + \\left(x + 2\\right) e^{x} + \\left(2 x - 3\\right) \\cos{\\left(\\left(- 3 x^{2} + 3 x\\right) + 3 \\right)} + e^{x}$", "final_answer": "- 3 \\left(1 - 2 x\\right) \\left(x \\left(x - 3\\right) - 2\\right) \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} + \\left(x + 2\\right) e^{x} + \\left(2 x - 3\\right) \\cos{\\left(\\left(- 3 x^{2} + 3 x\\right) + 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "4cff65a04a227438", "problem_id": "cf28b8d8d06a25fb", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "You want 64 liters of a 30% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 64−x liters is 70% solution.\n2) Salt equation: 25x + 70(64−x) = 30·64.\n3) (25-70)x = (30-70)·64 ⇒ x = 512/9.\n4) Then 64−x = 64/9.\nFinal answer: $512/9 L of 25% and 64/9 L of 70%$", "final_answer": "512/9 L of 25% and 64/9 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7f8661746203da93", "problem_id": "8b610851d00548d3", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Worker A finishes a job in 16 hours and Worker B finishes it in 13 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/16 job/hour. B's rate = 1/13 job/hour.\n2) Combined rate = 1/16 + 1/13 = 29/208 job/hour.\n3) Time = 1 ÷ (combined rate) = 208/29 hours.\nFinal answer: $\\frac{208}{29}$", "final_answer": "\\frac{208}{29}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "05bd5635be5bb3c9", "problem_id": "0398c0b744502010", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 2 x^{3} - 3 x^{2} + 8 x - 27$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} - 3 x^{2} + 7 x - 27.\n3) Then the limit is -129.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -132.\nFinal answer: $-132$", "final_answer": "-132", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "19e5c7aeb3fb20a4", "problem_id": "23cfb90923e0ef5d", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the multiplicative inverse of 11 modulo 19.", "output": "1) We want x such that 11x ≡ 1 (mod 19).\n2) Use extended Euclid to find s,t with 11s + 19t = 1.\n3) One Bezout identity is: 11·(7) + 19·(-4) = 1.\n4) Reduce s modulo 19: x ≡ 7 (mod 19).\nFinal answer: $x\\equiv 7\\pmod{19}$", "final_answer": "x\\equiv 7\\pmod{19}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4bda074ba37a0043", "problem_id": "23cfb90923e0ef5d", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 11 modulo 19.", "output": "We want x such that 11x ≡ 1 (mod 19).\nUse extended Euclid to find s,t with 11s + 19t = 1.\nOne Bezout identity is: 11·(7) + 19·(-4) = 1.\nReduce s modulo 19: x ≡ 7 (mod 19).\nCheck: 11·7 mod 19 = 1.\nFinal answer: $x\\equiv 7\\pmod{19}$", "final_answer": "x\\equiv 7\\pmod{19}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5fe0b895dad91597", "problem_id": "7cc8adc96dbb3772", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Worker A finishes a job in 8 hours and Worker B finishes it in 14 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/8 job/hour. B's rate = 1/14 job/hour.\n2) Combined rate = 1/8 + 1/14 = 11/56 job/hour.\n3) Time = 1 ÷ (combined rate) = 56/11 hours.\nFinal answer: $\\frac{56}{11}$", "final_answer": "\\frac{56}{11}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c7b57b101bd04d73", "problem_id": "7cc8adc96dbb3772", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Worker A finishes a job in 8 hours and Worker B finishes it in 14 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/8 job/hour. B's rate = 1/14 job/hour.\nCombined rate = 1/8 + 1/14 = 11/56 job/hour.\nTime = 1 ÷ (combined rate) = 56/11 hours.\nCheck: (combined rate)×(time) = 11/56×56/11 = 1.\nFinal answer: $\\frac{56}{11}$", "final_answer": "\\frac{56}{11}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9acada3136f89ac1", "problem_id": "53d3088cbaf9b163", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "You want 36 liters of a 25% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 36−x liters is 70% solution.\n2) Salt equation: 20x + 70(36−x) = 25·36.\n3) (20-70)x = (25-70)·36 ⇒ x = 162/5.\n4) Then 36−x = 18/5.\nFinal answer: $162/5 L of 20% and 18/5 L of 70%$", "final_answer": "162/5 L of 20% and 18/5 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2e5b628d266c0eeb", "problem_id": "53d3088cbaf9b163", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "You want 36 liters of a 25% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 36−x liters is 70% solution.\nSalt equation: 20x + 70(36−x) = 25·36.\n(20-70)x = (25-70)·36 ⇒ x = 162/5.\nFinal answer: $162/5 L of 20% and 18/5 L of 70%$", "final_answer": "162/5 L of 20% and 18/5 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "307aec75bc07c20d", "problem_id": "888006c9c982bed3", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the multiplicative inverse of 14 modulo 23.", "output": "1) We want x such that 14x ≡ 1 (mod 23).\n2) Use extended Euclid to find s,t with 14s + 23t = 1.\n3) One Bezout identity is: 14·(5) + 23·(-3) = 1.\n4) Reduce s modulo 23: x ≡ 5 (mod 23).\nFinal answer: $x\\equiv 5\\pmod{23}$", "final_answer": "x\\equiv 5\\pmod{23}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dd94af9195cd59c0", "problem_id": "888006c9c982bed3", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 14 modulo 23.", "output": "We want x such that 14x ≡ 1 (mod 23).\nUse extended Euclid to find s,t with 14s + 23t = 1.\nOne Bezout identity is: 14·(5) + 23·(-3) = 1.\nReduce s modulo 23: x ≡ 5 (mod 23).\nCheck: 14·5 mod 23 = 1.\nFinal answer: $x\\equiv 5\\pmod{23}$", "final_answer": "x\\equiv 5\\pmod{23}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bff2e8f2c890111d", "problem_id": "accd71a03ccba435", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$- 3 x$ = $-6$\n$- 8 x - 3 y$ = $-34$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-8$: $24 x$ = $48$.\n3) Multiply the second equation by $-3$: $24 x + 9 y$ = $102$.\n4) Subtract to eliminate x: $- 9 y$ = $-54$.\n5) Solve for y: y = $6$.\n6) Substitute back to find x: x = $2$.\nFinal answer: $(x,y)=\\left(2, 6\\right)$", "final_answer": "(x,y)=\\left(2, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e93218caa13f6c57", "problem_id": "accd71a03ccba435", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$- 3 x$ = $-6$\n$- 8 x - 3 y$ = $-34$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-8$: $24 x$ = $48$.\nMultiply the second equation by $-3$: $24 x + 9 y$ = $102$.\nSubtract to eliminate x: $- 9 y$ = $-54$.\nSolve for y: y = $6$.\nSubstitute back to find x: x = $2$.\nCheck: plug (x,y)=(2,6) into both equations.\nFinal answer: $(x,y)=\\left(2, 6\\right)$", "final_answer": "(x,y)=\\left(2, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4fe5a6ce0c8035f1", "problem_id": "942a23a2565e128e", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "A cyclist rides at 37 miles per hour for 3/2 hours, then at 49 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 37 × 3/2 = 111/2 miles.\nSecond leg distance = 49 × 2 = 98 miles.\nTotal distance = 111/2 + 98 = 307/2 miles.\nFinal answer: $\\frac{307}{2}$", "final_answer": "\\frac{307}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f7336af57eeee508", "problem_id": "942a23a2565e128e", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A cyclist rides at 37 miles per hour for 3/2 hours, then at 49 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 37 × 3/2 = 111/2 miles.\n2) Second leg distance = 49 × 2 = 98 miles.\n3) Total distance = 111/2 + 98 = 307/2 miles.\nFinal answer: $\\frac{307}{2}$", "final_answer": "\\frac{307}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b557a838caddf086", "problem_id": "96281b58ed76def0", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Two friends split $1116 in the ratio 9:3. How much does each receive?", "output": "1) Total ratio parts = 9+3 = 12.\n2) Each part = 1116 ÷ 12 = 93.\n3) First share = 9 × 93 = 837.\n4) Second share = 3 × 93 = 279.\nFinal answer: $279$", "final_answer": "279", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "262ad4f20177378b", "problem_id": "96281b58ed76def0", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Two friends split $1116 in the ratio 9:3. How much does each receive?", "output": "Total ratio parts = 9+3 = 12.\nEach part = 1116 ÷ 12 = 93.\nFirst share = 9 × 93 = 837.\nFinal answer: $279$", "final_answer": "279", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "808c49472f293872", "problem_id": "9382fde7b02427a7", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$8 x - y$ = $54$\n$- 2 x + 2 y$ = $-10$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $- 16 x + 2 y$ = $-108$.\n3) Multiply the second equation by $8$: $- 16 x + 16 y$ = $-80$.\n4) Subtract to eliminate x: $- 14 y$ = $-28$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $7$.\nFinal answer: $(x,y)=\\left(7, 2\\right)$", "final_answer": "(x,y)=\\left(7, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "48a90efb064553e9", "problem_id": "9382fde7b02427a7", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the linear system:\n$8 x - y$ = $54$\n$- 2 x + 2 y$ = $-10$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $- 16 x + 2 y$ = $-108$.\nMultiply the second equation by $8$: $- 16 x + 16 y$ = $-80$.\nFinal answer: $(x,y)=\\left(7, 2\\right)$", "final_answer": "(x,y)=\\left(7, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7b2baeadf8ba70d1", "problem_id": "bbd90636f13742cb", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} x \\left(2 - 3 x\\right)$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} + x.\n3) Then the limit is -4.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: -5.\nFinal answer: $-5$", "final_answer": "-5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d2a51d383cd25ee1", "problem_id": "2076212cb81f8167", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$3 x - 4 y$ = $-41$\n$- x - 5 y$ = $-37$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $- 3 x + 4 y$ = $41$.\n3) Multiply the second equation by $3$: $- 3 x - 15 y$ = $-111$.\n4) Subtract to eliminate x: $19 y$ = $152$.\n5) Solve for y: y = $8$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, 8\\right)$", "final_answer": "(x,y)=\\left(-3, 8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c93741d319d3e7e7", "problem_id": "c4136ef43f2e39ef", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$4 x$ = $12$\n$4 x - 2 y$ = $16$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $16 x$ = $48$.\n3) Multiply the second equation by $4$: $16 x - 8 y$ = $64$.\n4) Subtract to eliminate x: $8 y$ = $-16$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, -2\\right)$", "final_answer": "(x,y)=\\left(3, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bdbb6306996f5090", "problem_id": "c4136ef43f2e39ef", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$4 x$ = $12$\n$4 x - 2 y$ = $16$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $16 x$ = $48$.\nMultiply the second equation by $4$: $16 x - 8 y$ = $64$.\nSubtract to eliminate x: $8 y$ = $-16$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $3$.\nCheck: plug (x,y)=(3,-2) into both equations.\nFinal answer: $(x,y)=\\left(3, -2\\right)$", "final_answer": "(x,y)=\\left(3, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "deff52ab346b0e1f", "problem_id": "d0cfcdf5ddb30a7d", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{7} x + \\frac{34 x}{15} - \\frac{3 \\sqrt{7}}{5} + 1$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{34}{15} - \\sqrt{7}$, c=$1 - \\frac{3 \\sqrt{7}}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{1831}{225} - \\frac{32 \\sqrt{7}}{15}$, so √Δ = $\\sqrt{\\frac{1831}{225} - \\frac{32 \\sqrt{7}}{15}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{17}{15} + \\frac{\\sqrt{1831 - 480 \\sqrt{7}}}{30} + \\frac{\\sqrt{7}}{2}$.\nx2 = $- \\frac{17}{15} - \\frac{\\sqrt{1831 - 480 \\sqrt{7}}}{30} + \\frac{\\sqrt{7}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{17}{15} + \\frac{\\sqrt{1831 - 480\\cdot \\sqrt{7}}}{30} + \\frac{\\sqrt{7}}{2}, - \\frac{17}{15} - \\frac{\\sqrt{1831 - 480\\cdot \\sqrt{7}}}{30} + \\frac{\\sqrt{7}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{17}{15} + \\frac{\\sqrt{1831 - 480\\cdot \\sqrt{7}}}{30} + \\frac{\\sqrt{7}}{2}, - \\frac{17}{15} - \\frac{\\sqrt{1831 - 480\\cdot \\sqrt{7}}}{30} + \\frac{\\sqrt{7}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "363a1a4cf9a803d6", "problem_id": "d0cfcdf5ddb30a7d", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{7} x + \\frac{34 x}{15} - \\frac{3 \\sqrt{7}}{5} + 1$ = 0.", "output": "$x \\in \\{- \\frac{17}{15} + \\frac{\\sqrt{1831 - 480\\cdot \\sqrt{7}}}{30} + \\frac{\\sqrt{7}}{2}, - \\frac{17}{15} - \\frac{\\sqrt{1831 - 480\\cdot \\sqrt{7}}}{30} + \\frac{\\sqrt{7}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{17}{15} + \\frac{\\sqrt{1831 - 480\\cdot \\sqrt{7}}}{30} + \\frac{\\sqrt{7}}{2}, - \\frac{17}{15} - \\frac{\\sqrt{1831 - 480\\cdot \\sqrt{7}}}{30} + \\frac{\\sqrt{7}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9583d837924d501f", "problem_id": "47ec0cb85e9a9fe8", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{2} + 2 x + 2\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $3 x^{2} + 2 x + 2$.\n3) Compute an antiderivative: F(x) = $x^{3} + x^{2} + 2 x$.\n4) Include the constant of integration C.\nFinal answer: $C + x^{3} + x^{2} + 2\\cdot x$", "final_answer": "C + x^{3} + x^{2} + 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6649e3846b7e1e03", "problem_id": "47ec0cb85e9a9fe8", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{2} + 2 x + 2\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $3 x^{2} + 2 x + 2$.\nCompute an antiderivative: F(x) = $x^{3} + x^{2} + 2 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + x^{3} + x^{2} + 2\\cdot x$", "final_answer": "C + x^{3} + x^{2} + 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "85d848a25938e252", "problem_id": "18597672e4a81b13", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$6 x + 2 y$ = $-2$\n$- 7 x + 9 y$ = $-43$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-7$: $- 42 x - 14 y$ = $14$.\n3) Multiply the second equation by $6$: $- 42 x + 54 y$ = $-258$.\n4) Subtract to eliminate x: $- 68 y$ = $272$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $1$.\nFinal answer: $(x,y)=\\left(1, -4\\right)$", "final_answer": "(x,y)=\\left(1, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "903e87e84409e58d", "problem_id": "18597672e4a81b13", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$6 x + 2 y$ = $-2$\n$- 7 x + 9 y$ = $-43$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-7$: $- 42 x - 14 y$ = $14$.\nMultiply the second equation by $6$: $- 42 x + 54 y$ = $-258$.\nSubtract to eliminate x: $- 68 y$ = $272$.\nSolve for y: y = $-4$.\nSubstitute back to find x: x = $1$.\nCheck: plug (x,y)=(1,-4) into both equations.\nFinal answer: $(x,y)=\\left(1, -4\\right)$", "final_answer": "(x,y)=\\left(1, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "75f4fedbf5949664", "problem_id": "573e5321515f3b67", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_10=39$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_10=19.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=19 and k=10.\n4) Compute C(28,9) = 6906900.\nFinal answer: $6906900$", "final_answer": "6906900", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6fb851232c61f535", "problem_id": "573e5321515f3b67", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_10=39$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_10=19.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=19 and k=10.\nCompute C(28,9) = 6906900.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $6906900$", "final_answer": "6906900", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8dff0afdd13696a0", "problem_id": "041888f2942c3f03", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_8=40$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=40 and k=8.\n4) Compute C(47,7) = 62891499.\nFinal answer: $62891499$", "final_answer": "62891499", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "794b10e83915f5bd", "problem_id": "041888f2942c3f03", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_8=40$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=40 and k=8.\nFinal answer: $62891499$", "final_answer": "62891499", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "190607a88d6e1c15", "problem_id": "93ab01f5f260cf48", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 12 red balls and 7 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(19,4).\nFavorable samples: choose k reds and n−k blues: C(12,1)·C(7,3).\nCompute totals: C(19,4) = 3876, and favorable = 420.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{35}{323}$.\nFinal answer: $\\frac{35}{323}$", "final_answer": "\\frac{35}{323}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "17f939f33bbc2d6b", "problem_id": "93ab01f5f260cf48", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 12 red balls and 7 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(19,4).\n3) Favorable samples: choose k reds and n−k blues: C(12,1)·C(7,3).\n4) Compute totals: C(19,4) = 3876, and favorable = 420.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{35}{323}$", "final_answer": "\\frac{35}{323}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ac1d9dab3cb353df", "problem_id": "4634818da7fa79fd", "style": "answer_only", "instruction": "Final answer only.", "input": "Find the multiplicative inverse of 12 modulo 17.", "output": "$x\\equiv 10\\pmod{17}$", "final_answer": "x\\equiv 10\\pmod{17}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec992cfbb67cde24", "problem_id": "4634818da7fa79fd", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the multiplicative inverse of 12 modulo 17.", "output": "1) We want x such that 12x ≡ 1 (mod 17).\n2) Use extended Euclid to find s,t with 12s + 17t = 1.\n3) One Bezout identity is: 12·(-7) + 17·(5) = 1.\n4) Reduce s modulo 17: x ≡ 10 (mod 17).\nFinal answer: $x\\equiv 10\\pmod{17}$", "final_answer": "x\\equiv 10\\pmod{17}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "30dbd733a79fc76f", "problem_id": "91202f8b99910ad4", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Worker A finishes a job in 22 hours and Worker B finishes it in 16 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/22 job/hour. B's rate = 1/16 job/hour.\n2) Combined rate = 1/22 + 1/16 = 19/176 job/hour.\n3) Time = 1 ÷ (combined rate) = 176/19 hours.\nFinal answer: $\\frac{176}{19}$", "final_answer": "\\frac{176}{19}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "873a8187f8a5768f", "problem_id": "5d6146d3e3d49489", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x \\sin{\\left(x \\right)}) = 2 x \\cos{\\left(x \\right)}$.\n3) So $f'(x) = 2 x^{2} \\cos{\\left(x \\right)} + 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x \\sin{\\left(x \\right)}) = 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 x^{2} \\cos{\\left(x \\right)} + 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$.\nFinal answer: $2 x^{2} \\cos{\\left(x \\right)} + 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$", "final_answer": "2 x^{2} \\cos{\\left(x \\right)} + 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "093e5ddc83420a31", "problem_id": "f7c0b0635d9ddf78", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} - 3 x - 3$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-3) = 0.\n3) d/dx(- 3 x) = -3.\n4) d/dx(- 2 x^{2}) = - 4 x.\n5) Combine: f'(x) = - 4 x - 3.\nFinal answer: $- 4\\cdot x - 3$", "final_answer": "- 4\\cdot x - 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b93b18a5f19ef377", "problem_id": "f7c0b0635d9ddf78", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} - 3 x - 3$.", "output": "Differentiate term-by-term:\nd/dx(-3) = 0.\nd/dx(- 3 x) = -3.\nd/dx(- 2 x^{2}) = - 4 x.\nCombine: f'(x) = - 4 x - 3.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 4\\cdot x - 3$", "final_answer": "- 4\\cdot x - 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b4a0e4916e594570", "problem_id": "ec34515e31646402", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Two friends split $462 in the ratio 2:9. How much does each receive?", "output": "Total ratio parts = 2+9 = 11.\nEach part = 462 ÷ 11 = 42.\nFirst share = 2 × 42 = 84.\nFinal answer: $378$", "final_answer": "378", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "711f7b1fa277afa1", "problem_id": "ec34515e31646402", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Two friends split $462 in the ratio 2:9. How much does each receive?", "output": "1) Total ratio parts = 2+9 = 11.\n2) Each part = 462 ÷ 11 = 42.\n3) First share = 2 × 42 = 84.\n4) Second share = 9 × 42 = 378.\nFinal answer: $378$", "final_answer": "378", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ae9a368dc63020d4", "problem_id": "f0d6ed07ce7fe486", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many binary strings of length 25 contain exactly 8 ones with no two ones adjacent?", "output": "1) Reserve 7 mandatory zeros between the 8 ones.\n2) This leaves n−k+1 = 18 valid slots for the ones.\n3) Choose the slots: C(18,8) = 43758.\nFinal answer: $43758$", "final_answer": "43758", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d8086e7b797382ef", "problem_id": "f0d6ed07ce7fe486", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many binary strings of length 25 contain exactly 8 ones with no two ones adjacent?", "output": "Reserve 7 mandatory zeros between the 8 ones.\nThis leaves n−k+1 = 18 valid slots for the ones.\nChoose the slots: C(18,8) = 43758.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $43758$", "final_answer": "43758", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5c11b2cc9290bbe5", "problem_id": "90b5c9913946f531", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- x^{2} - x - 2\\right) \\cos{\\left(2 x^{2} + 2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\n3) So $f'(x) = 4 x^{3} \\sin{\\left(2 x^{2} + 2 x \\right)} + 6 x^{2} \\sin{\\left(2 x^{2} + 2 x \\right)} + 2 x e^{x} + 10 x \\sin{\\left(2 x^{2} + 2 x \\right)} - 2 x \\cos{\\left(2 x^{2} + 2 x \\right)} + 4 e^{x} + 4 \\sin{\\left(2 x^{2} + 2 x \\right)} - \\cos{\\left(2 x^{2} + 2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = \\left(- 2 x - 1\\right) \\cos{\\left(2 x^{2} + 2 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 2 \\left(2 x + 1\\right) \\left(x \\left(- x - 1\\right) - 2\\right) \\sin{\\left(2 x^{2} + 2 x \\right)} + 2 e^{x}$.\nFinal answer: $\\left(- 2 x - 1\\right) \\cos{\\left(2 x^{2} + 2 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 2 \\left(2 x + 1\\right) \\left(x \\left(- x - 1\\right) - 2\\right) \\sin{\\left(2 x^{2} + 2 x \\right)} + 2 e^{x}$", "final_answer": "\\left(- 2 x - 1\\right) \\cos{\\left(2 x^{2} + 2 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 2 \\left(2 x + 1\\right) \\left(x \\left(- x - 1\\right) - 2\\right) \\sin{\\left(2 x^{2} + 2 x \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "59753295b9d99acb", "problem_id": "090af0f020fa0579", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} - 2 x^{4} + 4 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{4} + 4 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\n3) Subtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{\\pi^{5}}{80} - \\pi + \\frac{3 \\pi^{2}}{8} + 5 + \\frac{\\pi^{3}}{6}$.\n4) Simplify the result.\nFinal answer: $- \\frac{\\pi^{5}}{80} - \\pi + \\frac{3\\cdot \\pi^{2}}{8} + 5 + \\frac{\\pi^{3}}{6}$", "final_answer": "- \\frac{\\pi^{5}}{80} - \\pi + \\frac{3\\cdot \\pi^{2}}{8} + 5 + \\frac{\\pi^{3}}{6}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ba7d2353b8e682c3", "problem_id": "820730d4577c68b3", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 30 red balls and 21 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(51,7).\n3) Favorable samples: choose k reds and n−k blues: C(30,2)·C(21,5).\n4) Compute totals: C(51,7) = 115775100, and favorable = 8851815.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1653}{21620}$", "final_answer": "\\frac{1653}{21620}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "59e7b28b565099f3", "problem_id": "820730d4577c68b3", "style": "answer_only", "instruction": "Final answer only.", "input": "An urn has 30 red balls and 21 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "$\\frac{1653}{21620}$", "final_answer": "\\frac{1653}{21620}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a4da56fe4f70c97e", "problem_id": "f0bbc6c10c63ddf9", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the congruences: x≡6 (mod 9), x≡2 (mod 11). Give the solution modulo 99.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 99.\n3) Compute the CRT solution to get x ≡ 24 (mod 99).\nFinal answer: $x\\equiv 24\\pmod{99}$", "final_answer": "x\\equiv 24\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ea3d363654a6dfc5", "problem_id": "f0bbc6c10c63ddf9", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡6 (mod 9), x≡2 (mod 11). Give the solution modulo 99.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 99.\nCompute the CRT solution to get x ≡ 24 (mod 99).\nCheck: 24 mod 9=6 and mod 11=2.\nFinal answer: $x\\equiv 24\\pmod{99}$", "final_answer": "x\\equiv 24\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3c98f77d553ac3d7", "problem_id": "e3604c65fc8af973", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A cyclist rides at 28 miles per hour for 3/2 hours, then at 57 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 28 × 3/2 = 42 miles.\n2) Second leg distance = 57 × 2 = 114 miles.\n3) Total distance = 42 + 114 = 156 miles.\nFinal answer: $156$", "final_answer": "156", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "da75e487dc7a7bb4", "problem_id": "e94b9007718cfc98", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} x - 5$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to -5.\n3) Then the limit is -5.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -8.\nFinal answer: $-8$", "final_answer": "-8", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "75ead86fb1f7a8ff", "problem_id": "9cc6ff7beb2ca484", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$4 x - 2 y$ = $18$\n$- 3 x + 2 y$ = $-15$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $- 12 x + 6 y$ = $-54$.\n3) Multiply the second equation by $4$: $- 12 x + 8 y$ = $-60$.\n4) Subtract to eliminate x: $- 2 y$ = $6$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, -3\\right)$", "final_answer": "(x,y)=\\left(3, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "029c32757d83103a", "problem_id": "9cc6ff7beb2ca484", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$4 x - 2 y$ = $18$\n$- 3 x + 2 y$ = $-15$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $- 12 x + 6 y$ = $-54$.\nMultiply the second equation by $4$: $- 12 x + 8 y$ = $-60$.\nSubtract to eliminate x: $- 2 y$ = $6$.\nSolve for y: y = $-3$.\nSubstitute back to find x: x = $3$.\nCheck: plug (x,y)=(3,-3) into both equations.\nFinal answer: $(x,y)=\\left(3, -3\\right)$", "final_answer": "(x,y)=\\left(3, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "efbb3cd208314213", "problem_id": "1229884fde180898", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "You want 15 liters of a 25% salt solution. You have a 15% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 15−x liters is 50% solution.\n2) Salt equation: 15x + 50(15−x) = 25·15.\n3) (15-50)x = (25-50)·15 ⇒ x = 75/7.\n4) Then 15−x = 30/7.\nFinal answer: $75/7 L of 15% and 30/7 L of 50%$", "final_answer": "75/7 L of 15% and 30/7 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "538dc6393ee27756", "problem_id": "1229884fde180898", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "You want 15 liters of a 25% salt solution. You have a 15% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 15−x liters is 50% solution.\nSalt equation: 15x + 50(15−x) = 25·15.\n(15-50)x = (25-50)·15 ⇒ x = 75/7.\nFinal answer: $75/7 L of 15% and 30/7 L of 50%$", "final_answer": "75/7 L of 15% and 30/7 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "323ab9ba8eaf6f14", "problem_id": "25b55ac20e661b0f", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$2 x - 6 y$ = $-20$\n$- 6 x + 8 y$ = $40$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-6$: $- 12 x + 36 y$ = $120$.\n3) Multiply the second equation by $2$: $- 12 x + 16 y$ = $80$.\n4) Subtract to eliminate x: $20 y$ = $40$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $-4$.\nFinal answer: $(x,y)=\\left(-4, 2\\right)$", "final_answer": "(x,y)=\\left(-4, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4c634ecba1e1b5fa", "problem_id": "25b55ac20e661b0f", "style": "answer_only", "instruction": "Final answer only.", "input": "Solve the linear system:\n$2 x - 6 y$ = $-20$\n$- 6 x + 8 y$ = $40$", "output": "$(x,y)=\\left(-4, 2\\right)$", "final_answer": "(x,y)=\\left(-4, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "763d408b0d768bb8", "problem_id": "7d441d4fa9f7e32e", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 2 x^{3} + 2 x^{2} + x - 1$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 2 x^{4} + 3 x^{2}$ and a = -1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 8 x^{3} + 6 x$.\n5) Evaluate at a: p'(-1) = $2$.\nFinal answer: $2$", "final_answer": "2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d2c1442cc2310e90", "problem_id": "7d441d4fa9f7e32e", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 2 x^{3} + 2 x^{2} + x - 1$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 2 x^{4} + 3 x^{2}$ and a = -1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $2$", "final_answer": "2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "675c8cb30522ce9f", "problem_id": "e05370ec025e3e88", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 70 miles per hour for 1/2 hours, then at 57 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 70 × 1/2 = 35 miles.\nSecond leg distance = 57 × 3/2 = 171/2 miles.\nTotal distance = 35 + 171/2 = 241/2 miles.\nCheck: adding both legs gives 241/2.\nFinal answer: $\\frac{241}{2}$", "final_answer": "\\frac{241}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6e6ec8420d368d26", "problem_id": "e05370ec025e3e88", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A cyclist rides at 70 miles per hour for 1/2 hours, then at 57 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 70 × 1/2 = 35 miles.\n2) Second leg distance = 57 × 3/2 = 171/2 miles.\n3) Total distance = 35 + 171/2 = 241/2 miles.\nFinal answer: $\\frac{241}{2}$", "final_answer": "\\frac{241}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "208329a5146402cd", "problem_id": "409283c0dfac821a", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Find the multiplicative inverse of 6 modulo 11.", "output": "We want x such that 6x ≡ 1 (mod 11).\nUse extended Euclid to find s,t with 6s + 11t = 1.\nOne Bezout identity is: 6·(2) + 11·(-1) = 1.\nFinal answer: $x\\equiv 2\\pmod{11}$", "final_answer": "x\\equiv 2\\pmod{11}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72f431001f1e0a1e", "problem_id": "409283c0dfac821a", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the multiplicative inverse of 6 modulo 11.", "output": "1) We want x such that 6x ≡ 1 (mod 11).\n2) Use extended Euclid to find s,t with 6s + 11t = 1.\n3) One Bezout identity is: 6·(2) + 11·(-1) = 1.\n4) Reduce s modulo 11: x ≡ 2 (mod 11).\nFinal answer: $x\\equiv 2\\pmod{11}$", "final_answer": "x\\equiv 2\\pmod{11}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "03e4883f47e1504c", "problem_id": "e725b9b3a9f46c53", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(2 x^{2} + 2 x + 1\\right) \\cos{\\left(2 x^{2} + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x e^{x}) = x e^{x}$.\n3) So $f'(x) = - 8 x^{3} \\sin{\\left(2 x^{2} + 3 \\right)} - 8 x^{2} \\sin{\\left(2 x^{2} + 3 \\right)} + x e^{x} - 4 x \\sin{\\left(2 x^{2} + 3 \\right)} + 4 x \\cos{\\left(2 x^{2} + 3 \\right)} + 2 e^{x} + 2 \\cos{\\left(2 x^{2} + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x e^{x}) = x e^{x} + e^{x}$.\nTherefore the correct derivative is $f'(x) = - 4 x \\left(2 x \\left(x + 1\\right) + 1\\right) \\sin{\\left(2 x^{2} + 3 \\right)} + \\left(x + 2\\right) e^{x} + 2 \\left(2 x + 1\\right) \\cos{\\left(2 x^{2} + 3 \\right)} + e^{x}$.\nFinal answer: $- 4 x \\left(2 x \\left(x + 1\\right) + 1\\right) \\sin{\\left(2 x^{2} + 3 \\right)} + \\left(x + 2\\right) e^{x} + 2 \\left(2 x + 1\\right) \\cos{\\left(2 x^{2} + 3 \\right)} + e^{x}$", "final_answer": "- 4 x \\left(2 x \\left(x + 1\\right) + 1\\right) \\sin{\\left(2 x^{2} + 3 \\right)} + \\left(x + 2\\right) e^{x} + 2 \\left(2 x + 1\\right) \\cos{\\left(2 x^{2} + 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a2107f0b416e83af", "problem_id": "3535149b3022afb9", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "How many binary strings of length 15 contain exactly 5 ones with no two ones adjacent?", "output": "1) Reserve 4 mandatory zeros between the 5 ones.\n2) This leaves n−k+1 = 11 valid slots for the ones.\n3) Choose the slots: C(11,5) = 462.\nFinal answer: $462$", "final_answer": "462", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6aa93cfc9216dd5a", "problem_id": "5528e6789b3b5ac9", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} - x^{4} - 3 x^{2} + 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{4} - 3 x^{2} + 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{33 \\pi^{5}}{160} - \\frac{9 \\pi^{3}}{8} - \\frac{3 \\pi^{2}}{4} + 5 + 6 \\pi$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{33\\cdot \\pi^{5}}{160} - \\frac{9\\cdot \\pi^{3}}{8} - \\frac{3\\cdot \\pi^{2}}{4} + 5 + 6\\cdot \\pi$", "final_answer": "- \\frac{33\\cdot \\pi^{5}}{160} - \\frac{9\\cdot \\pi^{3}}{8} - \\frac{3\\cdot \\pi^{2}}{4} + 5 + 6\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0c2eb39a288bd82a", "problem_id": "5528e6789b3b5ac9", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} - x^{4} - 3 x^{2} + 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{4} - 3 x^{2} + 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{33 \\pi^{5}}{160} - \\frac{9 \\pi^{3}}{8} - \\frac{3 \\pi^{2}}{4} + 5 + 6 \\pi$.\n4) Simplify the result.\nFinal answer: $- \\frac{33\\cdot \\pi^{5}}{160} - \\frac{9\\cdot \\pi^{3}}{8} - \\frac{3\\cdot \\pi^{2}}{4} + 5 + 6\\cdot \\pi$", "final_answer": "- \\frac{33\\cdot \\pi^{5}}{160} - \\frac{9\\cdot \\pi^{3}}{8} - \\frac{3\\cdot \\pi^{2}}{4} + 5 + 6\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "56e66aa90fd1afb3", "problem_id": "9fd93e05f75ea9dd", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(2 x^{2} + 2 x - 3\\right) \\cos{\\left(3 x^{2} + 2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(e^{x}) = e^{x}$.\n3) So $f'(x) = - 12 x^{3} \\sin{\\left(3 x^{2} + 2 x \\right)} - 16 x^{2} \\sin{\\left(3 x^{2} + 2 x \\right)} + x e^{x} + 14 x \\sin{\\left(3 x^{2} + 2 x \\right)} + 4 x \\cos{\\left(3 x^{2} + 2 x \\right)} + 2 e^{x} + 6 \\sin{\\left(3 x^{2} + 2 x \\right)} + 2 \\cos{\\left(3 x^{2} + 2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(e^{x}) = e^{x}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 1\\right) e^{x} + 2 \\left(2 x + 1\\right) \\cos{\\left(3 x^{2} + 2 x \\right)} - 2 \\left(3 x + 1\\right) \\left(2 x \\left(x + 1\\right) - 3\\right) \\sin{\\left(3 x^{2} + 2 x \\right)} + e^{x}$.\nFinal answer: $\\left(x + 1\\right) e^{x} + 2 \\left(2 x + 1\\right) \\cos{\\left(3 x^{2} + 2 x \\right)} - 2 \\left(3 x + 1\\right) \\left(2 x \\left(x + 1\\right) - 3\\right) \\sin{\\left(3 x^{2} + 2 x \\right)} + e^{x}$", "final_answer": "\\left(x + 1\\right) e^{x} + 2 \\left(2 x + 1\\right) \\cos{\\left(3 x^{2} + 2 x \\right)} - 2 \\left(3 x + 1\\right) \\left(2 x \\left(x + 1\\right) - 3\\right) \\sin{\\left(3 x^{2} + 2 x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "62c8f9b9630893a0", "problem_id": "ea09c8316899d64e", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$- 4 x - 6 y$ = $-4$\n$- 7 x - 2 y$ = $27$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-7$: $28 x + 42 y$ = $28$.\n3) Multiply the second equation by $-4$: $28 x + 8 y$ = $-108$.\n4) Subtract to eliminate x: $34 y$ = $136$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, 4\\right)$", "final_answer": "(x,y)=\\left(-5, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "267d99bb9af5b34b", "problem_id": "f6335ed4fa22f026", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A store sells a backpack for $180. It is discounted by 30%. You buy 4 backpacks. What is the total cost?", "output": "1) Discounted price = 180 × (1 − 30/100) = 180 × 70/100 = 126.\n2) Subtotal = 4 × 126 = 504.\nFinal answer: $504$", "final_answer": "504", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0cf1583c801264ff", "problem_id": "f6335ed4fa22f026", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $180. It is discounted by 30%. You buy 4 backpacks. What is the total cost?", "output": "Discounted price = 180 × (1 − 30/100) = 180 × 70/100 = 126.\nSubtotal = 4 × 126 = 504.\nCheck: recomputing the formula yields 504.\nFinal answer: $504$", "final_answer": "504", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d2d232bfe7381f6", "problem_id": "f06d089f7839d132", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} + 2 x^{2} - 2 x$ at $x=0$.", "output": "1) Differentiate: f'(x) = $- 9 x^{2} + 4 x - 2$.\n2) Slope at x=0: m = f'(0) = $-2$.\n3) Point on curve: (0, f(0)) = (0, $0$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 0 = -2(x − 0).\n6) Optionally expand: y = $- 2 x$.\nFinal answer: $y = - 2\\cdot x$", "final_answer": "y = - 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "194494f9538d1ce9", "problem_id": "f06d089f7839d132", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} + 2 x^{2} - 2 x$ at $x=0$.", "output": "Differentiate: f'(x) = $- 9 x^{2} + 4 x - 2$.\nSlope at x=0: m = f'(0) = $-2$.\nPoint on curve: (0, f(0)) = (0, $0$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 0 = -2(x − 0).\nOptionally expand: y = $- 2 x$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 2\\cdot x$", "final_answer": "y = - 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c944ea7f64d65edb", "problem_id": "7fc358cfb0cf78b3", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Two friends split $616 in the ratio 7:4. How much does each receive?", "output": "1) Total ratio parts = 7+4 = 11.\n2) Each part = 616 ÷ 11 = 56.\n3) First share = 7 × 56 = 392.\n4) Second share = 4 × 56 = 224.\nFinal answer: $224$", "final_answer": "224", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ab7cf78335488f51", "problem_id": "3ecb21c9d55ee786", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 2 x^{3} + 3 x^{2} + 2 x + \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{3 x - 5}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 5}) = - \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 5}$.\n3) So $f'(x) = - \\frac{24 x^{3}}{18 x^{2} - 60 x + 50} + \\frac{9 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} - 60 x + 50} + \\frac{18 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} - 60 x + 50} + \\frac{78 x^{2}}{18 x^{2} - 60 x + 50} - \\frac{30 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} - 60 x + 50} - \\frac{60 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} - 60 x + 50} - \\frac{60 x}{18 x^{2} - 60 x + 50} + \\frac{25 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} - 60 x + 50} + \\frac{50 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} - 60 x + 50} - \\frac{2}{18 x^{2} - 60 x + 50} - \\frac{15 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{2} - 30 x + 25} + \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x - 10} - \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 5} + \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 5}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 5}) = - \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(3 x - 5\\right)} - \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 5} + \\frac{15 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(3 x - 5\\right)^{2}}$.\nTherefore the correct derivative is $f'(x) = \\frac{12 x^{3} - 18 x^{2} - 12 x + \\left(3 x - 5\\right) \\left(- 12 x^{2} + 12 x + \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 6 \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 18}{2 \\left(3 x - 5\\right)^{2}}$.\nFinal answer: $\\frac{12 x^{3} - 18 x^{2} - 12 x + \\left(3 x - 5\\right) \\left(- 12 x^{2} + 12 x + \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 6 \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 18}{2 \\left(3 x - 5\\right)^{2}}$", "final_answer": "\\frac{12 x^{3} - 18 x^{2} - 12 x + \\left(3 x - 5\\right) \\left(- 12 x^{2} + 12 x + \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 6 \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 18}{2 \\left(3 x - 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0ca1ee77743d911d", "problem_id": "ac9009291e4b41c0", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{2} - 3$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 6 x$.\n2) Slope at x=2: m = f'(2) = $-12$.\n3) Point on curve: (2, f(2)) = (2, $-15$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -15 = -12(x − 2).\n6) Optionally expand: y = $9 - 12 x$.\nFinal answer: $y = 9 - 12\\cdot x$", "final_answer": "y = 9 - 12\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e65f61a9cd36b2bc", "problem_id": "ac9009291e4b41c0", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{2} - 3$ at $x=2$.", "output": "Differentiate: f'(x) = $- 6 x$.\nSlope at x=2: m = f'(2) = $-12$.\nPoint on curve: (2, f(2)) = (2, $-15$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -15 = -12(x − 2).\nOptionally expand: y = $9 - 12 x$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 9 - 12\\cdot x$", "final_answer": "y = 9 - 12\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e50fafcf902af39a", "problem_id": "07f11e83d308b461", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 12 red balls and 15 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(27,6).\nFavorable samples: choose k reds and n−k blues: C(12,3)·C(15,3).\nCompute totals: C(27,6) = 296010, and favorable = 100100.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{70}{207}$.\nFinal answer: $\\frac{70}{207}$", "final_answer": "\\frac{70}{207}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bdc1a4690195ffdd", "problem_id": "07f11e83d308b461", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 12 red balls and 15 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(27,6).\n3) Favorable samples: choose k reds and n−k blues: C(12,3)·C(15,3).\n4) Compute totals: C(27,6) = 296010, and favorable = 100100.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{70}{207}$", "final_answer": "\\frac{70}{207}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "04b6809af8fadc2e", "problem_id": "0f222f9eed9e2c27", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{7}$. If it is flipped 11 times, what is the probability of exactly 9 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=11, k=9, p=$\\frac{4}{7}$ and simplify.\nHere C(11,9) = 55.\nCheck: expression simplifies to $\\frac{129761280}{1977326743}$.\nFinal answer: $\\frac{129761280}{1977326743}$", "final_answer": "\\frac{129761280}{1977326743}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4003be25e119c97f", "problem_id": "0f222f9eed9e2c27", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{7}$. If it is flipped 11 times, what is the probability of exactly 9 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=9, p=$\\frac{4}{7}$ and simplify.\n4) Here C(11,9) = 55.\nFinal answer: $\\frac{129761280}{1977326743}$", "final_answer": "\\frac{129761280}{1977326743}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "44a666e32c7aa9e3", "problem_id": "b1e80ed01f6f52e2", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=28$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=28 and k=5.\n4) Compute C(32,4) = 35960.\nFinal answer: $35960$", "final_answer": "35960", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "83bdad1837b1d2bc", "problem_id": "5f137f0736c7c320", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 3 x^{2} + 8 x - 24$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} + 7 x - 24.\n3) Then the limit is -72.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -75.\nFinal answer: $-75$", "final_answer": "-75", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f917aeccd11e8bb1", "problem_id": "2016f91680a51875", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $3 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3$.\n3) Compute an antiderivative: F(x) = $x^{3} - x^{2} - 3 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + x^{3} - x^{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + x^{3} - x^{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3a651e75942f6288", "problem_id": "2016f91680a51875", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $3 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3$.\nCompute an antiderivative: F(x) = $x^{3} - x^{2} - 3 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + x^{3} - x^{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + x^{3} - x^{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cef69977474c69ed", "problem_id": "efe9ac657ea2e78a", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$- 3 x$ = $-15$\n$2 x - 3 y$ = $4$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $- 6 x$ = $-30$.\n3) Multiply the second equation by $-3$: $- 6 x + 9 y$ = $-12$.\n4) Subtract to eliminate x: $- 9 y$ = $-18$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, 2\\right)$", "final_answer": "(x,y)=\\left(5, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "55a3d197d91b43b2", "problem_id": "efe9ac657ea2e78a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$- 3 x$ = $-15$\n$2 x - 3 y$ = $4$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $2$: $- 6 x$ = $-30$.\nMultiply the second equation by $-3$: $- 6 x + 9 y$ = $-12$.\nSubtract to eliminate x: $- 9 y$ = $-18$.\nSolve for y: y = $2$.\nSubstitute back to find x: x = $5$.\nCheck: plug (x,y)=(5,2) into both equations.\nFinal answer: $(x,y)=\\left(5, 2\\right)$", "final_answer": "(x,y)=\\left(5, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d4c8fd31ef8d4abd", "problem_id": "346b43bae6e83905", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A cyclist rides at 44 miles per hour for 3/2 hours, then at 64 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 44 × 3/2 = 66 miles.\n2) Second leg distance = 64 × 2 = 128 miles.\n3) Total distance = 66 + 128 = 194 miles.\nFinal answer: $194$", "final_answer": "194", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ebb5fc7481413b77", "problem_id": "346b43bae6e83905", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A cyclist rides at 44 miles per hour for 3/2 hours, then at 64 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 44 × 3/2 = 66 miles.\nSecond leg distance = 64 × 2 = 128 miles.\nTotal distance = 66 + 128 = 194 miles.\nCheck: adding both legs gives 194.\nFinal answer: $194$", "final_answer": "194", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "51c82b95bde9f9dc", "problem_id": "47e1295ae4057c61", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$6 x + y$ = $-26$\n$6 y$ = $-48$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $6$: $36 y$ = $-288$.\n4) Subtract to eliminate x: $- 36 y$ = $288$.\n5) Solve for y: y = $-8$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, -8\\right)$", "final_answer": "(x,y)=\\left(-3, -8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2c82aaec2388353a", "problem_id": "47e1295ae4057c61", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$6 x + y$ = $-26$\n$6 y$ = $-48$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $6$: $36 y$ = $-288$.\nSubtract to eliminate x: $- 36 y$ = $288$.\nSolve for y: y = $-8$.\nSubstitute back to find x: x = $-3$.\nCheck: plug (x,y)=(-3,-8) into both equations.\nFinal answer: $(x,y)=\\left(-3, -8\\right)$", "final_answer": "(x,y)=\\left(-3, -8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "044f938e114be95a", "problem_id": "1dbc0cf5cfe77ee9", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $133. It is discounted by 35%. You buy 4 backpacks. What is the total cost?", "output": "Discounted price = 133 × (1 − 35/100) = 133 × 65/100 = 1729/20.\nSubtotal = 4 × 1729/20 = 1729/5.\nCheck: recomputing the formula yields 1729/5.\nFinal answer: $\\frac{1729}{5}$", "final_answer": "\\frac{1729}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "13cd952a40ec59ab", "problem_id": "1dbc0cf5cfe77ee9", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A store sells a backpack for $133. It is discounted by 35%. You buy 4 backpacks. What is the total cost?", "output": "1) Discounted price = 133 × (1 − 35/100) = 133 × 65/100 = 1729/20.\n2) Subtotal = 4 × 1729/20 = 1729/5.\nFinal answer: $\\frac{1729}{5}$", "final_answer": "\\frac{1729}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "896d8d4893f72362", "problem_id": "276c94ad624a95fe", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x^{3} - 4 x^{2} - 7 x - 12$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{4} - 2 x^{3} + x^{2} + 2 x$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 4 x^{3} - 6 x^{2} + 2 x + 2$.\nEvaluate at a: p'(2) = $-50$.\nCheck: the identity gives $-50$.\nFinal answer: $-50$", "final_answer": "-50", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0818965ef33ac895", "problem_id": "276c94ad624a95fe", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x^{3} - 4 x^{2} - 7 x - 12$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{4} - 2 x^{3} + x^{2} + 2 x$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 4 x^{3} - 6 x^{2} + 2 x + 2$.\n5) Evaluate at a: p'(2) = $-50$.\nFinal answer: $-50$", "final_answer": "-50", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c43b0058ec4bb15c", "problem_id": "c803e60da26349d1", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the multiplicative inverse of 4 modulo 53.", "output": "1) We want x such that 4x ≡ 1 (mod 53).\n2) Use extended Euclid to find s,t with 4s + 53t = 1.\n3) One Bezout identity is: 4·(-13) + 53·(1) = 1.\n4) Reduce s modulo 53: x ≡ 40 (mod 53).\nFinal answer: $x\\equiv 40\\pmod{53}$", "final_answer": "x\\equiv 40\\pmod{53}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8b3f4e0a997f3da1", "problem_id": "c803e60da26349d1", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Find the multiplicative inverse of 4 modulo 53.", "output": "We want x such that 4x ≡ 1 (mod 53).\nUse extended Euclid to find s,t with 4s + 53t = 1.\nOne Bezout identity is: 4·(-13) + 53·(1) = 1.\nFinal answer: $x\\equiv 40\\pmod{53}$", "final_answer": "x\\equiv 40\\pmod{53}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c329ca776807f998", "problem_id": "6c3230c6f9b915a9", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{3 \\pi}{2} - \\frac{3 \\pi^{2}}{8} - 2 + \\frac{3 \\pi^{3}}{8}$.\n4) Simplify the result.\nFinal answer: $- \\frac{3\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{2}}{8} - 2 + \\frac{3\\cdot \\pi^{3}}{8}$", "final_answer": "- \\frac{3\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{2}}{8} - 2 + \\frac{3\\cdot \\pi^{3}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ded63ad27afff5af", "problem_id": "6c3230c6f9b915a9", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{3 \\pi}{2} - \\frac{3 \\pi^{2}}{8} - 2 + \\frac{3 \\pi^{3}}{8}$.\nFinal answer: $- \\frac{3\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{2}}{8} - 2 + \\frac{3\\cdot \\pi^{3}}{8}$", "final_answer": "- \\frac{3\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{2}}{8} - 2 + \\frac{3\\cdot \\pi^{3}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3849b7052ebddc43", "problem_id": "9d3d8ef2e9afe43e", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Worker A finishes a job in 22 hours and Worker B finishes it in 13 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/22 job/hour. B's rate = 1/13 job/hour.\nCombined rate = 1/22 + 1/13 = 35/286 job/hour.\nTime = 1 ÷ (combined rate) = 286/35 hours.\nCheck: (combined rate)×(time) = 35/286×286/35 = 1.\nFinal answer: $\\frac{286}{35}$", "final_answer": "\\frac{286}{35}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e3fa3fb63e8b423c", "problem_id": "9d3d8ef2e9afe43e", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Worker A finishes a job in 22 hours and Worker B finishes it in 13 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/22 job/hour. B's rate = 1/13 job/hour.\n2) Combined rate = 1/22 + 1/13 = 35/286 job/hour.\n3) Time = 1 ÷ (combined rate) = 286/35 hours.\nFinal answer: $\\frac{286}{35}$", "final_answer": "\\frac{286}{35}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6fc3e65d80db60bf", "problem_id": "322eac5380ac1f46", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Solve the linear system:\n$- 3 x + 3 y$ = $-21$\n$- 3 x + 4 y$ = $-25$", "output": "$(x,y)=\\left(3, -4\\right)$", "final_answer": "(x,y)=\\left(3, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "33db2a092427fa58", "problem_id": "322eac5380ac1f46", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$- 3 x + 3 y$ = $-21$\n$- 3 x + 4 y$ = $-25$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $9 x - 9 y$ = $63$.\n3) Multiply the second equation by $-3$: $9 x - 12 y$ = $75$.\n4) Subtract to eliminate x: $3 y$ = $-12$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, -4\\right)$", "final_answer": "(x,y)=\\left(3, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cadb092308dbd0cd", "problem_id": "1a749f2a7899da76", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡2 (mod 11), x≡4 (mod 7). Give the solution modulo 77.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 77.\n3) Compute the CRT solution to get x ≡ 46 (mod 77).\nFinal answer: $x\\equiv 46\\pmod{77}$", "final_answer": "x\\equiv 46\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "327f32809c160862", "problem_id": "1a749f2a7899da76", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the congruences: x≡2 (mod 11), x≡4 (mod 7). Give the solution modulo 77.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 77.\nCompute the CRT solution to get x ≡ 46 (mod 77).\nFinal answer: $x\\equiv 46\\pmod{77}$", "final_answer": "x\\equiv 46\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d6cb6958becf8286", "problem_id": "396828af72e33035", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Two friends split $1305 in the ratio 1:8. How much does each receive?", "output": "1) Total ratio parts = 1+8 = 9.\n2) Each part = 1305 ÷ 9 = 145.\n3) First share = 1 × 145 = 145.\n4) Second share = 8 × 145 = 1160.\nFinal answer: $1160$", "final_answer": "1160", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dc6fcd5187688309", "problem_id": "396828af72e33035", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $1305 in the ratio 1:8. How much does each receive?", "output": "Total ratio parts = 1+8 = 9.\nEach part = 1305 ÷ 9 = 145.\nFirst share = 1 × 145 = 145.\nSecond share = 8 × 145 = 1160.\nCheck: 145+1160=1305.\nFinal answer: $1160$", "final_answer": "1160", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec77d33456c8f00f", "problem_id": "0f187a002787f6ec", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{10} x}{2} + \\frac{15 x}{4} - \\frac{9 \\sqrt{10}}{8} + \\frac{9}{4}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{15}{4} - \\frac{3 \\sqrt{10}}{2}$, c=$\\frac{9}{4} - \\frac{9 \\sqrt{10}}{8}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{441}{16} - \\frac{27 \\sqrt{10}}{4}$, so √Δ = $\\sqrt{\\frac{441}{16} - \\frac{27 \\sqrt{10}}{4}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{15}{8} + \\frac{3 \\sqrt{49 - 12 \\sqrt{10}}}{8} + \\frac{3 \\sqrt{10}}{4}$.\nx2 = $- \\frac{15}{8} - \\frac{3 \\sqrt{49 - 12 \\sqrt{10}}}{8} + \\frac{3 \\sqrt{10}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{15}{8} + \\frac{3\\cdot \\sqrt{49 - 12\\cdot \\sqrt{10}}}{8} + \\frac{3\\cdot \\sqrt{10}}{4}, - \\frac{15}{8} - \\frac{3\\cdot \\sqrt{49 - 12\\cdot \\sqrt{10}}}{8} + \\frac{3\\cdot \\sqrt{10}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{15}{8} + \\frac{3\\cdot \\sqrt{49 - 12\\cdot \\sqrt{10}}}{8} + \\frac{3\\cdot \\sqrt{10}}{4}, - \\frac{15}{8} - \\frac{3\\cdot \\sqrt{49 - 12\\cdot \\sqrt{10}}}{8} + \\frac{3\\cdot \\sqrt{10}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a7af21f16e7db5eb", "problem_id": "0f187a002787f6ec", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{10} x}{2} + \\frac{15 x}{4} - \\frac{9 \\sqrt{10}}{8} + \\frac{9}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{15}{4} - \\frac{3 \\sqrt{10}}{2}$, c=$\\frac{9}{4} - \\frac{9 \\sqrt{10}}{8}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{441}{16} - \\frac{27 \\sqrt{10}}{4}$, so √Δ = $\\sqrt{\\frac{441}{16} - \\frac{27 \\sqrt{10}}{4}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{15}{8} + \\frac{3 \\sqrt{49 - 12 \\sqrt{10}}}{8} + \\frac{3 \\sqrt{10}}{4}$.\n7) x2 = $- \\frac{15}{8} - \\frac{3 \\sqrt{49 - 12 \\sqrt{10}}}{8} + \\frac{3 \\sqrt{10}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{15}{8} + \\frac{3\\cdot \\sqrt{49 - 12\\cdot \\sqrt{10}}}{8} + \\frac{3\\cdot \\sqrt{10}}{4}, - \\frac{15}{8} - \\frac{3\\cdot \\sqrt{49 - 12\\cdot \\sqrt{10}}}{8} + \\frac{3\\cdot \\sqrt{10}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{15}{8} + \\frac{3\\cdot \\sqrt{49 - 12\\cdot \\sqrt{10}}}{8} + \\frac{3\\cdot \\sqrt{10}}{4}, - \\frac{15}{8} - \\frac{3\\cdot \\sqrt{49 - 12\\cdot \\sqrt{10}}}{8} + \\frac{3\\cdot \\sqrt{10}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ee82cff302af7c49", "problem_id": "4b514f6be0c5a071", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} + 3 x^{2} - x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $- 6 x^{2} + 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 1$.\n2) Slope at x=-1: m = f'(-1) = $-13 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\n3) Point on curve: (-1, f(-1)) = (-1, $\\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 7$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 7 = -13 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}(x − -1).\n6) Optionally expand: y = $- 13 x + \\frac{x}{2 e^{\\frac{1}{2}}} + 2 x \\sin{\\left(2 \\right)} - 6 + \\cos{\\left(2 \\right)} + \\frac{3}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = - 13\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 6 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 13\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 6 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7dfa73e3db73902f", "problem_id": "4b514f6be0c5a071", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} + 3 x^{2} - x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=-1$.", "output": "Differentiate: f'(x) = $- 6 x^{2} + 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 1$.\nSlope at x=-1: m = f'(-1) = $-13 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nPoint on curve: (-1, f(-1)) = (-1, $\\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 7$).\nFinal answer: $y = - 13\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 6 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 13\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 6 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cf98d2a9a4015e8e", "problem_id": "6ba99aa955fab2e0", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} + 4 x$.", "output": "1) Differentiate term-by-term:\n2) d/dx(- 2 x^{3}) = - 6 x^{2}.\n3) d/dx(4 x) = 4.\n4) Combine: f'(x) = 2 \\left(2 - 3 x^{2}\\right).\nFinal answer: $4 - 6\\cdot x^{2}$", "final_answer": "4 - 6\\cdot x^{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "74094dbc3305432e", "problem_id": "6ba99aa955fab2e0", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} + 4 x$.", "output": "Differentiate term-by-term:\nd/dx(- 2 x^{3}) = - 6 x^{2}.\nd/dx(4 x) = 4.\nCombine: f'(x) = 2 \\left(2 - 3 x^{2}\\right).\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $4 - 6\\cdot x^{2}$", "final_answer": "4 - 6\\cdot x^{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "92b6e27e18668b1f", "problem_id": "a2834fb9e51a07b3", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - x^{3} + x + 5$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{4} + 3 x^{3} + x^{2} + 2 x + 1$ and a = 3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 4 x^{3} + 9 x^{2} + 2 x + 2$.\nEvaluate at a: p'(3) = $-19$.\nCheck: the identity gives $-19$.\nFinal answer: $-19$", "final_answer": "-19", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c78c3b96fbc162de", "problem_id": "a2834fb9e51a07b3", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - x^{3} + x + 5$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{4} + 3 x^{3} + x^{2} + 2 x + 1$ and a = 3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 4 x^{3} + 9 x^{2} + 2 x + 2$.\n5) Evaluate at a: p'(3) = $-19$.\nFinal answer: $-19$", "final_answer": "-19", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6111c92cf028f877", "problem_id": "a62079d4419094b3", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many binary strings of length 17 contain exactly 4 ones with no two ones adjacent?", "output": "Reserve 3 mandatory zeros between the 4 ones.\nThis leaves n−k+1 = 14 valid slots for the ones.\nChoose the slots: C(14,4) = 1001.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $1001$", "final_answer": "1001", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c4296f687b0ba8df", "problem_id": "a62079d4419094b3", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "How many binary strings of length 17 contain exactly 4 ones with no two ones adjacent?", "output": "1) Reserve 3 mandatory zeros between the 4 ones.\n2) This leaves n−k+1 = 14 valid slots for the ones.\n3) Choose the slots: C(14,4) = 1001.\nFinal answer: $1001$", "final_answer": "1001", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a8aa3356928a68d", "problem_id": "4afa3afa78fe21ae", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many binary strings of length 22 contain exactly 5 ones with no two ones adjacent?", "output": "1) Reserve 4 mandatory zeros between the 5 ones.\n2) This leaves n−k+1 = 18 valid slots for the ones.\n3) Choose the slots: C(18,5) = 8568.\nFinal answer: $8568$", "final_answer": "8568", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7c16cbbc17a775ba", "problem_id": "4afa3afa78fe21ae", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "How many binary strings of length 22 contain exactly 5 ones with no two ones adjacent?", "output": "Reserve 4 mandatory zeros between the 5 ones.\nThis leaves n−k+1 = 18 valid slots for the ones.\nChoose the slots: C(18,5) = 8568.\nFinal answer: $8568$", "final_answer": "8568", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "12aefd4f071dbb0f", "problem_id": "d4508b3a9beae98c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{2} - 3 x + \\left(x^{2} - 3 x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3 x + 5}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3 x + 5}) = \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3 x + 5}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} - \\frac{6 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} - \\frac{12 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} + \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 6 x^{3} + 19 x^{2} - 30 x + 25} + \\frac{19 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} + \\frac{38 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} - \\frac{3 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 6 x^{3} + 19 x^{2} - 30 x + 25} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 6 x + 10} - \\frac{30 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} - \\frac{60 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} + \\frac{20 x}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} - \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3 x + 5} + \\frac{25 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} + \\frac{50 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} - \\frac{30}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3 x + 5}) = \\frac{x^{2} \\left(3 - 2 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - 3 x + 5\\right)^{2}} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x^{2} - 3 x + 5\\right)} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3 x + 5} + \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3 x + 5}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(3 - 2 x\\right) \\left(x^{2} - 3 x + \\left(x \\left(x - 3\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x \\left(x - 3\\right) + 5\\right) \\left(4 x + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 3\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 3\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x \\left(x - 3\\right) + 5\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(3 - 2 x\\right) \\left(x^{2} - 3 x + \\left(x \\left(x - 3\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x \\left(x - 3\\right) + 5\\right) \\left(4 x + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 3\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 3\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x \\left(x - 3\\right) + 5\\right)^{2}}$", "final_answer": "\\frac{2 \\left(3 - 2 x\\right) \\left(x^{2} - 3 x + \\left(x \\left(x - 3\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x \\left(x - 3\\right) + 5\\right) \\left(4 x + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 3\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 3\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x \\left(x - 3\\right) + 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b289091a6b03a55d", "problem_id": "cd269ecf733cf0ff", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} 3 x^{3} + 3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{3} + 3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1$.\nEvaluate at the bounds: F(2 \\pi) and F(0).\nSubtract: F(2 \\pi) − F(0) = $2 \\pi \\left(- 3 \\pi - 1 + 4 \\pi^{2} + 6 \\pi^{3}\\right)$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $2\\cdot \\pi\\cdot \\left(- 3\\cdot \\pi - 1 + 4\\cdot \\pi^{2} + 6\\cdot \\pi^{3}\\right)$", "final_answer": "2\\cdot \\pi\\cdot \\left(- 3\\cdot \\pi - 1 + 4\\cdot \\pi^{2} + 6\\cdot \\pi^{3}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d497e784b4c5e85", "problem_id": "05117b8348af7246", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} - 2 x^{3} + x^{2} - x - 4$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 2 x^{4} + 3 x^{3} - 2 x^{2} - 3 x + 1$ and a = 1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 8 x^{3} + 9 x^{2} - 4 x - 3$.\nEvaluate at a: p'(1) = $-6$.\nCheck: the identity gives $-6$.\nFinal answer: $-6$", "final_answer": "-6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "107fb1304da88c5d", "problem_id": "05117b8348af7246", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} - 2 x^{3} + x^{2} - x - 4$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 2 x^{4} + 3 x^{3} - 2 x^{2} - 3 x + 1$ and a = 1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 8 x^{3} + 9 x^{2} - 4 x - 3$.\n5) Evaluate at a: p'(1) = $-6$.\nFinal answer: $-6$", "final_answer": "-6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "83b82dcf10122748", "problem_id": "2ba925ca048c1e9e", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $760. It is discounted by 40%. You buy 2 backpacks. What is the total cost?", "output": "Discounted price = 760 × (1 − 40/100) = 760 × 60/100 = 456.\nSubtotal = 2 × 456 = 912.\nCheck: recomputing the formula yields 912.\nFinal answer: $912$", "final_answer": "912", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "df41ff834e40b966", "problem_id": "2ba925ca048c1e9e", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A store sells a backpack for $760. It is discounted by 40%. You buy 2 backpacks. What is the total cost?", "output": "1) Discounted price = 760 × (1 − 40/100) = 760 × 60/100 = 456.\n2) Subtotal = 2 × 456 = 912.\nFinal answer: $912$", "final_answer": "912", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1f452b4320d06342", "problem_id": "65e0903ca32c7e9a", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$3 y$ = $-12$\n$3 x - 4 y$ = $28$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $3$: $9 y$ = $-36$.\n3) Multiply the second equation by $0$: $0$ = $0$.\n4) Subtract to eliminate x: $9 y$ = $-36$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, -4\\right)$", "final_answer": "(x,y)=\\left(4, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6c5dcfd5d41bc78a", "problem_id": "65e0903ca32c7e9a", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$3 y$ = $-12$\n$3 x - 4 y$ = $28$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $9 y$ = $-36$.\nMultiply the second equation by $0$: $0$ = $0$.\nSubtract to eliminate x: $9 y$ = $-36$.\nSolve for y: y = $-4$.\nSubstitute back to find x: x = $4$.\nCheck: plug (x,y)=(4,-4) into both equations.\nFinal answer: $(x,y)=\\left(4, -4\\right)$", "final_answer": "(x,y)=\\left(4, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1e8bcddcccb80c5f", "problem_id": "f3dcf3f671f9ae79", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "You want 39 liters of a 40% salt solution. You have a 15% solution and a 60% solution. How many liters of each should you mix?", "output": "52/3 L of 15% and 65/3 L of 60%", "final_answer": "52/3 L of 15% and 65/3 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ad9948fdb9fde715", "problem_id": "f3dcf3f671f9ae79", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 39 liters of a 40% salt solution. You have a 15% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 39−x liters is 60% solution.\nSalt equation: 15x + 60(39−x) = 40·39.\n(15-60)x = (40-60)·39 ⇒ x = 52/3.\nThen 39−x = 65/3.\nCheck: 15·52/3 + 60·65/3 = 40·39.\nFinal answer: $52/3 L of 15% and 65/3 L of 60%$", "final_answer": "52/3 L of 15% and 65/3 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dd4e8eb700675fb0", "problem_id": "70f3d1d7a3038a3d", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $x^{2}\\cdot \\cos{\\left(x \\right)} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - \\cos{\\left(x \\right)}$", "final_answer": "x^{2}\\cdot \\cos{\\left(x \\right)} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e5ed4942da777a8e", "problem_id": "70f3d1d7a3038a3d", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.\nFinal answer: $x^{2}\\cdot \\cos{\\left(x \\right)} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - \\cos{\\left(x \\right)}$", "final_answer": "x^{2}\\cdot \\cos{\\left(x \\right)} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d1639efb8c17befd", "problem_id": "23eafac488bc3cdd", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} x - 2$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{2} - 3$ and a = -2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $2 x$.\n5) Evaluate at a: p'(-2) = $-4$.\nFinal answer: $-4$", "final_answer": "-4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f5596ba0862df243", "problem_id": "23eafac488bc3cdd", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} x - 2$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $x^{2} - 3$ and a = -2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $2 x$.\nEvaluate at a: p'(-2) = $-4$.\nCheck: the identity gives $-4$.\nFinal answer: $-4$", "final_answer": "-4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e6ab0a39e3b56750", "problem_id": "fd9addc1eae6aaca", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Two friends split $1020 in the ratio 2:8. How much does each receive?", "output": "Total ratio parts = 2+8 = 10.\nEach part = 1020 ÷ 10 = 102.\nFirst share = 2 × 102 = 204.\nSecond share = 8 × 102 = 816.\nCheck: 204+816=1020.\nFinal answer: $816$", "final_answer": "816", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0773f591e3b9736a", "problem_id": "fd9addc1eae6aaca", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Two friends split $1020 in the ratio 2:8. How much does each receive?", "output": "1) Total ratio parts = 2+8 = 10.\n2) Each part = 1020 ÷ 10 = 102.\n3) First share = 2 × 102 = 204.\n4) Second share = 8 × 102 = 816.\nFinal answer: $816$", "final_answer": "816", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "82acbad6735224b8", "problem_id": "f055dfa871d94818", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} - 3 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $- 9 \\pi^{3} - 3 \\pi^{2} - 4$.\n4) Simplify the result.\nFinal answer: $- 9\\cdot \\pi^{3} - 3\\cdot \\pi^{2} - 4$", "final_answer": "- 9\\cdot \\pi^{3} - 3\\cdot \\pi^{2} - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c28b1e21f2617cde", "problem_id": "f055dfa871d94818", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} - 3 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $- 9 \\pi^{3} - 3 \\pi^{2} - 4$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- 9\\cdot \\pi^{3} - 3\\cdot \\pi^{2} - 4$", "final_answer": "- 9\\cdot \\pi^{3} - 3\\cdot \\pi^{2} - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "303bd98327ae653c", "problem_id": "fe687de8f4f16c6d", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "A store sells a backpack for $21. It is discounted by 35%. You buy 10 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 21 × (1 − 35/100) = 21 × 65/100 = 273/20.\nSubtotal = 10 × 273/20 = 273/2.\nTotal with tax = 273/2 × 110/100 = 3003/20.\nFinal answer: $\\frac{3003}{20}$", "final_answer": "\\frac{3003}{20}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7755cb800060a670", "problem_id": "fe687de8f4f16c6d", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A store sells a backpack for $21. It is discounted by 35%. You buy 10 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 21 × (1 − 35/100) = 21 × 65/100 = 273/20.\n2) Subtotal = 10 × 273/20 = 273/2.\n3) Total with tax = 273/2 × 110/100 = 3003/20.\nFinal answer: $\\frac{3003}{20}$", "final_answer": "\\frac{3003}{20}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8895e889ff9255b1", "problem_id": "33fe00b7153dbe84", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the congruences: x≡2 (mod 4), x≡8 (mod 9). Give the solution modulo 36.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 36.\n3) Compute the CRT solution to get x ≡ 26 (mod 36).\nFinal answer: $x\\equiv 26\\pmod{36}$", "final_answer": "x\\equiv 26\\pmod{36}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a1cdeff3dbb809f", "problem_id": "33fe00b7153dbe84", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 4), x≡8 (mod 9). Give the solution modulo 36.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 36.\nCompute the CRT solution to get x ≡ 26 (mod 36).\nCheck: 26 mod 4=2 and mod 9=8.\nFinal answer: $x\\equiv 26\\pmod{36}$", "final_answer": "x\\equiv 26\\pmod{36}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4ffefa5ec6dcc76a", "problem_id": "f3cb433250a67a45", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Find the multiplicative inverse of 11 modulo 17.", "output": "1) We want x such that 11x ≡ 1 (mod 17).\n2) Use extended Euclid to find s,t with 11s + 17t = 1.\n3) One Bezout identity is: 11·(-3) + 17·(2) = 1.\n4) Reduce s modulo 17: x ≡ 14 (mod 17).\nFinal answer: $x\\equiv 14\\pmod{17}$", "final_answer": "x\\equiv 14\\pmod{17}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "46b9995071ac3839", "problem_id": "f3cb433250a67a45", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the multiplicative inverse of 11 modulo 17.", "output": "We want x such that 11x ≡ 1 (mod 17).\nUse extended Euclid to find s,t with 11s + 17t = 1.\nOne Bezout identity is: 11·(-3) + 17·(2) = 1.\nReduce s modulo 17: x ≡ 14 (mod 17).\nCheck: 11·14 mod 17 = 1.\nFinal answer: $x\\equiv 14\\pmod{17}$", "final_answer": "x\\equiv 14\\pmod{17}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d2f321b913dc854", "problem_id": "14ba1a5b0fab0a6a", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} - x^{2} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=1$.", "output": "1) Differentiate: f'(x) = $- 3 x^{2} - 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\n2) Slope at x=1: m = f'(1) = $-7 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}$.\n3) Point on curve: (1, f(1)) = (1, $-5 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -5 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} = -7 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}(x − 1).\n6) Optionally expand: y = $- 7 x - 2 x \\sin{\\left(2 \\right)} + \\frac{x e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2 \\sin{\\left(2 \\right)} + 2$.\nFinal answer: $y = - 7\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 2$", "final_answer": "y = - 7\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dde7fd4c5c7d6986", "problem_id": "14ba1a5b0fab0a6a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} - x^{2} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=1$.", "output": "Differentiate: f'(x) = $- 3 x^{2} - 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\nSlope at x=1: m = f'(1) = $-7 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}$.\nPoint on curve: (1, f(1)) = (1, $-5 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -5 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} = -7 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}(x − 1).\nOptionally expand: y = $- 7 x - 2 x \\sin{\\left(2 \\right)} + \\frac{x e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2 \\sin{\\left(2 \\right)} + 2$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 7\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 2$", "final_answer": "y = - 7\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fbdc23e7d59a26b0", "problem_id": "aec7571f85e64a2a", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$- 5 x - 4 y$ = $-61$\n$2 x + 7 y$ = $46$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $- 10 x - 8 y$ = $-122$.\n3) Multiply the second equation by $-5$: $- 10 x - 35 y$ = $-230$.\n4) Subtract to eliminate x: $27 y$ = $108$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $9$.\nFinal answer: $(x,y)=\\left(9, 4\\right)$", "final_answer": "(x,y)=\\left(9, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0027e944aa8fb6c6", "problem_id": "aec7571f85e64a2a", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Solve the linear system:\n$- 5 x - 4 y$ = $-61$\n$2 x + 7 y$ = $46$", "output": "$(x,y)=\\left(9, 4\\right)$", "final_answer": "(x,y)=\\left(9, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4b5a694e1e2356d3", "problem_id": "c966cec6ac26c65a", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_7=25$?", "output": "Let y_i=x_i-3, then y_i≥0 and y_1+…+y_7=4.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=4 and k=7.\nCompute C(10,6) = 210.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $210$", "final_answer": "210", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "910db6bb7c0979f2", "problem_id": "c966cec6ac26c65a", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_7=25$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_7=4.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=4 and k=7.\n4) Compute C(10,6) = 210.\nFinal answer: $210$", "final_answer": "210", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e0a018175c82acb6", "problem_id": "de39b2fb1e2c08d4", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 18 x - 21$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} - 18 x - 21$ = $3 \\left(x - 7\\right) \\left(x + 1\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $7$.\n5) From the second factor: x = $-1$.\nFinal answer: $x \\in \\{7, -1\\}$", "final_answer": "x \\in \\{7, -1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "199f89e872f270a7", "problem_id": "de39b2fb1e2c08d4", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 18 x - 21$ = 0.", "output": "$x \\in \\{7, -1\\}$", "final_answer": "x \\in \\{7, -1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2911f6bf7d33e802", "problem_id": "3740881f707050d6", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 11 liters of a 42% salt solution. You have a 25% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 11−x liters is 60% solution.\nSalt equation: 25x + 60(11−x) = 42·11.\n(25-60)x = (42-60)·11 ⇒ x = 198/35.\nThen 11−x = 187/35.\nCheck: 25·198/35 + 60·187/35 = 42·11.\nFinal answer: $198/35 L of 25% and 187/35 L of 60%$", "final_answer": "198/35 L of 25% and 187/35 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "23c74f61b24368ab", "problem_id": "3740881f707050d6", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "You want 11 liters of a 42% salt solution. You have a 25% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 11−x liters is 60% solution.\n2) Salt equation: 25x + 60(11−x) = 42·11.\n3) (25-60)x = (42-60)·11 ⇒ x = 198/35.\n4) Then 11−x = 187/35.\nFinal answer: $198/35 L of 25% and 187/35 L of 60%$", "final_answer": "198/35 L of 25% and 187/35 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8ddc4945f8259d55", "problem_id": "e29dbb553995d761", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "You want 47 liters of a 37% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 47−x liters is 50% solution.\n2) Salt equation: 25x + 50(47−x) = 37·47.\n3) (25-50)x = (37-50)·47 ⇒ x = 611/25.\n4) Then 47−x = 564/25.\nFinal answer: $611/25 L of 25% and 564/25 L of 50%$", "final_answer": "611/25 L of 25% and 564/25 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ecbaa0d9356ecd93", "problem_id": "e29dbb553995d761", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 47 liters of a 37% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 47−x liters is 50% solution.\nSalt equation: 25x + 50(47−x) = 37·47.\n(25-50)x = (37-50)·47 ⇒ x = 611/25.\nThen 47−x = 564/25.\nCheck: 25·611/25 + 50·564/25 = 37·47.\nFinal answer: $611/25 L of 25% and 564/25 L of 50%$", "final_answer": "611/25 L of 25% and 564/25 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dd93c6cdec9af9e2", "problem_id": "faadbf65af7d7776", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the equation of the tangent line to $y=- 3 x^{2} + 2 \\sin{\\left(x \\right)} - 3$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $- 6 x + 2 \\cos{\\left(x \\right)}$.\n2) Slope at x=-2: m = f'(-2) = $2 \\cos{\\left(2 \\right)} + 12$.\n3) Point on curve: (-2, f(-2)) = (-2, $-15 - 2 \\sin{\\left(2 \\right)}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -15 - 2 \\sin{\\left(2 \\right)} = 2 \\cos{\\left(2 \\right)} + 12(x − -2).\n6) Optionally expand: y = $2 x \\cos{\\left(2 \\right)} + 12 x - 2 \\sin{\\left(2 \\right)} + 4 \\cos{\\left(2 \\right)} + 9$.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 12\\cdot x - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)} + 9$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 12\\cdot x - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)} + 9", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a4a96aeab8d17ee8", "problem_id": "faadbf65af7d7776", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{2} + 2 \\sin{\\left(x \\right)} - 3$ at $x=-2$.", "output": "Differentiate: f'(x) = $- 6 x + 2 \\cos{\\left(x \\right)}$.\nSlope at x=-2: m = f'(-2) = $2 \\cos{\\left(2 \\right)} + 12$.\nPoint on curve: (-2, f(-2)) = (-2, $-15 - 2 \\sin{\\left(2 \\right)}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -15 - 2 \\sin{\\left(2 \\right)} = 2 \\cos{\\left(2 \\right)} + 12(x − -2).\nOptionally expand: y = $2 x \\cos{\\left(2 \\right)} + 12 x - 2 \\sin{\\left(2 \\right)} + 4 \\cos{\\left(2 \\right)} + 9$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 12\\cdot x - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)} + 9$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 12\\cdot x - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)} + 9", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b7cf070dde42d10f", "problem_id": "487c3b7a47820420", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡3 (mod 9), x≡3 (mod 11). Give the solution modulo 99.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 99.\n3) Compute the CRT solution to get x ≡ 3 (mod 99).\nFinal answer: $x\\equiv 3\\pmod{99}$", "final_answer": "x\\equiv 3\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1ea56a9a1e5d0f18", "problem_id": "487c3b7a47820420", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the congruences: x≡3 (mod 9), x≡3 (mod 11). Give the solution modulo 99.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 99.\nCompute the CRT solution to get x ≡ 3 (mod 99).\nFinal answer: $x\\equiv 3\\pmod{99}$", "final_answer": "x\\equiv 3\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1bfe5b588329bd28", "problem_id": "c3f9aefa21961f35", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡3 (mod 9), x≡0 (mod 7). Give the solution modulo 63.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 63.\nCompute the CRT solution to get x ≡ 21 (mod 63).\nCheck: 21 mod 9=3 and mod 7=0.\nFinal answer: $x\\equiv 21\\pmod{63}$", "final_answer": "x\\equiv 21\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aee4d9ec92d067ba", "problem_id": "c3f9aefa21961f35", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the congruences: x≡3 (mod 9), x≡0 (mod 7). Give the solution modulo 63.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 63.\n3) Compute the CRT solution to get x ≡ 21 (mod 63).\nFinal answer: $x\\equiv 21\\pmod{63}$", "final_answer": "x\\equiv 21\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ce725a041c18eb8", "problem_id": "22a3089edc84880c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x - 3$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 2 x - 3.\n3) Then the limit is -7.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -5.\nFinal answer: $-5$", "final_answer": "-5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "99507379111553b5", "problem_id": "e663070afe083fd0", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 18 red balls and 28 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(46,8).\n3) Favorable samples: choose k reds and n−k blues: C(18,2)·C(28,6).\n4) Compute totals: C(46,8) = 260932815, and favorable = 57641220.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{4284}{19393}$", "final_answer": "\\frac{4284}{19393}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ccd04eb40e01087d", "problem_id": "bd8139b48e3e2388", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A cyclist rides at 69 miles per hour for 1/2 hours, then at 60 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 69 × 1/2 = 69/2 miles.\n2) Second leg distance = 60 × 3 = 180 miles.\n3) Total distance = 69/2 + 180 = 429/2 miles.\nFinal answer: $\\frac{429}{2}$", "final_answer": "\\frac{429}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ad838789a9486506", "problem_id": "bd8139b48e3e2388", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 69 miles per hour for 1/2 hours, then at 60 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 69 × 1/2 = 69/2 miles.\nSecond leg distance = 60 × 3 = 180 miles.\nTotal distance = 69/2 + 180 = 429/2 miles.\nCheck: adding both legs gives 429/2.\nFinal answer: $\\frac{429}{2}$", "final_answer": "\\frac{429}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "91e5190220071cbb", "problem_id": "71d1452e82026193", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_3=17$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=17 and k=3.\n4) Compute C(19,2) = 171.\nFinal answer: $171$", "final_answer": "171", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a5a189cb51a26068", "problem_id": "561de89f8d614f10", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{2} + 3 x + \\left(x^{2} + 3 x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 3 x - 6}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 6}) = - \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3 x - 6}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 6 x^{2} - 72 x + 72} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 6 x^{2} - 72 x + 72} + \\frac{6 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 6 x^{2} - 72 x + 72} + \\frac{12 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 6 x^{2} - 72 x + 72} - \\frac{3 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 6 x^{2} - 72 x + 72} - \\frac{6 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 6 x^{2} - 72 x + 72} - \\frac{36 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 6 x^{2} - 72 x + 72} - \\frac{72 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 6 x^{2} - 72 x + 72} - \\frac{24 x}{2 x^{4} + 12 x^{3} - 6 x^{2} - 72 x + 72} - \\frac{12 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 6 x^{3} - 3 x^{2} - 36 x + 36} + \\frac{36 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 6 x^{2} - 72 x + 72} + \\frac{72 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 6 x^{2} - 72 x + 72} - \\frac{36}{2 x^{4} + 12 x^{3} - 6 x^{2} - 72 x + 72} - \\frac{18 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 6 x^{3} - 3 x^{2} - 36 x + 36} - \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3 x - 6} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 6} + \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 6}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 6}) = - \\frac{6 \\left(- 2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(x^{2} + 3 x\\right) - 6\\right)^{2}} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 6} - \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 6}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(- 2 x - 3\\right) \\left(x^{2} + 3 x + \\left(x \\left(x + 3\\right) - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x \\left(x + 3\\right) - 6\\right) \\left(4 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x + 3\\right) - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x + 3\\right) - 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6\\right)}{2 \\left(x \\left(x + 3\\right) - 6\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(- 2 x - 3\\right) \\left(x^{2} + 3 x + \\left(x \\left(x + 3\\right) - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x \\left(x + 3\\right) - 6\\right) \\left(4 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x + 3\\right) - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x + 3\\right) - 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6\\right)}{2 \\left(x \\left(x + 3\\right) - 6\\right)^{2}}$", "final_answer": "\\frac{2 \\left(- 2 x - 3\\right) \\left(x^{2} + 3 x + \\left(x \\left(x + 3\\right) - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x \\left(x + 3\\right) - 6\\right) \\left(4 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x + 3\\right) - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x + 3\\right) - 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6\\right)}{2 \\left(x \\left(x + 3\\right) - 6\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e179748bcb2ef80d", "problem_id": "938d8ded7a32ddb3", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$- 6 x - 6 y$ = $-48$\n$- 8 x$ = $-72$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-8$: $48 x + 48 y$ = $384$.\n3) Multiply the second equation by $-6$: $48 x$ = $432$.\n4) Subtract to eliminate x: $48 y$ = $-48$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $9$.\nFinal answer: $(x,y)=\\left(9, -1\\right)$", "final_answer": "(x,y)=\\left(9, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cfc635487e9bf16c", "problem_id": "938d8ded7a32ddb3", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Solve the linear system:\n$- 6 x - 6 y$ = $-48$\n$- 8 x$ = $-72$", "output": "$(x,y)=\\left(9, -1\\right)$", "final_answer": "(x,y)=\\left(9, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "de54fe8725729212", "problem_id": "7865cabfd2df606c", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_4=19$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=19 and k=4.\n4) Compute C(22,3) = 1540.\nFinal answer: $1540$", "final_answer": "1540", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ca86fec1304a07be", "problem_id": "7865cabfd2df606c", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_4=19$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=19 and k=4.\nCompute C(22,3) = 1540.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $1540$", "final_answer": "1540", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ea7094232ff988d0", "problem_id": "d67caedba7c32adc", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 14 red balls and 29 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(43,8).\n3) Favorable samples: choose k reds and n−k blues: C(14,2)·C(29,6).\n4) Compute totals: C(43,8) = 145008513, and favorable = 43226820.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{369460}{1239389}$", "final_answer": "\\frac{369460}{1239389}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e6a69b113eb76fec", "problem_id": "d67caedba7c32adc", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "An urn has 14 red balls and 29 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "$\\frac{369460}{1239389}$", "final_answer": "\\frac{369460}{1239389}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0b978c030e9f5d66", "problem_id": "87da493b9638923e", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the multiplicative inverse of 5 modulo 29.", "output": "1) We want x such that 5x ≡ 1 (mod 29).\n2) Use extended Euclid to find s,t with 5s + 29t = 1.\n3) One Bezout identity is: 5·(6) + 29·(-1) = 1.\n4) Reduce s modulo 29: x ≡ 6 (mod 29).\nFinal answer: $x\\equiv 6\\pmod{29}$", "final_answer": "x\\equiv 6\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7d3d61b6e712d0f8", "problem_id": "87da493b9638923e", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the multiplicative inverse of 5 modulo 29.", "output": "We want x such that 5x ≡ 1 (mod 29).\nUse extended Euclid to find s,t with 5s + 29t = 1.\nOne Bezout identity is: 5·(6) + 29·(-1) = 1.\nReduce s modulo 29: x ≡ 6 (mod 29).\nCheck: 5·6 mod 29 = 1.\nFinal answer: $x\\equiv 6\\pmod{29}$", "final_answer": "x\\equiv 6\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "65f60b52036527ce", "problem_id": "5baffeeac5a4af65", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=- x^{2} - 4$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-4) = 0.\n3) d/dx(- x^{2}) = - 2 x.\n4) Combine: f'(x) = - 2 x.\nFinal answer: $- 2\\cdot x$", "final_answer": "- 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1531702059ee99dd", "problem_id": "5baffeeac5a4af65", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{2} - 4$.", "output": "Differentiate term-by-term:\nd/dx(-4) = 0.\nd/dx(- x^{2}) = - 2 x.\nCombine: f'(x) = - 2 x.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 2\\cdot x$", "final_answer": "- 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8f8fc958a60d5e5a", "problem_id": "858987b15db34fd9", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Two friends split $330 in the ratio 3:2. How much does each receive?", "output": "1) Total ratio parts = 3+2 = 5.\n2) Each part = 330 ÷ 5 = 66.\n3) First share = 3 × 66 = 198.\n4) Second share = 2 × 66 = 132.\nFinal answer: $132$", "final_answer": "132", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b0e49b8e8433302e", "problem_id": "858987b15db34fd9", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $330 in the ratio 3:2. How much does each receive?", "output": "Total ratio parts = 3+2 = 5.\nEach part = 330 ÷ 5 = 66.\nFirst share = 3 × 66 = 198.\nSecond share = 2 × 66 = 132.\nCheck: 198+132=330.\nFinal answer: $132$", "final_answer": "132", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "359ce6a0df315345", "problem_id": "6ff1499f7dc4b349", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int \\left(2 x + 1\\right) \\cos{\\left(x \\right)} + \\left(3 x + 2\\right) e^{3 x}\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $\\left(2 x + 1\\right) \\cos{\\left(x \\right)} + \\left(3 x + 2\\right) e^{3 x}$.\n3) Compute an antiderivative: F(x) = $2 x \\sin{\\left(x \\right)} + \\frac{\\left(3 x + 1\\right) e^{3 x}}{3} + \\sin{\\left(x \\right)} + 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\frac{\\left(3\\cdot x + 1\\right)\\cdot e^{3\\cdot x}}{3} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\frac{\\left(3\\cdot x + 1\\right)\\cdot e^{3\\cdot x}}{3} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db664924376f0207", "problem_id": "6ff1499f7dc4b349", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int \\left(2 x + 1\\right) \\cos{\\left(x \\right)} + \\left(3 x + 2\\right) e^{3 x}\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $\\left(2 x + 1\\right) \\cos{\\left(x \\right)} + \\left(3 x + 2\\right) e^{3 x}$.\nCompute an antiderivative: F(x) = $2 x \\sin{\\left(x \\right)} + \\frac{\\left(3 x + 1\\right) e^{3 x}}{3} + \\sin{\\left(x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $C + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\frac{\\left(3\\cdot x + 1\\right)\\cdot e^{3\\cdot x}}{3} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\frac{\\left(3\\cdot x + 1\\right)\\cdot e^{3\\cdot x}}{3} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9b4eb5f75eed5f8a", "problem_id": "30d0ff6d486eaa28", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many binary strings of length 13 contain exactly 5 ones with no two ones adjacent?", "output": "Reserve 4 mandatory zeros between the 5 ones.\nThis leaves n−k+1 = 9 valid slots for the ones.\nChoose the slots: C(9,5) = 126.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $126$", "final_answer": "126", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d09c9c02909eba98", "problem_id": "b6247d562fb05d69", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} - 4 x + 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(2 x^{2}) = 4 x^{2}$.\n3) So $f'(x) = 4 x^{2} - 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(2 x^{2}) = 4 x$.\nTherefore the correct derivative is $f'(x) = 4 \\left(x - 1\\right)$.\nFinal answer: $4 \\left(x - 1\\right)$", "final_answer": "4 \\left(x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8e25e13e9fa705e0", "problem_id": "951bbe06394fe4bd", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - 3 x^{3} + x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{3} + x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{765 \\pi^{4}}{64} - \\frac{15 \\pi}{2} - 1 + \\frac{45 \\pi^{2}}{8} + \\frac{65 \\pi^{3}}{24}$.\n4) Simplify the result.\nFinal answer: $- \\frac{765\\cdot \\pi^{4}}{64} - \\frac{15\\cdot \\pi}{2} - 1 + \\frac{45\\cdot \\pi^{2}}{8} + \\frac{65\\cdot \\pi^{3}}{24}$", "final_answer": "- \\frac{765\\cdot \\pi^{4}}{64} - \\frac{15\\cdot \\pi}{2} - 1 + \\frac{45\\cdot \\pi^{2}}{8} + \\frac{65\\cdot \\pi^{3}}{24}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bb8608763135fdfe", "problem_id": "951bbe06394fe4bd", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - 3 x^{3} + x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{3} + x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{765 \\pi^{4}}{64} - \\frac{15 \\pi}{2} - 1 + \\frac{45 \\pi^{2}}{8} + \\frac{65 \\pi^{3}}{24}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{765\\cdot \\pi^{4}}{64} - \\frac{15\\cdot \\pi}{2} - 1 + \\frac{45\\cdot \\pi^{2}}{8} + \\frac{65\\cdot \\pi^{3}}{24}$", "final_answer": "- \\frac{765\\cdot \\pi^{4}}{64} - \\frac{15\\cdot \\pi}{2} - 1 + \\frac{45\\cdot \\pi^{2}}{8} + \\frac{65\\cdot \\pi^{3}}{24}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "454dcac272d657b6", "problem_id": "26e818e30c39332b", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "You want 29 liters of a 30% salt solution. You have a 20% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 29−x liters is 50% solution.\n2) Salt equation: 20x + 50(29−x) = 30·29.\n3) (20-50)x = (30-50)·29 ⇒ x = 58/3.\n4) Then 29−x = 29/3.\nFinal answer: $58/3 L of 20% and 29/3 L of 50%$", "final_answer": "58/3 L of 20% and 29/3 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "895bc8830e9b38a8", "problem_id": "26e818e30c39332b", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 29 liters of a 30% salt solution. You have a 20% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 29−x liters is 50% solution.\nSalt equation: 20x + 50(29−x) = 30·29.\n(20-50)x = (30-50)·29 ⇒ x = 58/3.\nThen 29−x = 29/3.\nCheck: 20·58/3 + 50·29/3 = 30·29.\nFinal answer: $58/3 L of 20% and 29/3 L of 50%$", "final_answer": "58/3 L of 20% and 29/3 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b48a704bb490b039", "problem_id": "76758c19bf1aea60", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 13 red balls and 29 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(42,11).\n3) Favorable samples: choose k reds and n−k blues: C(13,5)·C(29,6).\n4) Compute totals: C(42,11) = 4280561376, and favorable = 611350740.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{559845}{3919928}$", "final_answer": "\\frac{559845}{3919928}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "20f6e9d27d64a0b6", "problem_id": "76758c19bf1aea60", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 13 red balls and 29 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(42,11).\nFavorable samples: choose k reds and n−k blues: C(13,5)·C(29,6).\nCompute totals: C(42,11) = 4280561376, and favorable = 611350740.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{559845}{3919928}$.\nFinal answer: $\\frac{559845}{3919928}$", "final_answer": "\\frac{559845}{3919928}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "be1aba777368a151", "problem_id": "93c66c9e736c066d", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A cyclist rides at 32 miles per hour for 1/2 hours, then at 90 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 32 × 1/2 = 16 miles.\n2) Second leg distance = 90 × 1 = 90 miles.\n3) Total distance = 16 + 90 = 106 miles.\nFinal answer: $106$", "final_answer": "106", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2f78c30a223c5620", "problem_id": "89d6d184019b9d06", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(3 x^{2} + 1\\right) \\cos{\\left(3 x + 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\cos{\\left(3 x + 2 \\right)}) = - \\sin{\\left(3 x + 2 \\right)}$.\n3) So $f'(x) = - 9 x^{2} \\sin{\\left(3 x + 2 \\right)} + x e^{x} + 6 x \\cos{\\left(3 x + 2 \\right)} - 2 e^{x} - \\sin{\\left(3 x + 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\cos{\\left(3 x + 2 \\right)}) = - 3 \\sin{\\left(3 x + 2 \\right)}$.\nTherefore the correct derivative is $f'(x) = 6 x \\cos{\\left(3 x + 2 \\right)} + \\left(x - 3\\right) e^{x} - 3 \\left(3 x^{2} + 1\\right) \\sin{\\left(3 x + 2 \\right)} + e^{x}$.\nFinal answer: $6 x \\cos{\\left(3 x + 2 \\right)} + \\left(x - 3\\right) e^{x} - 3 \\left(3 x^{2} + 1\\right) \\sin{\\left(3 x + 2 \\right)} + e^{x}$", "final_answer": "6 x \\cos{\\left(3 x + 2 \\right)} + \\left(x - 3\\right) e^{x} - 3 \\left(3 x^{2} + 1\\right) \\sin{\\left(3 x + 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5df9067a0a388ced", "problem_id": "6a22ab8fa06f7c57", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Two friends split $336 in the ratio 1:5. How much does each receive?", "output": "1) Total ratio parts = 1+5 = 6.\n2) Each part = 336 ÷ 6 = 56.\n3) First share = 1 × 56 = 56.\n4) Second share = 5 × 56 = 280.\nFinal answer: $280$", "final_answer": "280", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "45f5e0ac83db0d46", "problem_id": "aaa146a38fe8ea5e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{2} + 2 x - 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(4 x^{2}) = 8 x^{2}$.\n3) So $f'(x) = 8 x^{2} + 2$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(4 x^{2}) = 8 x$.\nTherefore the correct derivative is $f'(x) = 2 \\left(4 x + 1\\right)$.\nFinal answer: $2 \\left(4 x + 1\\right)$", "final_answer": "2 \\left(4 x + 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "eb4a7912172a8731", "problem_id": "a4898b82e7b4d45a", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(- 2 x^{3} - 2 x + 3\\right) \\cos{\\left(2 x + 1 \\right)}$.", "output": "1) Let f(x) = $2 \\left(x + 2\\right) e^{x} + \\left(- 2 x^{3} - 2 x + 3\\right) \\cos{\\left(2 x + 1 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $2 \\left(\\left(x + 2\\right) e^{x} + \\left(- 3 x^{2} - 1\\right) \\cos{\\left(2 x + 1 \\right)} - \\left(- 2 x^{3} - 2 x + 3\\right) \\sin{\\left(2 x + 1 \\right)} + e^{x}\\right)$.\nFinal answer: $2\\cdot \\left(x + 2\\right)\\cdot e^{x} + 2\\cdot \\left(- 3\\cdot x^{2} - 1\\right)\\cdot \\cos{\\left(2\\cdot x + 1 \\right)} - 2\\cdot \\left(- 2\\cdot x^{3} - 2\\cdot x + 3\\right)\\cdot \\sin{\\left(2\\cdot x + 1 \\right)} + 2\\cdot e^{x}$", "final_answer": "2\\cdot \\left(x + 2\\right)\\cdot e^{x} + 2\\cdot \\left(- 3\\cdot x^{2} - 1\\right)\\cdot \\cos{\\left(2\\cdot x + 1 \\right)} - 2\\cdot \\left(- 2\\cdot x^{3} - 2\\cdot x + 3\\right)\\cdot \\sin{\\left(2\\cdot x + 1 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2030ff3a11d0a948", "problem_id": "a4898b82e7b4d45a", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(- 2 x^{3} - 2 x + 3\\right) \\cos{\\left(2 x + 1 \\right)}$.", "output": "Let f(x) = $2 \\left(x + 2\\right) e^{x} + \\left(- 2 x^{3} - 2 x + 3\\right) \\cos{\\left(2 x + 1 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $2 \\left(\\left(x + 2\\right) e^{x} + \\left(- 3 x^{2} - 1\\right) \\cos{\\left(2 x + 1 \\right)} - \\left(- 2 x^{3} - 2 x + 3\\right) \\sin{\\left(2 x + 1 \\right)} + e^{x}\\right)$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $2\\cdot \\left(x + 2\\right)\\cdot e^{x} + 2\\cdot \\left(- 3\\cdot x^{2} - 1\\right)\\cdot \\cos{\\left(2\\cdot x + 1 \\right)} - 2\\cdot \\left(- 2\\cdot x^{3} - 2\\cdot x + 3\\right)\\cdot \\sin{\\left(2\\cdot x + 1 \\right)} + 2\\cdot e^{x}$", "final_answer": "2\\cdot \\left(x + 2\\right)\\cdot e^{x} + 2\\cdot \\left(- 3\\cdot x^{2} - 1\\right)\\cdot \\cos{\\left(2\\cdot x + 1 \\right)} - 2\\cdot \\left(- 2\\cdot x^{3} - 2\\cdot x + 3\\right)\\cdot \\sin{\\left(2\\cdot x + 1 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c3a6348abe4737f5", "problem_id": "1e9e175b9675bcea", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Worker A finishes a job in 15 hours and Worker B finishes it in 16 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/15 job/hour. B's rate = 1/16 job/hour.\n2) Combined rate = 1/15 + 1/16 = 31/240 job/hour.\n3) Time = 1 ÷ (combined rate) = 240/31 hours.\nFinal answer: $\\frac{240}{31}$", "final_answer": "\\frac{240}{31}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2fbe221f62d28340", "problem_id": "1e9e175b9675bcea", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Worker A finishes a job in 15 hours and Worker B finishes it in 16 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/15 job/hour. B's rate = 1/16 job/hour.\nCombined rate = 1/15 + 1/16 = 31/240 job/hour.\nTime = 1 ÷ (combined rate) = 240/31 hours.\nFinal answer: $\\frac{240}{31}$", "final_answer": "\\frac{240}{31}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f749352eeee05c4c", "problem_id": "5d448eba11a66396", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $92. It is discounted by 25%. You buy 7 backpacks. What is the total cost?", "output": "Discounted price = 92 × (1 − 25/100) = 92 × 75/100 = 69.\nSubtotal = 7 × 69 = 483.\nCheck: recomputing the formula yields 483.\nFinal answer: $483$", "final_answer": "483", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f14d532693f72a8b", "problem_id": "5d448eba11a66396", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A store sells a backpack for $92. It is discounted by 25%. You buy 7 backpacks. What is the total cost?", "output": "1) Discounted price = 92 × (1 − 25/100) = 92 × 75/100 = 69.\n2) Subtotal = 7 × 69 = 483.\nFinal answer: $483$", "final_answer": "483", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5f3ab432cf63a933", "problem_id": "b766449c464082e9", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$- x + 2 y$ = $-6$\n$- 3 x + 3 y$ = $-12$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $3 x - 6 y$ = $18$.\n3) Multiply the second equation by $-1$: $3 x - 3 y$ = $12$.\n4) Subtract to eliminate x: $- 3 y$ = $6$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $2$.\nFinal answer: $(x,y)=\\left(2, -2\\right)$", "final_answer": "(x,y)=\\left(2, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9137fd24a4394889", "problem_id": "b766449c464082e9", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the linear system:\n$- x + 2 y$ = $-6$\n$- 3 x + 3 y$ = $-12$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $3 x - 6 y$ = $18$.\nMultiply the second equation by $-1$: $3 x - 3 y$ = $12$.\nFinal answer: $(x,y)=\\left(2, -2\\right)$", "final_answer": "(x,y)=\\left(2, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7f524c98cf7f79bf", "problem_id": "6a9a7b43150449f1", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Worker A finishes a job in 8 hours and Worker B finishes it in 16 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/8 job/hour. B's rate = 1/16 job/hour.\n2) Combined rate = 1/8 + 1/16 = 3/16 job/hour.\n3) Time = 1 ÷ (combined rate) = 16/3 hours.\nFinal answer: $\\frac{16}{3}$", "final_answer": "\\frac{16}{3}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dcaf40c21a2c8572", "problem_id": "6a9a7b43150449f1", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Worker A finishes a job in 8 hours and Worker B finishes it in 16 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/8 job/hour. B's rate = 1/16 job/hour.\nCombined rate = 1/8 + 1/16 = 3/16 job/hour.\nTime = 1 ÷ (combined rate) = 16/3 hours.\nCheck: (combined rate)×(time) = 3/16×16/3 = 1.\nFinal answer: $\\frac{16}{3}$", "final_answer": "\\frac{16}{3}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "47fe0873a56a30e6", "problem_id": "c16dc9d9015fb31e", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$2 x - 2 y$ = $-14$\n$- 2 x + 4 y$ = $22$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $- 4 x + 4 y$ = $28$.\n3) Multiply the second equation by $2$: $- 4 x + 8 y$ = $44$.\n4) Subtract to eliminate x: $- 4 y$ = $-16$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, 4\\right)$", "final_answer": "(x,y)=\\left(-3, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d1f54129bb62ff08", "problem_id": "c16dc9d9015fb31e", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$2 x - 2 y$ = $-14$\n$- 2 x + 4 y$ = $22$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $- 4 x + 4 y$ = $28$.\nMultiply the second equation by $2$: $- 4 x + 8 y$ = $44$.\nSubtract to eliminate x: $- 4 y$ = $-16$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $-3$.\nCheck: plug (x,y)=(-3,4) into both equations.\nFinal answer: $(x,y)=\\left(-3, 4\\right)$", "final_answer": "(x,y)=\\left(-3, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fcb66b7aabb1e9a9", "problem_id": "a04878faf758f286", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - x^{2} + 2 x - 4\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- x^{2} + 2 x - 4$.\n3) Compute an antiderivative: F(x) = $- \\frac{x^{3}}{3} + x^{2} - 4 x$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{x^{3}}{3} + x^{2} - 4\\cdot x$", "final_answer": "C - \\frac{x^{3}}{3} + x^{2} - 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "85abd8dd0c4ddb44", "problem_id": "a04878faf758f286", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - x^{2} + 2 x - 4\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- x^{2} + 2 x - 4$.\nCompute an antiderivative: F(x) = $- \\frac{x^{3}}{3} + x^{2} - 4 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - \\frac{x^{3}}{3} + x^{2} - 4\\cdot x$", "final_answer": "C - \\frac{x^{3}}{3} + x^{2} - 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4cb360d5fc73e2da", "problem_id": "313ecf397b04a1b6", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - 4 x - 4$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $- 9 x^{2} - 4$.\n2) Slope at x=-2: m = f'(-2) = $-40$.\n3) Point on curve: (-2, f(-2)) = (-2, $28$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 28 = -40(x − -2).\n6) Optionally expand: y = $- 40 x - 52$.\nFinal answer: $y = - 40\\cdot x - 52$", "final_answer": "y = - 40\\cdot x - 52", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aa199202eb3ce8c9", "problem_id": "313ecf397b04a1b6", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - 4 x - 4$ at $x=-2$.", "output": "Differentiate: f'(x) = $- 9 x^{2} - 4$.\nSlope at x=-2: m = f'(-2) = $-40$.\nPoint on curve: (-2, f(-2)) = (-2, $28$).\nFinal answer: $y = - 40\\cdot x - 52$", "final_answer": "y = - 40\\cdot x - 52", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fadd55ea20070406", "problem_id": "c4da89996467b06f", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "You want 49 liters of a 35% salt solution. You have a 20% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 49−x liters is 40% solution.\n2) Salt equation: 20x + 40(49−x) = 35·49.\n3) (20-40)x = (35-40)·49 ⇒ x = 49/4.\n4) Then 49−x = 147/4.\nFinal answer: $49/4 L of 20% and 147/4 L of 40%$", "final_answer": "49/4 L of 20% and 147/4 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3ee06cde5c3f6d8b", "problem_id": "c4da89996467b06f", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "You want 49 liters of a 35% salt solution. You have a 20% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 49−x liters is 40% solution.\nSalt equation: 20x + 40(49−x) = 35·49.\n(20-40)x = (35-40)·49 ⇒ x = 49/4.\nFinal answer: $49/4 L of 20% and 147/4 L of 40%$", "final_answer": "49/4 L of 20% and 147/4 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3458cb4cb7071ec2", "problem_id": "4cd1dba140d595f2", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 13 x + 40$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} - 13 x + 40$ = $\\left(x - 8\\right) \\left(x - 5\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $8$.\nFrom the second factor: x = $5$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{8, 5\\}$", "final_answer": "x \\in \\{8, 5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a0d86dcddeb6bda0", "problem_id": "4cd1dba140d595f2", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 13 x + 40$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} - 13 x + 40$ = $\\left(x - 8\\right) \\left(x - 5\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $8$.\n5) From the second factor: x = $5$.\nFinal answer: $x \\in \\{8, 5\\}$", "final_answer": "x \\in \\{8, 5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6405fd22371dfa10", "problem_id": "bdbed06a35be6153", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "An urn has 11 red balls and 12 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(23,5).\n3) Favorable samples: choose k reds and n−k blues: C(11,2)·C(12,3).\n4) Compute totals: C(23,5) = 33649, and favorable = 12100.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1100}{3059}$", "final_answer": "\\frac{1100}{3059}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d352ea64c30147a9", "problem_id": "bdbed06a35be6153", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 11 red balls and 12 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(23,5).\nFavorable samples: choose k reds and n−k blues: C(11,2)·C(12,3).\nCompute totals: C(23,5) = 33649, and favorable = 12100.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1100}{3059}$.\nFinal answer: $\\frac{1100}{3059}$", "final_answer": "\\frac{1100}{3059}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ae8e05adcf6358c", "problem_id": "5f78dadf30fa1e65", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$2 x - 3 y$ = $17$\n$- 3 x - 3 y$ = $27$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $- 6 x + 9 y$ = $-51$.\n3) Multiply the second equation by $2$: $- 6 x - 6 y$ = $54$.\n4) Subtract to eliminate x: $15 y$ = $-105$.\n5) Solve for y: y = $-7$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, -7\\right)$", "final_answer": "(x,y)=\\left(-2, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "55a3e875c0c75ae0", "problem_id": "5f78dadf30fa1e65", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$2 x - 3 y$ = $17$\n$- 3 x - 3 y$ = $27$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $- 6 x + 9 y$ = $-51$.\nMultiply the second equation by $2$: $- 6 x - 6 y$ = $54$.\nSubtract to eliminate x: $15 y$ = $-105$.\nSolve for y: y = $-7$.\nSubstitute back to find x: x = $-2$.\nCheck: plug (x,y)=(-2,-7) into both equations.\nFinal answer: $(x,y)=\\left(-2, -7\\right)$", "final_answer": "(x,y)=\\left(-2, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "05dfa2108ac02eb5", "problem_id": "843349b4a9a28fa9", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$- 4 x$ = $-16$\n$- 3 x - 3 y$ = $0$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $12 x$ = $48$.\n3) Multiply the second equation by $-4$: $12 x + 12 y$ = $0$.\n4) Subtract to eliminate x: $- 12 y$ = $48$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, -4\\right)$", "final_answer": "(x,y)=\\left(4, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9faf0ae9861d5280", "problem_id": "843349b4a9a28fa9", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$- 4 x$ = $-16$\n$- 3 x - 3 y$ = $0$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $12 x$ = $48$.\nMultiply the second equation by $-4$: $12 x + 12 y$ = $0$.\nSubtract to eliminate x: $- 12 y$ = $48$.\nSolve for y: y = $-4$.\nSubstitute back to find x: x = $4$.\nCheck: plug (x,y)=(4,-4) into both equations.\nFinal answer: $(x,y)=\\left(4, -4\\right)$", "final_answer": "(x,y)=\\left(4, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2ba13fbd26654e6e", "problem_id": "e43a6cea78b1b7fa", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A disease has prevalence $\\frac{3}{100}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{77}{1000}$.\n3) Compute P(D|+) = $\\frac{57}{154}$.\nFinal answer: $\\frac{57}{154}$", "final_answer": "\\frac{57}{154}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7eb4c2ed508ce50f", "problem_id": "e43a6cea78b1b7fa", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A disease has prevalence $\\frac{3}{100}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{77}{1000}$.\nCompute P(D|+) = $\\frac{57}{154}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{57}{154}$", "final_answer": "\\frac{57}{154}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "367e110b4c33847d", "problem_id": "6ba1913ad8ec628a", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A store sells a backpack for $394. It is discounted by 35%. You buy 7 backpacks. What is the total cost?", "output": "Discounted price = 394 × (1 − 35/100) = 394 × 65/100 = 2561/10.\nSubtotal = 7 × 2561/10 = 17927/10.\nCheck: recomputing the formula yields 17927/10.\nFinal answer: $\\frac{17927}{10}$", "final_answer": "\\frac{17927}{10}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "24dc04cf8bd3d004", "problem_id": "6ba1913ad8ec628a", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A store sells a backpack for $394. It is discounted by 35%. You buy 7 backpacks. What is the total cost?", "output": "1) Discounted price = 394 × (1 − 35/100) = 394 × 65/100 = 2561/10.\n2) Subtotal = 7 × 2561/10 = 17927/10.\nFinal answer: $\\frac{17927}{10}$", "final_answer": "\\frac{17927}{10}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2b82e44ca3874759", "problem_id": "5861ff0d0cf37320", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} - 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $- 9 \\pi^{3} - 9 \\pi - 4 + 3 \\pi^{2}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- 9\\cdot \\pi^{3} - 9\\cdot \\pi - 4 + 3\\cdot \\pi^{2}$", "final_answer": "- 9\\cdot \\pi^{3} - 9\\cdot \\pi - 4 + 3\\cdot \\pi^{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "16e607a0af11c6ae", "problem_id": "5861ff0d0cf37320", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} - 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $- 9 \\pi^{3} - 9 \\pi - 4 + 3 \\pi^{2}$.\n4) Simplify the result.\nFinal answer: $- 9\\cdot \\pi^{3} - 9\\cdot \\pi - 4 + 3\\cdot \\pi^{2}$", "final_answer": "- 9\\cdot \\pi^{3} - 9\\cdot \\pi - 4 + 3\\cdot \\pi^{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6c32777549bca4a8", "problem_id": "c2ddf89cc4f955d5", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $160. It is discounted by 35%. You buy 4 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 160 × (1 − 35/100) = 160 × 65/100 = 104.\nSubtotal = 4 × 104 = 416.\nTotal with tax = 416 × 105/100 = 2184/5.\nCheck: recomputing the formula yields 2184/5.\nFinal answer: $\\frac{2184}{5}$", "final_answer": "\\frac{2184}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "00b727e922f3176e", "problem_id": "c2ddf89cc4f955d5", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A store sells a backpack for $160. It is discounted by 35%. You buy 4 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 160 × (1 − 35/100) = 160 × 65/100 = 104.\n2) Subtotal = 4 × 104 = 416.\n3) Total with tax = 416 × 105/100 = 2184/5.\nFinal answer: $\\frac{2184}{5}$", "final_answer": "\\frac{2184}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9333b8683e8ce845", "problem_id": "e9da09f6c9bd9d8c", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{4}{7}$. If it is flipped 6 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=6, k=4, p=$\\frac{4}{7}$ and simplify.\n4) Here C(6,4) = 15.\nFinal answer: $\\frac{34560}{117649}$", "final_answer": "\\frac{34560}{117649}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "245b347e0da2e04a", "problem_id": "e9da09f6c9bd9d8c", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{7}$. If it is flipped 6 times, what is the probability of exactly 4 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=6, k=4, p=$\\frac{4}{7}$ and simplify.\nHere C(6,4) = 15.\nCheck: expression simplifies to $\\frac{34560}{117649}$.\nFinal answer: $\\frac{34560}{117649}$", "final_answer": "\\frac{34560}{117649}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0f4cfac02a623f69", "problem_id": "41dec57a289dba4b", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$4 x + 2 y$ = $26$\n$4 x + 3 y$ = $31$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $16 x + 8 y$ = $104$.\nMultiply the second equation by $4$: $16 x + 12 y$ = $124$.\nSubtract to eliminate x: $- 4 y$ = $-20$.\nSolve for y: y = $5$.\nSubstitute back to find x: x = $4$.\nCheck: plug (x,y)=(4,5) into both equations.\nFinal answer: $(x,y)=\\left(4, 5\\right)$", "final_answer": "(x,y)=\\left(4, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8557503c7436831f", "problem_id": "41dec57a289dba4b", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the linear system:\n$4 x + 2 y$ = $26$\n$4 x + 3 y$ = $31$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $16 x + 8 y$ = $104$.\n3) Multiply the second equation by $4$: $16 x + 12 y$ = $124$.\n4) Subtract to eliminate x: $- 4 y$ = $-20$.\n5) Solve for y: y = $5$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, 5\\right)$", "final_answer": "(x,y)=\\left(4, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "05fce0696497861f", "problem_id": "af77447aa117d950", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the congruences: x≡0 (mod 9), x≡4 (mod 5). Give the solution modulo 45.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 45.\n3) Compute the CRT solution to get x ≡ 9 (mod 45).\nFinal answer: $x\\equiv 9\\pmod{45}$", "final_answer": "x\\equiv 9\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "450d5d41c9492660", "problem_id": "30f0a13b5f4c0c0d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- 2 x^{2} + 2 x + 3\\right) \\cos{\\left(x^{2} - 3 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\n3) So $f'(x) = 4 x^{3} \\sin{\\left(x^{2} - 3 x \\right)} - 10 x^{2} \\sin{\\left(x^{2} - 3 x \\right)} + 2 x e^{x} - 4 x \\cos{\\left(x^{2} - 3 x \\right)} + 4 e^{x} + 9 \\sin{\\left(x^{2} - 3 x \\right)} + 2 \\cos{\\left(x^{2} - 3 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(1 - 2 x\\right) \\cos{\\left(x^{2} - 3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - \\left(2 x - 3\\right) \\left(2 x \\left(1 - x\\right) + 3\\right) \\sin{\\left(x^{2} - 3 x \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(1 - 2 x\\right) \\cos{\\left(x^{2} - 3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - \\left(2 x - 3\\right) \\left(2 x \\left(1 - x\\right) + 3\\right) \\sin{\\left(x^{2} - 3 x \\right)} + 2 e^{x}$", "final_answer": "2 \\left(1 - 2 x\\right) \\cos{\\left(x^{2} - 3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - \\left(2 x - 3\\right) \\left(2 x \\left(1 - x\\right) + 3\\right) \\sin{\\left(x^{2} - 3 x \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "023fa46db5bc1081", "problem_id": "bd4ddd233b870ecb", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{37 x}{20} + \\frac{2}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{37}{20}$, c=$\\frac{2}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{8}{5}, \\frac{1}{4}\\}$", "final_answer": "x \\in \\{\\frac{8}{5}, \\frac{1}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a7cc62cdf4f7b570", "problem_id": "bd4ddd233b870ecb", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{37 x}{20} + \\frac{2}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{37}{20}$, c=$\\frac{2}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{729}{400}$, so √Δ = $\\frac{27}{20}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{8}{5}$.\nx2 = $\\frac{1}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{8}{5}, \\frac{1}{4}\\}$", "final_answer": "x \\in \\{\\frac{8}{5}, \\frac{1}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "223bcfe0c0037b85", "problem_id": "d68b28234432cf75", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Two friends split $315 in the ratio 4:3. How much does each receive?", "output": "Total ratio parts = 4+3 = 7.\nEach part = 315 ÷ 7 = 45.\nFirst share = 4 × 45 = 180.\nSecond share = 3 × 45 = 135.\nCheck: 180+135=315.\nFinal answer: $135$", "final_answer": "135", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b283c69af09081a3", "problem_id": "d68b28234432cf75", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Two friends split $315 in the ratio 4:3. How much does each receive?", "output": "1) Total ratio parts = 4+3 = 7.\n2) Each part = 315 ÷ 7 = 45.\n3) First share = 4 × 45 = 180.\n4) Second share = 3 × 45 = 135.\nFinal answer: $135$", "final_answer": "135", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b30fa53a3b26f893", "problem_id": "a4b290edabf15b2f", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=26$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=26 and k=6.\nCompute C(31,5) = 169911.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $169911$", "final_answer": "169911", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dcae000b473794e0", "problem_id": "a4b290edabf15b2f", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=26$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=26 and k=6.\n4) Compute C(31,5) = 169911.\nFinal answer: $169911$", "final_answer": "169911", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "307e48ce54900554", "problem_id": "f63f7a44d91b8951", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 2 x^{2} + 3 x + \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 6}$.", "output": "1) Let f(x) = $\\frac{- 2 x^{2} + 3 x + \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 6}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{4 x^{2} - 6 x + \\left(x - 6\\right) \\left(- 8 x + \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 2 \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x - 6\\right)^{2}}$.\nFinal answer: $\\frac{4\\cdot x^{2} - 6\\cdot x + \\left(x - 6\\right)\\cdot \\left(- 8\\cdot x + \\left(x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6\\right) - \\left(2\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)}}{2\\cdot \\left(x - 6\\right)^{2}}$", "final_answer": "\\frac{4\\cdot x^{2} - 6\\cdot x + \\left(x - 6\\right)\\cdot \\left(- 8\\cdot x + \\left(x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6\\right) - \\left(2\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)}}{2\\cdot \\left(x - 6\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a28c5018954156d7", "problem_id": "f63f7a44d91b8951", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 2 x^{2} + 3 x + \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 6}$.", "output": "Let f(x) = $\\frac{- 2 x^{2} + 3 x + \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 6}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{4 x^{2} - 6 x + \\left(x - 6\\right) \\left(- 8 x + \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 2 \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x - 6\\right)^{2}}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\frac{4\\cdot x^{2} - 6\\cdot x + \\left(x - 6\\right)\\cdot \\left(- 8\\cdot x + \\left(x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6\\right) - \\left(2\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)}}{2\\cdot \\left(x - 6\\right)^{2}}$", "final_answer": "\\frac{4\\cdot x^{2} - 6\\cdot x + \\left(x - 6\\right)\\cdot \\left(- 8\\cdot x + \\left(x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6\\right) - \\left(2\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)}}{2\\cdot \\left(x - 6\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6b0ccdf76a661ad5", "problem_id": "5377178775cdd578", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{2} - 4 x$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 4 x) = - 4 x$.\n3) So $f'(x) = - 6 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 4 x) = -4$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- x - 2\\right)$.\nFinal answer: $2 \\left(- x - 2\\right)$", "final_answer": "2 \\left(- x - 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6fd4b77c26f8f07a", "problem_id": "3fe9dff49680ce4d", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 1$.\n3) Compute an antiderivative: F(x) = $- x^{3} + x^{2} + x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C - x^{3} + x^{2} + x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - x^{3} + x^{2} + x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "58cef68d91db48ab", "problem_id": "99292125e67e7608", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A disease has prevalence $\\frac{1}{25}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{167}{2500}$.\n3) Compute P(D|+) = $\\frac{95}{167}$.\nFinal answer: $\\frac{95}{167}$", "final_answer": "\\frac{95}{167}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ce725a041c18eb8", "problem_id": "22a3089edc84880c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x - 3$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 2 x - 3.\n3) Then the limit is -7.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -5.\nFinal answer: $-5$", "final_answer": "-5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "12f73cad75acbeaf", "problem_id": "3bc14bd823a47818", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the congruences: x≡5 (mod 11), x≡2 (mod 10). Give the solution modulo 110.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 110.\n3) Compute the CRT solution to get x ≡ 82 (mod 110).\nFinal answer: $x\\equiv 82\\pmod{110}$", "final_answer": "x\\equiv 82\\pmod{110}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "20b93b758a41ce21", "problem_id": "8b1a348fa710b7f2", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} - 2 x + \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{2 x - 5}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 5}) = \\frac{2 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 5}$.\n3) So $f'(x) = \\frac{4 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{2} - 40 x + 50} + \\frac{8 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{2} - 40 x + 50} - \\frac{4 x^{2}}{8 x^{2} - 40 x + 50} - \\frac{20 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{2} - 40 x + 50} - \\frac{40 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{2} - 40 x + 50} + \\frac{20 x}{8 x^{2} - 40 x + 50} + \\frac{4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{4 x^{2} - 20 x + 25} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 5} - \\frac{2 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 5} + \\frac{2 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 5} + \\frac{25 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{2} - 40 x + 50} + \\frac{50 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{2} - 40 x + 50} + \\frac{12}{8 x^{2} - 40 x + 50} - \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 5}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 5}) = \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 5} + \\frac{2 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 5} - \\frac{4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x - 5\\right)^{2}} + \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 5}$.\nTherefore the correct derivative is $f'(x) = \\frac{4 x^{2} + 8 x + \\left(2 x - 5\\right) \\left(- 4 x + \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4\\right) - 4 \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8}{2 \\left(2 x - 5\\right)^{2}}$.\nFinal answer: $\\frac{4 x^{2} + 8 x + \\left(2 x - 5\\right) \\left(- 4 x + \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4\\right) - 4 \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8}{2 \\left(2 x - 5\\right)^{2}}$", "final_answer": "\\frac{4 x^{2} + 8 x + \\left(2 x - 5\\right) \\left(- 4 x + \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4\\right) - 4 \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8}{2 \\left(2 x - 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "03fb8e904e95a2ef", "problem_id": "8652fa8cc8e82d40", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int \\left(x + 1\\right) e^{x} + \\left(2 x + 1\\right) \\cos{\\left(x \\right)}\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $\\left(x + 1\\right) e^{x} + \\left(2 x + 1\\right) \\cos{\\left(x \\right)}$.\nCompute an antiderivative: F(x) = $x e^{x} + 2 x \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + x\\cdot e^{x} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + x\\cdot e^{x} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b44feeaa99232902", "problem_id": "8652fa8cc8e82d40", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int \\left(x + 1\\right) e^{x} + \\left(2 x + 1\\right) \\cos{\\left(x \\right)}\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $\\left(x + 1\\right) e^{x} + \\left(2 x + 1\\right) \\cos{\\left(x \\right)}$.\nCompute an antiderivative: F(x) = $x e^{x} + 2 x \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $C + x\\cdot e^{x} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + x\\cdot e^{x} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "935362752371286c", "problem_id": "e07c8f3c1d5f8a5b", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - 3 x^{3} - 12 x^{2} - 38 x - 113$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 3 x^{4} - 3 x^{3} - 2 x^{2} + x - 1$ and a = 3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 12 x^{3} - 9 x^{2} - 4 x + 1$.\nEvaluate at a: p'(3) = $-416$.\nCheck: the identity gives $-416$.\nFinal answer: $-416$", "final_answer": "-416", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f5be1ee6c1180b49", "problem_id": "e07c8f3c1d5f8a5b", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - 3 x^{3} - 12 x^{2} - 38 x - 113$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 3 x^{4} - 3 x^{3} - 2 x^{2} + x - 1$ and a = 3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-416$", "final_answer": "-416", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "545b84843e2d3782", "problem_id": "1cf9a02b1c7bca0a", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 15 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=15, k=3, p=$\\frac{2}{7}$ and simplify.\n4) Here C(15,3) = 455.\nFinal answer: $\\frac{126953125000}{678223072849}$", "final_answer": "\\frac{126953125000}{678223072849}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6423704647d67c29", "problem_id": "1cf9a02b1c7bca0a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 15 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=15, k=3, p=$\\frac{2}{7}$ and simplify.\nHere C(15,3) = 455.\nCheck: expression simplifies to $\\frac{126953125000}{678223072849}$.\nFinal answer: $\\frac{126953125000}{678223072849}$", "final_answer": "\\frac{126953125000}{678223072849}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0f06a7a89f204589", "problem_id": "7bd8d1f9f5440d40", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 28 x + 80$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} + 28 x + 80$ = $2 \\left(x + 4\\right) \\left(x + 10\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $-4$.\nFrom the second factor: x = $-10$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-4, -10\\}$", "final_answer": "x \\in \\{-4, -10\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a992080ac429cd7", "problem_id": "7bd8d1f9f5440d40", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 28 x + 80$ = 0.", "output": "$x \\in \\{-4, -10\\}$", "final_answer": "x \\in \\{-4, -10\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ce2429e382d2afb0", "problem_id": "5b278782940c7238", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A store sells a backpack for $840. It is discounted by 20%. You buy 7 backpacks. What is the total cost?", "output": "1) Discounted price = 840 × (1 − 20/100) = 840 × 80/100 = 672.\n2) Subtotal = 7 × 672 = 4704.\nFinal answer: $4704$", "final_answer": "4704", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7a451beee4003483", "problem_id": "5b278782940c7238", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A store sells a backpack for $840. It is discounted by 20%. You buy 7 backpacks. What is the total cost?", "output": "Discounted price = 840 × (1 − 20/100) = 840 × 80/100 = 672.\nSubtotal = 7 × 672 = 4704.\nCheck: recomputing the formula yields 4704.\nFinal answer: $4704$", "final_answer": "4704", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e85e1a1504ae56b9", "problem_id": "18afd1cb1d1dfedf", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Two friends split $1515 in the ratio 7:8. How much does each receive?", "output": "1) Total ratio parts = 7+8 = 15.\n2) Each part = 1515 ÷ 15 = 101.\n3) First share = 7 × 101 = 707.\n4) Second share = 8 × 101 = 808.\nFinal answer: $808$", "final_answer": "808", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "81c3981712ff3d0c", "problem_id": "18afd1cb1d1dfedf", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Two friends split $1515 in the ratio 7:8. How much does each receive?", "output": "Total ratio parts = 7+8 = 15.\nEach part = 1515 ÷ 15 = 101.\nFirst share = 7 × 101 = 707.\nSecond share = 8 × 101 = 808.\nCheck: 707+808=1515.\nFinal answer: $808$", "final_answer": "808", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "961fe8ed8cfe4cca", "problem_id": "e8ace00224efba55", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Two friends split $1256 in the ratio 4:4. How much does each receive?", "output": "1) Total ratio parts = 4+4 = 8.\n2) Each part = 1256 ÷ 8 = 157.\n3) First share = 4 × 157 = 628.\n4) Second share = 4 × 157 = 628.\nFinal answer: $628$", "final_answer": "628", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "41be58c6553ac222", "problem_id": "e8ace00224efba55", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Two friends split $1256 in the ratio 4:4. How much does each receive?", "output": "$628$", "final_answer": "628", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e5c0275dbc7219fa", "problem_id": "b26a3efd62ffc603", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the linear system:\n$3 x + 3 y$ = $0$\n$- 3 x - y$ = $-6$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $- 9 x - 9 y$ = $0$.\n3) Multiply the second equation by $3$: $- 9 x - 3 y$ = $-18$.\n4) Subtract to eliminate x: $- 6 y$ = $18$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, -3\\right)$", "final_answer": "(x,y)=\\left(3, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8bac87fe537c29b7", "problem_id": "b26a3efd62ffc603", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$3 x + 3 y$ = $0$\n$- 3 x - y$ = $-6$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $- 9 x - 9 y$ = $0$.\nMultiply the second equation by $3$: $- 9 x - 3 y$ = $-18$.\nSubtract to eliminate x: $- 6 y$ = $18$.\nSolve for y: y = $-3$.\nSubstitute back to find x: x = $3$.\nCheck: plug (x,y)=(3,-3) into both equations.\nFinal answer: $(x,y)=\\left(3, -3\\right)$", "final_answer": "(x,y)=\\left(3, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "48fc398c20107bfc", "problem_id": "26848d99ad3514f1", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$4 x + 7 y$ = $-45$\n$6 x - 4 y$ = $34$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $6$: $24 x + 42 y$ = $-270$.\n3) Multiply the second equation by $4$: $24 x - 16 y$ = $136$.\n4) Subtract to eliminate x: $58 y$ = $-406$.\n5) Solve for y: y = $-7$.\n6) Substitute back to find x: x = $1$.\nFinal answer: $(x,y)=\\left(1, -7\\right)$", "final_answer": "(x,y)=\\left(1, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "25e6b6b73bda27db", "problem_id": "26848d99ad3514f1", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the linear system:\n$4 x + 7 y$ = $-45$\n$6 x - 4 y$ = $34$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $6$: $24 x + 42 y$ = $-270$.\nMultiply the second equation by $4$: $24 x - 16 y$ = $136$.\nFinal answer: $(x,y)=\\left(1, -7\\right)$", "final_answer": "(x,y)=\\left(1, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ef0192bb681a076f", "problem_id": "0e94698edb22bdb2", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 2 x - 7$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to x - 7.\n3) Then the limit is -10.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -13.\nFinal answer: $-13$", "final_answer": "-13", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "035129af20632e0a", "problem_id": "cac773660b9c899c", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A cyclist rides at 89 miles per hour for 1 hours, then at 76 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 89 × 1 = 89 miles.\n2) Second leg distance = 76 × 3 = 228 miles.\n3) Total distance = 89 + 228 = 317 miles.\nFinal answer: $317$", "final_answer": "317", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9a162b1321699da0", "problem_id": "f99e5f6dd02096d5", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Worker A finishes a job in 16 hours and Worker B finishes it in 9 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/16 job/hour. B's rate = 1/9 job/hour.\n2) Combined rate = 1/16 + 1/9 = 25/144 job/hour.\n3) Time = 1 ÷ (combined rate) = 144/25 hours.\nFinal answer: $\\frac{144}{25}$", "final_answer": "\\frac{144}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bbeb36ba84cbce74", "problem_id": "70f3d1d7a3038a3d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\left(x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)}\\right) - \\sin{\\left(x \\right)}) = \\left(x^{2} + x - 1\\right) \\cos{\\left(x \\right)}$.\n3) So $f'(x) = x^{2} \\cos{\\left(x \\right)} + x \\cos{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\left(x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)}\\right) - \\sin{\\left(x \\right)}) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - \\cos{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.\nFinal answer: $x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$", "final_answer": "x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "919dad5717e9b2d6", "problem_id": "a8b7f4ccc19b327f", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 2 x^{2} + 3 x - 12$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 2 x^{3} - 3 x^{2} - 3 x - 2$ and a = -3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 6 x^{2} - 6 x - 3$.\nEvaluate at a: p'(-3) = $-39$.\nCheck: the identity gives $-39$.\nFinal answer: $-39$", "final_answer": "-39", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "57e368b2bd4719ae", "problem_id": "a8b7f4ccc19b327f", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 2 x^{2} + 3 x - 12$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 2 x^{3} - 3 x^{2} - 3 x - 2$ and a = -3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-39$", "final_answer": "-39", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1768b01510229c34", "problem_id": "691c0a592e4b8d7d", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 31 modulo 37.", "output": "We want x such that 31x ≡ 1 (mod 37).\nUse extended Euclid to find s,t with 31s + 37t = 1.\nOne Bezout identity is: 31·(6) + 37·(-5) = 1.\nReduce s modulo 37: x ≡ 6 (mod 37).\nCheck: 31·6 mod 37 = 1.\nFinal answer: $x\\equiv 6\\pmod{37}$", "final_answer": "x\\equiv 6\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f2c3c0a52171e593", "problem_id": "691c0a592e4b8d7d", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the multiplicative inverse of 31 modulo 37.", "output": "1) We want x such that 31x ≡ 1 (mod 37).\n2) Use extended Euclid to find s,t with 31s + 37t = 1.\n3) One Bezout identity is: 31·(6) + 37·(-5) = 1.\n4) Reduce s modulo 37: x ≡ 6 (mod 37).\nFinal answer: $x\\equiv 6\\pmod{37}$", "final_answer": "x\\equiv 6\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "124bf3d0c785d5f1", "problem_id": "a59d1cb316de1858", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the linear system:\n$- 6 x - 7 y$ = $23$\n$- 4 x + y$ = $21$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $24 x + 28 y$ = $-92$.\n3) Multiply the second equation by $-6$: $24 x - 6 y$ = $-126$.\n4) Subtract to eliminate x: $34 y$ = $34$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, 1\\right)$", "final_answer": "(x,y)=\\left(-5, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6e6bc71c8ab162aa", "problem_id": "a59d1cb316de1858", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$- 6 x - 7 y$ = $23$\n$- 4 x + y$ = $21$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $24 x + 28 y$ = $-92$.\nMultiply the second equation by $-6$: $24 x - 6 y$ = $-126$.\nSubtract to eliminate x: $34 y$ = $34$.\nSolve for y: y = $1$.\nSubstitute back to find x: x = $-5$.\nCheck: plug (x,y)=(-5,1) into both equations.\nFinal answer: $(x,y)=\\left(-5, 1\\right)$", "final_answer": "(x,y)=\\left(-5, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a413646134a6cdc2", "problem_id": "04b237553172114c", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Two friends split $1078 in the ratio 4:3. How much does each receive?", "output": "1) Total ratio parts = 4+3 = 7.\n2) Each part = 1078 ÷ 7 = 154.\n3) First share = 4 × 154 = 616.\n4) Second share = 3 × 154 = 462.\nFinal answer: $462$", "final_answer": "462", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4fcae9d28026b88f", "problem_id": "04b237553172114c", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Two friends split $1078 in the ratio 4:3. How much does each receive?", "output": "$462$", "final_answer": "462", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4f8c6f06a33dbb67", "problem_id": "9d43e998652ea3a7", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} - 2 x^{3} + 3 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{3} + 3 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{15 \\pi^{4}}{32} + 2 + \\frac{3 \\pi^{2}}{8} + 3 \\pi + \\frac{9 \\pi^{3}}{8}$.\n4) Simplify the result.\nFinal answer: $- \\frac{15\\cdot \\pi^{4}}{32} + 2 + \\frac{3\\cdot \\pi^{2}}{8} + 3\\cdot \\pi + \\frac{9\\cdot \\pi^{3}}{8}$", "final_answer": "- \\frac{15\\cdot \\pi^{4}}{32} + 2 + \\frac{3\\cdot \\pi^{2}}{8} + 3\\cdot \\pi + \\frac{9\\cdot \\pi^{3}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3ffe942cdcb6f078", "problem_id": "9d43e998652ea3a7", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} - 2 x^{3} + 3 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{3} + 3 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\nEvaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{15 \\pi^{4}}{32} + 2 + \\frac{3 \\pi^{2}}{8} + 3 \\pi + \\frac{9 \\pi^{3}}{8}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{15\\cdot \\pi^{4}}{32} + 2 + \\frac{3\\cdot \\pi^{2}}{8} + 3\\cdot \\pi + \\frac{9\\cdot \\pi^{3}}{8}$", "final_answer": "- \\frac{15\\cdot \\pi^{4}}{32} + 2 + \\frac{3\\cdot \\pi^{2}}{8} + 3\\cdot \\pi + \\frac{9\\cdot \\pi^{3}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "25631123c135f0cc", "problem_id": "d2266df0b4a0f3d1", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(2 x^{3} - 3 x^{2} - 3 x\\right) \\cos{\\left(3 x \\right)}$.", "output": "1) Let f(x) = $2 \\left(x - 3\\right) e^{x} + \\left(2 x^{3} - 3 x^{2} - 3 x\\right) \\cos{\\left(3 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- 3 x \\left(2 x^{2} - 3 x - 3\\right) \\sin{\\left(3 x \\right)} + 2 \\left(x - 3\\right) e^{x} + 3 \\left(2 x^{2} - 2 x - 1\\right) \\cos{\\left(3 x \\right)} + 2 e^{x}$.\nFinal answer: $- 3\\cdot x\\cdot \\left(2\\cdot x^{2} - 3\\cdot x - 3\\right)\\cdot \\sin{\\left(3\\cdot x \\right)} + \\left(2\\cdot x - 6\\right)\\cdot e^{x} + \\left(6\\cdot x^{2} - 6\\cdot x - 3\\right)\\cdot \\cos{\\left(3\\cdot x \\right)} + 2\\cdot e^{x}$", "final_answer": "- 3\\cdot x\\cdot \\left(2\\cdot x^{2} - 3\\cdot x - 3\\right)\\cdot \\sin{\\left(3\\cdot x \\right)} + \\left(2\\cdot x - 6\\right)\\cdot e^{x} + \\left(6\\cdot x^{2} - 6\\cdot x - 3\\right)\\cdot \\cos{\\left(3\\cdot x \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0cadb907a1e5da83", "problem_id": "d2266df0b4a0f3d1", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(2 x^{3} - 3 x^{2} - 3 x\\right) \\cos{\\left(3 x \\right)}$.", "output": "Let f(x) = $2 \\left(x - 3\\right) e^{x} + \\left(2 x^{3} - 3 x^{2} - 3 x\\right) \\cos{\\left(3 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- 3 x \\left(2 x^{2} - 3 x - 3\\right) \\sin{\\left(3 x \\right)} + 2 \\left(x - 3\\right) e^{x} + 3 \\left(2 x^{2} - 2 x - 1\\right) \\cos{\\left(3 x \\right)} + 2 e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 3\\cdot x\\cdot \\left(2\\cdot x^{2} - 3\\cdot x - 3\\right)\\cdot \\sin{\\left(3\\cdot x \\right)} + \\left(2\\cdot x - 6\\right)\\cdot e^{x} + \\left(6\\cdot x^{2} - 6\\cdot x - 3\\right)\\cdot \\cos{\\left(3\\cdot x \\right)} + 2\\cdot e^{x}$", "final_answer": "- 3\\cdot x\\cdot \\left(2\\cdot x^{2} - 3\\cdot x - 3\\right)\\cdot \\sin{\\left(3\\cdot x \\right)} + \\left(2\\cdot x - 6\\right)\\cdot e^{x} + \\left(6\\cdot x^{2} - 6\\cdot x - 3\\right)\\cdot \\cos{\\left(3\\cdot x \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e32e75d2a602f9bb", "problem_id": "01f74b0638da6921", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡4 (mod 11). Give the solution modulo 33.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 33.\n3) Compute the CRT solution to get x ≡ 4 (mod 33).\nFinal answer: $x\\equiv 4\\pmod{33}$", "final_answer": "x\\equiv 4\\pmod{33}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2c7ac21af6b4f451", "problem_id": "01f74b0638da6921", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡4 (mod 11). Give the solution modulo 33.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 33.\nCompute the CRT solution to get x ≡ 4 (mod 33).\nCheck: 4 mod 3=1 and mod 11=4.\nFinal answer: $x\\equiv 4\\pmod{33}$", "final_answer": "x\\equiv 4\\pmod{33}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d223a5ecc44e16cd", "problem_id": "192c30ef456c4825", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $- 2 x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} + 3\\cdot x\\cdot \\cos{\\left(x \\right)} + 3\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - 3\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} + 3\\cdot x\\cdot \\cos{\\left(x \\right)} + 3\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - 3\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "169b0425db406440", "problem_id": "192c30ef456c4825", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $- 2 x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} + 3\\cdot x\\cdot \\cos{\\left(x \\right)} + 3\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - 3\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} + 3\\cdot x\\cdot \\cos{\\left(x \\right)} + 3\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - 3\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d6a9c97ca3bef3c8", "problem_id": "c07fbae28cd7b8fa", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - x - 3$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$-1$, c=$-3$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $25$, so √Δ = $5$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{3}{2}$.\nx2 = $-1$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{3}{2}, -1\\}$", "final_answer": "x \\in \\{\\frac{3}{2}, -1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ee83ccf45554ffa5", "problem_id": "c07fbae28cd7b8fa", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - x - 3$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$-1$, c=$-3$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{3}{2}, -1\\}$", "final_answer": "x \\in \\{\\frac{3}{2}, -1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cd273e10c28cc1d5", "problem_id": "f63d2fc4bae6405e", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=24$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=24 and k=6.\n4) Compute C(29,5) = 118755.\nFinal answer: $118755$", "final_answer": "118755", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f3206b7e31134fb9", "problem_id": "f63d2fc4bae6405e", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=24$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=24 and k=6.\nCompute C(29,5) = 118755.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $118755$", "final_answer": "118755", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "15681abebeaa6acf", "problem_id": "0d1ed7df721c833d", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(3 x^{2} + x + 1\\right) \\cos{\\left(x + 3 \\right)}$.", "output": "1) Let f(x) = $2 \\left(x - 3\\right) e^{x} + \\left(3 x^{2} + x + 1\\right) \\cos{\\left(x + 3 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $2 \\left(x - 3\\right) e^{x} + \\left(6 x + 1\\right) \\cos{\\left(x + 3 \\right)} - \\left(3 x^{2} + x + 1\\right) \\sin{\\left(x + 3 \\right)} + 2 e^{x}$.\nFinal answer: $\\left(2\\cdot x - 6\\right)\\cdot e^{x} + \\left(6\\cdot x + 1\\right)\\cdot \\cos{\\left(x + 3 \\right)} - \\left(3\\cdot x^{2} + x + 1\\right)\\cdot \\sin{\\left(x + 3 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x - 6\\right)\\cdot e^{x} + \\left(6\\cdot x + 1\\right)\\cdot \\cos{\\left(x + 3 \\right)} - \\left(3\\cdot x^{2} + x + 1\\right)\\cdot \\sin{\\left(x + 3 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6f560c440795b619", "problem_id": "0d1ed7df721c833d", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(3 x^{2} + x + 1\\right) \\cos{\\left(x + 3 \\right)}$.", "output": "Let f(x) = $2 \\left(x - 3\\right) e^{x} + \\left(3 x^{2} + x + 1\\right) \\cos{\\left(x + 3 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $2 \\left(x - 3\\right) e^{x} + \\left(6 x + 1\\right) \\cos{\\left(x + 3 \\right)} - \\left(3 x^{2} + x + 1\\right) \\sin{\\left(x + 3 \\right)} + 2 e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\left(2\\cdot x - 6\\right)\\cdot e^{x} + \\left(6\\cdot x + 1\\right)\\cdot \\cos{\\left(x + 3 \\right)} - \\left(3\\cdot x^{2} + x + 1\\right)\\cdot \\sin{\\left(x + 3 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x - 6\\right)\\cdot e^{x} + \\left(6\\cdot x + 1\\right)\\cdot \\cos{\\left(x + 3 \\right)} - \\left(3\\cdot x^{2} + x + 1\\right)\\cdot \\sin{\\left(x + 3 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "323d5f59492a2c20", "problem_id": "1caf36e71918de2f", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{2} - 4 x + 1$.", "output": "Differentiate term-by-term:\nd/dx(1) = 0.\nd/dx(- 4 x) = -4.\nd/dx(4 x^{2}) = 8 x.\nCombine: f'(x) = 4 \\left(2 x - 1\\right).\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $8\\cdot x - 4$", "final_answer": "8\\cdot x - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f97104a2e701ec9a", "problem_id": "1caf36e71918de2f", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{2} - 4 x + 1$.", "output": "1) Differentiate term-by-term:\n2) d/dx(1) = 0.\n3) d/dx(- 4 x) = -4.\n4) d/dx(4 x^{2}) = 8 x.\n5) Combine: f'(x) = 4 \\left(2 x - 1\\right).\nFinal answer: $8\\cdot x - 4$", "final_answer": "8\\cdot x - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "11872581f58c2019", "problem_id": "e875bd97312a9ed6", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$- 3 x - 2 y$ = $-2$\n$- 3 x - 3 y$ = $-6$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $9 x + 6 y$ = $6$.\n3) Multiply the second equation by $-3$: $9 x + 9 y$ = $18$.\n4) Subtract to eliminate x: $- 3 y$ = $-12$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, 4\\right)$", "final_answer": "(x,y)=\\left(-2, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ef84a7225bae1c6a", "problem_id": "a1f650c53ce275bf", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 x^{2} - 9 x + 21$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{2} - 10 x + 21.\n3) Then the limit is 53.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 51.\nFinal answer: $51$", "final_answer": "51", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0535ccc7c58ff3b9", "problem_id": "0ea552b99e4bad29", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} 2 x^{3} + 4 x^{2} + 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{3} + 4 x^{2} + 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\pi).\n3) Subtract: F(\\pi) − F(- \\pi) = $\\frac{8 \\pi \\left(3 + \\pi^{2}\\right)}{3}$.\n4) Simplify the result.\nFinal answer: $\\frac{8\\cdot \\pi\\cdot \\left(3 + \\pi^{2}\\right)}{3}$", "final_answer": "\\frac{8\\cdot \\pi\\cdot \\left(3 + \\pi^{2}\\right)}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ffe77b5d8b7f8be", "problem_id": "0ea552b99e4bad29", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} 2 x^{3} + 4 x^{2} + 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $2 x^{3} + 4 x^{2} + 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4$.\nEvaluate at the bounds: F(\\pi) and F(- \\pi).\nSubtract: F(\\pi) − F(- \\pi) = $\\frac{8 \\pi \\left(3 + \\pi^{2}\\right)}{3}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{8\\cdot \\pi\\cdot \\left(3 + \\pi^{2}\\right)}{3}$", "final_answer": "\\frac{8\\cdot \\pi\\cdot \\left(3 + \\pi^{2}\\right)}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9fa58b2202cc8620", "problem_id": "3e77595f0de7c486", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} x - 7$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{2} - 3 x - 3$ and a = -4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $2 x - 3$.\n5) Evaluate at a: p'(-4) = $-11$.\nFinal answer: $-11$", "final_answer": "-11", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "01b89faeb5d45391", "problem_id": "3e77595f0de7c486", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} x - 7$.", "output": "$-11$", "final_answer": "-11", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4708f3c85e7612c5", "problem_id": "9e43959aff786845", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "An urn has 10 red balls and 15 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(25,8).\nFavorable samples: choose k reds and n−k blues: C(10,1)·C(15,7).\nFinal answer: $\\frac{26}{437}$", "final_answer": "\\frac{26}{437}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b54cdb775484314", "problem_id": "9e43959aff786845", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 10 red balls and 15 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(25,8).\nFavorable samples: choose k reds and n−k blues: C(10,1)·C(15,7).\nCompute totals: C(25,8) = 1081575, and favorable = 64350.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{26}{437}$.\nFinal answer: $\\frac{26}{437}$", "final_answer": "\\frac{26}{437}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "83d52a67601c3f9f", "problem_id": "d56b4742f07d3bd1", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 10 x + 7$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$-10$, c=$7$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $16$, so √Δ = $4$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{7}{3}$.\nx2 = $1$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{7}{3}, 1\\}$", "final_answer": "x \\in \\{\\frac{7}{3}, 1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0691999a3e1690d9", "problem_id": "d56b4742f07d3bd1", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 10 x + 7$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$-10$, c=$7$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $16$, so √Δ = $4$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{7}{3}$.\n7) x2 = $1$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{7}{3}, 1\\}$", "final_answer": "x \\in \\{\\frac{7}{3}, 1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "408ed09c96faa98e", "problem_id": "3da13dbfa44db97f", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the equation of the tangent line to $y=x^{2} - x + 2 \\sin{\\left(x \\right)} + 1$ at $x=1$.", "output": "1) Differentiate: f'(x) = $2 x + 2 \\cos{\\left(x \\right)} - 1$.\n2) Slope at x=1: m = f'(1) = $1 + 2 \\cos{\\left(1 \\right)}$.\n3) Point on curve: (1, f(1)) = (1, $1 + 2 \\sin{\\left(1 \\right)}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 1 + 2 \\sin{\\left(1 \\right)} = 1 + 2 \\cos{\\left(1 \\right)}(x − 1).\n6) Optionally expand: y = $x + 2 x \\cos{\\left(1 \\right)} - 2 \\cos{\\left(1 \\right)} + 2 \\sin{\\left(1 \\right)}$.\nFinal answer: $y = x + 2\\cdot x\\cdot \\cos{\\left(1 \\right)} - 2\\cdot \\cos{\\left(1 \\right)} + 2\\cdot \\sin{\\left(1 \\right)}$", "final_answer": "y = x + 2\\cdot x\\cdot \\cos{\\left(1 \\right)} - 2\\cdot \\cos{\\left(1 \\right)} + 2\\cdot \\sin{\\left(1 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d99ff54a5da1566d", "problem_id": "3da13dbfa44db97f", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the equation of the tangent line to $y=x^{2} - x + 2 \\sin{\\left(x \\right)} + 1$ at $x=1$.", "output": "Differentiate: f'(x) = $2 x + 2 \\cos{\\left(x \\right)} - 1$.\nSlope at x=1: m = f'(1) = $1 + 2 \\cos{\\left(1 \\right)}$.\nPoint on curve: (1, f(1)) = (1, $1 + 2 \\sin{\\left(1 \\right)}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 1 + 2 \\sin{\\left(1 \\right)} = 1 + 2 \\cos{\\left(1 \\right)}(x − 1).\nOptionally expand: y = $x + 2 x \\cos{\\left(1 \\right)} - 2 \\cos{\\left(1 \\right)} + 2 \\sin{\\left(1 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = x + 2\\cdot x\\cdot \\cos{\\left(1 \\right)} - 2\\cdot \\cos{\\left(1 \\right)} + 2\\cdot \\sin{\\left(1 \\right)}$", "final_answer": "y = x + 2\\cdot x\\cdot \\cos{\\left(1 \\right)} - 2\\cdot \\cos{\\left(1 \\right)} + 2\\cdot \\sin{\\left(1 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e4a7e3a03d9d160e", "problem_id": "eccd02fdd5e69b5f", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A store sells a backpack for $54. It is discounted by 25%. You buy 6 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 54 × (1 − 25/100) = 54 × 75/100 = 81/2.\n2) Subtotal = 6 × 81/2 = 243.\n3) Total with tax = 243 × 107/100 = 26001/100.\nFinal answer: $\\frac{26001}{100}$", "final_answer": "\\frac{26001}{100}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3c995a2d486950e1", "problem_id": "eccd02fdd5e69b5f", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $54. It is discounted by 25%. You buy 6 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 54 × (1 − 25/100) = 54 × 75/100 = 81/2.\nSubtotal = 6 × 81/2 = 243.\nTotal with tax = 243 × 107/100 = 26001/100.\nCheck: recomputing the formula yields 26001/100.\nFinal answer: $\\frac{26001}{100}$", "final_answer": "\\frac{26001}{100}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9382732d6b630211", "problem_id": "3ac39cf0c13ec5a0", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the linear system:\n$- 2 x - y$ = $-4$\n$- 3 x$ = $-12$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $6 x + 3 y$ = $12$.\nMultiply the second equation by $-2$: $6 x$ = $24$.\nFinal answer: $(x,y)=\\left(4, -4\\right)$", "final_answer": "(x,y)=\\left(4, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a06dc46e75edd939", "problem_id": "3ac39cf0c13ec5a0", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$- 2 x - y$ = $-4$\n$- 3 x$ = $-12$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $6 x + 3 y$ = $12$.\n3) Multiply the second equation by $-2$: $6 x$ = $24$.\n4) Subtract to eliminate x: $3 y$ = $-12$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, -4\\right)$", "final_answer": "(x,y)=\\left(4, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1a86de0d890045a5", "problem_id": "2e69b74b03d3b447", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} 2 x^{4} - 3 x^{3} - 3 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{4} - 3 x^{3} - 3 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{45 \\pi^{4}}{64} - \\frac{9 \\pi^{3}}{8} - 6 \\pi - 1 + \\frac{9 \\pi^{2}}{8} + \\frac{33 \\pi^{5}}{80}$.\n4) Simplify the result.\nFinal answer: $- \\frac{45\\cdot \\pi^{4}}{64} - \\frac{9\\cdot \\pi^{3}}{8} - 6\\cdot \\pi - 1 + \\frac{9\\cdot \\pi^{2}}{8} + \\frac{33\\cdot \\pi^{5}}{80}$", "final_answer": "- \\frac{45\\cdot \\pi^{4}}{64} - \\frac{9\\cdot \\pi^{3}}{8} - 6\\cdot \\pi - 1 + \\frac{9\\cdot \\pi^{2}}{8} + \\frac{33\\cdot \\pi^{5}}{80}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a1a2cc4add43740c", "problem_id": "2e69b74b03d3b447", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} 2 x^{4} - 3 x^{3} - 3 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $2 x^{4} - 3 x^{3} - 3 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4$.\nEvaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{45 \\pi^{4}}{64} - \\frac{9 \\pi^{3}}{8} - 6 \\pi - 1 + \\frac{9 \\pi^{2}}{8} + \\frac{33 \\pi^{5}}{80}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{45\\cdot \\pi^{4}}{64} - \\frac{9\\cdot \\pi^{3}}{8} - 6\\cdot \\pi - 1 + \\frac{9\\cdot \\pi^{2}}{8} + \\frac{33\\cdot \\pi^{5}}{80}$", "final_answer": "- \\frac{45\\cdot \\pi^{4}}{64} - \\frac{9\\cdot \\pi^{3}}{8} - 6\\cdot \\pi - 1 + \\frac{9\\cdot \\pi^{2}}{8} + \\frac{33\\cdot \\pi^{5}}{80}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0c3d7cc551539797", "problem_id": "ac99f005bc63d3b3", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A disease has prevalence $\\frac{2}{25}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{239}{2500}$.\n3) Compute P(D|+) = $\\frac{170}{239}$.\nFinal answer: $\\frac{170}{239}$", "final_answer": "\\frac{170}{239}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6fe4934d6356e245", "problem_id": "ac99f005bc63d3b3", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "A disease has prevalence $\\frac{2}{25}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{239}{2500}$.\nCompute P(D|+) = $\\frac{170}{239}$.\nFinal answer: $\\frac{170}{239}$", "final_answer": "\\frac{170}{239}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dbff9d06ac6d1ca2", "problem_id": "caaf7e1bda75d8b2", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A store sells a backpack for $316. It is discounted by 25%. You buy 8 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 316 × (1 − 25/100) = 316 × 75/100 = 237.\n2) Subtotal = 8 × 237 = 1896.\n3) Total with tax = 1896 × 107/100 = 50718/25.\nFinal answer: $\\frac{50718}{25}$", "final_answer": "\\frac{50718}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "15751cb7bb3d3dc4", "problem_id": "caaf7e1bda75d8b2", "style": "answer_only", "instruction": "Final answer only.", "input": "A store sells a backpack for $316. It is discounted by 25%. You buy 8 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "$\\frac{50718}{25}$", "final_answer": "\\frac{50718}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2acab51bd1298a28", "problem_id": "4293977e26942a05", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Worker A finishes a job in 11 hours and Worker B finishes it in 6 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/11 job/hour. B's rate = 1/6 job/hour.\n2) Combined rate = 1/11 + 1/6 = 17/66 job/hour.\n3) Time = 1 ÷ (combined rate) = 66/17 hours.\nFinal answer: $\\frac{66}{17}$", "final_answer": "\\frac{66}{17}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "836e70ba494326a7", "problem_id": "4293977e26942a05", "style": "answer_only", "instruction": "Final answer only.", "input": "Worker A finishes a job in 11 hours and Worker B finishes it in 6 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "$\\frac{66}{17}$", "final_answer": "\\frac{66}{17}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "90588250c9842afb", "problem_id": "eb9da7d2da870746", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "An urn has 18 red balls and 17 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(35,12).\n3) Favorable samples: choose k reds and n−k blues: C(18,1)·C(17,11).\n4) Compute totals: C(35,12) = 834451800, and favorable = 222768.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{6}{22475}$", "final_answer": "\\frac{6}{22475}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bbfe4f70cb6ecc53", "problem_id": "eb9da7d2da870746", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 18 red balls and 17 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(35,12).\nFavorable samples: choose k reds and n−k blues: C(18,1)·C(17,11).\nCompute totals: C(35,12) = 834451800, and favorable = 222768.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{6}{22475}$.\nFinal answer: $\\frac{6}{22475}$", "final_answer": "\\frac{6}{22475}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b0812e601f8d4cff", "problem_id": "7bde40268b8dba9a", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - 3 x^{2} + 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 3$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $- 9 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 2$.\n2) Slope at x=-1: m = f'(-1) = $-1 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\n3) Point on curve: (-1, f(-1)) = (-1, $\\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 1$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 1 = -1 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}(x − -1).\n6) Optionally expand: y = $- x + \\frac{x}{2 e^{\\frac{1}{2}}} + 2 x \\sin{\\left(2 \\right)} + \\cos{\\left(2 \\right)} + \\frac{3}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = - x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f7993f60c0df7f57", "problem_id": "7bde40268b8dba9a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - 3 x^{2} + 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 3$ at $x=-1$.", "output": "Differentiate: f'(x) = $- 9 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 2$.\nSlope at x=-1: m = f'(-1) = $-1 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nPoint on curve: (-1, f(-1)) = (-1, $\\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 1$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 1 = -1 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}(x − -1).\nOptionally expand: y = $- x + \\frac{x}{2 e^{\\frac{1}{2}}} + 2 x \\sin{\\left(2 \\right)} + \\cos{\\left(2 \\right)} + \\frac{3}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "07fbd61e1490e4df", "problem_id": "960630399ae4b845", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(- 3 x^{2} - x + 2\\right) \\cos{\\left(3 x - 1 \\right)}$.", "output": "1) Let f(x) = $\\left(x + 1\\right) e^{x} + \\left(- 3 x^{2} - x + 2\\right) \\cos{\\left(3 x - 1 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\left(- 6 x - 1\\right) \\cos{\\left(3 x - 1 \\right)} + \\left(x + 1\\right) e^{x} - 3 \\left(- 3 x^{2} - x + 2\\right) \\sin{\\left(3 x - 1 \\right)} + e^{x}$.\nFinal answer: $\\left(- 6\\cdot x - 1\\right)\\cdot \\cos{\\left(3\\cdot x - 1 \\right)} + \\left(x + 1\\right)\\cdot e^{x} - \\left(- 9\\cdot x^{2} - 3\\cdot x + 6\\right)\\cdot \\sin{\\left(3\\cdot x - 1 \\right)} + e^{x}$", "final_answer": "\\left(- 6\\cdot x - 1\\right)\\cdot \\cos{\\left(3\\cdot x - 1 \\right)} + \\left(x + 1\\right)\\cdot e^{x} - \\left(- 9\\cdot x^{2} - 3\\cdot x + 6\\right)\\cdot \\sin{\\left(3\\cdot x - 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "897100e0eb85e988", "problem_id": "4166a418f2c3a4dd", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x^{2} \\sin{\\left(x \\right)}) = x^{2} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = x^{2} \\cos{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x^{2} \\sin{\\left(x \\right)}) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$", "final_answer": "x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "bd8874181d27c4b1", "problem_id": "fa4026f8b556c449", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Worker A finishes a job in 15 hours and Worker B finishes it in 14 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/15 job/hour. B's rate = 1/14 job/hour.\n2) Combined rate = 1/15 + 1/14 = 29/210 job/hour.\n3) Time = 1 ÷ (combined rate) = 210/29 hours.\nFinal answer: $\\frac{210}{29}$", "final_answer": "\\frac{210}{29}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "758605b8248069b9", "problem_id": "fa4026f8b556c449", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Worker A finishes a job in 15 hours and Worker B finishes it in 14 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/15 job/hour. B's rate = 1/14 job/hour.\nCombined rate = 1/15 + 1/14 = 29/210 job/hour.\nTime = 1 ÷ (combined rate) = 210/29 hours.\nCheck: (combined rate)×(time) = 29/210×210/29 = 1.\nFinal answer: $\\frac{210}{29}$", "final_answer": "\\frac{210}{29}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "86f3ea6e1e4a527f", "problem_id": "7e767d587261d438", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} - 4 x^{4} - 2 x^{2} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 4 x^{4} - 2 x^{2} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{33 \\pi^{5}}{40} - \\frac{3 \\pi^{3}}{4} - \\frac{9 \\pi}{2} - 1 + \\frac{3 \\pi^{2}}{2}$.\n4) Simplify the result.\nFinal answer: $- \\frac{33\\cdot \\pi^{5}}{40} - \\frac{3\\cdot \\pi^{3}}{4} - \\frac{9\\cdot \\pi}{2} - 1 + \\frac{3\\cdot \\pi^{2}}{2}$", "final_answer": "- \\frac{33\\cdot \\pi^{5}}{40} - \\frac{3\\cdot \\pi^{3}}{4} - \\frac{9\\cdot \\pi}{2} - 1 + \\frac{3\\cdot \\pi^{2}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f64a6b113ef42b4f", "problem_id": "7e767d587261d438", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} - 4 x^{4} - 2 x^{2} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 4 x^{4} - 2 x^{2} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\nEvaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{33 \\pi^{5}}{40} - \\frac{3 \\pi^{3}}{4} - \\frac{9 \\pi}{2} - 1 + \\frac{3 \\pi^{2}}{2}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{33\\cdot \\pi^{5}}{40} - \\frac{3\\cdot \\pi^{3}}{4} - \\frac{9\\cdot \\pi}{2} - 1 + \\frac{3\\cdot \\pi^{2}}{2}$", "final_answer": "- \\frac{33\\cdot \\pi^{5}}{40} - \\frac{3\\cdot \\pi^{3}}{4} - \\frac{9\\cdot \\pi}{2} - 1 + \\frac{3\\cdot \\pi^{2}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "155e4a4f926de9f8", "problem_id": "4d36c0a7f9852ece", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the linear system:\n$- 8 x + y$ = $-47$\n$6 x - 6 y$ = $-12$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $6$: $- 48 x + 6 y$ = $-282$.\n3) Multiply the second equation by $-8$: $- 48 x + 48 y$ = $96$.\n4) Subtract to eliminate x: $- 42 y$ = $-378$.\n5) Solve for y: y = $9$.\n6) Substitute back to find x: x = $7$.\nFinal answer: $(x,y)=\\left(7, 9\\right)$", "final_answer": "(x,y)=\\left(7, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ca0a6a34bbe8e68d", "problem_id": "4d36c0a7f9852ece", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the linear system:\n$- 8 x + y$ = $-47$\n$6 x - 6 y$ = $-12$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $6$: $- 48 x + 6 y$ = $-282$.\nMultiply the second equation by $-8$: $- 48 x + 48 y$ = $96$.\nFinal answer: $(x,y)=\\left(7, 9\\right)$", "final_answer": "(x,y)=\\left(7, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ea990de2c4fdb39", "problem_id": "9e39d22ab1790e02", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + \\frac{39 x}{4} + \\frac{9}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$\\frac{39}{4}$, c=$\\frac{9}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{1089}{16}$, so √Δ = $\\frac{33}{4}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{1}{4}$.\n7) x2 = $-3$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{1}{4}, -3\\}$", "final_answer": "x \\in \\{- \\frac{1}{4}, -3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "31cb8fb89e69bf75", "problem_id": "a2834fb9e51a07b3", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - x^{3} + x + 5$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to 5 - x^{3}.\n3) Then the limit is -22.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: -19.\nFinal answer: $-19$", "final_answer": "-19", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7b3bb5ea697c637c", "problem_id": "365bb1f0a2393337", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A cyclist rides at 49 miles per hour for 1 hours, then at 37 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 49 × 1 = 49 miles.\nSecond leg distance = 37 × 1 = 37 miles.\nTotal distance = 49 + 37 = 86 miles.\nCheck: adding both legs gives 86.\nFinal answer: $86$", "final_answer": "86", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a252e0cc564d7eb4", "problem_id": "365bb1f0a2393337", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A cyclist rides at 49 miles per hour for 1 hours, then at 37 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 49 × 1 = 49 miles.\n2) Second leg distance = 37 × 1 = 37 miles.\n3) Total distance = 49 + 37 = 86 miles.\nFinal answer: $86$", "final_answer": "86", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4746de7965cbf404", "problem_id": "c9caa646a1f664d6", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} + 4 x - 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(2 x^{2}) = 4 x^{2}$.\n3) So $f'(x) = 4 x^{2} + 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(2 x^{2}) = 4 x$.\nTherefore the correct derivative is $f'(x) = 4 \\left(x + 1\\right)$.\nFinal answer: $4 \\left(x + 1\\right)$", "final_answer": "4 \\left(x + 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d2579187b66aba71", "problem_id": "235a655f86fa0551", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 42 liters of a 25% salt solution. You have a 15% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 42−x liters is 50% solution.\nSalt equation: 15x + 50(42−x) = 25·42.\n(15-50)x = (25-50)·42 ⇒ x = 30.\nThen 42−x = 12.\nCheck: 15·30 + 50·12 = 25·42.\nFinal answer: $30 L of 15% and 12 L of 50%$", "final_answer": "30 L of 15% and 12 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d4e10cff3aa887a6", "problem_id": "171507cacba2478d", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 2 x^{3} + x^{2} + 2 x + \\left(- 3 x^{2} + x + 8\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{- 3 x^{2} + x + 8}$.", "output": "1) Let f(x) = $\\frac{- 2 x^{3} + x^{2} + 2 x + \\left(- 3 x^{2} + x + 8\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{- 3 x^{2} + x + 8}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{2 \\left(6 x - 1\\right) \\left(- 2 x^{3} + x^{2} + 2 x + \\left(- 3 x^{2} + x + 8\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(- 3 x^{2} + x + 8\\right) \\left(- 12 x^{2} + 4 x + 2 \\left(1 - 6 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(- 3 x^{2} + x + 8\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(- 3 x^{2} + x + 8\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(- 3 x^{2} + x + 8\\right)^{2}}$.\nFinal answer: $\\frac{\\left(12\\cdot x - 2\\right)\\cdot \\left(- 2\\cdot x^{3} + x^{2} + 2\\cdot x + \\left(- 3\\cdot x^{2} + x + 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 3\\right) + \\left(- 3\\cdot x^{2} + x + 8\\right)\\cdot \\left(- 12\\cdot x^{2} + 4\\cdot x + \\left(2 - 12\\cdot x\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(- 6\\cdot x^{2} + 2\\cdot x + 16\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + \\left(- 3\\cdot x^{2} + x + 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 4\\right)}{2\\cdot \\left(- 3\\cdot x^{2} + x + 8\\right)^{2}}$", "final_answer": "\\frac{\\left(12\\cdot x - 2\\right)\\cdot \\left(- 2\\cdot x^{3} + x^{2} + 2\\cdot x + \\left(- 3\\cdot x^{2} + x + 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 3\\right) + \\left(- 3\\cdot x^{2} + x + 8\\right)\\cdot \\left(- 12\\cdot x^{2} + 4\\cdot x + \\left(2 - 12\\cdot x\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(- 6\\cdot x^{2} + 2\\cdot x + 16\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + \\left(- 3\\cdot x^{2} + x + 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 4\\right)}{2\\cdot \\left(- 3\\cdot x^{2} + x + 8\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "914769d2faaffb74", "problem_id": "5bf1c679d39ecb2e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + x + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{3 x + 4}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 4}) = \\frac{4 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 4}$.\n3) So $f'(x) = \\frac{9 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{18 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{24 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{48 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{16 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{32 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} + 48 x + 32} - \\frac{4}{18 x^{2} + 48 x + 32} + \\frac{12 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{2} + 24 x + 16} + \\frac{4 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 4} - \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 4} - \\frac{4 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 4}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 4}) = \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 4} + \\frac{4 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 4} - \\frac{12 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(3 x + 4\\right)^{2}}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 18 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6 x + \\left(3 x + 4\\right) \\left(3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 10 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right) - 24 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 12}{2 \\left(3 x + 4\\right)^{2}}$.\nFinal answer: $\\frac{- 18 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6 x + \\left(3 x + 4\\right) \\left(3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 10 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right) - 24 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 12}{2 \\left(3 x + 4\\right)^{2}}$", "final_answer": "\\frac{- 18 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6 x + \\left(3 x + 4\\right) \\left(3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 10 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right) - 24 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 12}{2 \\left(3 x + 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7658ec7178f0aac0", "problem_id": "7335a600aa0e9c0f", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{7} x - \\frac{13 x}{5} + \\frac{6}{5} + \\frac{3 \\sqrt{7}}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\sqrt{7} - \\frac{13}{5}$, c=$\\frac{6}{5} + \\frac{3 \\sqrt{7}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{14 \\sqrt{7}}{5} + \\frac{224}{25}$, so √Δ = $\\sqrt{\\frac{14 \\sqrt{7}}{5} + \\frac{224}{25}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{13}{10} + \\frac{\\sqrt{7}}{2} + \\frac{\\sqrt{70 \\sqrt{7} + 224}}{10}$.\n7) x2 = $- \\frac{\\sqrt{70 \\sqrt{7} + 224}}{10} + \\frac{13}{10} + \\frac{\\sqrt{7}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{13}{10} + \\frac{\\sqrt{7}}{2} + \\frac{\\sqrt{70\\cdot \\sqrt{7} + 224}}{10}, - \\frac{\\sqrt{70\\cdot \\sqrt{7} + 224}}{10} + \\frac{13}{10} + \\frac{\\sqrt{7}}{2}\\}$", "final_answer": "x \\in \\{\\frac{13}{10} + \\frac{\\sqrt{7}}{2} + \\frac{\\sqrt{70\\cdot \\sqrt{7} + 224}}{10}, - \\frac{\\sqrt{70\\cdot \\sqrt{7} + 224}}{10} + \\frac{13}{10} + \\frac{\\sqrt{7}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "904d91dd6c96db88", "problem_id": "7335a600aa0e9c0f", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{7} x - \\frac{13 x}{5} + \\frac{6}{5} + \\frac{3 \\sqrt{7}}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\sqrt{7} - \\frac{13}{5}$, c=$\\frac{6}{5} + \\frac{3 \\sqrt{7}}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{14 \\sqrt{7}}{5} + \\frac{224}{25}$, so √Δ = $\\sqrt{\\frac{14 \\sqrt{7}}{5} + \\frac{224}{25}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{13}{10} + \\frac{\\sqrt{7}}{2} + \\frac{\\sqrt{70 \\sqrt{7} + 224}}{10}$.\nx2 = $- \\frac{\\sqrt{70 \\sqrt{7} + 224}}{10} + \\frac{13}{10} + \\frac{\\sqrt{7}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{13}{10} + \\frac{\\sqrt{7}}{2} + \\frac{\\sqrt{70\\cdot \\sqrt{7} + 224}}{10}, - \\frac{\\sqrt{70\\cdot \\sqrt{7} + 224}}{10} + \\frac{13}{10} + \\frac{\\sqrt{7}}{2}\\}$", "final_answer": "x \\in \\{\\frac{13}{10} + \\frac{\\sqrt{7}}{2} + \\frac{\\sqrt{70\\cdot \\sqrt{7} + 224}}{10}, - \\frac{\\sqrt{70\\cdot \\sqrt{7} + 224}}{10} + \\frac{13}{10} + \\frac{\\sqrt{7}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "abc1d38598ee19a9", "problem_id": "f039ef399824ebe3", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Find the multiplicative inverse of 11 modulo 13.", "output": "1) We want x such that 11x ≡ 1 (mod 13).\n2) Use extended Euclid to find s,t with 11s + 13t = 1.\n3) One Bezout identity is: 11·(6) + 13·(-5) = 1.\n4) Reduce s modulo 13: x ≡ 6 (mod 13).\nFinal answer: $x\\equiv 6\\pmod{13}$", "final_answer": "x\\equiv 6\\pmod{13}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f76cee3f4d91eda4", "problem_id": "f039ef399824ebe3", "style": "answer_only", "instruction": "Final answer only.", "input": "Find the multiplicative inverse of 11 modulo 13.", "output": "$x\\equiv 6\\pmod{13}$", "final_answer": "x\\equiv 6\\pmod{13}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "340a07f365ce0a5a", "problem_id": "4ca1963d6057596f", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} x^{3} + x^{2} - x + 1$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $x^{4} + 3 x^{3} + x^{2} - x + 1$ and a = -2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $4 x^{3} + 9 x^{2} + 2 x - 1$.\nEvaluate at a: p'(-2) = $-1$.\nCheck: the identity gives $-1$.\nFinal answer: $-1$", "final_answer": "-1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "10103bfe407d50ba", "problem_id": "4ca1963d6057596f", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} x^{3} + x^{2} - x + 1$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $x^{4} + 3 x^{3} + x^{2} - x + 1$ and a = -2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-1$", "final_answer": "-1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bd55df1aeb689f81", "problem_id": "ae5da39416e1608b", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A cyclist rides at 46 miles per hour for 2 hours, then at 48 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 46 × 2 = 92 miles.\nSecond leg distance = 48 × 1 = 48 miles.\nTotal distance = 92 + 48 = 140 miles.\nCheck: adding both legs gives 140.\nFinal answer: $140$", "final_answer": "140", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "715ae50b6fadcbe7", "problem_id": "ae5da39416e1608b", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A cyclist rides at 46 miles per hour for 2 hours, then at 48 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 46 × 2 = 92 miles.\n2) Second leg distance = 48 × 1 = 48 miles.\n3) Total distance = 92 + 48 = 140 miles.\nFinal answer: $140$", "final_answer": "140", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e666dae9c96f15ab", "problem_id": "02c6ef7a19331f9c", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the multiplicative inverse of 6 modulo 17.", "output": "We want x such that 6x ≡ 1 (mod 17).\nUse extended Euclid to find s,t with 6s + 17t = 1.\nOne Bezout identity is: 6·(3) + 17·(-1) = 1.\nReduce s modulo 17: x ≡ 3 (mod 17).\nCheck: 6·3 mod 17 = 1.\nFinal answer: $x\\equiv 3\\pmod{17}$", "final_answer": "x\\equiv 3\\pmod{17}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e444751b104174c7", "problem_id": "02c6ef7a19331f9c", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the multiplicative inverse of 6 modulo 17.", "output": "1) We want x such that 6x ≡ 1 (mod 17).\n2) Use extended Euclid to find s,t with 6s + 17t = 1.\n3) One Bezout identity is: 6·(3) + 17·(-1) = 1.\n4) Reduce s modulo 17: x ≡ 3 (mod 17).\nFinal answer: $x\\equiv 3\\pmod{17}$", "final_answer": "x\\equiv 3\\pmod{17}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3646c81cda11a4ae", "problem_id": "ec4e06b47d8a56e5", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Worker A finishes a job in 9 hours and Worker B finishes it in 14 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/9 job/hour. B's rate = 1/14 job/hour.\nCombined rate = 1/9 + 1/14 = 23/126 job/hour.\nTime = 1 ÷ (combined rate) = 126/23 hours.\nFinal answer: $\\frac{126}{23}$", "final_answer": "\\frac{126}{23}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db67cc5514541259", "problem_id": "ec4e06b47d8a56e5", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Worker A finishes a job in 9 hours and Worker B finishes it in 14 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/9 job/hour. B's rate = 1/14 job/hour.\n2) Combined rate = 1/9 + 1/14 = 23/126 job/hour.\n3) Time = 1 ÷ (combined rate) = 126/23 hours.\nFinal answer: $\\frac{126}{23}$", "final_answer": "\\frac{126}{23}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5322b98fa77a0e37", "problem_id": "c39ada8a0b196d62", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{2} + x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $3 x^{2} + x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3$.\nCompute an antiderivative: F(x) = $x^{3} + \\frac{x^{2}}{2} - 3 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + x^{3} + \\frac{x^{2}}{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + x^{3} + \\frac{x^{2}}{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c291c75d463cd19c", "problem_id": "c39ada8a0b196d62", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{2} + x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $3 x^{2} + x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3$.\n3) Compute an antiderivative: F(x) = $x^{3} + \\frac{x^{2}}{2} - 3 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + x^{3} + \\frac{x^{2}}{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + x^{3} + \\frac{x^{2}}{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cf7f50bde3f75edb", "problem_id": "83e1b559710e6d56", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 2 \\sqrt{13} x - \\frac{23 x}{5} - \\frac{4 \\sqrt{13}}{5} - 2$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- 2 \\sqrt{13} - \\frac{23}{5}$, c=$- \\frac{4 \\sqrt{13}}{5} - 2$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{108 \\sqrt{13}}{5} + \\frac{2029}{25}$, so √Δ = $\\sqrt{\\frac{108 \\sqrt{13}}{5} + \\frac{2029}{25}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{23}{10} + \\sqrt{13} + \\frac{\\sqrt{540 \\sqrt{13} + 2029}}{10}$.\n7) x2 = $- \\frac{\\sqrt{540 \\sqrt{13} + 2029}}{10} + \\frac{23}{10} + \\sqrt{13}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{23}{10} + \\sqrt{13} + \\frac{\\sqrt{540\\cdot \\sqrt{13} + 2029}}{10}, - \\frac{\\sqrt{540\\cdot \\sqrt{13} + 2029}}{10} + \\frac{23}{10} + \\sqrt{13}\\}$", "final_answer": "x \\in \\{\\frac{23}{10} + \\sqrt{13} + \\frac{\\sqrt{540\\cdot \\sqrt{13} + 2029}}{10}, - \\frac{\\sqrt{540\\cdot \\sqrt{13} + 2029}}{10} + \\frac{23}{10} + \\sqrt{13}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "80b6301bf1cac41e", "problem_id": "83e1b559710e6d56", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 2 \\sqrt{13} x - \\frac{23 x}{5} - \\frac{4 \\sqrt{13}}{5} - 2$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- 2 \\sqrt{13} - \\frac{23}{5}$, c=$- \\frac{4 \\sqrt{13}}{5} - 2$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{23}{10} + \\sqrt{13} + \\frac{\\sqrt{540\\cdot \\sqrt{13} + 2029}}{10}, - \\frac{\\sqrt{540\\cdot \\sqrt{13} + 2029}}{10} + \\frac{23}{10} + \\sqrt{13}\\}$", "final_answer": "x \\in \\{\\frac{23}{10} + \\sqrt{13} + \\frac{\\sqrt{540\\cdot \\sqrt{13} + 2029}}{10}, - \\frac{\\sqrt{540\\cdot \\sqrt{13} + 2029}}{10} + \\frac{23}{10} + \\sqrt{13}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "212b30f03a4b13f0", "problem_id": "c0033663ed9ab62c", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$3 x + y$ = $-2$\n$- 4 x - 3 y$ = $1$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $- 12 x - 4 y$ = $8$.\n3) Multiply the second equation by $3$: $- 12 x - 9 y$ = $3$.\n4) Subtract to eliminate x: $5 y$ = $5$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, 1\\right)$", "final_answer": "(x,y)=\\left(-1, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "92c10b3489e38956", "problem_id": "c0033663ed9ab62c", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$3 x + y$ = $-2$\n$- 4 x - 3 y$ = $1$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $- 12 x - 4 y$ = $8$.\nMultiply the second equation by $3$: $- 12 x - 9 y$ = $3$.\nSubtract to eliminate x: $5 y$ = $5$.\nSolve for y: y = $1$.\nSubstitute back to find x: x = $-1$.\nCheck: plug (x,y)=(-1,1) into both equations.\nFinal answer: $(x,y)=\\left(-1, 1\\right)$", "final_answer": "(x,y)=\\left(-1, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a5f775864c404156", "problem_id": "04b51c0dc0f9416d", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "You want 10 liters of a 32% salt solution. You have a 25% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 10−x liters is 40% solution.\n2) Salt equation: 25x + 40(10−x) = 32·10.\n3) (25-40)x = (32-40)·10 ⇒ x = 16/3.\n4) Then 10−x = 14/3.\nFinal answer: $16/3 L of 25% and 14/3 L of 40%$", "final_answer": "16/3 L of 25% and 14/3 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dc01cb811ede1e29", "problem_id": "04b51c0dc0f9416d", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "You want 10 liters of a 32% salt solution. You have a 25% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 10−x liters is 40% solution.\nSalt equation: 25x + 40(10−x) = 32·10.\n(25-40)x = (32-40)·10 ⇒ x = 16/3.\nFinal answer: $16/3 L of 25% and 14/3 L of 40%$", "final_answer": "16/3 L of 25% and 14/3 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ce55cb0f0bdca31d", "problem_id": "d8edf4225f6f4261", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} + 3 x^{2} + x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 3$ at $x=-1$.", "output": "Differentiate: f'(x) = $- 3 x^{2} + 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 1$.\nSlope at x=-1: m = f'(-1) = $-8 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nPoint on curve: (-1, f(-1)) = (-1, $\\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 6$).\nFinal answer: $y = - 8\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 2 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 8\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 2 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "97690744752a325f", "problem_id": "d8edf4225f6f4261", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} + 3 x^{2} + x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 3$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $- 3 x^{2} + 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 1$.\n2) Slope at x=-1: m = f'(-1) = $-8 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\n3) Point on curve: (-1, f(-1)) = (-1, $\\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 6$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 6 = -8 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}(x − -1).\n6) Optionally expand: y = $- 8 x + \\frac{x}{2 e^{\\frac{1}{2}}} + 2 x \\sin{\\left(2 \\right)} - 2 + \\cos{\\left(2 \\right)} + \\frac{3}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = - 8\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 2 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 8\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 2 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d6eb2fcca88134cd", "problem_id": "3e88438da6c9bc72", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the congruences: x≡3 (mod 5), x≡7 (mod 9). Give the solution modulo 45.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 45.\n3) Compute the CRT solution to get x ≡ 43 (mod 45).\nFinal answer: $x\\equiv 43\\pmod{45}$", "final_answer": "x\\equiv 43\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8f31981d2d6dec20", "problem_id": "3e88438da6c9bc72", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡3 (mod 5), x≡7 (mod 9). Give the solution modulo 45.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 45.\nCompute the CRT solution to get x ≡ 43 (mod 45).\nCheck: 43 mod 5=3 and mod 9=7.\nFinal answer: $x\\equiv 43\\pmod{45}$", "final_answer": "x\\equiv 43\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8d5ae62d2120f1fc", "problem_id": "19a3d96a87a983cb", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "An urn has 16 red balls and 15 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(31,8).\n3) Favorable samples: choose k reds and n−k blues: C(16,3)·C(15,5).\n4) Compute totals: C(31,8) = 7888725, and favorable = 1681680.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{8624}{40455}$", "final_answer": "\\frac{8624}{40455}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ce5153d324c62c5", "problem_id": "660f83bc358400a6", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{3} - 3 x^{2} - 2$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(4 x^{3}) = 12 x^{3}$.\n3) So $f'(x) = 12 x^{3} - 6 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(4 x^{3}) = 12 x^{2}$.\nTherefore the correct derivative is $f'(x) = 6 x \\left(2 x - 1\\right)$.\nFinal answer: $6 x \\left(2 x - 1\\right)$", "final_answer": "6 x \\left(2 x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "4eab0239e109bc7b", "problem_id": "3c8834a65fc582b9", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Worker A finishes a job in 5 hours and Worker B finishes it in 19 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/5 job/hour. B's rate = 1/19 job/hour.\n2) Combined rate = 1/5 + 1/19 = 24/95 job/hour.\n3) Time = 1 ÷ (combined rate) = 95/24 hours.\nFinal answer: $\\frac{95}{24}$", "final_answer": "\\frac{95}{24}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dec93dfdb14c008e", "problem_id": "3c8834a65fc582b9", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Worker A finishes a job in 5 hours and Worker B finishes it in 19 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/5 job/hour. B's rate = 1/19 job/hour.\nCombined rate = 1/5 + 1/19 = 24/95 job/hour.\nTime = 1 ÷ (combined rate) = 95/24 hours.\nFinal answer: $\\frac{95}{24}$", "final_answer": "\\frac{95}{24}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d90b5586f42543a5", "problem_id": "0da41088b7251389", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{5} x + \\frac{24 x}{5} - \\frac{27 \\sqrt{5}}{5} + \\frac{27}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{24}{5} - 3 \\sqrt{5}$, c=$\\frac{27}{5} - \\frac{27 \\sqrt{5}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{1161}{25} - \\frac{36 \\sqrt{5}}{5}$, so √Δ = $\\sqrt{\\frac{1161}{25} - \\frac{36 \\sqrt{5}}{5}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{12}{5} + \\frac{3 \\sqrt{129 - 20 \\sqrt{5}}}{10} + \\frac{3 \\sqrt{5}}{2}$.\n7) x2 = $- \\frac{3 \\sqrt{129 - 20 \\sqrt{5}}}{10} - \\frac{12}{5} + \\frac{3 \\sqrt{5}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{12}{5} + \\frac{3\\cdot \\sqrt{129 - 20\\cdot \\sqrt{5}}}{10} + \\frac{3\\cdot \\sqrt{5}}{2}, - \\frac{3\\cdot \\sqrt{129 - 20\\cdot \\sqrt{5}}}{10} - \\frac{12}{5} + \\frac{3\\cdot \\sqrt{5}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{12}{5} + \\frac{3\\cdot \\sqrt{129 - 20\\cdot \\sqrt{5}}}{10} + \\frac{3\\cdot \\sqrt{5}}{2}, - \\frac{3\\cdot \\sqrt{129 - 20\\cdot \\sqrt{5}}}{10} - \\frac{12}{5} + \\frac{3\\cdot \\sqrt{5}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "da73160df4065fb4", "problem_id": "0da41088b7251389", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{5} x + \\frac{24 x}{5} - \\frac{27 \\sqrt{5}}{5} + \\frac{27}{5}$ = 0.", "output": "$x \\in \\{- \\frac{12}{5} + \\frac{3\\cdot \\sqrt{129 - 20\\cdot \\sqrt{5}}}{10} + \\frac{3\\cdot \\sqrt{5}}{2}, - \\frac{3\\cdot \\sqrt{129 - 20\\cdot \\sqrt{5}}}{10} - \\frac{12}{5} + \\frac{3\\cdot \\sqrt{5}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{12}{5} + \\frac{3\\cdot \\sqrt{129 - 20\\cdot \\sqrt{5}}}{10} + \\frac{3\\cdot \\sqrt{5}}{2}, - \\frac{3\\cdot \\sqrt{129 - 20\\cdot \\sqrt{5}}}{10} - \\frac{12}{5} + \\frac{3\\cdot \\sqrt{5}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ba2666b959170c94", "problem_id": "fa43667357a8f0ea", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$x + 2 y$ = $-12$\n$- x - 8 y$ = $36$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $- x - 2 y$ = $12$.\n3) Multiply the second equation by $1$: $- x - 8 y$ = $36$.\n4) Subtract to eliminate x: $6 y$ = $-24$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $-4$.\nFinal answer: $(x,y)=\\left(-4, -4\\right)$", "final_answer": "(x,y)=\\left(-4, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "291e3475756b8684", "problem_id": "fa43667357a8f0ea", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$x + 2 y$ = $-12$\n$- x - 8 y$ = $36$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $- x - 2 y$ = $12$.\nMultiply the second equation by $1$: $- x - 8 y$ = $36$.\nSubtract to eliminate x: $6 y$ = $-24$.\nSolve for y: y = $-4$.\nSubstitute back to find x: x = $-4$.\nCheck: plug (x,y)=(-4,-4) into both equations.\nFinal answer: $(x,y)=\\left(-4, -4\\right)$", "final_answer": "(x,y)=\\left(-4, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9510d3d283ea467e", "problem_id": "8dfd558916bddb04", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Two friends split $648 in the ratio 7:1. How much does each receive?", "output": "Total ratio parts = 7+1 = 8.\nEach part = 648 ÷ 8 = 81.\nFirst share = 7 × 81 = 567.\nSecond share = 1 × 81 = 81.\nCheck: 567+81=648.\nFinal answer: $81$", "final_answer": "81", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d9302c73d23de738", "problem_id": "8dfd558916bddb04", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Two friends split $648 in the ratio 7:1. How much does each receive?", "output": "1) Total ratio parts = 7+1 = 8.\n2) Each part = 648 ÷ 8 = 81.\n3) First share = 7 × 81 = 567.\n4) Second share = 1 × 81 = 81.\nFinal answer: $81$", "final_answer": "81", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db655db7eabf6968", "problem_id": "f7d4f6af5c069b4a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{2} + 3 x - 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- x^{2}) = - 2 x^{2}$.\n3) So $f'(x) = 3 - 2 x^{2}$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- x^{2}) = - 2 x$.\nTherefore the correct derivative is $f'(x) = 3 - 2 x$.\nFinal answer: $3 - 2 x$", "final_answer": "3 - 2 x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ff023a8ccb386131", "problem_id": "55c612265a4b68a7", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A cyclist rides at 71 miles per hour for 3/2 hours, then at 71 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 71 × 3/2 = 213/2 miles.\nSecond leg distance = 71 × 3/2 = 213/2 miles.\nTotal distance = 213/2 + 213/2 = 213 miles.\nCheck: adding both legs gives 213.\nFinal answer: $213$", "final_answer": "213", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a1df1272622a9d2", "problem_id": "c5d1cd436aa93ea2", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "An urn has 8 red balls and 6 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(14,4).\n3) Favorable samples: choose k reds and n−k blues: C(8,2)·C(6,2).\n4) Compute totals: C(14,4) = 1001, and favorable = 420.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{60}{143}$", "final_answer": "\\frac{60}{143}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1f5a97be554db026", "problem_id": "c5d1cd436aa93ea2", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "An urn has 8 red balls and 6 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(14,4).\nFavorable samples: choose k reds and n−k blues: C(8,2)·C(6,2).\nFinal answer: $\\frac{60}{143}$", "final_answer": "\\frac{60}{143}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2966f9c1564605ce", "problem_id": "bcfa8996e233ae56", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} - 3 x^{3} - x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{3} - x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3$.\nEvaluate at the bounds: F(2 \\pi) and F(0).\nSubtract: F(2 \\pi) − F(0) = $\\frac{2 \\pi \\left(- 18 \\pi^{3} - 4 \\pi^{2} - 9 \\pi + 9\\right)}{3}$.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(- 18\\cdot \\pi^{3} - 4\\cdot \\pi^{2} - 9\\cdot \\pi + 9\\right)}{3}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(- 18\\cdot \\pi^{3} - 4\\cdot \\pi^{2} - 9\\cdot \\pi + 9\\right)}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fef5c968206f9567", "problem_id": "bcfa8996e233ae56", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} - 3 x^{3} - x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{3} - x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3$.\nEvaluate at the bounds: F(2 \\pi) and F(0).\nSubtract: F(2 \\pi) − F(0) = $\\frac{2 \\pi \\left(- 18 \\pi^{3} - 4 \\pi^{2} - 9 \\pi + 9\\right)}{3}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(- 18\\cdot \\pi^{3} - 4\\cdot \\pi^{2} - 9\\cdot \\pi + 9\\right)}{3}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(- 18\\cdot \\pi^{3} - 4\\cdot \\pi^{2} - 9\\cdot \\pi + 9\\right)}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4406bd57ba17fae6", "problem_id": "abf148f647f1622c", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "You want 62 liters of a 35% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 62−x liters is 70% solution.\n2) Salt equation: 25x + 70(62−x) = 35·62.\n3) (25-70)x = (35-70)·62 ⇒ x = 434/9.\n4) Then 62−x = 124/9.\nFinal answer: $434/9 L of 25% and 124/9 L of 70%$", "final_answer": "434/9 L of 25% and 124/9 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b50f3c27dde86ea0", "problem_id": "abf148f647f1622c", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "You want 62 liters of a 35% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 62−x liters is 70% solution.\nSalt equation: 25x + 70(62−x) = 35·62.\n(25-70)x = (35-70)·62 ⇒ x = 434/9.\nThen 62−x = 124/9.\nCheck: 25·434/9 + 70·124/9 = 35·62.\nFinal answer: $434/9 L of 25% and 124/9 L of 70%$", "final_answer": "434/9 L of 25% and 124/9 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "19f6c53a78b4f9b7", "problem_id": "1408609af3a0ebfc", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - x^{3} - x^{2} - 5 x - 17$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{4} + 3 x^{3} - x^{2} + 3 x$ and a = 4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 4 x^{3} + 9 x^{2} - 2 x + 3$.\n5) Evaluate at a: p'(4) = $-117$.\nFinal answer: $-117$", "final_answer": "-117", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "157733d4a03bb250", "problem_id": "1408609af3a0ebfc", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - x^{3} - x^{2} - 5 x - 17$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{4} + 3 x^{3} - x^{2} + 3 x$ and a = 4.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 4 x^{3} + 9 x^{2} - 2 x + 3$.\nEvaluate at a: p'(4) = $-117$.\nCheck: the identity gives $-117$.\nFinal answer: $-117$", "final_answer": "-117", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "356b99262da47215", "problem_id": "21e0eed989dce55f", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A store sells a backpack for $885. It is discounted by 20%. You buy 12 backpacks. What is the total cost?", "output": "1) Discounted price = 885 × (1 − 20/100) = 885 × 80/100 = 708.\n2) Subtotal = 12 × 708 = 8496.\nFinal answer: $8496$", "final_answer": "8496", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5f6e6487328f9d28", "problem_id": "21e0eed989dce55f", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A store sells a backpack for $885. It is discounted by 20%. You buy 12 backpacks. What is the total cost?", "output": "Discounted price = 885 × (1 − 20/100) = 885 × 80/100 = 708.\nSubtotal = 12 × 708 = 8496.\nCheck: recomputing the formula yields 8496.\nFinal answer: $8496$", "final_answer": "8496", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a4796df985d62788", "problem_id": "a0e416b0b7175957", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 13 red balls and 30 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(43,9).\n3) Favorable samples: choose k reds and n−k blues: C(13,4)·C(30,5).\n4) Compute totals: C(43,9) = 563921995, and favorable = 101891790.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{223938}{1239389}$", "final_answer": "\\frac{223938}{1239389}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2e55baedc68e3fb6", "problem_id": "a0e416b0b7175957", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "An urn has 13 red balls and 30 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(43,9).\nFavorable samples: choose k reds and n−k blues: C(13,4)·C(30,5).\nFinal answer: $\\frac{223938}{1239389}$", "final_answer": "\\frac{223938}{1239389}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "61abed5a55a56dc7", "problem_id": "a776b2e03eb96679", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 3 x^{3} - 10 x^{2} + 28 x - 82$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $3 x^{4} - x^{3} - 2 x^{2} + 2 x + 1$ and a = -3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $12 x^{3} - 3 x^{2} - 4 x + 2$.\nEvaluate at a: p'(-3) = $-337$.\nCheck: the identity gives $-337$.\nFinal answer: $-337$", "final_answer": "-337", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a3d3c5106f9eb958", "problem_id": "a776b2e03eb96679", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 3 x^{3} - 10 x^{2} + 28 x - 82$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{4} - x^{3} - 2 x^{2} + 2 x + 1$ and a = -3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $12 x^{3} - 3 x^{2} - 4 x + 2$.\n5) Evaluate at a: p'(-3) = $-337$.\nFinal answer: $-337$", "final_answer": "-337", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d60dfec2e2b12d4", "problem_id": "4e5e51f53bf24457", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} 3 x^{3} + 10 x^{2} + 39 x + 153$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} + 10 x^{2} + 38 x + 153.\n3) Then the limit is 657.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: 661.\nFinal answer: $661$", "final_answer": "661", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7a5208b3509daf75", "problem_id": "9ba994734ceb1226", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 14 x - 60$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} - 14 x - 60$ = $2 \\left(x - 10\\right) \\left(x + 3\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $10$.\nFrom the second factor: x = $-3$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{10, -3\\}$", "final_answer": "x \\in \\{10, -3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "364095db507e5ce5", "problem_id": "6116e17f404a4160", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} - x^{2} + 7 x - 29$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{3} + 3 x^{2} - x$ and a = -4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 3 x^{2} + 6 x - 1$.\n5) Evaluate at a: p'(-4) = $-73$.\nFinal answer: $-73$", "final_answer": "-73", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3fa72221208eeca9", "problem_id": "6116e17f404a4160", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} - x^{2} + 7 x - 29$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{3} + 3 x^{2} - x$ and a = -4.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-73$", "final_answer": "-73", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c6bcf965f941095f", "problem_id": "b708677091c5e5b4", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(2 x^{3} - x^{2} + x - 2\\right) \\cos{\\left(2 x + 3 \\right)}$.", "output": "1) Let f(x) = $2 \\left(x + 2\\right) e^{x} + \\left(2 x^{3} - x^{2} + x - 2\\right) \\cos{\\left(2 x + 3 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $2 \\left(x + 2\\right) e^{x} + \\left(6 x^{2} - 2 x + 1\\right) \\cos{\\left(2 x + 3 \\right)} - 2 \\left(2 x^{3} - x^{2} + x - 2\\right) \\sin{\\left(2 x + 3 \\right)} + 2 e^{x}$.\nFinal answer: $\\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(6\\cdot x^{2} - 2\\cdot x + 1\\right)\\cdot \\cos{\\left(2\\cdot x + 3 \\right)} - \\left(4\\cdot x^{3} - 2\\cdot x^{2} + 2\\cdot x - 4\\right)\\cdot \\sin{\\left(2\\cdot x + 3 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(6\\cdot x^{2} - 2\\cdot x + 1\\right)\\cdot \\cos{\\left(2\\cdot x + 3 \\right)} - \\left(4\\cdot x^{3} - 2\\cdot x^{2} + 2\\cdot x - 4\\right)\\cdot \\sin{\\left(2\\cdot x + 3 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6bee3501b65f472e", "problem_id": "b708677091c5e5b4", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(2 x^{3} - x^{2} + x - 2\\right) \\cos{\\left(2 x + 3 \\right)}$.", "output": "Let f(x) = $2 \\left(x + 2\\right) e^{x} + \\left(2 x^{3} - x^{2} + x - 2\\right) \\cos{\\left(2 x + 3 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $2 \\left(x + 2\\right) e^{x} + \\left(6 x^{2} - 2 x + 1\\right) \\cos{\\left(2 x + 3 \\right)} - 2 \\left(2 x^{3} - x^{2} + x - 2\\right) \\sin{\\left(2 x + 3 \\right)} + 2 e^{x}$.\nFinal answer: $\\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(6\\cdot x^{2} - 2\\cdot x + 1\\right)\\cdot \\cos{\\left(2\\cdot x + 3 \\right)} - \\left(4\\cdot x^{3} - 2\\cdot x^{2} + 2\\cdot x - 4\\right)\\cdot \\sin{\\left(2\\cdot x + 3 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(6\\cdot x^{2} - 2\\cdot x + 1\\right)\\cdot \\cos{\\left(2\\cdot x + 3 \\right)} - \\left(4\\cdot x^{3} - 2\\cdot x^{2} + 2\\cdot x - 4\\right)\\cdot \\sin{\\left(2\\cdot x + 3 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1dc723041bd30b81", "problem_id": "c645f6736c50278d", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 49 miles per hour for 1 hours, then at 28 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 49 × 1 = 49 miles.\nSecond leg distance = 28 × 1 = 28 miles.\nTotal distance = 49 + 28 = 77 miles.\nCheck: adding both legs gives 77.\nFinal answer: $77$", "final_answer": "77", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "08e357228ec42389", "problem_id": "c645f6736c50278d", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "A cyclist rides at 49 miles per hour for 1 hours, then at 28 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 49 × 1 = 49 miles.\nSecond leg distance = 28 × 1 = 28 miles.\nTotal distance = 49 + 28 = 77 miles.\nFinal answer: $77$", "final_answer": "77", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e21dcce584c7f5d3", "problem_id": "959406ec0ac6c2d0", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} - x^{3} - 2 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{3} - 2 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3$.\nEvaluate at the bounds: F(\\pi) and F(0).\nSubtract: F(\\pi) − F(0) = $\\frac{\\pi \\left(- 3 \\pi^{3} - 8 \\pi^{2} + 36\\right)}{12}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{\\pi\\cdot \\left(- 3\\cdot \\pi^{3} - 8\\cdot \\pi^{2} + 36\\right)}{12}$", "final_answer": "\\frac{\\pi\\cdot \\left(- 3\\cdot \\pi^{3} - 8\\cdot \\pi^{2} + 36\\right)}{12}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "41b3b1aeb0f5b543", "problem_id": "959406ec0ac6c2d0", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} - x^{3} - 2 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{3} - 2 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3$.\n2) Evaluate at the bounds: F(\\pi) and F(0).\n3) Subtract: F(\\pi) − F(0) = $\\frac{\\pi \\left(- 3 \\pi^{3} - 8 \\pi^{2} + 36\\right)}{12}$.\n4) Simplify the result.\nFinal answer: $\\frac{\\pi\\cdot \\left(- 3\\cdot \\pi^{3} - 8\\cdot \\pi^{2} + 36\\right)}{12}$", "final_answer": "\\frac{\\pi\\cdot \\left(- 3\\cdot \\pi^{3} - 8\\cdot \\pi^{2} + 36\\right)}{12}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ab76296f599c3d33", "problem_id": "ce1fc64968c8f4d3", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} - x^{3} + 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{3} + 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{9 \\pi^{2}}{8} - 2 + \\frac{15 \\pi^{4}}{64} + \\frac{3 \\pi^{3}}{4}$.\n4) Simplify the result.\nFinal answer: $- \\frac{9\\cdot \\pi^{2}}{8} - 2 + \\frac{15\\cdot \\pi^{4}}{64} + \\frac{3\\cdot \\pi^{3}}{4}$", "final_answer": "- \\frac{9\\cdot \\pi^{2}}{8} - 2 + \\frac{15\\cdot \\pi^{4}}{64} + \\frac{3\\cdot \\pi^{3}}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3fed6f69886556eb", "problem_id": "ce1fc64968c8f4d3", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} - x^{3} + 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{3} + 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{9 \\pi^{2}}{8} - 2 + \\frac{15 \\pi^{4}}{64} + \\frac{3 \\pi^{3}}{4}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{9\\cdot \\pi^{2}}{8} - 2 + \\frac{15\\cdot \\pi^{4}}{64} + \\frac{3\\cdot \\pi^{3}}{4}$", "final_answer": "- \\frac{9\\cdot \\pi^{2}}{8} - 2 + \\frac{15\\cdot \\pi^{4}}{64} + \\frac{3\\cdot \\pi^{3}}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6bbbfc195f272ccf", "problem_id": "21f2bf50890565cc", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\n2) Evaluate at the bounds: F(\\pi) and F(0).\n3) Subtract: F(\\pi) − F(0) = $4 + 2 \\pi + \\frac{3 \\pi^{2}}{2} + \\frac{2 \\pi^{3}}{3}$.\n4) Simplify the result.\nFinal answer: $4 + 2\\cdot \\pi + \\frac{3\\cdot \\pi^{2}}{2} + \\frac{2\\cdot \\pi^{3}}{3}$", "final_answer": "4 + 2\\cdot \\pi + \\frac{3\\cdot \\pi^{2}}{2} + \\frac{2\\cdot \\pi^{3}}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "43e2883aa48f4a83", "problem_id": "d62e0e0ca7029ce5", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} - x - 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x^{2}) = 2 x^{2}$.\n3) So $f'(x) = 2 x^{2} - 1$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x^{2}) = 2 x$.\nTherefore the correct derivative is $f'(x) = 2 x - 1$.\nFinal answer: $2 x - 1$", "final_answer": "2 x - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "29b23d79112d9514", "problem_id": "b20058bb503a43ff", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Two friends split $630 in the ratio 7:8. How much does each receive?", "output": "1) Total ratio parts = 7+8 = 15.\n2) Each part = 630 ÷ 15 = 42.\n3) First share = 7 × 42 = 294.\n4) Second share = 8 × 42 = 336.\nFinal answer: $336$", "final_answer": "336", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7266ec87fc1a93fb", "problem_id": "b20058bb503a43ff", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $630 in the ratio 7:8. How much does each receive?", "output": "Total ratio parts = 7+8 = 15.\nEach part = 630 ÷ 15 = 42.\nFirst share = 7 × 42 = 294.\nSecond share = 8 × 42 = 336.\nCheck: 294+336=630.\nFinal answer: $336$", "final_answer": "336", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e80e40a5206df8b2", "problem_id": "4722070b7decbc79", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$- x + 3 y$ = $-18$\n$3 x - 5 y$ = $30$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $3$: $- 3 x + 9 y$ = $-54$.\n3) Multiply the second equation by $-1$: $- 3 x + 5 y$ = $-30$.\n4) Subtract to eliminate x: $4 y$ = $-24$.\n5) Solve for y: y = $-6$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, -6\\right)$", "final_answer": "(x,y)=\\left(0, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cb1904ece6590ce5", "problem_id": "6b0e3c2fd06ac29e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x \\sin{\\left(x \\right)}) = x \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x \\sin{\\left(x \\right)}) = x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.\nFinal answer: $- 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$", "final_answer": "- 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "1097814781934015", "problem_id": "6d4f3d6b6affd527", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{10} x}{2} + \\frac{19 x}{10} - \\frac{3}{2} + \\frac{3 \\sqrt{10}}{10}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{19}{10} - \\frac{\\sqrt{10}}{2}$, c=$- \\frac{3}{2} + \\frac{3 \\sqrt{10}}{10}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{1211}{100} - \\frac{31 \\sqrt{10}}{10}$, so √Δ = $\\sqrt{\\frac{1211}{100} - \\frac{31 \\sqrt{10}}{10}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{19}{20} + \\frac{\\sqrt{1211 - 310 \\sqrt{10}}}{20} + \\frac{\\sqrt{10}}{4}$.\nx2 = $- \\frac{19}{20} - \\frac{\\sqrt{1211 - 310 \\sqrt{10}}}{20} + \\frac{\\sqrt{10}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{19}{20} + \\frac{\\sqrt{1211 - 310\\cdot \\sqrt{10}}}{20} + \\frac{\\sqrt{10}}{4}, - \\frac{19}{20} - \\frac{\\sqrt{1211 - 310\\cdot \\sqrt{10}}}{20} + \\frac{\\sqrt{10}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{19}{20} + \\frac{\\sqrt{1211 - 310\\cdot \\sqrt{10}}}{20} + \\frac{\\sqrt{10}}{4}, - \\frac{19}{20} - \\frac{\\sqrt{1211 - 310\\cdot \\sqrt{10}}}{20} + \\frac{\\sqrt{10}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e973f313d9f3a2c2", "problem_id": "6d4f3d6b6affd527", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{10} x}{2} + \\frac{19 x}{10} - \\frac{3}{2} + \\frac{3 \\sqrt{10}}{10}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{19}{10} - \\frac{\\sqrt{10}}{2}$, c=$- \\frac{3}{2} + \\frac{3 \\sqrt{10}}{10}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{1211}{100} - \\frac{31 \\sqrt{10}}{10}$, so √Δ = $\\sqrt{\\frac{1211}{100} - \\frac{31 \\sqrt{10}}{10}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{19}{20} + \\frac{\\sqrt{1211 - 310 \\sqrt{10}}}{20} + \\frac{\\sqrt{10}}{4}$.\n7) x2 = $- \\frac{19}{20} - \\frac{\\sqrt{1211 - 310 \\sqrt{10}}}{20} + \\frac{\\sqrt{10}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{19}{20} + \\frac{\\sqrt{1211 - 310\\cdot \\sqrt{10}}}{20} + \\frac{\\sqrt{10}}{4}, - \\frac{19}{20} - \\frac{\\sqrt{1211 - 310\\cdot \\sqrt{10}}}{20} + \\frac{\\sqrt{10}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{19}{20} + \\frac{\\sqrt{1211 - 310\\cdot \\sqrt{10}}}{20} + \\frac{\\sqrt{10}}{4}, - \\frac{19}{20} - \\frac{\\sqrt{1211 - 310\\cdot \\sqrt{10}}}{20} + \\frac{\\sqrt{10}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b051eb323e08c4f2", "problem_id": "2e34f9f7382fa7c6", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 20 x + 48$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} - 20 x + 48$ = $2 \\left(x - 6\\right) \\left(x - 4\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $6$.\nFrom the second factor: x = $4$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{6, 4\\}$", "final_answer": "x \\in \\{6, 4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a7b0783639cd8087", "problem_id": "2e34f9f7382fa7c6", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 20 x + 48$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 20 x + 48$ = $2 \\left(x - 6\\right) \\left(x - 4\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $6$.\n5) From the second factor: x = $4$.\nFinal answer: $x \\in \\{6, 4\\}$", "final_answer": "x \\in \\{6, 4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "91b1c808d0ad2985", "problem_id": "aaf3e8a53c54a189", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A cyclist rides at 79 miles per hour for 2 hours, then at 54 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 79 × 2 = 158 miles.\n2) Second leg distance = 54 × 3/2 = 81 miles.\n3) Total distance = 158 + 81 = 239 miles.\nFinal answer: $239$", "final_answer": "239", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c825e0c035a6d0d1", "problem_id": "aaf3e8a53c54a189", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "A cyclist rides at 79 miles per hour for 2 hours, then at 54 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 79 × 2 = 158 miles.\nSecond leg distance = 54 × 3/2 = 81 miles.\nTotal distance = 158 + 81 = 239 miles.\nFinal answer: $239$", "final_answer": "239", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7261db0651413a8d", "problem_id": "52e2d62e1df78cf9", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A cyclist rides at 25 miles per hour for 3/2 hours, then at 57 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 25 × 3/2 = 75/2 miles.\n2) Second leg distance = 57 × 2 = 114 miles.\n3) Total distance = 75/2 + 114 = 303/2 miles.\nFinal answer: $\\frac{303}{2}$", "final_answer": "\\frac{303}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "05582209bc1b2479", "problem_id": "0afa354efc24e7f3", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - 3 x^{2} - 11 x - 43$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} - 12 x - 43.\n3) Then the limit is -139.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: -135.\nFinal answer: $-135$", "final_answer": "-135", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "9abadac447c4e586", "problem_id": "1177be0060a63a57", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_7=27$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_7=20.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=20 and k=7.\n4) Compute C(26,6) = 230230.\nFinal answer: $230230$", "final_answer": "230230", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2b0b8d95059298b4", "problem_id": "dfe45db6d235951c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x \\sin{\\left(x \\right)}) = - x \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x \\sin{\\left(x \\right)}) = - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$", "final_answer": "- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "52351196ac31e2c7", "problem_id": "c4aa75c753ae9dfe", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x^{2} - 5 x - 13$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 2 x^{3} - x^{2} - 3 x + 2$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 6 x^{2} - 2 x - 3$.\n5) Evaluate at a: p'(2) = $-31$.\nFinal answer: $-31$", "final_answer": "-31", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b2d5e0987d402094", "problem_id": "c4aa75c753ae9dfe", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x^{2} - 5 x - 13$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 2 x^{3} - x^{2} - 3 x + 2$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-31$", "final_answer": "-31", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0adadcbcf7bd8442", "problem_id": "056816228bbfd396", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the multiplicative inverse of 5 modulo 23.", "output": "1) We want x such that 5x ≡ 1 (mod 23).\n2) Use extended Euclid to find s,t with 5s + 23t = 1.\n3) One Bezout identity is: 5·(-9) + 23·(2) = 1.\n4) Reduce s modulo 23: x ≡ 14 (mod 23).\nFinal answer: $x\\equiv 14\\pmod{23}$", "final_answer": "x\\equiv 14\\pmod{23}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "44089b93ca37d4bf", "problem_id": "056816228bbfd396", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the multiplicative inverse of 5 modulo 23.", "output": "We want x such that 5x ≡ 1 (mod 23).\nUse extended Euclid to find s,t with 5s + 23t = 1.\nOne Bezout identity is: 5·(-9) + 23·(2) = 1.\nReduce s modulo 23: x ≡ 14 (mod 23).\nCheck: 5·14 mod 23 = 1.\nFinal answer: $x\\equiv 14\\pmod{23}$", "final_answer": "x\\equiv 14\\pmod{23}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f4c66ce35192386e", "problem_id": "d499965c86bd4c17", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 16 times, what is the probability of exactly 7 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=16, k=7, p=$\\frac{1}{2}$ and simplify.\nHere C(16,7) = 11440.\nCheck: expression simplifies to $\\frac{715}{4096}$.\nFinal answer: $\\frac{715}{4096}$", "final_answer": "\\frac{715}{4096}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7a11ba583ca27626", "problem_id": "d499965c86bd4c17", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 16 times, what is the probability of exactly 7 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=16, k=7, p=$\\frac{1}{2}$ and simplify.\n4) Here C(16,7) = 11440.\nFinal answer: $\\frac{715}{4096}$", "final_answer": "\\frac{715}{4096}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "68c214df043dfb45", "problem_id": "f08410af077897e5", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 2 x - \\frac{2 \\sqrt{6} x}{3} + \\frac{5}{9} + \\frac{10 \\sqrt{6}}{9}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$-2 - \\frac{2 \\sqrt{6}}{3}$, c=$\\frac{5}{9} + \\frac{10 \\sqrt{6}}{9}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{40}{9} - \\frac{16 \\sqrt{6}}{9}$, so √Δ = $\\sqrt{\\frac{40}{9} - \\frac{16 \\sqrt{6}}{9}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{10 - 4 \\sqrt{6}}}{3} + \\frac{\\sqrt{6}}{3} + 1$.\n7) x2 = $- \\frac{\\sqrt{10 - 4 \\sqrt{6}}}{3} + \\frac{\\sqrt{6}}{3} + 1$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{10 - 4\\cdot \\sqrt{6}}}{3} + \\frac{\\sqrt{6}}{3} + 1, - \\frac{\\sqrt{10 - 4\\cdot \\sqrt{6}}}{3} + \\frac{\\sqrt{6}}{3} + 1\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{10 - 4\\cdot \\sqrt{6}}}{3} + \\frac{\\sqrt{6}}{3} + 1, - \\frac{\\sqrt{10 - 4\\cdot \\sqrt{6}}}{3} + \\frac{\\sqrt{6}}{3} + 1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0a04f16ab29b0483", "problem_id": "56ad6eb37304924e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 \\cos{\\left(2 x \\right)}) = - 2 \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 \\cos{\\left(2 x \\right)}) = - 4 \\sin{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$", "final_answer": "- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "1326031abba9d6d0", "problem_id": "683fef02f3e46d41", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 4 x^{3} - 2 x^{2} - 4 x + 4$.", "output": "1) Differentiate term-by-term:\n2) d/dx(4) = 0.\n3) d/dx(- 4 x) = -4.\n4) d/dx(- 4 x^{3}) = - 12 x^{2}.\n5) d/dx(- 2 x^{2}) = - 4 x.\n6) Combine: f'(x) = 4 \\left(- 3 x^{2} - x - 1\\right).\nFinal answer: $- 12\\cdot x^{2} - 4\\cdot x - 4$", "final_answer": "- 12\\cdot x^{2} - 4\\cdot x - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7f215dc33169393b", "problem_id": "683fef02f3e46d41", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 4 x^{3} - 2 x^{2} - 4 x + 4$.", "output": "$- 12\\cdot x^{2} - 4\\cdot x - 4$", "final_answer": "- 12\\cdot x^{2} - 4\\cdot x - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fc8e483e36bbf0c7", "problem_id": "15709b98c07e5cb9", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A store sells a backpack for $121. It is discounted by 25%. You buy 8 backpacks. What is the total cost?", "output": "1) Discounted price = 121 × (1 − 25/100) = 121 × 75/100 = 363/4.\n2) Subtotal = 8 × 363/4 = 726.\nFinal answer: $726$", "final_answer": "726", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a495c7c1773f932d", "problem_id": "15709b98c07e5cb9", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $121. It is discounted by 25%. You buy 8 backpacks. What is the total cost?", "output": "Discounted price = 121 × (1 − 25/100) = 121 × 75/100 = 363/4.\nSubtotal = 8 × 363/4 = 726.\nCheck: recomputing the formula yields 726.\nFinal answer: $726$", "final_answer": "726", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3e84d9283a3e62b6", "problem_id": "caaec681c642f172", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{2} x + \\frac{5 x}{2} - 2 \\sqrt{2} + 1$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{5}{2} - \\sqrt{2}$, c=$1 - 2 \\sqrt{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $3 \\sqrt{2} + \\frac{17}{4}$, so √Δ = $\\sqrt{3 \\sqrt{2} + \\frac{17}{4}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{5}{4} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{12 \\sqrt{2} + 17}}{4}$.\n7) x2 = $- \\frac{\\sqrt{12 \\sqrt{2} + 17}}{4} - \\frac{5}{4} + \\frac{\\sqrt{2}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{5}{4} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{12\\cdot \\sqrt{2} + 17}}{4}, - \\frac{\\sqrt{12\\cdot \\sqrt{2} + 17}}{4} - \\frac{5}{4} + \\frac{\\sqrt{2}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{5}{4} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{12\\cdot \\sqrt{2} + 17}}{4}, - \\frac{\\sqrt{12\\cdot \\sqrt{2} + 17}}{4} - \\frac{5}{4} + \\frac{\\sqrt{2}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3be6a468b2a70be0", "problem_id": "caaec681c642f172", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{2} x + \\frac{5 x}{2} - 2 \\sqrt{2} + 1$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{5}{2} - \\sqrt{2}$, c=$1 - 2 \\sqrt{2}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $3 \\sqrt{2} + \\frac{17}{4}$, so √Δ = $\\sqrt{3 \\sqrt{2} + \\frac{17}{4}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{5}{4} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{12 \\sqrt{2} + 17}}{4}$.\nx2 = $- \\frac{\\sqrt{12 \\sqrt{2} + 17}}{4} - \\frac{5}{4} + \\frac{\\sqrt{2}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{5}{4} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{12\\cdot \\sqrt{2} + 17}}{4}, - \\frac{\\sqrt{12\\cdot \\sqrt{2} + 17}}{4} - \\frac{5}{4} + \\frac{\\sqrt{2}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{5}{4} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{12\\cdot \\sqrt{2} + 17}}{4}, - \\frac{\\sqrt{12\\cdot \\sqrt{2} + 17}}{4} - \\frac{5}{4} + \\frac{\\sqrt{2}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d56a3982fd51ce68", "problem_id": "72605bd07164b5a5", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{3} - 4 x^{2} + 4 x + 1\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 3 x^{3} - 4 x^{2} + 4 x + 1$.\n3) Compute an antiderivative: F(x) = $- \\frac{3 x^{4}}{4} - \\frac{4 x^{3}}{3} + 2 x^{2} + x$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{3\\cdot x^{4}}{4} - \\frac{4\\cdot x^{3}}{3} + 2\\cdot x^{2} + x$", "final_answer": "C - \\frac{3\\cdot x^{4}}{4} - \\frac{4\\cdot x^{3}}{3} + 2\\cdot x^{2} + x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "08b4cac1d4fe2ddb", "problem_id": "72605bd07164b5a5", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{3} - 4 x^{2} + 4 x + 1\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 3 x^{3} - 4 x^{2} + 4 x + 1$.\nCompute an antiderivative: F(x) = $- \\frac{3 x^{4}}{4} - \\frac{4 x^{3}}{3} + 2 x^{2} + x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - \\frac{3\\cdot x^{4}}{4} - \\frac{4\\cdot x^{3}}{3} + 2\\cdot x^{2} + x$", "final_answer": "C - \\frac{3\\cdot x^{4}}{4} - \\frac{4\\cdot x^{3}}{3} + 2\\cdot x^{2} + x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ef0192bb681a076f", "problem_id": "0e94698edb22bdb2", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 2 x - 7$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to x - 7.\n3) Then the limit is -10.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -13.\nFinal answer: $-13$", "final_answer": "-13", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a36c7bcacf5c2ade", "problem_id": "1fc70041284e54aa", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} - 2 x^{3} - 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{3} - 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $- \\frac{\\pi^{3}}{6}$.\n4) Simplify the result.\nFinal answer: $- \\frac{\\pi^{3}}{6}$", "final_answer": "- \\frac{\\pi^{3}}{6}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "520c2f64a1ba6398", "problem_id": "4515d91ae660cecc", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - 2 x^{2} + 3 x - 8$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 2 x^{3} - x^{2} - 2 x - 1$ and a = -2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 6 x^{2} - 2 x - 2$.\n5) Evaluate at a: p'(-2) = $-22$.\nFinal answer: $-22$", "final_answer": "-22", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2505c022a3ff4eca", "problem_id": "046f2807e22ccc9d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{3} \\sin{\\left(x \\right)} - x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 \\cos{\\left(2 x \\right)}) = - 2 \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = 3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 \\cos{\\left(2 x \\right)}) = - 4 \\sin{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = 3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$.\nFinal answer: $3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$", "final_answer": "3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "4342448aaedb7f7c", "problem_id": "e4bdbe5d6712457b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{3 x^{2} - x + \\left(- 3 x^{2} + 2 x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{- 3 x^{2} + 2 x + 5}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{3 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- 3 x^{2} + 2 x\\right) + 5}) = \\frac{3 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x^{2} - 2 x - 5}$.\n3) So $f'(x) = \\frac{9 x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} + \\frac{18 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} - \\frac{12 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} - \\frac{24 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} + \\frac{18 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{4} - 12 x^{3} - 26 x^{2} + 20 x + 25} - \\frac{26 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} - \\frac{52 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} + \\frac{6 x^{2}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} - \\frac{6 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{4} - 12 x^{3} - 26 x^{2} + 20 x + 25} + \\frac{3 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x^{2} - 2 x - 5} + \\frac{3 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{- 6 x^{2} + 4 x + 10} + \\frac{3 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(- 3 x^{2} + 2 x\\right) + 5} + \\frac{20 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} + \\frac{40 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} + \\frac{96 x}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} + \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- 3 x^{2} + 2 x\\right) + 5} + \\frac{25 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} + \\frac{50 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} - \\frac{22}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{3 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- 3 x^{2} + 2 x\\right) + 5}) = - \\frac{3 x^{2} \\left(6 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(- 3 x^{2} + 2 x\\right) + 5\\right)^{2}} - \\frac{3 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(- 3 x^{2} + 2 x\\right) + 5\\right)} - \\frac{3 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(- 3 x^{2} + 2 x\\right) + 5} - \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- 3 x^{2} + 2 x\\right) + 5}$.\nTherefore the correct derivative is $f'(x) = \\frac{4 \\left(3 x - 1\\right) \\left(3 x^{2} - x + \\left(x \\left(2 - 3 x\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(2 - 3 x\\right) + 5\\right) \\left(12 x + 4 \\left(1 - 3 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 - 3 x\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 - 3 x\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x \\left(2 - 3 x\\right) + 5\\right)^{2}}$.\nFinal answer: $\\frac{4 \\left(3 x - 1\\right) \\left(3 x^{2} - x + \\left(x \\left(2 - 3 x\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(2 - 3 x\\right) + 5\\right) \\left(12 x + 4 \\left(1 - 3 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 - 3 x\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 - 3 x\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x \\left(2 - 3 x\\right) + 5\\right)^{2}}$", "final_answer": "\\frac{4 \\left(3 x - 1\\right) \\left(3 x^{2} - x + \\left(x \\left(2 - 3 x\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(2 - 3 x\\right) + 5\\right) \\left(12 x + 4 \\left(1 - 3 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 - 3 x\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 - 3 x\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x \\left(2 - 3 x\\right) + 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ac43e4e29346821b", "problem_id": "7295015566a322d2", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the multiplicative inverse of 4 modulo 37.", "output": "1) We want x such that 4x ≡ 1 (mod 37).\n2) Use extended Euclid to find s,t with 4s + 37t = 1.\n3) One Bezout identity is: 4·(-9) + 37·(1) = 1.\n4) Reduce s modulo 37: x ≡ 28 (mod 37).\nFinal answer: $x\\equiv 28\\pmod{37}$", "final_answer": "x\\equiv 28\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eae89126bace20ed", "problem_id": "668ad1f643feb0a1", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "You want 40 liters of a 30% salt solution. You have a 10% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 40−x liters is 60% solution.\nSalt equation: 10x + 60(40−x) = 30·40.\n(10-60)x = (30-60)·40 ⇒ x = 24.\nFinal answer: $24 L of 10% and 16 L of 60%$", "final_answer": "24 L of 10% and 16 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "872a94204c5a45b0", "problem_id": "668ad1f643feb0a1", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "You want 40 liters of a 30% salt solution. You have a 10% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 40−x liters is 60% solution.\nSalt equation: 10x + 60(40−x) = 30·40.\n(10-60)x = (30-60)·40 ⇒ x = 24.\nThen 40−x = 16.\nCheck: 10·24 + 60·16 = 30·40.\nFinal answer: $24 L of 10% and 16 L of 60%$", "final_answer": "24 L of 10% and 16 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0916c35b1f241aa0", "problem_id": "68815efc994aa640", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{7 x}{3} - \\sqrt{5} x + \\frac{4}{3} + \\sqrt{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{7}{3} - \\sqrt{5}$, c=$\\frac{4}{3} + \\sqrt{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{2 \\sqrt{5}}{3} + \\frac{46}{9}$, so √Δ = $\\sqrt{\\frac{2 \\sqrt{5}}{3} + \\frac{46}{9}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{5}}{2} + \\frac{7}{6} + \\frac{\\sqrt{6 \\sqrt{5} + 46}}{6}$.\n7) x2 = $- \\frac{\\sqrt{6 \\sqrt{5} + 46}}{6} + \\frac{\\sqrt{5}}{2} + \\frac{7}{6}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{5}}{2} + \\frac{7}{6} + \\frac{\\sqrt{6\\cdot \\sqrt{5} + 46}}{6}, - \\frac{\\sqrt{6\\cdot \\sqrt{5} + 46}}{6} + \\frac{\\sqrt{5}}{2} + \\frac{7}{6}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{5}}{2} + \\frac{7}{6} + \\frac{\\sqrt{6\\cdot \\sqrt{5} + 46}}{6}, - \\frac{\\sqrt{6\\cdot \\sqrt{5} + 46}}{6} + \\frac{\\sqrt{5}}{2} + \\frac{7}{6}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a80c58bd4be1327f", "problem_id": "552ce9f80be21247", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{2} + 3 x + 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 3 x^{2}) = - 6 x^{2}$.\n3) So $f'(x) = 3 - 6 x^{2}$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 3 x^{2}) = - 6 x$.\nTherefore the correct derivative is $f'(x) = 3 \\left(1 - 2 x\\right)$.\nFinal answer: $3 \\left(1 - 2 x\\right)$", "final_answer": "3 \\left(1 - 2 x\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "37b8ef5ee122dce9", "problem_id": "b0ba41fbaa153da4", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "An urn has 14 red balls and 17 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "$\\frac{476476}{1550775}$", "final_answer": "\\frac{476476}{1550775}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "71edacbf6c83a9a4", "problem_id": "b0ba41fbaa153da4", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "An urn has 14 red balls and 17 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(31,9).\n3) Favorable samples: choose k reds and n−k blues: C(14,4)·C(17,5).\n4) Compute totals: C(31,9) = 20160075, and favorable = 6194188.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{476476}{1550775}$", "final_answer": "\\frac{476476}{1550775}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "95e124fce34aed38", "problem_id": "f815b7f0c064432c", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} 3 x^{4} - x^{3} - 3 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{4} - x^{3} - 3 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2$.\n2) Evaluate at the bounds: F(\\pi) and F(0).\n3) Subtract: F(\\pi) − F(0) = $\\frac{\\pi \\left(- 20 \\pi^{2} - 5 \\pi^{3} + 40 + 30 \\pi + 12 \\pi^{4}\\right)}{20}$.\n4) Simplify the result.\nFinal answer: $\\frac{\\pi\\cdot \\left(- 20\\cdot \\pi^{2} - 5\\cdot \\pi^{3} + 40 + 30\\cdot \\pi + 12\\cdot \\pi^{4}\\right)}{20}$", "final_answer": "\\frac{\\pi\\cdot \\left(- 20\\cdot \\pi^{2} - 5\\cdot \\pi^{3} + 40 + 30\\cdot \\pi + 12\\cdot \\pi^{4}\\right)}{20}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ceee08193528dc85", "problem_id": "f815b7f0c064432c", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} 3 x^{4} - x^{3} - 3 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{4} - x^{3} - 3 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2$.\nEvaluate at the bounds: F(\\pi) and F(0).\nSubtract: F(\\pi) − F(0) = $\\frac{\\pi \\left(- 20 \\pi^{2} - 5 \\pi^{3} + 40 + 30 \\pi + 12 \\pi^{4}\\right)}{20}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{\\pi\\cdot \\left(- 20\\cdot \\pi^{2} - 5\\cdot \\pi^{3} + 40 + 30\\cdot \\pi + 12\\cdot \\pi^{4}\\right)}{20}$", "final_answer": "\\frac{\\pi\\cdot \\left(- 20\\cdot \\pi^{2} - 5\\cdot \\pi^{3} + 40 + 30\\cdot \\pi + 12\\cdot \\pi^{4}\\right)}{20}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ffdfc35db89ec25f", "problem_id": "8d32dcda12ef8223", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 32 x + 126$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} + 32 x + 126$ = $2 \\left(x + 7\\right) \\left(x + 9\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-7$.\n5) From the second factor: x = $-9$.\nFinal answer: $x \\in \\{-7, -9\\}$", "final_answer": "x \\in \\{-7, -9\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c39313805492de3c", "problem_id": "8d32dcda12ef8223", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 32 x + 126$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} + 32 x + 126$ = $2 \\left(x + 7\\right) \\left(x + 9\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFinal answer: $x \\in \\{-7, -9\\}$", "final_answer": "x \\in \\{-7, -9\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cdc751a04fb36946", "problem_id": "6266a9e1f9700524", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{25 x}{3} - 3$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$- \\frac{25}{3}$, c=$-3$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{841}{9}$, so √Δ = $\\frac{29}{3}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{9}{2}$.\n7) x2 = $- \\frac{1}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{9}{2}, - \\frac{1}{3}\\}$", "final_answer": "x \\in \\{\\frac{9}{2}, - \\frac{1}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "51eedcb4119f8dc9", "problem_id": "6266a9e1f9700524", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{25 x}{3} - 3$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$- \\frac{25}{3}$, c=$-3$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{841}{9}$, so √Δ = $\\frac{29}{3}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{9}{2}$.\nx2 = $- \\frac{1}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{9}{2}, - \\frac{1}{3}\\}$", "final_answer": "x \\in \\{\\frac{9}{2}, - \\frac{1}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "86d62258a56b3a87", "problem_id": "1b3279da9aef3ca3", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{3} + 2 x^{2} + 2 x + 3 \\left(x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 \\left(x - 1\\right)}$.", "output": "1) Let f(x) = $\\frac{x^{3} + 2 x^{2} + 2 x + 3 \\left(x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 \\left(x - 1\\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{- 2 x^{3} - 4 x^{2} - 4 x + \\left(x - 1\\right) \\left(6 x^{2} + 8 x + 3 \\left(x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x - 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 6 \\left(x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 \\left(x - 1\\right)^{2}}$.\nFinal answer: $\\frac{- 2\\cdot x^{3} - 4\\cdot x^{2} - 4\\cdot x + \\left(x - 1\\right)\\cdot \\left(6\\cdot x^{2} + 8\\cdot x + \\left(3\\cdot x - 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 4\\right) - \\left(6\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)}}{6\\cdot \\left(x - 1\\right)^{2}}$", "final_answer": "\\frac{- 2\\cdot x^{3} - 4\\cdot x^{2} - 4\\cdot x + \\left(x - 1\\right)\\cdot \\left(6\\cdot x^{2} + 8\\cdot x + \\left(3\\cdot x - 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 4\\right) - \\left(6\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)}}{6\\cdot \\left(x - 1\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "af08e22514863887", "problem_id": "2f98b98631d9deb2", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 2 x^{3} - 5 x^{2} + 2 x - 5$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} - 5 x^{2} + x - 5.\n3) Then the limit is -13.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: -14.\nFinal answer: $-14$", "final_answer": "-14", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "dbb86821cf30e772", "problem_id": "caa2030059bd0932", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A cyclist rides at 90 miles per hour for 1/2 hours, then at 37 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 90 × 1/2 = 45 miles.\n2) Second leg distance = 37 × 5/2 = 185/2 miles.\n3) Total distance = 45 + 185/2 = 275/2 miles.\nFinal answer: $\\frac{275}{2}$", "final_answer": "\\frac{275}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1fc7a6131cc0ed59", "problem_id": "4303602b72d4f841", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} 3 x^{3} - 11 x^{2} + 47 x - 191$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} - 11 x^{2} + 46 x - 191.\n3) Then the limit is -743.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: -747.\nFinal answer: $-747$", "final_answer": "-747", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c17c93349ed114f8", "problem_id": "d4508b3a9beae98c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{2} - 3 x + \\left(x^{2} - 3 x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3 x + 5}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3 x + 5}) = - \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3 x + 5}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} - \\frac{6 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} - \\frac{12 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} + \\frac{19 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} + \\frac{38 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} - \\frac{6 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 6 x^{3} + 19 x^{2} - 30 x + 25} - \\frac{30 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} - \\frac{60 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} + \\frac{20 x}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} + \\frac{9 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 6 x^{3} + 19 x^{2} - 30 x + 25} + \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 6 x + 10} + \\frac{25 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} + \\frac{50 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} - \\frac{30}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3 x + 5}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3 x + 5}) = - \\frac{3 x \\left(3 - 2 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - 3 x + 5\\right)^{2}} - \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x^{2} - 3 x + 5\\right)} - \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3 x + 5} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3 x + 5}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(3 - 2 x\\right) \\left(x^{2} - 3 x + \\left(x \\left(x - 3\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x \\left(x - 3\\right) + 5\\right) \\left(4 x + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 3\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 3\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x \\left(x - 3\\right) + 5\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(3 - 2 x\\right) \\left(x^{2} - 3 x + \\left(x \\left(x - 3\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x \\left(x - 3\\right) + 5\\right) \\left(4 x + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 3\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 3\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x \\left(x - 3\\right) + 5\\right)^{2}}$", "final_answer": "\\frac{2 \\left(3 - 2 x\\right) \\left(x^{2} - 3 x + \\left(x \\left(x - 3\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x \\left(x - 3\\right) + 5\\right) \\left(4 x + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 3\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 3\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x \\left(x - 3\\right) + 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "095b449f5ba1c14d", "problem_id": "8f8d0bfaa94de307", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡1 (mod 7), x≡5 (mod 11). Give the solution modulo 77.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 77.\n3) Compute the CRT solution to get x ≡ 71 (mod 77).\nFinal answer: $x\\equiv 71\\pmod{77}$", "final_answer": "x\\equiv 71\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dffbcf5541ec93d2", "problem_id": "af85fb2283c708c2", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Worker A finishes a job in 6 hours and Worker B finishes it in 17 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/6 job/hour. B's rate = 1/17 job/hour.\n2) Combined rate = 1/6 + 1/17 = 23/102 job/hour.\n3) Time = 1 ÷ (combined rate) = 102/23 hours.\nFinal answer: $\\frac{102}{23}$", "final_answer": "\\frac{102}{23}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "436e170f91fdd129", "problem_id": "af85fb2283c708c2", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Worker A finishes a job in 6 hours and Worker B finishes it in 17 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/6 job/hour. B's rate = 1/17 job/hour.\nCombined rate = 1/6 + 1/17 = 23/102 job/hour.\nTime = 1 ÷ (combined rate) = 102/23 hours.\nCheck: (combined rate)×(time) = 23/102×102/23 = 1.\nFinal answer: $\\frac{102}{23}$", "final_answer": "\\frac{102}{23}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bd17be1ddc68e6fa", "problem_id": "8ad6637a51cd0757", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A store sells a backpack for $77. It is discounted by 10%. You buy 3 backpacks. What is the total cost?", "output": "1) Discounted price = 77 × (1 − 10/100) = 77 × 90/100 = 693/10.\n2) Subtotal = 3 × 693/10 = 2079/10.\nFinal answer: $\\frac{2079}{10}$", "final_answer": "\\frac{2079}{10}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4355e44fa85df775", "problem_id": "8ad6637a51cd0757", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $77. It is discounted by 10%. You buy 3 backpacks. What is the total cost?", "output": "Discounted price = 77 × (1 − 10/100) = 77 × 90/100 = 693/10.\nSubtotal = 3 × 693/10 = 2079/10.\nCheck: recomputing the formula yields 2079/10.\nFinal answer: $\\frac{2079}{10}$", "final_answer": "\\frac{2079}{10}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "23029be3ba96fca4", "problem_id": "465af7df0faf50fc", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "How many binary strings of length 21 contain exactly 3 ones with no two ones adjacent?", "output": "1) Reserve 2 mandatory zeros between the 3 ones.\n2) This leaves n−k+1 = 19 valid slots for the ones.\n3) Choose the slots: C(19,3) = 969.\nFinal answer: $969$", "final_answer": "969", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "94cc527d2b7649ea", "problem_id": "465af7df0faf50fc", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many binary strings of length 21 contain exactly 3 ones with no two ones adjacent?", "output": "Reserve 2 mandatory zeros between the 3 ones.\nThis leaves n−k+1 = 19 valid slots for the ones.\nChoose the slots: C(19,3) = 969.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $969$", "final_answer": "969", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "030bc560a1995809", "problem_id": "a062b15ed9c0c5a2", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 40 liters of a 30% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 40−x liters is 50% solution.\nSalt equation: 25x + 50(40−x) = 30·40.\n(25-50)x = (30-50)·40 ⇒ x = 32.\nThen 40−x = 8.\nCheck: 25·32 + 50·8 = 30·40.\nFinal answer: $32 L of 25% and 8 L of 50%$", "final_answer": "32 L of 25% and 8 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bc42ae6bbb256190", "problem_id": "a062b15ed9c0c5a2", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "You want 40 liters of a 30% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 40−x liters is 50% solution.\n2) Salt equation: 25x + 50(40−x) = 30·40.\n3) (25-50)x = (30-50)·40 ⇒ x = 32.\n4) Then 40−x = 8.\nFinal answer: $32 L of 25% and 8 L of 50%$", "final_answer": "32 L of 25% and 8 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1aea1f67b34c2eb3", "problem_id": "01c1b2320cfb64cc", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_9=36$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_9=18.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=18 and k=9.\n4) Compute C(26,8) = 1562275.\nFinal answer: $1562275$", "final_answer": "1562275", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e95a44a58244c7b6", "problem_id": "01c1b2320cfb64cc", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_9=36$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_9=18.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=18 and k=9.\nCompute C(26,8) = 1562275.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $1562275$", "final_answer": "1562275", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e4f8146ade4c7dc7", "problem_id": "c5e9a77041eb350d", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A disease has prevalence $\\frac{1}{25}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{43}{500}$.\n3) Compute P(D|+) = $\\frac{19}{43}$.\nFinal answer: $\\frac{19}{43}$", "final_answer": "\\frac{19}{43}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "944d63cb66ff3faf", "problem_id": "c5e9a77041eb350d", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A disease has prevalence $\\frac{1}{25}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{43}{500}$.\nCompute P(D|+) = $\\frac{19}{43}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{19}{43}$", "final_answer": "\\frac{19}{43}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "493b99497d34b4fc", "problem_id": "92b4aff7d4789ab6", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Two friends split $312 in the ratio 1:5. How much does each receive?", "output": "1) Total ratio parts = 1+5 = 6.\n2) Each part = 312 ÷ 6 = 52.\n3) First share = 1 × 52 = 52.\n4) Second share = 5 × 52 = 260.\nFinal answer: $260$", "final_answer": "260", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4a49c21852e22063", "problem_id": "d4508b3a9beae98c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{2} - 3 x + \\left(x^{2} - 3 x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3 x + 5}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3 x + 5}) = \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3 x + 5}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} - \\frac{6 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} - \\frac{12 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} + \\frac{19 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} + \\frac{38 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} - \\frac{30 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} - \\frac{60 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} + \\frac{20 x}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} + \\frac{10 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 6 x^{3} + 19 x^{2} - 30 x + 25} + \\frac{25 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} + \\frac{50 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} - \\frac{30}{2 x^{4} - 12 x^{3} + 38 x^{2} - 60 x + 50} - \\frac{15 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 6 x^{3} + 19 x^{2} - 30 x + 25} - \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 6 x + 10}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3 x + 5}) = \\frac{5 \\left(3 - 2 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - 3 x + 5\\right)^{2}} + \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x^{2} - 3 x + 5\\right)} + \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3 x + 5}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(3 - 2 x\\right) \\left(x^{2} - 3 x + \\left(x \\left(x - 3\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x \\left(x - 3\\right) + 5\\right) \\left(4 x + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 3\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 3\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x \\left(x - 3\\right) + 5\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(3 - 2 x\\right) \\left(x^{2} - 3 x + \\left(x \\left(x - 3\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x \\left(x - 3\\right) + 5\\right) \\left(4 x + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 3\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 3\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x \\left(x - 3\\right) + 5\\right)^{2}}$", "final_answer": "\\frac{2 \\left(3 - 2 x\\right) \\left(x^{2} - 3 x + \\left(x \\left(x - 3\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x \\left(x - 3\\right) + 5\\right) \\left(4 x + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 3\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 3\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x \\left(x - 3\\right) + 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "555b44c1aa0b1432", "problem_id": "0038ca07e2e1862d", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡8 (mod 9), x≡1 (mod 7). Give the solution modulo 63.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 63.\nCompute the CRT solution to get x ≡ 8 (mod 63).\nCheck: 8 mod 9=8 and mod 7=1.\nFinal answer: $x\\equiv 8\\pmod{63}$", "final_answer": "x\\equiv 8\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ffc7af87f91f795e", "problem_id": "0038ca07e2e1862d", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡8 (mod 9), x≡1 (mod 7). Give the solution modulo 63.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 63.\n3) Compute the CRT solution to get x ≡ 8 (mod 63).\nFinal answer: $x\\equiv 8\\pmod{63}$", "final_answer": "x\\equiv 8\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c9c7407d59d67f1a", "problem_id": "c31a8f9f742bfc57", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 10 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=3, p=$\\frac{2}{5}$ and simplify.\n4) Here C(10,3) = 120.\nFinal answer: $\\frac{419904}{1953125}$", "final_answer": "\\frac{419904}{1953125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e35662aed3501c5b", "problem_id": "b82e99176332d679", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} x \\left(- x - 2\\right)$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} - 3 x.\n3) Then the limit is 2.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: 1.\nFinal answer: $1$", "final_answer": "1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "48ae7b80bd76b3cd", "problem_id": "52ef31e37f8e0ab4", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the linear system:\n$2 x - 3 y$ = $-17$\n$- 4 x + 5 y$ = $33$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $- 8 x + 12 y$ = $68$.\nMultiply the second equation by $2$: $- 8 x + 10 y$ = $66$.\nFinal answer: $(x,y)=\\left(-7, 1\\right)$", "final_answer": "(x,y)=\\left(-7, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "017f43b7a7e926d1", "problem_id": "52ef31e37f8e0ab4", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$2 x - 3 y$ = $-17$\n$- 4 x + 5 y$ = $33$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $- 8 x + 12 y$ = $68$.\n3) Multiply the second equation by $2$: $- 8 x + 10 y$ = $66$.\n4) Subtract to eliminate x: $2 y$ = $2$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $-7$.\nFinal answer: $(x,y)=\\left(-7, 1\\right)$", "final_answer": "(x,y)=\\left(-7, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aa0fd9b1a447390c", "problem_id": "10fffa9c4b01f48a", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "You want 24 liters of a 25% salt solution. You have a 15% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 24−x liters is 60% solution.\n2) Salt equation: 15x + 60(24−x) = 25·24.\n3) (15-60)x = (25-60)·24 ⇒ x = 56/3.\n4) Then 24−x = 16/3.\nFinal answer: $56/3 L of 15% and 16/3 L of 60%$", "final_answer": "56/3 L of 15% and 16/3 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b42916884596f65", "problem_id": "20e6e34ac6bb1843", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x - 1$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 2 x - 1.\n3) Then the limit is -5.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -3.\nFinal answer: $-3$", "final_answer": "-3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "537c1ab8067c594f", "problem_id": "9aca98c0ac96ff71", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$7 x + 3 y$ = $-23$\n$2 x + 7 y$ = $18$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $14 x + 6 y$ = $-46$.\n3) Multiply the second equation by $7$: $14 x + 49 y$ = $126$.\n4) Subtract to eliminate x: $- 43 y$ = $-172$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, 4\\right)$", "final_answer": "(x,y)=\\left(-5, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "abb8544a1400b8b3", "problem_id": "9aca98c0ac96ff71", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$7 x + 3 y$ = $-23$\n$2 x + 7 y$ = $18$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $2$: $14 x + 6 y$ = $-46$.\nMultiply the second equation by $7$: $14 x + 49 y$ = $126$.\nSubtract to eliminate x: $- 43 y$ = $-172$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $-5$.\nCheck: plug (x,y)=(-5,4) into both equations.\nFinal answer: $(x,y)=\\left(-5, 4\\right)$", "final_answer": "(x,y)=\\left(-5, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "824841ab75a0b2a9", "problem_id": "186c57c46eccb36b", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 16 x + 30$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 16 x + 30$ = $2 \\left(x - 5\\right) \\left(x - 3\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $5$.\n5) From the second factor: x = $3$.\nFinal answer: $x \\in \\{5, 3\\}$", "final_answer": "x \\in \\{5, 3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "68508fc54294000d", "problem_id": "186c57c46eccb36b", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 16 x + 30$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} - 16 x + 30$ = $2 \\left(x - 5\\right) \\left(x - 3\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFinal answer: $x \\in \\{5, 3\\}$", "final_answer": "x \\in \\{5, 3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d3a96d4707064b2b", "problem_id": "1d65258c9bb0cacd", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 10 times, what is the probability of exactly 7 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=7, p=$\\frac{2}{3}$ and simplify.\nHere C(10,7) = 120.\nCheck: expression simplifies to $\\frac{5120}{19683}$.\nFinal answer: $\\frac{5120}{19683}$", "final_answer": "\\frac{5120}{19683}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d045114c8418721e", "problem_id": "1d65258c9bb0cacd", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 10 times, what is the probability of exactly 7 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=7, p=$\\frac{2}{3}$ and simplify.\n4) Here C(10,7) = 120.\nFinal answer: $\\frac{5120}{19683}$", "final_answer": "\\frac{5120}{19683}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a8ee2792c26809f2", "problem_id": "bd1d561289b8bc9e", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the multiplicative inverse of 49 modulo 59.", "output": "1) We want x such that 49x ≡ 1 (mod 59).\n2) Use extended Euclid to find s,t with 49s + 59t = 1.\n3) One Bezout identity is: 49·(-6) + 59·(5) = 1.\n4) Reduce s modulo 59: x ≡ 53 (mod 59).\nFinal answer: $x\\equiv 53\\pmod{59}$", "final_answer": "x\\equiv 53\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "24b3ab499fbf5666", "problem_id": "bd1d561289b8bc9e", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the multiplicative inverse of 49 modulo 59.", "output": "We want x such that 49x ≡ 1 (mod 59).\nUse extended Euclid to find s,t with 49s + 59t = 1.\nOne Bezout identity is: 49·(-6) + 59·(5) = 1.\nReduce s modulo 59: x ≡ 53 (mod 59).\nCheck: 49·53 mod 59 = 1.\nFinal answer: $x\\equiv 53\\pmod{59}$", "final_answer": "x\\equiv 53\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "536f31ff98b00583", "problem_id": "95ca0867cbad07c2", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A cyclist rides at 58 miles per hour for 1 hours, then at 45 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 58 × 1 = 58 miles.\n2) Second leg distance = 45 × 3 = 135 miles.\n3) Total distance = 58 + 135 = 193 miles.\nFinal answer: $193$", "final_answer": "193", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ea39895f7d8e568", "problem_id": "95ca0867cbad07c2", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 58 miles per hour for 1 hours, then at 45 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 58 × 1 = 58 miles.\nSecond leg distance = 45 × 3 = 135 miles.\nTotal distance = 58 + 135 = 193 miles.\nCheck: adding both legs gives 193.\nFinal answer: $193$", "final_answer": "193", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1c9c1324c8dae37d", "problem_id": "54bdd991148d5114", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_4=22$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=22 and k=4.\n4) Compute C(25,3) = 2300.\nFinal answer: $2300$", "final_answer": "2300", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "16290dc10d8578b8", "problem_id": "54bdd991148d5114", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_4=22$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=22 and k=4.\nFinal answer: $2300$", "final_answer": "2300", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1fe92f06f844a5e3", "problem_id": "1098d15c3341cb10", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 8 x - 24$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 8 x - 24$ = $2 \\left(x - 6\\right) \\left(x + 2\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $6$.\n5) From the second factor: x = $-2$.\nFinal answer: $x \\in \\{6, -2\\}$", "final_answer": "x \\in \\{6, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9b19620747b441af", "problem_id": "1098d15c3341cb10", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 8 x - 24$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} - 8 x - 24$ = $2 \\left(x - 6\\right) \\left(x + 2\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $6$.\nFrom the second factor: x = $-2$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{6, -2\\}$", "final_answer": "x \\in \\{6, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "76bb4fa89cd70b6e", "problem_id": "fcd59c703425746a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A cyclist rides at 33 miles per hour for 3/2 hours, then at 46 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 33 × 3/2 = 99/2 miles.\nSecond leg distance = 46 × 3/2 = 69 miles.\nTotal distance = 99/2 + 69 = 237/2 miles.\nCheck: adding both legs gives 237/2.\nFinal answer: $\\frac{237}{2}$", "final_answer": "\\frac{237}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "40f1eac332fef7cc", "problem_id": "fcd59c703425746a", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A cyclist rides at 33 miles per hour for 3/2 hours, then at 46 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 33 × 3/2 = 99/2 miles.\n2) Second leg distance = 46 × 3/2 = 69 miles.\n3) Total distance = 99/2 + 69 = 237/2 miles.\nFinal answer: $\\frac{237}{2}$", "final_answer": "\\frac{237}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8197d2c7fda94570", "problem_id": "421bd844a79ee631", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "An urn has 24 red balls and 23 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(47,11).\nFavorable samples: choose k reds and n−k blues: C(24,1)·C(23,10).\nFinal answer: $\\frac{62832}{39856141}$", "final_answer": "\\frac{62832}{39856141}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4a3bfb964d68ae3c", "problem_id": "421bd844a79ee631", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 24 red balls and 23 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(47,11).\n3) Favorable samples: choose k reds and n−k blues: C(24,1)·C(23,10).\n4) Compute totals: C(47,11) = 17417133617, and favorable = 27457584.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{62832}{39856141}$", "final_answer": "\\frac{62832}{39856141}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2b1a7d7fe02e34b1", "problem_id": "c20fa888ea2c8da1", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{7} x}{2} + x - 3 \\sqrt{7} - 2$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$1 - \\frac{3 \\sqrt{7}}{2}$, c=$- 3 \\sqrt{7} - 2$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $9 \\sqrt{7} + \\frac{99}{4}$, so √Δ = $\\sqrt{9 \\sqrt{7} + \\frac{99}{4}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{1}{2} + \\frac{3 \\sqrt{7}}{4} + \\frac{3 \\sqrt{4 \\sqrt{7} + 11}}{4}$.\n7) x2 = $- \\frac{3 \\sqrt{4 \\sqrt{7} + 11}}{4} - \\frac{1}{2} + \\frac{3 \\sqrt{7}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{1}{2} + \\frac{3\\cdot \\sqrt{7}}{4} + \\frac{3\\cdot \\sqrt{4\\cdot \\sqrt{7} + 11}}{4}, - \\frac{3\\cdot \\sqrt{4\\cdot \\sqrt{7} + 11}}{4} - \\frac{1}{2} + \\frac{3\\cdot \\sqrt{7}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{1}{2} + \\frac{3\\cdot \\sqrt{7}}{4} + \\frac{3\\cdot \\sqrt{4\\cdot \\sqrt{7} + 11}}{4}, - \\frac{3\\cdot \\sqrt{4\\cdot \\sqrt{7} + 11}}{4} - \\frac{1}{2} + \\frac{3\\cdot \\sqrt{7}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f73b7ac480a39138", "problem_id": "c20fa888ea2c8da1", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{7} x}{2} + x - 3 \\sqrt{7} - 2$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$1 - \\frac{3 \\sqrt{7}}{2}$, c=$- 3 \\sqrt{7} - 2$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $9 \\sqrt{7} + \\frac{99}{4}$, so √Δ = $\\sqrt{9 \\sqrt{7} + \\frac{99}{4}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{1}{2} + \\frac{3 \\sqrt{7}}{4} + \\frac{3 \\sqrt{4 \\sqrt{7} + 11}}{4}$.\nx2 = $- \\frac{3 \\sqrt{4 \\sqrt{7} + 11}}{4} - \\frac{1}{2} + \\frac{3 \\sqrt{7}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{1}{2} + \\frac{3\\cdot \\sqrt{7}}{4} + \\frac{3\\cdot \\sqrt{4\\cdot \\sqrt{7} + 11}}{4}, - \\frac{3\\cdot \\sqrt{4\\cdot \\sqrt{7} + 11}}{4} - \\frac{1}{2} + \\frac{3\\cdot \\sqrt{7}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{1}{2} + \\frac{3\\cdot \\sqrt{7}}{4} + \\frac{3\\cdot \\sqrt{4\\cdot \\sqrt{7} + 11}}{4}, - \\frac{3\\cdot \\sqrt{4\\cdot \\sqrt{7} + 11}}{4} - \\frac{1}{2} + \\frac{3\\cdot \\sqrt{7}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "537d0100bbf34784", "problem_id": "da83ba5868cb53ad", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 24 x + 70$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} + 24 x + 70$ = $2 \\left(x + 5\\right) \\left(x + 7\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-5$.\n5) From the second factor: x = $-7$.\nFinal answer: $x \\in \\{-5, -7\\}$", "final_answer": "x \\in \\{-5, -7\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "776bec5963e6decc", "problem_id": "da83ba5868cb53ad", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 24 x + 70$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} + 24 x + 70$ = $2 \\left(x + 5\\right) \\left(x + 7\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFinal answer: $x \\in \\{-5, -7\\}$", "final_answer": "x \\in \\{-5, -7\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "914f3d37ebcbe318", "problem_id": "c14403102a9e1b8a", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A store sells a backpack for $179. It is discounted by 10%. You buy 5 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 179 × (1 − 10/100) = 179 × 90/100 = 1611/10.\n2) Subtotal = 5 × 1611/10 = 1611/2.\n3) Total with tax = 1611/2 × 107/100 = 172377/200.\nFinal answer: $\\frac{172377}{200}$", "final_answer": "\\frac{172377}{200}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d1c53c412f7d65a0", "problem_id": "c14403102a9e1b8a", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "A store sells a backpack for $179. It is discounted by 10%. You buy 5 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 179 × (1 − 10/100) = 179 × 90/100 = 1611/10.\nSubtotal = 5 × 1611/10 = 1611/2.\nTotal with tax = 1611/2 × 107/100 = 172377/200.\nFinal answer: $\\frac{172377}{200}$", "final_answer": "\\frac{172377}{200}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1c4577755c524b99", "problem_id": "b5b15efb741c63f1", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{9 x}{10} + \\frac{1}{10}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$- \\frac{9}{10}$, c=$\\frac{1}{10}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{1}{100}$, so √Δ = $\\frac{1}{10}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{1}{4}$.\n7) x2 = $\\frac{1}{5}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{1}{4}, \\frac{1}{5}\\}$", "final_answer": "x \\in \\{\\frac{1}{4}, \\frac{1}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b7e0324399da0ae", "problem_id": "b5b15efb741c63f1", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{9 x}{10} + \\frac{1}{10}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$- \\frac{9}{10}$, c=$\\frac{1}{10}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{1}{100}$, so √Δ = $\\frac{1}{10}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{1}{4}$.\nx2 = $\\frac{1}{5}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{1}{4}, \\frac{1}{5}\\}$", "final_answer": "x \\in \\{\\frac{1}{4}, \\frac{1}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9f3f64df745b7211", "problem_id": "2b30468f45c2f6d5", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "You want 62 liters of a 30% salt solution. You have a 25% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 62−x liters is 40% solution.\n2) Salt equation: 25x + 40(62−x) = 30·62.\n3) (25-40)x = (30-40)·62 ⇒ x = 124/3.\n4) Then 62−x = 62/3.\nFinal answer: $124/3 L of 25% and 62/3 L of 40%$", "final_answer": "124/3 L of 25% and 62/3 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7fc348facccaad65", "problem_id": "2b30468f45c2f6d5", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "You want 62 liters of a 30% salt solution. You have a 25% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 62−x liters is 40% solution.\nSalt equation: 25x + 40(62−x) = 30·62.\n(25-40)x = (30-40)·62 ⇒ x = 124/3.\nThen 62−x = 62/3.\nCheck: 25·124/3 + 40·62/3 = 30·62.\nFinal answer: $124/3 L of 25% and 62/3 L of 40%$", "final_answer": "124/3 L of 25% and 62/3 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "75653095b2466d08", "problem_id": "c9458b3266694140", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$- x - 6 y$ = $6$\n$- 8 x - 7 y$ = $7$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-8$: $8 x + 48 y$ = $-48$.\n3) Multiply the second equation by $-1$: $8 x + 7 y$ = $-7$.\n4) Subtract to eliminate x: $41 y$ = $-41$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, -1\\right)$", "final_answer": "(x,y)=\\left(0, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a603ddef879cbf4", "problem_id": "c9458b3266694140", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the linear system:\n$- x - 6 y$ = $6$\n$- 8 x - 7 y$ = $7$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-8$: $8 x + 48 y$ = $-48$.\nMultiply the second equation by $-1$: $8 x + 7 y$ = $-7$.\nFinal answer: $(x,y)=\\left(0, -1\\right)$", "final_answer": "(x,y)=\\left(0, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "492491abb77e265c", "problem_id": "2b81fad1535dc76c", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} x^{3} + 4 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $x^{3} + 4 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $-1 + \\frac{3 \\pi^{2}}{8} + \\frac{15 \\pi^{4}}{64} + \\frac{3 \\pi^{3}}{2}$.\n4) Simplify the result.\nFinal answer: $-1 + \\frac{3\\cdot \\pi^{2}}{8} + \\frac{15\\cdot \\pi^{4}}{64} + \\frac{3\\cdot \\pi^{3}}{2}$", "final_answer": "-1 + \\frac{3\\cdot \\pi^{2}}{8} + \\frac{15\\cdot \\pi^{4}}{64} + \\frac{3\\cdot \\pi^{3}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "142cf276a25f330e", "problem_id": "0dc2f8488645f051", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{2} + 3 x + 5$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{2} + 2 x + 5.\n3) Then the limit is 17.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 19.\nFinal answer: $19$", "final_answer": "19", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "4611177b2ff257bf", "problem_id": "03e3b9d193aa37fb", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡4 (mod 5), x≡9 (mod 13). Give the solution modulo 65.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 65.\n3) Compute the CRT solution to get x ≡ 9 (mod 65).\nFinal answer: $x\\equiv 9\\pmod{65}$", "final_answer": "x\\equiv 9\\pmod{65}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "47baa2a85f6ce6a9", "problem_id": "03e3b9d193aa37fb", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡4 (mod 5), x≡9 (mod 13). Give the solution modulo 65.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 65.\nCompute the CRT solution to get x ≡ 9 (mod 65).\nCheck: 9 mod 5=4 and mod 13=9.\nFinal answer: $x\\equiv 9\\pmod{65}$", "final_answer": "x\\equiv 9\\pmod{65}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b5c79e835a4536e3", "problem_id": "e195fae0a699ab0c", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 2 x - 80$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} - 2 x - 80$ = $\\left(x - 10\\right) \\left(x + 8\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $10$.\n5) From the second factor: x = $-8$.\nFinal answer: $x \\in \\{10, -8\\}$", "final_answer": "x \\in \\{10, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "88f84e50cfc4bbe8", "problem_id": "e195fae0a699ab0c", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 2 x - 80$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} - 2 x - 80$ = $\\left(x - 10\\right) \\left(x + 8\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $10$.\nFrom the second factor: x = $-8$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{10, -8\\}$", "final_answer": "x \\in \\{10, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a1ec3da34506a4e", "problem_id": "1e4596ebc4c4a8a8", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the linear system:\n$- x + 5 y$ = $-26$\n$2 x + 2 y$ = $4$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $- 2 x + 10 y$ = $-52$.\n3) Multiply the second equation by $-1$: $- 2 x - 2 y$ = $-4$.\n4) Subtract to eliminate x: $12 y$ = $-48$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $6$.\nFinal answer: $(x,y)=\\left(6, -4\\right)$", "final_answer": "(x,y)=\\left(6, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9a9739fe48c6ded7", "problem_id": "1e4596ebc4c4a8a8", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$- x + 5 y$ = $-26$\n$2 x + 2 y$ = $4$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $2$: $- 2 x + 10 y$ = $-52$.\nMultiply the second equation by $-1$: $- 2 x - 2 y$ = $-4$.\nSubtract to eliminate x: $12 y$ = $-48$.\nSolve for y: y = $-4$.\nSubstitute back to find x: x = $6$.\nCheck: plug (x,y)=(6,-4) into both equations.\nFinal answer: $(x,y)=\\left(6, -4\\right)$", "final_answer": "(x,y)=\\left(6, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c07dc6458307851f", "problem_id": "7eb9a1dfdbb5c0ce", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_3=13$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=13 and k=3.\nFinal answer: $105$", "final_answer": "105", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cae1536017372280", "problem_id": "7eb9a1dfdbb5c0ce", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_3=13$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=13 and k=3.\n4) Compute C(15,2) = 105.\nFinal answer: $105$", "final_answer": "105", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a06ed3317f4ccc37", "problem_id": "e51c96a37949518c", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "You want 64 liters of a 30% salt solution. You have a 10% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 64−x liters is 40% solution.\n2) Salt equation: 10x + 40(64−x) = 30·64.\n3) (10-40)x = (30-40)·64 ⇒ x = 64/3.\n4) Then 64−x = 128/3.\nFinal answer: $64/3 L of 10% and 128/3 L of 40%$", "final_answer": "64/3 L of 10% and 128/3 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a41d5c43e05028d6", "problem_id": "e51c96a37949518c", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "You want 64 liters of a 30% salt solution. You have a 10% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 64−x liters is 40% solution.\nSalt equation: 10x + 40(64−x) = 30·64.\n(10-40)x = (30-40)·64 ⇒ x = 64/3.\nThen 64−x = 128/3.\nCheck: 10·64/3 + 40·128/3 = 30·64.\nFinal answer: $64/3 L of 10% and 128/3 L of 40%$", "final_answer": "64/3 L of 10% and 128/3 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ea1212282f50479f", "problem_id": "80e5a33d3f17803a", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $931 in the ratio 3:4. How much does each receive?", "output": "Total ratio parts = 3+4 = 7.\nEach part = 931 ÷ 7 = 133.\nFirst share = 3 × 133 = 399.\nSecond share = 4 × 133 = 532.\nCheck: 399+532=931.\nFinal answer: $532$", "final_answer": "532", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bf040ee262f15721", "problem_id": "80e5a33d3f17803a", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Two friends split $931 in the ratio 3:4. How much does each receive?", "output": "1) Total ratio parts = 3+4 = 7.\n2) Each part = 931 ÷ 7 = 133.\n3) First share = 3 × 133 = 399.\n4) Second share = 4 × 133 = 532.\nFinal answer: $532$", "final_answer": "532", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a6dd207a0210445b", "problem_id": "d7c0d921be707479", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{7}$. If it is flipped 8 times, what is the probability of exactly 4 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=8, k=4, p=$\\frac{4}{7}$ and simplify.\nHere C(8,4) = 70.\nCheck: expression simplifies to $\\frac{207360}{823543}$.\nFinal answer: $\\frac{207360}{823543}$", "final_answer": "\\frac{207360}{823543}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "52736509e17b7f25", "problem_id": "d7c0d921be707479", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{7}$. If it is flipped 8 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=8, k=4, p=$\\frac{4}{7}$ and simplify.\n4) Here C(8,4) = 70.\nFinal answer: $\\frac{207360}{823543}$", "final_answer": "\\frac{207360}{823543}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "66935337b29a19e6", "problem_id": "ac75bcf0d8746705", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 4 x^{3} + 3 x^{2} + 4\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 4 x^{3} + 3 x^{2} + 4$.\n3) Compute an antiderivative: F(x) = $- x^{4} + x^{3} + 4 x$.\n4) Include the constant of integration C.\nFinal answer: $C - x^{4} + x^{3} + 4\\cdot x$", "final_answer": "C - x^{4} + x^{3} + 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cf5f8cce0f3d83a7", "problem_id": "ac75bcf0d8746705", "style": "answer_only", "instruction": "Final answer only.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 4 x^{3} + 3 x^{2} + 4\\,dx$.", "output": "$C - x^{4} + x^{3} + 4\\cdot x$", "final_answer": "C - x^{4} + x^{3} + 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e5e6c7ab76330e10", "problem_id": "aed1a0eaae1d9cef", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{3} + x^{2} + 4 x + 6$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $2 x^{4} - 3 x^{3} + 2 x^{2} - 2 x - 2$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $8 x^{3} - 9 x^{2} + 4 x - 2$.\n5) Evaluate at a: p'(2) = $34$.\nFinal answer: $34$", "final_answer": "34", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9b1af8f5b6b5e45c", "problem_id": "aed1a0eaae1d9cef", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{3} + x^{2} + 4 x + 6$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{4} - 3 x^{3} + 2 x^{2} - 2 x - 2$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $8 x^{3} - 9 x^{2} + 4 x - 2$.\nEvaluate at a: p'(2) = $34$.\nCheck: the identity gives $34$.\nFinal answer: $34$", "final_answer": "34", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "68b4802067b5f0ba", "problem_id": "331742ad7366406c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 3 x^{2} - 5 x - 12$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} - 6 x - 12.\n3) Then the limit is -36.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -34.\nFinal answer: $-34$", "final_answer": "-34", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0433aad0fde1af21", "problem_id": "b551c15f2315f7f0", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 22 red balls and 16 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(38,9).\nFavorable samples: choose k reds and n−k blues: C(22,1)·C(16,8).\nCompute totals: C(38,9) = 163011640, and favorable = 283140.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1287}{740962}$.\nFinal answer: $\\frac{1287}{740962}$", "final_answer": "\\frac{1287}{740962}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b4ac8ec5582e9ea6", "problem_id": "44aabb3a07457b8e", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $34. It is discounted by 10%. You buy 5 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 34 × (1 − 10/100) = 34 × 90/100 = 153/5.\nSubtotal = 5 × 153/5 = 153.\nTotal with tax = 153 × 108/100 = 4131/25.\nCheck: recomputing the formula yields 4131/25.\nFinal answer: $\\frac{4131}{25}$", "final_answer": "\\frac{4131}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4b4ab61b722229ee", "problem_id": "44aabb3a07457b8e", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A store sells a backpack for $34. It is discounted by 10%. You buy 5 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 34 × (1 − 10/100) = 34 × 90/100 = 153/5.\n2) Subtotal = 5 × 153/5 = 153.\n3) Total with tax = 153 × 108/100 = 4131/25.\nFinal answer: $\\frac{4131}{25}$", "final_answer": "\\frac{4131}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9829ebd8bf7534b0", "problem_id": "8058e84ae73ff75a", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the congruences: x≡1 (mod 3), x≡2 (mod 8). Give the solution modulo 24.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 24.\n3) Compute the CRT solution to get x ≡ 10 (mod 24).\nFinal answer: $x\\equiv 10\\pmod{24}$", "final_answer": "x\\equiv 10\\pmod{24}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "89b7d590bed29e1c", "problem_id": "8058e84ae73ff75a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡2 (mod 8). Give the solution modulo 24.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 24.\nCompute the CRT solution to get x ≡ 10 (mod 24).\nCheck: 10 mod 3=1 and mod 8=2.\nFinal answer: $x\\equiv 10\\pmod{24}$", "final_answer": "x\\equiv 10\\pmod{24}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "167859619c6f3b12", "problem_id": "4f061a43467c72ef", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{3} - 4 x^{2} - 2 x + 2\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $2 x^{3} - 4 x^{2} - 2 x + 2$.\n3) Compute an antiderivative: F(x) = $\\frac{x^{4}}{2} - \\frac{4 x^{3}}{3} - x^{2} + 2 x$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{x^{4}}{2} - \\frac{4\\cdot x^{3}}{3} - x^{2} + 2\\cdot x$", "final_answer": "C + \\frac{x^{4}}{2} - \\frac{4\\cdot x^{3}}{3} - x^{2} + 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3232c37322d9dfcc", "problem_id": "4f061a43467c72ef", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{3} - 4 x^{2} - 2 x + 2\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $2 x^{3} - 4 x^{2} - 2 x + 2$.\nCompute an antiderivative: F(x) = $\\frac{x^{4}}{2} - \\frac{4 x^{3}}{3} - x^{2} + 2 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{x^{4}}{2} - \\frac{4\\cdot x^{3}}{3} - x^{2} + 2\\cdot x$", "final_answer": "C + \\frac{x^{4}}{2} - \\frac{4\\cdot x^{3}}{3} - x^{2} + 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2901569eeaf305a6", "problem_id": "34391e20268483f5", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A cyclist rides at 72 miles per hour for 2 hours, then at 33 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 72 × 2 = 144 miles.\n2) Second leg distance = 33 × 3 = 99 miles.\n3) Total distance = 144 + 99 = 243 miles.\nFinal answer: $243$", "final_answer": "243", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9fff1903e404b678", "problem_id": "34391e20268483f5", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A cyclist rides at 72 miles per hour for 2 hours, then at 33 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 72 × 2 = 144 miles.\nSecond leg distance = 33 × 3 = 99 miles.\nTotal distance = 144 + 99 = 243 miles.\nCheck: adding both legs gives 243.\nFinal answer: $243$", "final_answer": "243", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9a6d354fec8232a3", "problem_id": "b17f549445b64e44", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡7 (mod 8), x≡5 (mod 9). Give the solution modulo 72.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 72.\n3) Compute the CRT solution to get x ≡ 23 (mod 72).\nFinal answer: $x\\equiv 23\\pmod{72}$", "final_answer": "x\\equiv 23\\pmod{72}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7492df5c11b1f7ca", "problem_id": "b17f549445b64e44", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡7 (mod 8), x≡5 (mod 9). Give the solution modulo 72.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 72.\nCompute the CRT solution to get x ≡ 23 (mod 72).\nCheck: 23 mod 8=7 and mod 9=5.\nFinal answer: $x\\equiv 23\\pmod{72}$", "final_answer": "x\\equiv 23\\pmod{72}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f8a256a3953b32b", "problem_id": "0fe6a17bb4c761d3", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 2 x^{4} + x^{3} - 2 x^{2} - 4 x + 3\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 2 x^{4} + x^{3} - 2 x^{2} - 4 x + 3$.\n3) Compute an antiderivative: F(x) = $- \\frac{2 x^{5}}{5} + \\frac{x^{4}}{4} - \\frac{2 x^{3}}{3} - 2 x^{2} + 3 x$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{2\\cdot x^{5}}{5} + \\frac{x^{4}}{4} - \\frac{2\\cdot x^{3}}{3} - 2\\cdot x^{2} + 3\\cdot x$", "final_answer": "C - \\frac{2\\cdot x^{5}}{5} + \\frac{x^{4}}{4} - \\frac{2\\cdot x^{3}}{3} - 2\\cdot x^{2} + 3\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1f54236c281cda14", "problem_id": "0fe6a17bb4c761d3", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 2 x^{4} + x^{3} - 2 x^{2} - 4 x + 3\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 2 x^{4} + x^{3} - 2 x^{2} - 4 x + 3$.\nCompute an antiderivative: F(x) = $- \\frac{2 x^{5}}{5} + \\frac{x^{4}}{4} - \\frac{2 x^{3}}{3} - 2 x^{2} + 3 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - \\frac{2\\cdot x^{5}}{5} + \\frac{x^{4}}{4} - \\frac{2\\cdot x^{3}}{3} - 2\\cdot x^{2} + 3\\cdot x$", "final_answer": "C - \\frac{2\\cdot x^{5}}{5} + \\frac{x^{4}}{4} - \\frac{2\\cdot x^{3}}{3} - 2\\cdot x^{2} + 3\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4880803822fd9e14", "problem_id": "ebd4d06bb365050c", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A cyclist rides at 55 miles per hour for 2 hours, then at 76 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 55 × 2 = 110 miles.\n2) Second leg distance = 76 × 2 = 152 miles.\n3) Total distance = 110 + 152 = 262 miles.\nFinal answer: $262$", "final_answer": "262", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "840cfa31c51e8be4", "problem_id": "ebd4d06bb365050c", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 55 miles per hour for 2 hours, then at 76 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 55 × 2 = 110 miles.\nSecond leg distance = 76 × 2 = 152 miles.\nTotal distance = 110 + 152 = 262 miles.\nCheck: adding both legs gives 262.\nFinal answer: $262$", "final_answer": "262", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3dd2c5309686e94c", "problem_id": "4bb1a0716fd0613b", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Two friends split $824 in the ratio 4:4. How much does each receive?", "output": "1) Total ratio parts = 4+4 = 8.\n2) Each part = 824 ÷ 8 = 103.\n3) First share = 4 × 103 = 412.\n4) Second share = 4 × 103 = 412.\nFinal answer: $412$", "final_answer": "412", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c14453da0efd9751", "problem_id": "02f5ba068679179f", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(- x^{3} + 2 x^{2} - 2 x - 2\\right) \\cos{\\left(2 x - 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 3 e^{x}) = - 3 e^{x}$.\n3) So $f'(x) = 2 x^{3} \\sin{\\left(2 x - 2 \\right)} - 4 x^{2} \\sin{\\left(2 x - 2 \\right)} - 3 x^{2} \\cos{\\left(2 x - 2 \\right)} + x e^{x} + 4 x \\sin{\\left(2 x - 2 \\right)} + 4 x \\cos{\\left(2 x - 2 \\right)} - 2 e^{x} + 4 \\sin{\\left(2 x - 2 \\right)} - 2 \\cos{\\left(2 x - 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 3 e^{x}) = - 3 e^{x}$.\nTherefore the correct derivative is $f'(x) = \\left(x - 3\\right) e^{x} - 2 \\left(x \\left(x \\left(2 - x\\right) - 2\\right) - 2\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(- 3 x^{2} + 4 x - 2\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}$.\nFinal answer: $\\left(x - 3\\right) e^{x} - 2 \\left(x \\left(x \\left(2 - x\\right) - 2\\right) - 2\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(- 3 x^{2} + 4 x - 2\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}$", "final_answer": "\\left(x - 3\\right) e^{x} - 2 \\left(x \\left(x \\left(2 - x\\right) - 2\\right) - 2\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(- 3 x^{2} + 4 x - 2\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8b0f6aa2f4c3dfad", "problem_id": "edc00fcb23b7330a", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} x^{3} + 7 x^{2} + 27 x + 109$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{4} + 3 x^{3} - x^{2} + x - 3$ and a = 4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $4 x^{3} + 9 x^{2} - 2 x + 1$.\n5) Evaluate at a: p'(4) = $393$.\nFinal answer: $393$", "final_answer": "393", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f47d31d1a84e1da", "problem_id": "edc00fcb23b7330a", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} x^{3} + 7 x^{2} + 27 x + 109$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $x^{4} + 3 x^{3} - x^{2} + x - 3$ and a = 4.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $393$", "final_answer": "393", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "86c4c57930b958d8", "problem_id": "38bf4dafbc7d6312", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=41$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=41 and k=6.\nCompute C(46,5) = 1370754.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $1370754$", "final_answer": "1370754", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5915069a5caaa99f", "problem_id": "38bf4dafbc7d6312", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=41$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=41 and k=6.\n4) Compute C(46,5) = 1370754.\nFinal answer: $1370754$", "final_answer": "1370754", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5702d4b2135c0143", "problem_id": "e648e7a3759dd340", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the multiplicative inverse of 3 modulo 31.", "output": "1) We want x such that 3x ≡ 1 (mod 31).\n2) Use extended Euclid to find s,t with 3s + 31t = 1.\n3) One Bezout identity is: 3·(-10) + 31·(1) = 1.\n4) Reduce s modulo 31: x ≡ 21 (mod 31).\nFinal answer: $x\\equiv 21\\pmod{31}$", "final_answer": "x\\equiv 21\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "028a1822cc0763d2", "problem_id": "e648e7a3759dd340", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Find the multiplicative inverse of 3 modulo 31.", "output": "We want x such that 3x ≡ 1 (mod 31).\nUse extended Euclid to find s,t with 3s + 31t = 1.\nOne Bezout identity is: 3·(-10) + 31·(1) = 1.\nFinal answer: $x\\equiv 21\\pmod{31}$", "final_answer": "x\\equiv 21\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3b86a842894a1d9c", "problem_id": "5f524d6ba83f5e9e", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=34$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=34 and k=7.\nCompute C(40,6) = 3838380.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $3838380$", "final_answer": "3838380", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c36f1c6fe4031024", "problem_id": "5f524d6ba83f5e9e", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=34$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=34 and k=7.\n4) Compute C(40,6) = 3838380.\nFinal answer: $3838380$", "final_answer": "3838380", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "97cc1a225032b8df", "problem_id": "94e79a67d6fe5450", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the congruences: x≡6 (mod 7), x≡0 (mod 5). Give the solution modulo 35.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 35.\nCompute the CRT solution to get x ≡ 20 (mod 35).\nFinal answer: $x\\equiv 20\\pmod{35}$", "final_answer": "x\\equiv 20\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b079ec27547d88db", "problem_id": "94e79a67d6fe5450", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡6 (mod 7), x≡0 (mod 5). Give the solution modulo 35.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 35.\n3) Compute the CRT solution to get x ≡ 20 (mod 35).\nFinal answer: $x\\equiv 20\\pmod{35}$", "final_answer": "x\\equiv 20\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5677d7730c8d1eb1", "problem_id": "b372db22b682634d", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "You want 63 liters of a 32% salt solution. You have a 25% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 63−x liters is 40% solution.\n2) Salt equation: 25x + 40(63−x) = 32·63.\n3) (25-40)x = (32-40)·63 ⇒ x = 168/5.\n4) Then 63−x = 147/5.\nFinal answer: $168/5 L of 25% and 147/5 L of 40%$", "final_answer": "168/5 L of 25% and 147/5 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3d61d332ec9ea4c7", "problem_id": "b372db22b682634d", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 63 liters of a 32% salt solution. You have a 25% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 63−x liters is 40% solution.\nSalt equation: 25x + 40(63−x) = 32·63.\n(25-40)x = (32-40)·63 ⇒ x = 168/5.\nThen 63−x = 147/5.\nCheck: 25·168/5 + 40·147/5 = 32·63.\nFinal answer: $168/5 L of 25% and 147/5 L of 40%$", "final_answer": "168/5 L of 25% and 147/5 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f99e36f86471b532", "problem_id": "ff3e283e33dcb829", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$- 2 x - 4 y$ = $2$\n$- 3 y$ = $9$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $-2$: $6 y$ = $-18$.\n4) Subtract to eliminate x: $- 6 y$ = $18$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, -3\\right)$", "final_answer": "(x,y)=\\left(5, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ecba63e7f383d60", "problem_id": "ff3e283e33dcb829", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the linear system:\n$- 2 x - 4 y$ = $2$\n$- 3 y$ = $9$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $-2$: $6 y$ = $-18$.\nFinal answer: $(x,y)=\\left(5, -3\\right)$", "final_answer": "(x,y)=\\left(5, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8b1f13d3edbb235f", "problem_id": "9e84074b80b7df79", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(- x^{2} - 2 x - 1\\right) \\cos{\\left(x^{2} + x - 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\n3) So $f'(x) = 2 x^{3} \\sin{\\left(x^{2} + x - 1 \\right)} + 5 x^{2} \\sin{\\left(x^{2} + x - 1 \\right)} + x e^{x} + 4 x \\sin{\\left(x^{2} + x - 1 \\right)} - 2 x \\cos{\\left(\\left(x^{2} + x\\right) - 1 \\right)} + 3 e^{x} + \\sin{\\left(x^{2} + x - 1 \\right)} - 2 \\cos{\\left(\\left(x^{2} + x\\right) - 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- x - 1\\right) \\cos{\\left(\\left(x^{2} + x\\right) - 1 \\right)} + \\left(x + 2\\right) e^{x} - \\left(2 x + 1\\right) \\left(x \\left(- x - 2\\right) - 1\\right) \\sin{\\left(x^{2} + x - 1 \\right)} + e^{x}$.\nFinal answer: $2 \\left(- x - 1\\right) \\cos{\\left(\\left(x^{2} + x\\right) - 1 \\right)} + \\left(x + 2\\right) e^{x} - \\left(2 x + 1\\right) \\left(x \\left(- x - 2\\right) - 1\\right) \\sin{\\left(x^{2} + x - 1 \\right)} + e^{x}$", "final_answer": "2 \\left(- x - 1\\right) \\cos{\\left(\\left(x^{2} + x\\right) - 1 \\right)} + \\left(x + 2\\right) e^{x} - \\left(2 x + 1\\right) \\left(x \\left(- x - 2\\right) - 1\\right) \\sin{\\left(x^{2} + x - 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3a50004088edc3e6", "problem_id": "e9b17fd721e6f8eb", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{2 \\sqrt{6} x}{3} + \\frac{13 x}{5} - \\frac{2 \\sqrt{6}}{5} + \\frac{6}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{13}{5} - \\frac{2 \\sqrt{6}}{3}$, c=$\\frac{6}{5} - \\frac{2 \\sqrt{6}}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{347}{75} - \\frac{28 \\sqrt{6}}{15}$, so √Δ = $\\sqrt{\\frac{347}{75} - \\frac{28 \\sqrt{6}}{15}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{13}{10} + \\frac{\\sqrt{1041 - 420 \\sqrt{6}}}{30} + \\frac{\\sqrt{6}}{3}$.\nx2 = $- \\frac{13}{10} - \\frac{\\sqrt{1041 - 420 \\sqrt{6}}}{30} + \\frac{\\sqrt{6}}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{13}{10} + \\frac{\\sqrt{1041 - 420\\cdot \\sqrt{6}}}{30} + \\frac{\\sqrt{6}}{3}, - \\frac{13}{10} - \\frac{\\sqrt{1041 - 420\\cdot \\sqrt{6}}}{30} + \\frac{\\sqrt{6}}{3}\\}$", "final_answer": "x \\in \\{- \\frac{13}{10} + \\frac{\\sqrt{1041 - 420\\cdot \\sqrt{6}}}{30} + \\frac{\\sqrt{6}}{3}, - \\frac{13}{10} - \\frac{\\sqrt{1041 - 420\\cdot \\sqrt{6}}}{30} + \\frac{\\sqrt{6}}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "73625c88a83c02b1", "problem_id": "238e168e8a7214e3", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$7 x + 8 y$ = $-30$\n$x$ = $-2$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $7 x + 8 y$ = $-30$.\n3) Multiply the second equation by $7$: $7 x$ = $-14$.\n4) Subtract to eliminate x: $8 y$ = $-16$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, -2\\right)$", "final_answer": "(x,y)=\\left(-2, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dd6cd4f47befcd73", "problem_id": "238e168e8a7214e3", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$7 x + 8 y$ = $-30$\n$x$ = $-2$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $7 x + 8 y$ = $-30$.\nMultiply the second equation by $7$: $7 x$ = $-14$.\nSubtract to eliminate x: $8 y$ = $-16$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $-2$.\nCheck: plug (x,y)=(-2,-2) into both equations.\nFinal answer: $(x,y)=\\left(-2, -2\\right)$", "final_answer": "(x,y)=\\left(-2, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ee9eb2852b32beb9", "problem_id": "5b59a3006d4dc40a", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Two friends split $390 in the ratio 4:1. How much does each receive?", "output": "Total ratio parts = 4+1 = 5.\nEach part = 390 ÷ 5 = 78.\nFirst share = 4 × 78 = 312.\nSecond share = 1 × 78 = 78.\nCheck: 312+78=390.\nFinal answer: $78$", "final_answer": "78", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6ced6e799b89a149", "problem_id": "5b59a3006d4dc40a", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Two friends split $390 in the ratio 4:1. How much does each receive?", "output": "1) Total ratio parts = 4+1 = 5.\n2) Each part = 390 ÷ 5 = 78.\n3) First share = 4 × 78 = 312.\n4) Second share = 1 × 78 = 78.\nFinal answer: $78$", "final_answer": "78", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b435d5a3b56ebcaa", "problem_id": "39a036e8e4b3525b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 3 x - 12$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to 2 x - 12.\n3) Then the limit is -18.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -21.\nFinal answer: $-21$", "final_answer": "-21", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6fbe22cafff7dc3c", "problem_id": "4459e82e0c9509d4", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡4 (mod 8), x≡8 (mod 11). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 52 (mod 88).\nFinal answer: $x\\equiv 52\\pmod{88}$", "final_answer": "x\\equiv 52\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eade467aaae24901", "problem_id": "7f012fca59981999", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 16 times, what is the probability of exactly 14 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=16, k=14, p=$\\frac{2}{5}$ and simplify.\n4) Here C(16,14) = 120.\nFinal answer: $\\frac{3538944}{30517578125}$", "final_answer": "\\frac{3538944}{30517578125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "358a69ff237c3b34", "problem_id": "7f012fca59981999", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 16 times, what is the probability of exactly 14 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=16, k=14, p=$\\frac{2}{5}$ and simplify.\nHere C(16,14) = 120.\nCheck: expression simplifies to $\\frac{3538944}{30517578125}$.\nFinal answer: $\\frac{3538944}{30517578125}$", "final_answer": "\\frac{3538944}{30517578125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e4d9a0cab5508d1", "problem_id": "f7a54165424c6081", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{4} + x^{3} - 2 x^{2} + 3 x + 3\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 3 x^{4} + x^{3} - 2 x^{2} + 3 x + 3$.\n3) Compute an antiderivative: F(x) = $- \\frac{3 x^{5}}{5} + \\frac{x^{4}}{4} - \\frac{2 x^{3}}{3} + \\frac{3 x^{2}}{2} + 3 x$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{3\\cdot x^{5}}{5} + \\frac{x^{4}}{4} - \\frac{2\\cdot x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} + 3\\cdot x$", "final_answer": "C - \\frac{3\\cdot x^{5}}{5} + \\frac{x^{4}}{4} - \\frac{2\\cdot x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} + 3\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1f32b5d22d983eff", "problem_id": "f7a54165424c6081", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{4} + x^{3} - 2 x^{2} + 3 x + 3\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 3 x^{4} + x^{3} - 2 x^{2} + 3 x + 3$.\nCompute an antiderivative: F(x) = $- \\frac{3 x^{5}}{5} + \\frac{x^{4}}{4} - \\frac{2 x^{3}}{3} + \\frac{3 x^{2}}{2} + 3 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - \\frac{3\\cdot x^{5}}{5} + \\frac{x^{4}}{4} - \\frac{2\\cdot x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} + 3\\cdot x$", "final_answer": "C - \\frac{3\\cdot x^{5}}{5} + \\frac{x^{4}}{4} - \\frac{2\\cdot x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} + 3\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "70505fdc3e63ca59", "problem_id": "08fb03bed4cf9456", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$x + y$ = $-7$\n$6 x + 2 y$ = $-22$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $6$: $6 x + 6 y$ = $-42$.\n3) Multiply the second equation by $1$: $6 x + 2 y$ = $-22$.\n4) Subtract to eliminate x: $4 y$ = $-20$.\n5) Solve for y: y = $-5$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, -5\\right)$", "final_answer": "(x,y)=\\left(-2, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8b3070e861d29098", "problem_id": "08fb03bed4cf9456", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$x + y$ = $-7$\n$6 x + 2 y$ = $-22$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $6$: $6 x + 6 y$ = $-42$.\nMultiply the second equation by $1$: $6 x + 2 y$ = $-22$.\nSubtract to eliminate x: $4 y$ = $-20$.\nSolve for y: y = $-5$.\nSubstitute back to find x: x = $-2$.\nCheck: plug (x,y)=(-2,-5) into both equations.\nFinal answer: $(x,y)=\\left(-2, -5\\right)$", "final_answer": "(x,y)=\\left(-2, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3b90eb9db5e835e4", "problem_id": "2c546d650942108c", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{3} + 2 x^{2} - 4 x + 4\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $2 x^{3} + 2 x^{2} - 4 x + 4$.\nCompute an antiderivative: F(x) = $\\frac{x^{4}}{2} + \\frac{2 x^{3}}{3} - 2 x^{2} + 4 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{x^{4}}{2} + \\frac{2\\cdot x^{3}}{3} - 2\\cdot x^{2} + 4\\cdot x$", "final_answer": "C + \\frac{x^{4}}{2} + \\frac{2\\cdot x^{3}}{3} - 2\\cdot x^{2} + 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e1c4cfc189509abd", "problem_id": "2c546d650942108c", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{3} + 2 x^{2} - 4 x + 4\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $2 x^{3} + 2 x^{2} - 4 x + 4$.\n3) Compute an antiderivative: F(x) = $\\frac{x^{4}}{2} + \\frac{2 x^{3}}{3} - 2 x^{2} + 4 x$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{x^{4}}{2} + \\frac{2\\cdot x^{3}}{3} - 2\\cdot x^{2} + 4\\cdot x$", "final_answer": "C + \\frac{x^{4}}{2} + \\frac{2\\cdot x^{3}}{3} - 2\\cdot x^{2} + 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "125fbd2e6860b76e", "problem_id": "4d7ab17f5dd739dc", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$x + 5 y$ = $5$\n$- 8 x + 5 y$ = $50$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-8$: $- 8 x - 40 y$ = $-40$.\nMultiply the second equation by $1$: $- 8 x + 5 y$ = $50$.\nSubtract to eliminate x: $- 45 y$ = $-90$.\nSolve for y: y = $2$.\nSubstitute back to find x: x = $-5$.\nCheck: plug (x,y)=(-5,2) into both equations.\nFinal answer: $(x,y)=\\left(-5, 2\\right)$", "final_answer": "(x,y)=\\left(-5, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b83cc3f5cb98d835", "problem_id": "4d7ab17f5dd739dc", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the linear system:\n$x + 5 y$ = $5$\n$- 8 x + 5 y$ = $50$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-8$: $- 8 x - 40 y$ = $-40$.\nMultiply the second equation by $1$: $- 8 x + 5 y$ = $50$.\nFinal answer: $(x,y)=\\left(-5, 2\\right)$", "final_answer": "(x,y)=\\left(-5, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "523401667e7d2a10", "problem_id": "a59e887d6a343b73", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "You want 23 liters of a 25% salt solution. You have a 10% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 23−x liters is 40% solution.\nSalt equation: 10x + 40(23−x) = 25·23.\n(10-40)x = (25-40)·23 ⇒ x = 23/2.\nFinal answer: $23/2 L of 10% and 23/2 L of 40%$", "final_answer": "23/2 L of 10% and 23/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db04da71a977d0f6", "problem_id": "a59e887d6a343b73", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "You want 23 liters of a 25% salt solution. You have a 10% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 23−x liters is 40% solution.\n2) Salt equation: 10x + 40(23−x) = 25·23.\n3) (10-40)x = (25-40)·23 ⇒ x = 23/2.\n4) Then 23−x = 23/2.\nFinal answer: $23/2 L of 10% and 23/2 L of 40%$", "final_answer": "23/2 L of 10% and 23/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "096ef2813651dd86", "problem_id": "e5935cf523f3be3b", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "You want 57 liters of a 35% salt solution. You have a 25% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 57−x liters is 60% solution.\n2) Salt equation: 25x + 60(57−x) = 35·57.\n3) (25-60)x = (35-60)·57 ⇒ x = 285/7.\n4) Then 57−x = 114/7.\nFinal answer: $285/7 L of 25% and 114/7 L of 60%$", "final_answer": "285/7 L of 25% and 114/7 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6d79c97ba16448e1", "problem_id": "b25da1383a89c6c7", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{2} - 3 x + \\left(- x^{2} + 2 x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{- x^{2} + 2 x + 6}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- x^{2} + 2 x\\right) + 6}) = \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 2 x - 6}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{4 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{8 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} + \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 4 x^{3} - 8 x^{2} + 24 x + 36} - \\frac{8 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{16 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{18 x^{2}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{2 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 4 x^{3} - 8 x^{2} + 24 x + 36} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 2 x - 6} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{- 2 x^{2} + 4 x + 12} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(- x^{2} + 2 x\\right) + 6} + \\frac{24 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} + \\frac{48 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{68 x}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} + \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- x^{2} + 2 x\\right) + 6} + \\frac{36 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} + \\frac{72 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{40}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- x^{2} + 2 x\\right) + 6}) = - \\frac{x^{2} \\left(2 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(- x^{2} + 2 x\\right) + 6\\right)^{2}} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(- x^{2} + 2 x\\right) + 6\\right)} - \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(- x^{2} + 2 x\\right) + 6} - \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- x^{2} + 2 x\\right) + 6}$.\nTherefore the correct derivative is $f'(x) = \\frac{4 \\left(x - 1\\right) \\left(- 3 x^{2} - 3 x + \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(x \\left(2 - x\\right) + 6\\right) \\left(- 12 x + 4 \\left(1 - x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x \\left(2 - x\\right) + 6\\right)^{2}}$.\nFinal answer: $\\frac{4 \\left(x - 1\\right) \\left(- 3 x^{2} - 3 x + \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(x \\left(2 - x\\right) + 6\\right) \\left(- 12 x + 4 \\left(1 - x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x \\left(2 - x\\right) + 6\\right)^{2}}$", "final_answer": "\\frac{4 \\left(x - 1\\right) \\left(- 3 x^{2} - 3 x + \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(x \\left(2 - x\\right) + 6\\right) \\left(- 12 x + 4 \\left(1 - x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x \\left(2 - x\\right) + 6\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "24c216bf5e310738", "problem_id": "d0e3b2ad80716f03", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "How many binary strings of length 10 contain exactly 4 ones with no two ones adjacent?", "output": "1) Reserve 3 mandatory zeros between the 4 ones.\n2) This leaves n−k+1 = 7 valid slots for the ones.\n3) Choose the slots: C(7,4) = 35.\nFinal answer: $35$", "final_answer": "35", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e2b98bd501eced2e", "problem_id": "d0e3b2ad80716f03", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many binary strings of length 10 contain exactly 4 ones with no two ones adjacent?", "output": "Reserve 3 mandatory zeros between the 4 ones.\nThis leaves n−k+1 = 7 valid slots for the ones.\nChoose the slots: C(7,4) = 35.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $35$", "final_answer": "35", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "92aeead660212ac8", "problem_id": "bb172e26d47c263b", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡3 (mod 7), x≡5 (mod 13). Give the solution modulo 91.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 91.\nCompute the CRT solution to get x ≡ 31 (mod 91).\nCheck: 31 mod 7=3 and mod 13=5.\nFinal answer: $x\\equiv 31\\pmod{91}$", "final_answer": "x\\equiv 31\\pmod{91}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f9f9acab23456bcf", "problem_id": "bb172e26d47c263b", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡3 (mod 7), x≡5 (mod 13). Give the solution modulo 91.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 91.\n3) Compute the CRT solution to get x ≡ 31 (mod 91).\nFinal answer: $x\\equiv 31\\pmod{91}$", "final_answer": "x\\equiv 31\\pmod{91}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bef4db22341d2374", "problem_id": "6b716e9d880bd1c9", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{11} x}{2} - \\frac{5 x}{6} - \\frac{5 \\sqrt{11}}{2} - \\frac{25}{6}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{3 \\sqrt{11}}{2} - \\frac{5}{6}$, c=$- \\frac{5 \\sqrt{11}}{2} - \\frac{25}{6}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{25 \\sqrt{11}}{2} + \\frac{379}{9}$, so √Δ = $\\sqrt{\\frac{25 \\sqrt{11}}{2} + \\frac{379}{9}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{5}{12} + \\frac{3 \\sqrt{11}}{4} + \\frac{\\sqrt{450 \\sqrt{11} + 1516}}{12}$.\n7) x2 = $- \\frac{\\sqrt{450 \\sqrt{11} + 1516}}{12} + \\frac{5}{12} + \\frac{3 \\sqrt{11}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{5}{12} + \\frac{3\\cdot \\sqrt{11}}{4} + \\frac{\\sqrt{450\\cdot \\sqrt{11} + 1516}}{12}, - \\frac{\\sqrt{450\\cdot \\sqrt{11} + 1516}}{12} + \\frac{5}{12} + \\frac{3\\cdot \\sqrt{11}}{4}\\}$", "final_answer": "x \\in \\{\\frac{5}{12} + \\frac{3\\cdot \\sqrt{11}}{4} + \\frac{\\sqrt{450\\cdot \\sqrt{11} + 1516}}{12}, - \\frac{\\sqrt{450\\cdot \\sqrt{11} + 1516}}{12} + \\frac{5}{12} + \\frac{3\\cdot \\sqrt{11}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ea72f4e9e469625", "problem_id": "6b716e9d880bd1c9", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{11} x}{2} - \\frac{5 x}{6} - \\frac{5 \\sqrt{11}}{2} - \\frac{25}{6}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{3 \\sqrt{11}}{2} - \\frac{5}{6}$, c=$- \\frac{5 \\sqrt{11}}{2} - \\frac{25}{6}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{25 \\sqrt{11}}{2} + \\frac{379}{9}$, so √Δ = $\\sqrt{\\frac{25 \\sqrt{11}}{2} + \\frac{379}{9}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{5}{12} + \\frac{3 \\sqrt{11}}{4} + \\frac{\\sqrt{450 \\sqrt{11} + 1516}}{12}$.\nx2 = $- \\frac{\\sqrt{450 \\sqrt{11} + 1516}}{12} + \\frac{5}{12} + \\frac{3 \\sqrt{11}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{5}{12} + \\frac{3\\cdot \\sqrt{11}}{4} + \\frac{\\sqrt{450\\cdot \\sqrt{11} + 1516}}{12}, - \\frac{\\sqrt{450\\cdot \\sqrt{11} + 1516}}{12} + \\frac{5}{12} + \\frac{3\\cdot \\sqrt{11}}{4}\\}$", "final_answer": "x \\in \\{\\frac{5}{12} + \\frac{3\\cdot \\sqrt{11}}{4} + \\frac{\\sqrt{450\\cdot \\sqrt{11} + 1516}}{12}, - \\frac{\\sqrt{450\\cdot \\sqrt{11} + 1516}}{12} + \\frac{5}{12} + \\frac{3\\cdot \\sqrt{11}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5eba9cffe5851c2d", "problem_id": "621d108fc24e97b8", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$3 x - 2 y$ = $0$\n$- 4 x + y$ = $0$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $- 12 x + 8 y$ = $0$.\n3) Multiply the second equation by $3$: $- 12 x + 3 y$ = $0$.\n4) Subtract to eliminate x: $5 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, 0\\right)$", "final_answer": "(x,y)=\\left(0, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a65da61688b06f14", "problem_id": "621d108fc24e97b8", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$3 x - 2 y$ = $0$\n$- 4 x + y$ = $0$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $- 12 x + 8 y$ = $0$.\nMultiply the second equation by $3$: $- 12 x + 3 y$ = $0$.\nSubtract to eliminate x: $5 y$ = $0$.\nSolve for y: y = $0$.\nSubstitute back to find x: x = $0$.\nCheck: plug (x,y)=(0,0) into both equations.\nFinal answer: $(x,y)=\\left(0, 0\\right)$", "final_answer": "(x,y)=\\left(0, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ade835c6c6c94c66", "problem_id": "61ce874c0fb5f868", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Two friends split $2159 in the ratio 9:8. How much does each receive?", "output": "1) Total ratio parts = 9+8 = 17.\n2) Each part = 2159 ÷ 17 = 127.\n3) First share = 9 × 127 = 1143.\n4) Second share = 8 × 127 = 1016.\nFinal answer: $1016$", "final_answer": "1016", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d6289a65b9e09b1f", "problem_id": "61ce874c0fb5f868", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $2159 in the ratio 9:8. How much does each receive?", "output": "Total ratio parts = 9+8 = 17.\nEach part = 2159 ÷ 17 = 127.\nFirst share = 9 × 127 = 1143.\nSecond share = 8 × 127 = 1016.\nCheck: 1143+1016=2159.\nFinal answer: $1016$", "final_answer": "1016", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "20828357e0573348", "problem_id": "837223f26bd43c1a", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{10} x - \\frac{x}{2} - 3 + 2 \\sqrt{10}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\sqrt{10} - \\frac{1}{2}$, c=$-3 + 2 \\sqrt{10}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{89}{4} - 7 \\sqrt{10}$, so √Δ = $\\sqrt{\\frac{89}{4} - 7 \\sqrt{10}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{89 - 28 \\sqrt{10}}}{4} + \\frac{1}{4} + \\frac{\\sqrt{10}}{2}$.\n7) x2 = $- \\frac{\\sqrt{89 - 28 \\sqrt{10}}}{4} + \\frac{1}{4} + \\frac{\\sqrt{10}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{89 - 28\\cdot \\sqrt{10}}}{4} + \\frac{1}{4} + \\frac{\\sqrt{10}}{2}, - \\frac{\\sqrt{89 - 28\\cdot \\sqrt{10}}}{4} + \\frac{1}{4} + \\frac{\\sqrt{10}}{2}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{89 - 28\\cdot \\sqrt{10}}}{4} + \\frac{1}{4} + \\frac{\\sqrt{10}}{2}, - \\frac{\\sqrt{89 - 28\\cdot \\sqrt{10}}}{4} + \\frac{1}{4} + \\frac{\\sqrt{10}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c797bc14d98d6215", "problem_id": "837223f26bd43c1a", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{10} x - \\frac{x}{2} - 3 + 2 \\sqrt{10}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\sqrt{10} - \\frac{1}{2}$, c=$-3 + 2 \\sqrt{10}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{\\sqrt{89 - 28\\cdot \\sqrt{10}}}{4} + \\frac{1}{4} + \\frac{\\sqrt{10}}{2}, - \\frac{\\sqrt{89 - 28\\cdot \\sqrt{10}}}{4} + \\frac{1}{4} + \\frac{\\sqrt{10}}{2}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{89 - 28\\cdot \\sqrt{10}}}{4} + \\frac{1}{4} + \\frac{\\sqrt{10}}{2}, - \\frac{\\sqrt{89 - 28\\cdot \\sqrt{10}}}{4} + \\frac{1}{4} + \\frac{\\sqrt{10}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b204ff8b70facfa7", "problem_id": "c47bf12191a7d484", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{86 x}{15} + \\frac{16}{15}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$- \\frac{86}{15}$, c=$\\frac{16}{15}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{5476}{225}$, so √Δ = $\\frac{74}{15}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{8}{3}$.\n7) x2 = $\\frac{1}{5}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{8}{3}, \\frac{1}{5}\\}$", "final_answer": "x \\in \\{\\frac{8}{3}, \\frac{1}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f475842386682bb1", "problem_id": "683f51bf6c7a9c8e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{4} - x^{3} - x^{2} - 2 x - 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x^{4}) = 4 x^{4}$.\n3) So $f'(x) = 4 x^{4} - 3 x^{2} - 2 x - 2$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x^{4}) = 4 x^{3}$.\nTherefore the correct derivative is $f'(x) = 4 x^{3} - 3 x^{2} - 2 x - 2$.\nFinal answer: $4 x^{3} - 3 x^{2} - 2 x - 2$", "final_answer": "4 x^{3} - 3 x^{2} - 2 x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7e01aa534b07f9ba", "problem_id": "6d23b3463fb3e5c7", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡7 (mod 8), x≡3 (mod 11). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 47 (mod 88).\nFinal answer: $x\\equiv 47\\pmod{88}$", "final_answer": "x\\equiv 47\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "156be077a63ad983", "problem_id": "6d23b3463fb3e5c7", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the congruences: x≡7 (mod 8), x≡3 (mod 11). Give the solution modulo 88.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 88.\nCompute the CRT solution to get x ≡ 47 (mod 88).\nFinal answer: $x\\equiv 47\\pmod{88}$", "final_answer": "x\\equiv 47\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3ab6e91c1890e982", "problem_id": "2a90a264286f48dd", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 63 miles per hour for 1 hours, then at 72 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 63 × 1 = 63 miles.\nSecond leg distance = 72 × 1 = 72 miles.\nTotal distance = 63 + 72 = 135 miles.\nCheck: adding both legs gives 135.\nFinal answer: $135$", "final_answer": "135", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6cb80201cd480323", "problem_id": "2a90a264286f48dd", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A cyclist rides at 63 miles per hour for 1 hours, then at 72 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 63 × 1 = 63 miles.\n2) Second leg distance = 72 × 1 = 72 miles.\n3) Total distance = 63 + 72 = 135 miles.\nFinal answer: $135$", "final_answer": "135", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fd84c4247e0ef5d6", "problem_id": "1112bba91c41d4e8", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡3 (mod 5), x≡5 (mod 8). Give the solution modulo 40.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 40.\n3) Compute the CRT solution to get x ≡ 13 (mod 40).\nFinal answer: $x\\equiv 13\\pmod{40}$", "final_answer": "x\\equiv 13\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "53096227fa9545e2", "problem_id": "1112bba91c41d4e8", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the congruences: x≡3 (mod 5), x≡5 (mod 8). Give the solution modulo 40.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 40.\nCompute the CRT solution to get x ≡ 13 (mod 40).\nFinal answer: $x\\equiv 13\\pmod{40}$", "final_answer": "x\\equiv 13\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f663958718dd33c2", "problem_id": "8ab909c6f31e8114", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the linear system:\n$2 x - 5 y$ = $-51$\n$- 4 x + 5 y$ = $57$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $- 8 x + 20 y$ = $204$.\nMultiply the second equation by $2$: $- 8 x + 10 y$ = $114$.\nFinal answer: $(x,y)=\\left(-3, 9\\right)$", "final_answer": "(x,y)=\\left(-3, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5ef5f7f87e9759a5", "problem_id": "8ab909c6f31e8114", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$2 x - 5 y$ = $-51$\n$- 4 x + 5 y$ = $57$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $- 8 x + 20 y$ = $204$.\nMultiply the second equation by $2$: $- 8 x + 10 y$ = $114$.\nSubtract to eliminate x: $10 y$ = $90$.\nSolve for y: y = $9$.\nSubstitute back to find x: x = $-3$.\nCheck: plug (x,y)=(-3,9) into both equations.\nFinal answer: $(x,y)=\\left(-3, 9\\right)$", "final_answer": "(x,y)=\\left(-3, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f360fc963867a56", "problem_id": "6b03e32bb281dfe1", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 2 x^{2} + 4 x - 6$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 2 x^{3} + 2 x^{2} - 2 x + 1$ and a = -1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 6 x^{2} + 4 x - 2$.\nEvaluate at a: p'(-1) = $-12$.\nCheck: the identity gives $-12$.\nFinal answer: $-12$", "final_answer": "-12", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f3895d097691dca7", "problem_id": "6b03e32bb281dfe1", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 2 x^{2} + 4 x - 6$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 2 x^{3} + 2 x^{2} - 2 x + 1$ and a = -1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 6 x^{2} + 4 x - 2$.\n5) Evaluate at a: p'(-1) = $-12$.\nFinal answer: $-12$", "final_answer": "-12", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8fd0f4347133eb4f", "problem_id": "842491a7a4806959", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$2 x - 3 y$ = $22$\n$- x + 4 y$ = $-21$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $- 2 x + 3 y$ = $-22$.\nMultiply the second equation by $2$: $- 2 x + 8 y$ = $-42$.\nSubtract to eliminate x: $- 5 y$ = $20$.\nSolve for y: y = $-4$.\nSubstitute back to find x: x = $5$.\nCheck: plug (x,y)=(5,-4) into both equations.\nFinal answer: $(x,y)=\\left(5, -4\\right)$", "final_answer": "(x,y)=\\left(5, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7a97a1eda50a14ea", "problem_id": "842491a7a4806959", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$2 x - 3 y$ = $22$\n$- x + 4 y$ = $-21$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $- 2 x + 3 y$ = $-22$.\n3) Multiply the second equation by $2$: $- 2 x + 8 y$ = $-42$.\n4) Subtract to eliminate x: $- 5 y$ = $20$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, -4\\right)$", "final_answer": "(x,y)=\\left(5, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "74e76615c933154f", "problem_id": "e114e303440f1d83", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A store sells a backpack for $365. It is discounted by 25%. You buy 10 backpacks. What is the total cost?", "output": "Discounted price = 365 × (1 − 25/100) = 365 × 75/100 = 1095/4.\nSubtotal = 10 × 1095/4 = 5475/2.\nFinal answer: $\\frac{5475}{2}$", "final_answer": "\\frac{5475}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "92538bf17fbd7954", "problem_id": "e114e303440f1d83", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A store sells a backpack for $365. It is discounted by 25%. You buy 10 backpacks. What is the total cost?", "output": "1) Discounted price = 365 × (1 − 25/100) = 365 × 75/100 = 1095/4.\n2) Subtotal = 10 × 1095/4 = 5475/2.\nFinal answer: $\\frac{5475}{2}$", "final_answer": "\\frac{5475}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7a035cd04911eb8e", "problem_id": "fedc4ad544934b72", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the multiplicative inverse of 24 modulo 59.", "output": "1) We want x such that 24x ≡ 1 (mod 59).\n2) Use extended Euclid to find s,t with 24s + 59t = 1.\n3) One Bezout identity is: 24·(-27) + 59·(11) = 1.\n4) Reduce s modulo 59: x ≡ 32 (mod 59).\nFinal answer: $x\\equiv 32\\pmod{59}$", "final_answer": "x\\equiv 32\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "21ccf3c2afb4b71a", "problem_id": "fedc4ad544934b72", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the multiplicative inverse of 24 modulo 59.", "output": "We want x such that 24x ≡ 1 (mod 59).\nUse extended Euclid to find s,t with 24s + 59t = 1.\nOne Bezout identity is: 24·(-27) + 59·(11) = 1.\nReduce s modulo 59: x ≡ 32 (mod 59).\nCheck: 24·32 mod 59 = 1.\nFinal answer: $x\\equiv 32\\pmod{59}$", "final_answer": "x\\equiv 32\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "538e13f38549396c", "problem_id": "5c4d171cb85a355c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - 3 x^{3} + 9 x^{2} - 15 x + 31$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{3} + 9 x^{2} - 16 x + 31.\n3) Then the limit is 123.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 121.\nFinal answer: $121$", "final_answer": "121", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "302e241117b23bfc", "problem_id": "89561e6b9ccca5cb", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 4 x^{4} - 2 x^{3} - 2 x^{2} - 4 x + 3\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $4 x^{4} - 2 x^{3} - 2 x^{2} - 4 x + 3$.\n3) Compute an antiderivative: F(x) = $\\frac{4 x^{5}}{5} - \\frac{x^{4}}{2} - \\frac{2 x^{3}}{3} - 2 x^{2} + 3 x$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{4\\cdot x^{5}}{5} - \\frac{x^{4}}{2} - \\frac{2\\cdot x^{3}}{3} - 2\\cdot x^{2} + 3\\cdot x$", "final_answer": "C + \\frac{4\\cdot x^{5}}{5} - \\frac{x^{4}}{2} - \\frac{2\\cdot x^{3}}{3} - 2\\cdot x^{2} + 3\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0cfcf69c41b01659", "problem_id": "e54de4d1a7b1963c", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 3 x^{2} - 9 x - 21$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 3 x^{3} - 3 x^{2} - 3 x - 1$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 9 x^{2} - 6 x - 3$.\nEvaluate at a: p'(2) = $-51$.\nCheck: the identity gives $-51$.\nFinal answer: $-51$", "final_answer": "-51", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "44dd3fb13f5ce7a9", "problem_id": "e54de4d1a7b1963c", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 3 x^{2} - 9 x - 21$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 3 x^{3} - 3 x^{2} - 3 x - 1$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 9 x^{2} - 6 x - 3$.\n5) Evaluate at a: p'(2) = $-51$.\nFinal answer: $-51$", "final_answer": "-51", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "65fdf413986f661d", "problem_id": "21058c703c076b8b", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$3 x + y$ = $4$\n$- x + y$ = $0$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $- 3 x - y$ = $-4$.\n3) Multiply the second equation by $3$: $- 3 x + 3 y$ = $0$.\n4) Subtract to eliminate x: $- 4 y$ = $-4$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $1$.\nFinal answer: $(x,y)=\\left(1, 1\\right)$", "final_answer": "(x,y)=\\left(1, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec6abd39207e779a", "problem_id": "21058c703c076b8b", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the linear system:\n$3 x + y$ = $4$\n$- x + y$ = $0$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $- 3 x - y$ = $-4$.\nMultiply the second equation by $3$: $- 3 x + 3 y$ = $0$.\nFinal answer: $(x,y)=\\left(1, 1\\right)$", "final_answer": "(x,y)=\\left(1, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1cbad1b990d958e3", "problem_id": "f1bd35b1b314d724", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} 3 x^{4} - 2 x^{3} - 4 x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{4} - 2 x^{3} - 4 x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\nEvaluate at the bounds: F(\\pi) and F(0).\nSubtract: F(\\pi) − F(0) = $\\frac{\\pi \\left(- 15 \\pi^{3} - 40 \\pi^{2} - 90 - 15 \\pi + 18 \\pi^{4}\\right)}{30}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{\\pi\\cdot \\left(- 15\\cdot \\pi^{3} - 40\\cdot \\pi^{2} - 90 - 15\\cdot \\pi + 18\\cdot \\pi^{4}\\right)}{30}$", "final_answer": "\\frac{\\pi\\cdot \\left(- 15\\cdot \\pi^{3} - 40\\cdot \\pi^{2} - 90 - 15\\cdot \\pi + 18\\cdot \\pi^{4}\\right)}{30}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "acebadb8341ff3ac", "problem_id": "f1bd35b1b314d724", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} 3 x^{4} - 2 x^{3} - 4 x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{4} - 2 x^{3} - 4 x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\nEvaluate at the bounds: F(\\pi) and F(0).\nSubtract: F(\\pi) − F(0) = $\\frac{\\pi \\left(- 15 \\pi^{3} - 40 \\pi^{2} - 90 - 15 \\pi + 18 \\pi^{4}\\right)}{30}$.\nFinal answer: $\\frac{\\pi\\cdot \\left(- 15\\cdot \\pi^{3} - 40\\cdot \\pi^{2} - 90 - 15\\cdot \\pi + 18\\cdot \\pi^{4}\\right)}{30}$", "final_answer": "\\frac{\\pi\\cdot \\left(- 15\\cdot \\pi^{3} - 40\\cdot \\pi^{2} - 90 - 15\\cdot \\pi + 18\\cdot \\pi^{4}\\right)}{30}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4b3a3b84b0f37ef4", "problem_id": "54ceb785ab461c3c", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many binary strings of length 21 contain exactly 10 ones with no two ones adjacent?", "output": "1) Reserve 9 mandatory zeros between the 10 ones.\n2) This leaves n−k+1 = 12 valid slots for the ones.\n3) Choose the slots: C(12,10) = 66.\nFinal answer: $66$", "final_answer": "66", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3e14f2b9d6b5e5b4", "problem_id": "54ceb785ab461c3c", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "How many binary strings of length 21 contain exactly 10 ones with no two ones adjacent?", "output": "Reserve 9 mandatory zeros between the 10 ones.\nThis leaves n−k+1 = 12 valid slots for the ones.\nChoose the slots: C(12,10) = 66.\nFinal answer: $66$", "final_answer": "66", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0576876e00891cf8", "problem_id": "1a00fcd3a4c19656", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{15 x}{4} - \\sqrt{5} x - 1 - \\frac{\\sqrt{5}}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{15}{4} - \\sqrt{5}$, c=$-1 - \\frac{\\sqrt{5}}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{17 \\sqrt{5}}{2} + \\frac{369}{16}$, so √Δ = $\\sqrt{\\frac{17 \\sqrt{5}}{2} + \\frac{369}{16}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{5}}{2} + \\frac{15}{8} + \\frac{\\sqrt{136 \\sqrt{5} + 369}}{8}$.\n7) x2 = $- \\frac{\\sqrt{136 \\sqrt{5} + 369}}{8} + \\frac{\\sqrt{5}}{2} + \\frac{15}{8}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{5}}{2} + \\frac{15}{8} + \\frac{\\sqrt{136\\cdot \\sqrt{5} + 369}}{8}, - \\frac{\\sqrt{136\\cdot \\sqrt{5} + 369}}{8} + \\frac{\\sqrt{5}}{2} + \\frac{15}{8}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{5}}{2} + \\frac{15}{8} + \\frac{\\sqrt{136\\cdot \\sqrt{5} + 369}}{8}, - \\frac{\\sqrt{136\\cdot \\sqrt{5} + 369}}{8} + \\frac{\\sqrt{5}}{2} + \\frac{15}{8}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5300d2227a5fc764", "problem_id": "1a00fcd3a4c19656", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{15 x}{4} - \\sqrt{5} x - 1 - \\frac{\\sqrt{5}}{4}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{15}{4} - \\sqrt{5}$, c=$-1 - \\frac{\\sqrt{5}}{4}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{17 \\sqrt{5}}{2} + \\frac{369}{16}$, so √Δ = $\\sqrt{\\frac{17 \\sqrt{5}}{2} + \\frac{369}{16}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{5}}{2} + \\frac{15}{8} + \\frac{\\sqrt{136 \\sqrt{5} + 369}}{8}$.\nx2 = $- \\frac{\\sqrt{136 \\sqrt{5} + 369}}{8} + \\frac{\\sqrt{5}}{2} + \\frac{15}{8}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{5}}{2} + \\frac{15}{8} + \\frac{\\sqrt{136\\cdot \\sqrt{5} + 369}}{8}, - \\frac{\\sqrt{136\\cdot \\sqrt{5} + 369}}{8} + \\frac{\\sqrt{5}}{2} + \\frac{15}{8}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{5}}{2} + \\frac{15}{8} + \\frac{\\sqrt{136\\cdot \\sqrt{5} + 369}}{8}, - \\frac{\\sqrt{136\\cdot \\sqrt{5} + 369}}{8} + \\frac{\\sqrt{5}}{2} + \\frac{15}{8}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cb666b5a4b0584d4", "problem_id": "c2c7aa94ca3b0644", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A disease has prevalence $\\frac{1}{100}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{217}{2000}$.\n3) Compute P(D|+) = $\\frac{19}{217}$.\nFinal answer: $\\frac{19}{217}$", "final_answer": "\\frac{19}{217}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2ad079c5fa00c9e8", "problem_id": "c2c7aa94ca3b0644", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A disease has prevalence $\\frac{1}{100}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{217}{2000}$.\nCompute P(D|+) = $\\frac{19}{217}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{19}{217}$", "final_answer": "\\frac{19}{217}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "af291aa4d6eac353", "problem_id": "d559cfa8e57a172d", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $3 x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $3 \\left(x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}\\right)$.\nFinal answer: $3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x\\cdot \\sin{\\left(x \\right)} + 3\\cdot x\\cdot \\cos{\\left(x \\right)} + 3\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}$", "final_answer": "3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x\\cdot \\sin{\\left(x \\right)} + 3\\cdot x\\cdot \\cos{\\left(x \\right)} + 3\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e39ad35967264cd4", "problem_id": "d559cfa8e57a172d", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $3 x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $3 \\left(x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}\\right)$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x\\cdot \\sin{\\left(x \\right)} + 3\\cdot x\\cdot \\cos{\\left(x \\right)} + 3\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}$", "final_answer": "3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x\\cdot \\sin{\\left(x \\right)} + 3\\cdot x\\cdot \\cos{\\left(x \\right)} + 3\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "41c3b14e74dd14ea", "problem_id": "ff91a298247203d7", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 3 x$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{2} + 3 x + 3$ and a = -1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $6 x + 3$.\n5) Evaluate at a: p'(-1) = $-3$.\nFinal answer: $-3$", "final_answer": "-3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f32ba02c344507b", "problem_id": "ff91a298247203d7", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 3 x$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $3 x^{2} + 3 x + 3$ and a = -1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-3$", "final_answer": "-3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2fea9a57f17c96b5", "problem_id": "402dcca88aac6349", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the multiplicative inverse of 6 modulo 19.", "output": "1) We want x such that 6x ≡ 1 (mod 19).\n2) Use extended Euclid to find s,t with 6s + 19t = 1.\n3) One Bezout identity is: 6·(-3) + 19·(1) = 1.\n4) Reduce s modulo 19: x ≡ 16 (mod 19).\nFinal answer: $x\\equiv 16\\pmod{19}$", "final_answer": "x\\equiv 16\\pmod{19}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "754fa2d9117acbdd", "problem_id": "402dcca88aac6349", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Find the multiplicative inverse of 6 modulo 19.", "output": "We want x such that 6x ≡ 1 (mod 19).\nUse extended Euclid to find s,t with 6s + 19t = 1.\nOne Bezout identity is: 6·(-3) + 19·(1) = 1.\nFinal answer: $x\\equiv 16\\pmod{19}$", "final_answer": "x\\equiv 16\\pmod{19}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "669a1daa5d07497b", "problem_id": "2db202662d503cca", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{10}$. If it is flipped 19 times, what is the probability of exactly 12 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=19, k=12, p=$\\frac{3}{10}$ and simplify.\n4) Here C(19,12) = 50388.\nFinal answer: $\\frac{5513259901287411}{2500000000000000000}$", "final_answer": "\\frac{5513259901287411}{2500000000000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0810f5d26b959f26", "problem_id": "2db202662d503cca", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{10}$. If it is flipped 19 times, what is the probability of exactly 12 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=19, k=12, p=$\\frac{3}{10}$ and simplify.\nHere C(19,12) = 50388.\nCheck: expression simplifies to $\\frac{5513259901287411}{2500000000000000000}$.\nFinal answer: $\\frac{5513259901287411}{2500000000000000000}$", "final_answer": "\\frac{5513259901287411}{2500000000000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2ee5ce255b13edbb", "problem_id": "a7c1d521c334eb8b", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{3} x + \\frac{x}{2} - \\frac{15}{2} + \\frac{5 \\sqrt{3}}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{1}{2} - \\sqrt{3}$, c=$- \\frac{15}{2} + \\frac{5 \\sqrt{3}}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{133}{4} - 11 \\sqrt{3}$, so √Δ = $\\sqrt{\\frac{133}{4} - 11 \\sqrt{3}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{1}{4} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{133 - 44 \\sqrt{3}}}{4}$.\nx2 = $- \\frac{\\sqrt{133 - 44 \\sqrt{3}}}{4} - \\frac{1}{4} + \\frac{\\sqrt{3}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{1}{4} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{133 - 44\\cdot \\sqrt{3}}}{4}, - \\frac{\\sqrt{133 - 44\\cdot \\sqrt{3}}}{4} - \\frac{1}{4} + \\frac{\\sqrt{3}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{1}{4} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{133 - 44\\cdot \\sqrt{3}}}{4}, - \\frac{\\sqrt{133 - 44\\cdot \\sqrt{3}}}{4} - \\frac{1}{4} + \\frac{\\sqrt{3}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3bb5068530ec79d8", "problem_id": "a7c1d521c334eb8b", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{3} x + \\frac{x}{2} - \\frac{15}{2} + \\frac{5 \\sqrt{3}}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{1}{2} - \\sqrt{3}$, c=$- \\frac{15}{2} + \\frac{5 \\sqrt{3}}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{133}{4} - 11 \\sqrt{3}$, so √Δ = $\\sqrt{\\frac{133}{4} - 11 \\sqrt{3}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{1}{4} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{133 - 44 \\sqrt{3}}}{4}$.\n7) x2 = $- \\frac{\\sqrt{133 - 44 \\sqrt{3}}}{4} - \\frac{1}{4} + \\frac{\\sqrt{3}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{1}{4} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{133 - 44\\cdot \\sqrt{3}}}{4}, - \\frac{\\sqrt{133 - 44\\cdot \\sqrt{3}}}{4} - \\frac{1}{4} + \\frac{\\sqrt{3}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{1}{4} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{133 - 44\\cdot \\sqrt{3}}}{4}, - \\frac{\\sqrt{133 - 44\\cdot \\sqrt{3}}}{4} - \\frac{1}{4} + \\frac{\\sqrt{3}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e7318b5f12b6b041", "problem_id": "ad425eaeb3daac3e", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $- 2 x^{2} \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $2 \\left(- x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}\\right)$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8f8a09e3e2b5085e", "problem_id": "ad425eaeb3daac3e", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $- 2 x^{2} \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $2 \\left(- x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}\\right)$.\nFinal answer: $- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6d99c2fd9436bc1c", "problem_id": "157f3c589e7935d3", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A store sells a backpack for $152. It is discounted by 25%. You buy 11 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 152 × (1 − 25/100) = 152 × 75/100 = 114.\n2) Subtotal = 11 × 114 = 1254.\n3) Total with tax = 1254 × 108/100 = 33858/25.\nFinal answer: $\\frac{33858}{25}$", "final_answer": "\\frac{33858}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2c85e917bc33bdd0", "problem_id": "157f3c589e7935d3", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $152. It is discounted by 25%. You buy 11 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 152 × (1 − 25/100) = 152 × 75/100 = 114.\nSubtotal = 11 × 114 = 1254.\nTotal with tax = 1254 × 108/100 = 33858/25.\nCheck: recomputing the formula yields 33858/25.\nFinal answer: $\\frac{33858}{25}$", "final_answer": "\\frac{33858}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff5ffb1c59909f7d", "problem_id": "906f869ecfe49945", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Worker A finishes a job in 7 hours and Worker B finishes it in 17 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/7 job/hour. B's rate = 1/17 job/hour.\n2) Combined rate = 1/7 + 1/17 = 24/119 job/hour.\n3) Time = 1 ÷ (combined rate) = 119/24 hours.\nFinal answer: $\\frac{119}{24}$", "final_answer": "\\frac{119}{24}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "960582b6c3acc104", "problem_id": "906f869ecfe49945", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Worker A finishes a job in 7 hours and Worker B finishes it in 17 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/7 job/hour. B's rate = 1/17 job/hour.\nCombined rate = 1/7 + 1/17 = 24/119 job/hour.\nTime = 1 ÷ (combined rate) = 119/24 hours.\nFinal answer: $\\frac{119}{24}$", "final_answer": "\\frac{119}{24}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cc3306af547c17d2", "problem_id": "824b38c523023e3e", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + \\frac{x}{2} - \\frac{1}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$\\frac{1}{2}$, c=$- \\frac{1}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{1}{3}, - \\frac{1}{2}\\}$", "final_answer": "x \\in \\{\\frac{1}{3}, - \\frac{1}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5f1b27ca7264ef3d", "problem_id": "824b38c523023e3e", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + \\frac{x}{2} - \\frac{1}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$\\frac{1}{2}$, c=$- \\frac{1}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{25}{4}$, so √Δ = $\\frac{5}{2}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{1}{3}$.\n7) x2 = $- \\frac{1}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{1}{3}, - \\frac{1}{2}\\}$", "final_answer": "x \\in \\{\\frac{1}{3}, - \\frac{1}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2bcb320b7b2de980", "problem_id": "1bfcd9762bea7c93", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 33 x + 54$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} + 33 x + 54$ = $3 \\left(x + 2\\right) \\left(x + 9\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $-2$.\nFrom the second factor: x = $-9$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-2, -9\\}$", "final_answer": "x \\in \\{-2, -9\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fbf519d6b1fcf827", "problem_id": "1bfcd9762bea7c93", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 33 x + 54$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} + 33 x + 54$ = $3 \\left(x + 2\\right) \\left(x + 9\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFinal answer: $x \\in \\{-2, -9\\}$", "final_answer": "x \\in \\{-2, -9\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0cb52f4553f277e1", "problem_id": "74907aac6bf30030", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{7} x}{2} + 2 x - \\sqrt{7}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$2 - \\frac{\\sqrt{7}}{2}$, c=$- \\sqrt{7}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $2 \\sqrt{7} + \\frac{23}{4}$, so √Δ = $\\sqrt{2 \\sqrt{7} + \\frac{23}{4}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $-1 + \\frac{\\sqrt{7}}{4} + \\frac{\\sqrt{8 \\sqrt{7} + 23}}{4}$.\n7) x2 = $- \\frac{\\sqrt{8 \\sqrt{7} + 23}}{4} - 1 + \\frac{\\sqrt{7}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{-1 + \\frac{\\sqrt{7}}{4} + \\frac{\\sqrt{8\\cdot \\sqrt{7} + 23}}{4}, - \\frac{\\sqrt{8\\cdot \\sqrt{7} + 23}}{4} - 1 + \\frac{\\sqrt{7}}{4}\\}$", "final_answer": "x \\in \\{-1 + \\frac{\\sqrt{7}}{4} + \\frac{\\sqrt{8\\cdot \\sqrt{7} + 23}}{4}, - \\frac{\\sqrt{8\\cdot \\sqrt{7} + 23}}{4} - 1 + \\frac{\\sqrt{7}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "946ec64e8ed99f33", "problem_id": "74907aac6bf30030", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{7} x}{2} + 2 x - \\sqrt{7}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$2 - \\frac{\\sqrt{7}}{2}$, c=$- \\sqrt{7}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{-1 + \\frac{\\sqrt{7}}{4} + \\frac{\\sqrt{8\\cdot \\sqrt{7} + 23}}{4}, - \\frac{\\sqrt{8\\cdot \\sqrt{7} + 23}}{4} - 1 + \\frac{\\sqrt{7}}{4}\\}$", "final_answer": "x \\in \\{-1 + \\frac{\\sqrt{7}}{4} + \\frac{\\sqrt{8\\cdot \\sqrt{7} + 23}}{4}, - \\frac{\\sqrt{8\\cdot \\sqrt{7} + 23}}{4} - 1 + \\frac{\\sqrt{7}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "20dd245f4c8f04e4", "problem_id": "c1c67160e53ffdd2", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "You want 22 liters of a 45% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 22−x liters is 70% solution.\n2) Salt equation: 20x + 70(22−x) = 45·22.\n3) (20-70)x = (45-70)·22 ⇒ x = 11.\n4) Then 22−x = 11.\nFinal answer: $11 L of 20% and 11 L of 70%$", "final_answer": "11 L of 20% and 11 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ab543c91ec237c17", "problem_id": "c1c67160e53ffdd2", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 22 liters of a 45% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 22−x liters is 70% solution.\nSalt equation: 20x + 70(22−x) = 45·22.\n(20-70)x = (45-70)·22 ⇒ x = 11.\nThen 22−x = 11.\nCheck: 20·11 + 70·11 = 45·22.\nFinal answer: $11 L of 20% and 11 L of 70%$", "final_answer": "11 L of 20% and 11 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5322d92c7f64574a", "problem_id": "5434139a06b7a3ee", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A disease has prevalence $\\frac{3}{50}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{213}{2500}$.\n3) Compute P(D|+) = $\\frac{95}{142}$.\nFinal answer: $\\frac{95}{142}$", "final_answer": "\\frac{95}{142}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "15ee826a7002c699", "problem_id": "19d224a19f416658", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 x^{3} - 7 x^{2} + 11 x - 23$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} - 7 x^{2} + 10 x - 23.\n3) Then the limit is -95.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -97.\nFinal answer: $-97$", "final_answer": "-97", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "45bbb422752d84b5", "problem_id": "030811549e621b42", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$2 x - 4 y$ = $-12$\n$- 6 y$ = $-30$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $2$: $- 12 y$ = $-60$.\nSubtract to eliminate x: $12 y$ = $60$.\nSolve for y: y = $5$.\nSubstitute back to find x: x = $4$.\nCheck: plug (x,y)=(4,5) into both equations.\nFinal answer: $(x,y)=\\left(4, 5\\right)$", "final_answer": "(x,y)=\\left(4, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ef6f97155f057062", "problem_id": "030811549e621b42", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$2 x - 4 y$ = $-12$\n$- 6 y$ = $-30$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $2$: $- 12 y$ = $-60$.\n4) Subtract to eliminate x: $12 y$ = $60$.\n5) Solve for y: y = $5$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, 5\\right)$", "final_answer": "(x,y)=\\left(4, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fab6af1677bbdcb8", "problem_id": "cc3fe49366e5cd52", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Worker A finishes a job in 10 hours and Worker B finishes it in 22 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/10 job/hour. B's rate = 1/22 job/hour.\n2) Combined rate = 1/10 + 1/22 = 8/55 job/hour.\n3) Time = 1 ÷ (combined rate) = 55/8 hours.\nFinal answer: $\\frac{55}{8}$", "final_answer": "\\frac{55}{8}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d27de9e438e12c5f", "problem_id": "cc3fe49366e5cd52", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Worker A finishes a job in 10 hours and Worker B finishes it in 22 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/10 job/hour. B's rate = 1/22 job/hour.\nCombined rate = 1/10 + 1/22 = 8/55 job/hour.\nTime = 1 ÷ (combined rate) = 55/8 hours.\nCheck: (combined rate)×(time) = 8/55×55/8 = 1.\nFinal answer: $\\frac{55}{8}$", "final_answer": "\\frac{55}{8}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5e7d56f40516a4a4", "problem_id": "17bdf8859a6d2f03", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 10 red balls and 14 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(24,5).\n3) Favorable samples: choose k reds and n−k blues: C(10,1)·C(14,4).\n4) Compute totals: C(24,5) = 42504, and favorable = 10010.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{65}{276}$", "final_answer": "\\frac{65}{276}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1cbc2c0a77be8c0d", "problem_id": "17bdf8859a6d2f03", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 10 red balls and 14 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(24,5).\nFavorable samples: choose k reds and n−k blues: C(10,1)·C(14,4).\nCompute totals: C(24,5) = 42504, and favorable = 10010.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{65}{276}$.\nFinal answer: $\\frac{65}{276}$", "final_answer": "\\frac{65}{276}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d59b4bed1b675335", "problem_id": "a8b2a3debe7a08b3", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Worker A finishes a job in 11 hours and Worker B finishes it in 9 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/11 job/hour. B's rate = 1/9 job/hour.\nCombined rate = 1/11 + 1/9 = 20/99 job/hour.\nTime = 1 ÷ (combined rate) = 99/20 hours.\nCheck: (combined rate)×(time) = 20/99×99/20 = 1.\nFinal answer: $\\frac{99}{20}$", "final_answer": "\\frac{99}{20}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a4adc48f2dc9abac", "problem_id": "a8b2a3debe7a08b3", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Worker A finishes a job in 11 hours and Worker B finishes it in 9 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/11 job/hour. B's rate = 1/9 job/hour.\n2) Combined rate = 1/11 + 1/9 = 20/99 job/hour.\n3) Time = 1 ÷ (combined rate) = 99/20 hours.\nFinal answer: $\\frac{99}{20}$", "final_answer": "\\frac{99}{20}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "92a9cb7e8566f46b", "problem_id": "f09d2c6886d3af5a", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} + 3 x^{2} + x + 2$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $- 9 x^{2} + 6 x + 1$.\n2) Slope at x=-2: m = f'(-2) = $-47$.\n3) Point on curve: (-2, f(-2)) = (-2, $36$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 36 = -47(x − -2).\n6) Optionally expand: y = $- 47 x - 58$.\nFinal answer: $y = - 47\\cdot x - 58$", "final_answer": "y = - 47\\cdot x - 58", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f34153337fe6d039", "problem_id": "f09d2c6886d3af5a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} + 3 x^{2} + x + 2$ at $x=-2$.", "output": "Differentiate: f'(x) = $- 9 x^{2} + 6 x + 1$.\nSlope at x=-2: m = f'(-2) = $-47$.\nPoint on curve: (-2, f(-2)) = (-2, $36$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 36 = -47(x − -2).\nOptionally expand: y = $- 47 x - 58$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 47\\cdot x - 58$", "final_answer": "y = - 47\\cdot x - 58", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "78f424e0171ed352", "problem_id": "11efe46b11fc43ff", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$5 x - 6 y$ = $-15$\n$4 x + 5 y$ = $37$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $20 x - 24 y$ = $-60$.\n3) Multiply the second equation by $5$: $20 x + 25 y$ = $185$.\n4) Subtract to eliminate x: $- 49 y$ = $-245$.\n5) Solve for y: y = $5$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, 5\\right)$", "final_answer": "(x,y)=\\left(3, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d0546ea3ecda85ea", "problem_id": "87e1d1d1c6222c9e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(- 2 x^{2} - 2 x\\right) \\cos{\\left(- 3 x^{2} + 2 x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 2 x^{2} \\cos{\\left(\\left(- 3 x^{2} + 2 x\\right) + 3 \\right)}) = 2 x^{2} \\sin{\\left(- 3 x^{2} + 2 x + 3 \\right)}$.\n3) So $f'(x) = - 10 x^{2} \\sin{\\left(- 3 x^{2} + 2 x + 3 \\right)} + 2 x e^{x} + 4 x \\sin{\\left(- 3 x^{2} + 2 x + 3 \\right)} - 4 e^{x} - 2 \\cos{\\left(\\left(- 3 x^{2} + 2 x\\right) + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 2 x^{2} \\cos{\\left(\\left(- 3 x^{2} + 2 x\\right) + 3 \\right)}) = 2 x^{2} \\left(2 - 6 x\\right) \\sin{\\left(- 3 x^{2} + 2 x + 3 \\right)} - 4 x \\cos{\\left(\\left(- 3 x^{2} + 2 x\\right) + 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- 2 x \\left(1 - 3 x\\right) \\left(- x - 1\\right) \\sin{\\left(- 3 x^{2} + 2 x + 3 \\right)} + \\left(- 2 x - 1\\right) \\cos{\\left(\\left(- 3 x^{2} + 2 x\\right) + 3 \\right)} + \\left(x - 3\\right) e^{x} + e^{x}\\right)$.\nFinal answer: $2 \\left(- 2 x \\left(1 - 3 x\\right) \\left(- x - 1\\right) \\sin{\\left(- 3 x^{2} + 2 x + 3 \\right)} + \\left(- 2 x - 1\\right) \\cos{\\left(\\left(- 3 x^{2} + 2 x\\right) + 3 \\right)} + \\left(x - 3\\right) e^{x} + e^{x}\\right)$", "final_answer": "2 \\left(- 2 x \\left(1 - 3 x\\right) \\left(- x - 1\\right) \\sin{\\left(- 3 x^{2} + 2 x + 3 \\right)} + \\left(- 2 x - 1\\right) \\cos{\\left(\\left(- 3 x^{2} + 2 x\\right) + 3 \\right)} + \\left(x - 3\\right) e^{x} + e^{x}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3e4bd1bf46621ef8", "problem_id": "1b9494c58c720cac", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$4 x - 4 y$ = $-40$\n$- 6 x - 8 y$ = $32$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-6$: $- 24 x + 24 y$ = $240$.\nMultiply the second equation by $4$: $- 24 x - 32 y$ = $128$.\nSubtract to eliminate x: $56 y$ = $112$.\nSolve for y: y = $2$.\nSubstitute back to find x: x = $-8$.\nCheck: plug (x,y)=(-8,2) into both equations.\nFinal answer: $(x,y)=\\left(-8, 2\\right)$", "final_answer": "(x,y)=\\left(-8, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ef00611c7cace17", "problem_id": "1b9494c58c720cac", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$4 x - 4 y$ = $-40$\n$- 6 x - 8 y$ = $32$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-6$: $- 24 x + 24 y$ = $240$.\n3) Multiply the second equation by $4$: $- 24 x - 32 y$ = $128$.\n4) Subtract to eliminate x: $56 y$ = $112$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $-8$.\nFinal answer: $(x,y)=\\left(-8, 2\\right)$", "final_answer": "(x,y)=\\left(-8, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b3be1fa9a875d344", "problem_id": "d95fe3513bbe6e6e", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 6 x - 40$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} - 6 x - 40$ = $\\left(x - 10\\right) \\left(x + 4\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $10$.\n5) From the second factor: x = $-4$.\nFinal answer: $x \\in \\{10, -4\\}$", "final_answer": "x \\in \\{10, -4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0a97dff8044cc105", "problem_id": "d95fe3513bbe6e6e", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 6 x - 40$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} - 6 x - 40$ = $\\left(x - 10\\right) \\left(x + 4\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $10$.\nFrom the second factor: x = $-4$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{10, -4\\}$", "final_answer": "x \\in \\{10, -4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c70b2c9ac92f668e", "problem_id": "317baec7fb2a1c10", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the multiplicative inverse of 10 modulo 13.", "output": "1) We want x such that 10x ≡ 1 (mod 13).\n2) Use extended Euclid to find s,t with 10s + 13t = 1.\n3) One Bezout identity is: 10·(4) + 13·(-3) = 1.\n4) Reduce s modulo 13: x ≡ 4 (mod 13).\nFinal answer: $x\\equiv 4\\pmod{13}$", "final_answer": "x\\equiv 4\\pmod{13}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ac8a4ec782b97a78", "problem_id": "e725b9b3a9f46c53", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(2 x^{2} + 2 x + 1\\right) \\cos{\\left(2 x^{2} + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x \\cos{\\left(2 x^{2} + 3 \\right)}) = - 2 x \\sin{\\left(2 x^{2} + 3 \\right)}$.\n3) So $f'(x) = - 8 x^{3} \\sin{\\left(2 x^{2} + 3 \\right)} + x e^{x} - 2 x \\sin{\\left(2 x^{2} + 3 \\right)} - 4 x \\sin{\\left(2 x^{2} + 3 \\right)} + 4 x \\cos{\\left(2 x^{2} + 3 \\right)} + 3 e^{x}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x \\cos{\\left(2 x^{2} + 3 \\right)}) = - 8 x^{2} \\sin{\\left(2 x^{2} + 3 \\right)} + 2 \\cos{\\left(2 x^{2} + 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = - 4 x \\left(2 x \\left(x + 1\\right) + 1\\right) \\sin{\\left(2 x^{2} + 3 \\right)} + \\left(x + 2\\right) e^{x} + 2 \\left(2 x + 1\\right) \\cos{\\left(2 x^{2} + 3 \\right)} + e^{x}$.\nFinal answer: $- 4 x \\left(2 x \\left(x + 1\\right) + 1\\right) \\sin{\\left(2 x^{2} + 3 \\right)} + \\left(x + 2\\right) e^{x} + 2 \\left(2 x + 1\\right) \\cos{\\left(2 x^{2} + 3 \\right)} + e^{x}$", "final_answer": "- 4 x \\left(2 x \\left(x + 1\\right) + 1\\right) \\sin{\\left(2 x^{2} + 3 \\right)} + \\left(x + 2\\right) e^{x} + 2 \\left(2 x + 1\\right) \\cos{\\left(2 x^{2} + 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7c82d159e5b79b41", "problem_id": "aedb7edf51a91faf", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{3} + 3 x^{2} + 4 x - 4$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-4) = 0.\n3) d/dx(3 x^{2}) = 6 x.\n4) d/dx(3 x^{3}) = 9 x^{2}.\n5) d/dx(4 x) = 4.\n6) Combine: f'(x) = 9 x^{2} + 6 x + 4.\nFinal answer: $9\\cdot x^{2} + 6\\cdot x + 4$", "final_answer": "9\\cdot x^{2} + 6\\cdot x + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9dedd8f5cce7e65f", "problem_id": "aedb7edf51a91faf", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{3} + 3 x^{2} + 4 x - 4$.", "output": "Differentiate term-by-term:\nd/dx(-4) = 0.\nd/dx(3 x^{2}) = 6 x.\nd/dx(3 x^{3}) = 9 x^{2}.\nd/dx(4 x) = 4.\nCombine: f'(x) = 9 x^{2} + 6 x + 4.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $9\\cdot x^{2} + 6\\cdot x + 4$", "final_answer": "9\\cdot x^{2} + 6\\cdot x + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bbcb086b1193d2e3", "problem_id": "387588ab52eba925", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$- 7 x - y$ = $37$\n$8 x - 7 y$ = $31$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $8$: $- 56 x - 8 y$ = $296$.\n3) Multiply the second equation by $-7$: $- 56 x + 49 y$ = $-217$.\n4) Subtract to eliminate x: $- 57 y$ = $513$.\n5) Solve for y: y = $-9$.\n6) Substitute back to find x: x = $-4$.\nFinal answer: $(x,y)=\\left(-4, -9\\right)$", "final_answer": "(x,y)=\\left(-4, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "32bdb89e8a7eae03", "problem_id": "387588ab52eba925", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the linear system:\n$- 7 x - y$ = $37$\n$8 x - 7 y$ = $31$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $8$: $- 56 x - 8 y$ = $296$.\nMultiply the second equation by $-7$: $- 56 x + 49 y$ = $-217$.\nFinal answer: $(x,y)=\\left(-4, -9\\right)$", "final_answer": "(x,y)=\\left(-4, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c42dbd7ea0bd365d", "problem_id": "7eed6768bc0312ee", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_8=29$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_8=5.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=5 and k=8.\n4) Compute C(12,7) = 792.\nFinal answer: $792$", "final_answer": "792", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "70a55c9208d557c2", "problem_id": "7eed6768bc0312ee", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_8=29$?", "output": "Let y_i=x_i-3, then y_i≥0 and y_1+…+y_8=5.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=5 and k=8.\nCompute C(12,7) = 792.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $792$", "final_answer": "792", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f104851e21a06ed9", "problem_id": "90b9eeef6cff8cc8", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$- 2 x - 5 y$ = $-6$\n$- 3 x + 5 y$ = $-34$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $6 x + 15 y$ = $18$.\nMultiply the second equation by $-2$: $6 x - 10 y$ = $68$.\nSubtract to eliminate x: $25 y$ = $-50$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $8$.\nCheck: plug (x,y)=(8,-2) into both equations.\nFinal answer: $(x,y)=\\left(8, -2\\right)$", "final_answer": "(x,y)=\\left(8, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fecea2977871750f", "problem_id": "90b9eeef6cff8cc8", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$- 2 x - 5 y$ = $-6$\n$- 3 x + 5 y$ = $-34$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $6 x + 15 y$ = $18$.\n3) Multiply the second equation by $-2$: $6 x - 10 y$ = $68$.\n4) Subtract to eliminate x: $25 y$ = $-50$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $8$.\nFinal answer: $(x,y)=\\left(8, -2\\right)$", "final_answer": "(x,y)=\\left(8, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "169208a04940a715", "problem_id": "57f8f6797d509b74", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 x - 10$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} - 3 x - 10$ = $\\left(x - 5\\right) \\left(x + 2\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $5$.\n5) From the second factor: x = $-2$.\nFinal answer: $x \\in \\{5, -2\\}$", "final_answer": "x \\in \\{5, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a4678119c1c90fbb", "problem_id": "57f8f6797d509b74", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 x - 10$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} - 3 x - 10$ = $\\left(x - 5\\right) \\left(x + 2\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $5$.\nFrom the second factor: x = $-2$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{5, -2\\}$", "final_answer": "x \\in \\{5, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a778ca028516f639", "problem_id": "524d7d4752d24dce", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "You want 61 liters of a 45% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 61−x liters is 50% solution.\n2) Salt equation: 25x + 50(61−x) = 45·61.\n3) (25-50)x = (45-50)·61 ⇒ x = 61/5.\n4) Then 61−x = 244/5.\nFinal answer: $61/5 L of 25% and 244/5 L of 50%$", "final_answer": "61/5 L of 25% and 244/5 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e08ea17e9a71b4d6", "problem_id": "524d7d4752d24dce", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "You want 61 liters of a 45% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 61−x liters is 50% solution.\nSalt equation: 25x + 50(61−x) = 45·61.\n(25-50)x = (45-50)·61 ⇒ x = 61/5.\nThen 61−x = 244/5.\nCheck: 25·61/5 + 50·244/5 = 45·61.\nFinal answer: $61/5 L of 25% and 244/5 L of 50%$", "final_answer": "61/5 L of 25% and 244/5 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4d5b5cb6d8391a3b", "problem_id": "92f5f2869a81d577", "style": "answer_only", "instruction": "Final answer only.", "input": "Worker A finishes a job in 11 hours and Worker B finishes it in 12 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "$\\frac{132}{23}$", "final_answer": "\\frac{132}{23}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8bd5d4464f18171f", "problem_id": "92f5f2869a81d577", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Worker A finishes a job in 11 hours and Worker B finishes it in 12 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/11 job/hour. B's rate = 1/12 job/hour.\n2) Combined rate = 1/11 + 1/12 = 23/132 job/hour.\n3) Time = 1 ÷ (combined rate) = 132/23 hours.\nFinal answer: $\\frac{132}{23}$", "final_answer": "\\frac{132}{23}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "826a8fbce75be685", "problem_id": "29f1f9e398c80276", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 3 x^{3} + 9 x^{2} + 17 x + 31$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $3 x^{4} + 3 x^{3} - x^{2} - 3 x$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $12 x^{3} + 9 x^{2} - 2 x - 3$.\nEvaluate at a: p'(2) = $125$.\nCheck: the identity gives $125$.\nFinal answer: $125$", "final_answer": "125", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1f22b02802162bbe", "problem_id": "29f1f9e398c80276", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 3 x^{3} + 9 x^{2} + 17 x + 31$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{4} + 3 x^{3} - x^{2} - 3 x$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $12 x^{3} + 9 x^{2} - 2 x - 3$.\n5) Evaluate at a: p'(2) = $125$.\nFinal answer: $125$", "final_answer": "125", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dee1ef2a94a79837", "problem_id": "b09867c773e9973c", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A store sells a backpack for $54. It is discounted by 35%. You buy 8 backpacks. What is the total cost?", "output": "1) Discounted price = 54 × (1 − 35/100) = 54 × 65/100 = 351/10.\n2) Subtotal = 8 × 351/10 = 1404/5.\nFinal answer: $\\frac{1404}{5}$", "final_answer": "\\frac{1404}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "919e82ba7d1029c3", "problem_id": "b09867c773e9973c", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A store sells a backpack for $54. It is discounted by 35%. You buy 8 backpacks. What is the total cost?", "output": "Discounted price = 54 × (1 − 35/100) = 54 × 65/100 = 351/10.\nSubtotal = 8 × 351/10 = 1404/5.\nCheck: recomputing the formula yields 1404/5.\nFinal answer: $\\frac{1404}{5}$", "final_answer": "\\frac{1404}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f86008bb4e28acf4", "problem_id": "09726800bf81c01d", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} + 4 x^{2} + 3 x$.", "output": "1) Differentiate term-by-term:\n2) d/dx(x^{3}) = 3 x^{2}.\n3) d/dx(3 x) = 3.\n4) d/dx(4 x^{2}) = 8 x.\n5) Combine: f'(x) = 3 x^{2} + 8 x + 3.\nFinal answer: $3\\cdot x^{2} + 8\\cdot x + 3$", "final_answer": "3\\cdot x^{2} + 8\\cdot x + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4400bb347c25e157", "problem_id": "09726800bf81c01d", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} + 4 x^{2} + 3 x$.", "output": "Differentiate term-by-term:\nd/dx(x^{3}) = 3 x^{2}.\nd/dx(3 x) = 3.\nd/dx(4 x^{2}) = 8 x.\nCombine: f'(x) = 3 x^{2} + 8 x + 3.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $3\\cdot x^{2} + 8\\cdot x + 3$", "final_answer": "3\\cdot x^{2} + 8\\cdot x + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "644fe4ebbf35c9f4", "problem_id": "897efd1aec636764", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{5}$. If it is flipped 12 times, what is the probability of exactly 6 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=12, k=6, p=$\\frac{3}{5}$ and simplify.\nHere C(12,6) = 924.\nCheck: expression simplifies to $\\frac{43110144}{244140625}$.\nFinal answer: $\\frac{43110144}{244140625}$", "final_answer": "\\frac{43110144}{244140625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "207a11a62964d12f", "problem_id": "897efd1aec636764", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{5}$. If it is flipped 12 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=6, p=$\\frac{3}{5}$ and simplify.\n4) Here C(12,6) = 924.\nFinal answer: $\\frac{43110144}{244140625}$", "final_answer": "\\frac{43110144}{244140625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "509b3e1df0a5d223", "problem_id": "509350b15811ea37", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 8 red balls and 8 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(16,5).\n3) Favorable samples: choose k reds and n−k blues: C(8,2)·C(8,3).\n4) Compute totals: C(16,5) = 4368, and favorable = 1568.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{14}{39}$", "final_answer": "\\frac{14}{39}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c7ebe5542ce18dbe", "problem_id": "509350b15811ea37", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 8 red balls and 8 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(16,5).\nFavorable samples: choose k reds and n−k blues: C(8,2)·C(8,3).\nCompute totals: C(16,5) = 4368, and favorable = 1568.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{14}{39}$.\nFinal answer: $\\frac{14}{39}$", "final_answer": "\\frac{14}{39}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ecef4b1a8c72598a", "problem_id": "8daf07e61c2b5bc3", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many binary strings of length 20 contain exactly 10 ones with no two ones adjacent?", "output": "Reserve 9 mandatory zeros between the 10 ones.\nThis leaves n−k+1 = 11 valid slots for the ones.\nChoose the slots: C(11,10) = 11.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $11$", "final_answer": "11", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "792e4a52c7e7d531", "problem_id": "8daf07e61c2b5bc3", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "How many binary strings of length 20 contain exactly 10 ones with no two ones adjacent?", "output": "1) Reserve 9 mandatory zeros between the 10 ones.\n2) This leaves n−k+1 = 11 valid slots for the ones.\n3) Choose the slots: C(11,10) = 11.\nFinal answer: $11$", "final_answer": "11", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f84af179b33deefa", "problem_id": "ae54fea6a1adf8fe", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many binary strings of length 16 contain exactly 8 ones with no two ones adjacent?", "output": "1) Reserve 7 mandatory zeros between the 8 ones.\n2) This leaves n−k+1 = 9 valid slots for the ones.\n3) Choose the slots: C(9,8) = 9.\nFinal answer: $9$", "final_answer": "9", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f480cfbcd0d4f49a", "problem_id": "ae54fea6a1adf8fe", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "How many binary strings of length 16 contain exactly 8 ones with no two ones adjacent?", "output": "$9$", "final_answer": "9", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8c4829d1cd081ad3", "problem_id": "9e96964ccd4033b2", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Find the multiplicative inverse of 14 modulo 37.", "output": "1) We want x such that 14x ≡ 1 (mod 37).\n2) Use extended Euclid to find s,t with 14s + 37t = 1.\n3) One Bezout identity is: 14·(8) + 37·(-3) = 1.\n4) Reduce s modulo 37: x ≡ 8 (mod 37).\nFinal answer: $x\\equiv 8\\pmod{37}$", "final_answer": "x\\equiv 8\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "053220876e2470bf", "problem_id": "9e96964ccd4033b2", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the multiplicative inverse of 14 modulo 37.", "output": "We want x such that 14x ≡ 1 (mod 37).\nUse extended Euclid to find s,t with 14s + 37t = 1.\nOne Bezout identity is: 14·(8) + 37·(-3) = 1.\nReduce s modulo 37: x ≡ 8 (mod 37).\nCheck: 14·8 mod 37 = 1.\nFinal answer: $x\\equiv 8\\pmod{37}$", "final_answer": "x\\equiv 8\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5568cb25476b7cf7", "problem_id": "fcff4f4a763e446e", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "$-4 + 3\\cdot \\pi + \\frac{9\\cdot \\pi^{2}}{2} + 9\\cdot \\pi^{3}$", "final_answer": "-4 + 3\\cdot \\pi + \\frac{9\\cdot \\pi^{2}}{2} + 9\\cdot \\pi^{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "145206d82072907b", "problem_id": "fcff4f4a763e446e", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $-4 + 3 \\pi + \\frac{9 \\pi^{2}}{2} + 9 \\pi^{3}$.\nFinal answer: $-4 + 3\\cdot \\pi + \\frac{9\\cdot \\pi^{2}}{2} + 9\\cdot \\pi^{3}$", "final_answer": "-4 + 3\\cdot \\pi + \\frac{9\\cdot \\pi^{2}}{2} + 9\\cdot \\pi^{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "48935f01b89a72a1", "problem_id": "192cc84e7fa41147", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} x^{3} + 4 x^{2} + 17 x + 71$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{4} + x^{2} + 3 x - 2$ and a = 4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $4 x^{3} + 2 x + 3$.\n5) Evaluate at a: p'(4) = $267$.\nFinal answer: $267$", "final_answer": "267", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "943bdcbe3669a1f8", "problem_id": "192cc84e7fa41147", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} x^{3} + 4 x^{2} + 17 x + 71$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $x^{4} + x^{2} + 3 x - 2$ and a = 4.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $4 x^{3} + 2 x + 3$.\nEvaluate at a: p'(4) = $267$.\nCheck: the identity gives $267$.\nFinal answer: $267$", "final_answer": "267", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3a621c5b65c17743", "problem_id": "6f207e9759e1c11d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{3} - \\frac{3 x}{2} + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}) = - \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} - \\frac{2 x^{4}}{2 x^{4} - 12 x^{2} + 18} - \\frac{6 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} - \\frac{12 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} + \\frac{21 x^{2}}{2 x^{4} - 12 x^{2} + 18} - \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 6 x^{2} + 9} + \\frac{9 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} + \\frac{18 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} + \\frac{9}{2 x^{4} - 12 x^{2} + 18} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 6} + \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3} - \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}) = \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - 3\\right)^{2}} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x^{2} - 3\\right)} - \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 2 x \\left(- 2 x^{3} - 3 x + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 3\\right)}{2 \\left(x^{2} - 3\\right)^{2}}$.\nFinal answer: $\\frac{- 2 x \\left(- 2 x^{3} - 3 x + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 3\\right)}{2 \\left(x^{2} - 3\\right)^{2}}$", "final_answer": "\\frac{- 2 x \\left(- 2 x^{3} - 3 x + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 3\\right)}{2 \\left(x^{2} - 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5992c65a1361fd75", "problem_id": "f386e9fb5971c6b6", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Find the multiplicative inverse of 17 modulo 31.", "output": "1) We want x such that 17x ≡ 1 (mod 31).\n2) Use extended Euclid to find s,t with 17s + 31t = 1.\n3) One Bezout identity is: 17·(11) + 31·(-6) = 1.\n4) Reduce s modulo 31: x ≡ 11 (mod 31).\nFinal answer: $x\\equiv 11\\pmod{31}$", "final_answer": "x\\equiv 11\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5587b15726d25a5f", "problem_id": "efeea33b1dba98bf", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{2} - \\frac{2 x}{3} + \\left(x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - \\frac{1}{3}}{x + 1}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 1}) = \\frac{e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 1}$.\n3) So $f'(x) = \\frac{3 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{2} + 12 x + 6} + \\frac{6 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{2} + 12 x + 6} + \\frac{6 x^{2}}{6 x^{2} + 12 x + 6} + \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{2} + 12 x + 6} + \\frac{12 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{2} + 12 x + 6} + \\frac{12 x}{6 x^{2} + 12 x + 6} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{2} + 12 x + 6} + \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{2} + 12 x + 6} - \\frac{2}{6 x^{2} + 12 x + 6} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 2 x + 1} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x + 2} - \\frac{e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 1} + \\frac{e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 1}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 1}) = \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x + 1\\right)} + \\frac{e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 1} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x + 1\\right)^{2}}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 6 x^{2} + 4 x + \\left(x + 1\\right) \\left(12 x + 3 \\left(x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4\\right) - 6 \\left(x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{6 \\left(x + 1\\right)^{2}}$.\nFinal answer: $\\frac{- 6 x^{2} + 4 x + \\left(x + 1\\right) \\left(12 x + 3 \\left(x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4\\right) - 6 \\left(x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{6 \\left(x + 1\\right)^{2}}$", "final_answer": "\\frac{- 6 x^{2} + 4 x + \\left(x + 1\\right) \\left(12 x + 3 \\left(x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4\\right) - 6 \\left(x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{6 \\left(x + 1\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "825d65647e1086f4", "problem_id": "03dd599ccae83e7b", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{6} x - \\frac{x}{2} - \\frac{27 \\sqrt{6}}{2} - \\frac{45}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- 3 \\sqrt{6} - \\frac{1}{2}$, c=$- \\frac{27 \\sqrt{6}}{2} - \\frac{45}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $57 \\sqrt{6} + \\frac{577}{4}$, so √Δ = $\\sqrt{57 \\sqrt{6} + \\frac{577}{4}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{1}{4} + \\frac{3 \\sqrt{6}}{2} + \\frac{\\sqrt{228 \\sqrt{6} + 577}}{4}$.\n7) x2 = $- \\frac{\\sqrt{228 \\sqrt{6} + 577}}{4} + \\frac{1}{4} + \\frac{3 \\sqrt{6}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{1}{4} + \\frac{3\\cdot \\sqrt{6}}{2} + \\frac{\\sqrt{228\\cdot \\sqrt{6} + 577}}{4}, - \\frac{\\sqrt{228\\cdot \\sqrt{6} + 577}}{4} + \\frac{1}{4} + \\frac{3\\cdot \\sqrt{6}}{2}\\}$", "final_answer": "x \\in \\{\\frac{1}{4} + \\frac{3\\cdot \\sqrt{6}}{2} + \\frac{\\sqrt{228\\cdot \\sqrt{6} + 577}}{4}, - \\frac{\\sqrt{228\\cdot \\sqrt{6} + 577}}{4} + \\frac{1}{4} + \\frac{3\\cdot \\sqrt{6}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "765cc64b5efea724", "problem_id": "37b9f046bc503dbb", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "An urn has 24 red balls and 17 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(41,10).\n3) Favorable samples: choose k reds and n−k blues: C(24,1)·C(17,9).\n4) Compute totals: C(41,10) = 1121099408, and favorable = 583440.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{15}{28823}$", "final_answer": "\\frac{15}{28823}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cc7b0a1b762afc37", "problem_id": "45a199b89ea9943a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - 3 x^{2} - 10 x - 32$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} - 11 x - 32.\n3) Then the limit is -92.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: -89.\nFinal answer: $-89$", "final_answer": "-89", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "791d2771c71d1421", "problem_id": "686bb2946a7217ea", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} + 2 x - 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(2 x^{2}) = 4 x^{2}$.\n3) So $f'(x) = 4 x^{2} + 2$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(2 x^{2}) = 4 x$.\nTherefore the correct derivative is $f'(x) = 2 \\left(2 x + 1\\right)$.\nFinal answer: $2 \\left(2 x + 1\\right)$", "final_answer": "2 \\left(2 x + 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "bfc9bb4cc3e86452", "problem_id": "9fbc1be1123faccc", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A store sells a backpack for $66. It is discounted by 30%. You buy 4 backpacks. What is the total cost?", "output": "1) Discounted price = 66 × (1 − 30/100) = 66 × 70/100 = 231/5.\n2) Subtotal = 4 × 231/5 = 924/5.\nFinal answer: $\\frac{924}{5}$", "final_answer": "\\frac{924}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "489bd40d32132538", "problem_id": "b8cae5ef6bc135fa", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{20 x}{3} + \\frac{50}{9}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$- \\frac{20}{3}$, c=$\\frac{50}{9}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $0$, so √Δ = $0$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{5}{3}$.\nx2 = $\\frac{5}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{5}{3}, \\frac{5}{3}\\}$", "final_answer": "x \\in \\{\\frac{5}{3}, \\frac{5}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d27e0ac77184964e", "problem_id": "b8cae5ef6bc135fa", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{20 x}{3} + \\frac{50}{9}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$- \\frac{20}{3}$, c=$\\frac{50}{9}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $0$, so √Δ = $0$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{5}{3}$.\n7) x2 = $\\frac{5}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{5}{3}, \\frac{5}{3}\\}$", "final_answer": "x \\in \\{\\frac{5}{3}, \\frac{5}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b663fbaeda8182c", "problem_id": "00dc65daf1799f11", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 9 x - 12$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} + 9 x - 12$ = $3 \\left(x - 1\\right) \\left(x + 4\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $1$.\n5) From the second factor: x = $-4$.\nFinal answer: $x \\in \\{1, -4\\}$", "final_answer": "x \\in \\{1, -4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "edde8f8cd25d5aad", "problem_id": "00dc65daf1799f11", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 9 x - 12$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} + 9 x - 12$ = $3 \\left(x - 1\\right) \\left(x + 4\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $1$.\nFrom the second factor: x = $-4$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{1, -4\\}$", "final_answer": "x \\in \\{1, -4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "df568d2df8bd411a", "problem_id": "2bc3b804c686b53c", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A cyclist rides at 56 miles per hour for 1/2 hours, then at 20 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 56 × 1/2 = 28 miles.\n2) Second leg distance = 20 × 5/2 = 50 miles.\n3) Total distance = 28 + 50 = 78 miles.\nFinal answer: $78$", "final_answer": "78", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8480eadf07eb4fc3", "problem_id": "2bc3b804c686b53c", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 56 miles per hour for 1/2 hours, then at 20 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 56 × 1/2 = 28 miles.\nSecond leg distance = 20 × 5/2 = 50 miles.\nTotal distance = 28 + 50 = 78 miles.\nCheck: adding both legs gives 78.\nFinal answer: $78$", "final_answer": "78", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a899920630be9a8c", "problem_id": "f7d847bda32d4623", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Two friends split $348 in the ratio 5:7. How much does each receive?", "output": "1) Total ratio parts = 5+7 = 12.\n2) Each part = 348 ÷ 12 = 29.\n3) First share = 5 × 29 = 145.\n4) Second share = 7 × 29 = 203.\nFinal answer: $203$", "final_answer": "203", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7e039bb874be5183", "problem_id": "f7d847bda32d4623", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Two friends split $348 in the ratio 5:7. How much does each receive?", "output": "$203$", "final_answer": "203", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8a9a4b3daea111b4", "problem_id": "9e7d3374cde109f0", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A disease has prevalence $\\frac{1}{50}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{33}{500}$.\n3) Compute P(D|+) = $\\frac{17}{66}$.\nFinal answer: $\\frac{17}{66}$", "final_answer": "\\frac{17}{66}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ce3278c536fd6347", "problem_id": "dfe45db6d235951c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 2 x^{2} \\sin{\\left(x \\right)}) = - 2 x^{2} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 2 x^{2} \\sin{\\left(x \\right)}) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$", "final_answer": "- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e3871a628143e0e0", "problem_id": "4b0b818cf654e4df", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡2 (mod 5), x≡2 (mod 7). Give the solution modulo 35.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 35.\n3) Compute the CRT solution to get x ≡ 2 (mod 35).\nFinal answer: $x\\equiv 2\\pmod{35}$", "final_answer": "x\\equiv 2\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d15fdb9a4d19e2be", "problem_id": "5bf1c679d39ecb2e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + x + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{3 x + 4}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 4}) = \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 4}$.\n3) So $f'(x) = \\frac{9 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{18 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{24 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{48 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{9 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{2} + 24 x + 16} - \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x + 8} + \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 4} - \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 4} + \\frac{16 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{32 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} + 48 x + 32} - \\frac{4}{18 x^{2} + 48 x + 32} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 4}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 4}) = \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(3 x + 4\\right)} + \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 4} - \\frac{9 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(3 x + 4\\right)^{2}} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 4}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 18 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6 x + \\left(3 x + 4\\right) \\left(3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 10 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right) - 24 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 12}{2 \\left(3 x + 4\\right)^{2}}$.\nFinal answer: $\\frac{- 18 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6 x + \\left(3 x + 4\\right) \\left(3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 10 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right) - 24 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 12}{2 \\left(3 x + 4\\right)^{2}}$", "final_answer": "\\frac{- 18 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6 x + \\left(3 x + 4\\right) \\left(3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 10 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right) - 24 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 12}{2 \\left(3 x + 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6b973f6901dcbd35", "problem_id": "c9fb8b84088b56bd", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} - 3 x + \\left(x^{2} - x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{x^{2} - x + 6}$.", "output": "1) Let f(x) = $\\frac{- x^{2} - 3 x + \\left(x^{2} - x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{x^{2} - x + 6}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{2 \\left(1 - 2 x\\right) \\left(- x^{2} - 3 x + \\left(x^{2} - x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2\\right) + \\left(x^{2} - x + 6\\right) \\left(- 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} - x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - x + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x^{2} - x + 6\\right)^{2}}$.\nFinal answer: $\\frac{\\left(2 - 4\\cdot x\\right)\\cdot \\left(- x^{2} - 3\\cdot x + \\left(x^{2} - x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 2\\right) + \\left(x^{2} - x + 6\\right)\\cdot \\left(- 4\\cdot x + \\left(4\\cdot x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(x^{2} - x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} - 2\\cdot x + 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 6\\right)}{2\\cdot \\left(x^{2} - x + 6\\right)^{2}}$", "final_answer": "\\frac{\\left(2 - 4\\cdot x\\right)\\cdot \\left(- x^{2} - 3\\cdot x + \\left(x^{2} - x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 2\\right) + \\left(x^{2} - x + 6\\right)\\cdot \\left(- 4\\cdot x + \\left(4\\cdot x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(x^{2} - x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} - 2\\cdot x + 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 6\\right)}{2\\cdot \\left(x^{2} - x + 6\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "42ca42520a6dbd30", "problem_id": "c9fb8b84088b56bd", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} - 3 x + \\left(x^{2} - x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{x^{2} - x + 6}$.", "output": "Let f(x) = $\\frac{- x^{2} - 3 x + \\left(x^{2} - x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{x^{2} - x + 6}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{2 \\left(1 - 2 x\\right) \\left(- x^{2} - 3 x + \\left(x^{2} - x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2\\right) + \\left(x^{2} - x + 6\\right) \\left(- 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} - x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - x + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x^{2} - x + 6\\right)^{2}}$.\nFinal answer: $\\frac{\\left(2 - 4\\cdot x\\right)\\cdot \\left(- x^{2} - 3\\cdot x + \\left(x^{2} - x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 2\\right) + \\left(x^{2} - x + 6\\right)\\cdot \\left(- 4\\cdot x + \\left(4\\cdot x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(x^{2} - x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} - 2\\cdot x + 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 6\\right)}{2\\cdot \\left(x^{2} - x + 6\\right)^{2}}$", "final_answer": "\\frac{\\left(2 - 4\\cdot x\\right)\\cdot \\left(- x^{2} - 3\\cdot x + \\left(x^{2} - x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 2\\right) + \\left(x^{2} - x + 6\\right)\\cdot \\left(- 4\\cdot x + \\left(4\\cdot x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(x^{2} - x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} - 2\\cdot x + 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 6\\right)}{2\\cdot \\left(x^{2} - x + 6\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "39667b72f179b191", "problem_id": "d662bd4c1f1b165d", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "You want 40 liters of a 30% salt solution. You have a 20% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 40−x liters is 40% solution.\nSalt equation: 20x + 40(40−x) = 30·40.\n(20-40)x = (30-40)·40 ⇒ x = 20.\nFinal answer: $20 L of 20% and 20 L of 40%$", "final_answer": "20 L of 20% and 20 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6ebaafd4faff9e23", "problem_id": "d662bd4c1f1b165d", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "You want 40 liters of a 30% salt solution. You have a 20% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 40−x liters is 40% solution.\n2) Salt equation: 20x + 40(40−x) = 30·40.\n3) (20-40)x = (30-40)·40 ⇒ x = 20.\n4) Then 40−x = 20.\nFinal answer: $20 L of 20% and 20 L of 40%$", "final_answer": "20 L of 20% and 20 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "110fb87d9d24100d", "problem_id": "1b70471675249d44", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 65 miles per hour for 1 hours, then at 45 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 65 × 1 = 65 miles.\nSecond leg distance = 45 × 1 = 45 miles.\nTotal distance = 65 + 45 = 110 miles.\nCheck: adding both legs gives 110.\nFinal answer: $110$", "final_answer": "110", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "79741b3d9090d6df", "problem_id": "1b70471675249d44", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A cyclist rides at 65 miles per hour for 1 hours, then at 45 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 65 × 1 = 65 miles.\n2) Second leg distance = 45 × 1 = 45 miles.\n3) Total distance = 65 + 45 = 110 miles.\nFinal answer: $110$", "final_answer": "110", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "46e70ef69ef535c1", "problem_id": "8a9873304acddee2", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡0 (mod 5), x≡3 (mod 8). Give the solution modulo 40.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 40.\n3) Compute the CRT solution to get x ≡ 35 (mod 40).\nFinal answer: $x\\equiv 35\\pmod{40}$", "final_answer": "x\\equiv 35\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ad911111ae9f33c", "problem_id": "8a9873304acddee2", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡0 (mod 5), x≡3 (mod 8). Give the solution modulo 40.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 40.\nCompute the CRT solution to get x ≡ 35 (mod 40).\nCheck: 35 mod 5=0 and mod 8=3.\nFinal answer: $x\\equiv 35\\pmod{40}$", "final_answer": "x\\equiv 35\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "253c3940a8b6e732", "problem_id": "98bc298f05d21123", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Worker A finishes a job in 7 hours and Worker B finishes it in 12 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/7 job/hour. B's rate = 1/12 job/hour.\nCombined rate = 1/7 + 1/12 = 19/84 job/hour.\nTime = 1 ÷ (combined rate) = 84/19 hours.\nCheck: (combined rate)×(time) = 19/84×84/19 = 1.\nFinal answer: $\\frac{84}{19}$", "final_answer": "\\frac{84}{19}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "33d045a3c5260768", "problem_id": "98bc298f05d21123", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Worker A finishes a job in 7 hours and Worker B finishes it in 12 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/7 job/hour. B's rate = 1/12 job/hour.\n2) Combined rate = 1/7 + 1/12 = 19/84 job/hour.\n3) Time = 1 ÷ (combined rate) = 84/19 hours.\nFinal answer: $\\frac{84}{19}$", "final_answer": "\\frac{84}{19}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3cb2f7bd508710a3", "problem_id": "ee8ca69cc5ede5ab", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} x + 1$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 1.\n3) Then the limit is 1.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -1.\nFinal answer: $-1$", "final_answer": "-1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5c3b9a2ea9dedf61", "problem_id": "240d545a75e3ccd7", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Find the multiplicative inverse of 15 modulo 19.", "output": "1) We want x such that 15x ≡ 1 (mod 19).\n2) Use extended Euclid to find s,t with 15s + 19t = 1.\n3) One Bezout identity is: 15·(-5) + 19·(4) = 1.\n4) Reduce s modulo 19: x ≡ 14 (mod 19).\nFinal answer: $x\\equiv 14\\pmod{19}$", "final_answer": "x\\equiv 14\\pmod{19}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a0ba695c72153c01", "problem_id": "dcf6e63212761200", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Worker A finishes a job in 7 hours and Worker B finishes it in 6 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/7 job/hour. B's rate = 1/6 job/hour.\nCombined rate = 1/7 + 1/6 = 13/42 job/hour.\nTime = 1 ÷ (combined rate) = 42/13 hours.\nCheck: (combined rate)×(time) = 13/42×42/13 = 1.\nFinal answer: $\\frac{42}{13}$", "final_answer": "\\frac{42}{13}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fcf118108000d25a", "problem_id": "dcf6e63212761200", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Worker A finishes a job in 7 hours and Worker B finishes it in 6 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/7 job/hour. B's rate = 1/6 job/hour.\n2) Combined rate = 1/7 + 1/6 = 13/42 job/hour.\n3) Time = 1 ÷ (combined rate) = 42/13 hours.\nFinal answer: $\\frac{42}{13}$", "final_answer": "\\frac{42}{13}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cc0eb4325bbec311", "problem_id": "386e651d3eb0e274", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_5=30$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_5=20.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=20 and k=5.\nCompute C(24,4) = 10626.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $10626$", "final_answer": "10626", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0857436a64f8926b", "problem_id": "7d441d4fa9f7e32e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 2 x^{3} + 2 x^{2} + x - 1$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{3} + 2 x^{2} - 1.\n3) Then the limit is 3.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: 2.\nFinal answer: $2$", "final_answer": "2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "eb77da2699d92ada", "problem_id": "74a3f1342a746ed6", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} + x^{2} - 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=0$.", "output": "1) Differentiate: f'(x) = $- 9 x^{2} + 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 3$.\n2) Slope at x=0: m = f'(0) = $- \\frac{5}{2}$.\n3) Point on curve: (0, f(0)) = (0, $3$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 3 = - \\frac{5}{2}(x − 0).\n6) Optionally expand: y = $3 - \\frac{5 x}{2}$.\nFinal answer: $y = 3 - \\frac{5\\cdot x}{2}$", "final_answer": "y = 3 - \\frac{5\\cdot x}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fc850bcbd8006372", "problem_id": "74a3f1342a746ed6", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} + x^{2} - 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=0$.", "output": "Differentiate: f'(x) = $- 9 x^{2} + 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 3$.\nSlope at x=0: m = f'(0) = $- \\frac{5}{2}$.\nPoint on curve: (0, f(0)) = (0, $3$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 3 = - \\frac{5}{2}(x − 0).\nOptionally expand: y = $3 - \\frac{5 x}{2}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 3 - \\frac{5\\cdot x}{2}$", "final_answer": "y = 3 - \\frac{5\\cdot x}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7b16202c99ebe4b0", "problem_id": "db65c2dfe1e6cb7b", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Two friends split $273 in the ratio 1:2. How much does each receive?", "output": "1) Total ratio parts = 1+2 = 3.\n2) Each part = 273 ÷ 3 = 91.\n3) First share = 1 × 91 = 91.\n4) Second share = 2 × 91 = 182.\nFinal answer: $182$", "final_answer": "182", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ea216e5d7de583a2", "problem_id": "db65c2dfe1e6cb7b", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Two friends split $273 in the ratio 1:2. How much does each receive?", "output": "Total ratio parts = 1+2 = 3.\nEach part = 273 ÷ 3 = 91.\nFirst share = 1 × 91 = 91.\nSecond share = 2 × 91 = 182.\nCheck: 91+182=273.\nFinal answer: $182$", "final_answer": "182", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "59dea3327e34ab4f", "problem_id": "5d8ef2d76d0f0383", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} - 2 x^{4} + x^{3} + x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{4} + x^{3} + x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $- \\frac{\\pi^{5}}{40} + \\frac{\\pi^{3}}{12} + 4 + 4 \\pi$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{\\pi^{5}}{40} + \\frac{\\pi^{3}}{12} + 4 + 4\\cdot \\pi$", "final_answer": "- \\frac{\\pi^{5}}{40} + \\frac{\\pi^{3}}{12} + 4 + 4\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "111f09c6e60aa3bf", "problem_id": "5d8ef2d76d0f0383", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} - 2 x^{4} + x^{3} + x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{4} + x^{3} + x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $- \\frac{\\pi^{5}}{40} + \\frac{\\pi^{3}}{12} + 4 + 4 \\pi$.\n4) Simplify the result.\nFinal answer: $- \\frac{\\pi^{5}}{40} + \\frac{\\pi^{3}}{12} + 4 + 4\\cdot \\pi$", "final_answer": "- \\frac{\\pi^{5}}{40} + \\frac{\\pi^{3}}{12} + 4 + 4\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d84f5e0f4724e2ad", "problem_id": "77a4ef78e5dbe913", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the equation of the tangent line to $y=- x^{3} + x^{2} - x + 3$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $- 3 x^{2} + 2 x - 1$.\n2) Slope at x=-2: m = f'(-2) = $-17$.\n3) Point on curve: (-2, f(-2)) = (-2, $17$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 17 = -17(x − -2).\n6) Optionally expand: y = $- 17 x - 17$.\nFinal answer: $y = - 17\\cdot x - 17$", "final_answer": "y = - 17\\cdot x - 17", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d1190b842b8d3de7", "problem_id": "77a4ef78e5dbe913", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} + x^{2} - x + 3$ at $x=-2$.", "output": "Differentiate: f'(x) = $- 3 x^{2} + 2 x - 1$.\nSlope at x=-2: m = f'(-2) = $-17$.\nPoint on curve: (-2, f(-2)) = (-2, $17$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 17 = -17(x − -2).\nOptionally expand: y = $- 17 x - 17$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 17\\cdot x - 17$", "final_answer": "y = - 17\\cdot x - 17", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "80c4c838095132f0", "problem_id": "73cb392bf1a3ed99", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 2 x^{2} - 3\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 2 x^{2} - 3$.\n3) Compute an antiderivative: F(x) = $- \\frac{2 x^{3}}{3} - 3 x$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{2\\cdot x^{3}}{3} - 3\\cdot x$", "final_answer": "C - \\frac{2\\cdot x^{3}}{3} - 3\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fc20de3b822aaeb4", "problem_id": "73cb392bf1a3ed99", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 2 x^{2} - 3\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 2 x^{2} - 3$.\nCompute an antiderivative: F(x) = $- \\frac{2 x^{3}}{3} - 3 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - \\frac{2\\cdot x^{3}}{3} - 3\\cdot x$", "final_answer": "C - \\frac{2\\cdot x^{3}}{3} - 3\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5185f647543210f8", "problem_id": "124a4641026b2b18", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 26 red balls and 24 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(50,7).\nFavorable samples: choose k reds and n−k blues: C(26,2)·C(24,5).\nCompute totals: C(50,7) = 99884400, and favorable = 13813800.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{13}{94}$.\nFinal answer: $\\frac{13}{94}$", "final_answer": "\\frac{13}{94}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e338b7c1cd4f72b8", "problem_id": "124a4641026b2b18", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "An urn has 26 red balls and 24 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(50,7).\nFavorable samples: choose k reds and n−k blues: C(26,2)·C(24,5).\nFinal answer: $\\frac{13}{94}$", "final_answer": "\\frac{13}{94}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "88d80d8963fd6fd8", "problem_id": "0afa354efc24e7f3", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - 3 x^{2} - 11 x - 43$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 3 x^{3} + x^{2} + x + 1$ and a = 4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 9 x^{2} + 2 x + 1$.\n5) Evaluate at a: p'(4) = $-135$.\nFinal answer: $-135$", "final_answer": "-135", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "de624db25ea3fd24", "problem_id": "0afa354efc24e7f3", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - 3 x^{2} - 11 x - 43$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 3 x^{3} + x^{2} + x + 1$ and a = 4.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 9 x^{2} + 2 x + 1$.\nEvaluate at a: p'(4) = $-135$.\nCheck: the identity gives $-135$.\nFinal answer: $-135$", "final_answer": "-135", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e984a3a541de0d15", "problem_id": "192e44a45e2bafd4", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} - x + 3$.", "output": "1) Differentiate term-by-term:\n2) d/dx(3) = 0.\n3) d/dx(x^{3}) = 3 x^{2}.\n4) d/dx(- x) = -1.\n5) Combine: f'(x) = 3 x^{2} - 1.\nFinal answer: $3\\cdot x^{2} - 1$", "final_answer": "3\\cdot x^{2} - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "05abbcefbf41cf38", "problem_id": "192e44a45e2bafd4", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} - x + 3$.", "output": "Differentiate term-by-term:\nd/dx(3) = 0.\nd/dx(x^{3}) = 3 x^{2}.\nd/dx(- x) = -1.\nCombine: f'(x) = 3 x^{2} - 1.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $3\\cdot x^{2} - 1$", "final_answer": "3\\cdot x^{2} - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a380f5c22a5996db", "problem_id": "38e8567189a2d952", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{2} + x + 1$ at $x=2$.", "output": "1) Differentiate: f'(x) = $1 - 6 x$.\n2) Slope at x=2: m = f'(2) = $-11$.\n3) Point on curve: (2, f(2)) = (2, $-9$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -9 = -11(x − 2).\n6) Optionally expand: y = $13 - 11 x$.\nFinal answer: $y = 13 - 11\\cdot x$", "final_answer": "y = 13 - 11\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "656647d2ed6ef063", "problem_id": "ca1db2df9672f21b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x \\sin{\\left(x \\right)}) = x \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x \\sin{\\left(x \\right)}) = x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$.\nFinal answer: $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$", "final_answer": "- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d8ca5f3307db76b3", "problem_id": "b5434f5becb56603", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the multiplicative inverse of 17 modulo 19.", "output": "1) We want x such that 17x ≡ 1 (mod 19).\n2) Use extended Euclid to find s,t with 17s + 19t = 1.\n3) One Bezout identity is: 17·(9) + 19·(-8) = 1.\n4) Reduce s modulo 19: x ≡ 9 (mod 19).\nFinal answer: $x\\equiv 9\\pmod{19}$", "final_answer": "x\\equiv 9\\pmod{19}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8083f46798d1a65a", "problem_id": "b25da1383a89c6c7", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{2} - 3 x + \\left(- x^{2} + 2 x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{- x^{2} + 2 x + 6}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- x^{2} + 2 x\\right) + 6}) = \\frac{2 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{- x^{2} + 2 x + 6}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{4 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{8 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{8 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{16 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{18 x^{2}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{4 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 4 x^{3} - 8 x^{2} + 24 x + 36} + \\frac{24 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} + \\frac{48 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{68 x}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} + \\frac{4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 4 x^{3} - 8 x^{2} + 24 x + 36} + \\frac{2 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{- x^{2} + 2 x + 6} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- x^{2} + 2 x\\right) + 6} - \\frac{2 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(- x^{2} + 2 x\\right) + 6} + \\frac{36 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} + \\frac{72 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{40}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- x^{2} + 2 x\\right) + 6}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- x^{2} + 2 x\\right) + 6}) = \\frac{2 x \\left(2 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(- x^{2} + 2 x\\right) + 6\\right)^{2}} + \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- x^{2} + 2 x\\right) + 6} + \\frac{2 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(- x^{2} + 2 x\\right) + 6} + \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- x^{2} + 2 x\\right) + 6}$.\nTherefore the correct derivative is $f'(x) = \\frac{4 \\left(x - 1\\right) \\left(- 3 x^{2} - 3 x + \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(x \\left(2 - x\\right) + 6\\right) \\left(- 12 x + 4 \\left(1 - x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x \\left(2 - x\\right) + 6\\right)^{2}}$.\nFinal answer: $\\frac{4 \\left(x - 1\\right) \\left(- 3 x^{2} - 3 x + \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(x \\left(2 - x\\right) + 6\\right) \\left(- 12 x + 4 \\left(1 - x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x \\left(2 - x\\right) + 6\\right)^{2}}$", "final_answer": "\\frac{4 \\left(x - 1\\right) \\left(- 3 x^{2} - 3 x + \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(x \\left(2 - x\\right) + 6\\right) \\left(- 12 x + 4 \\left(1 - x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x \\left(2 - x\\right) + 6\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a4204fb2e6a2175b", "problem_id": "56e37d94f076f8f7", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Two friends split $477 in the ratio 2:7. How much does each receive?", "output": "1) Total ratio parts = 2+7 = 9.\n2) Each part = 477 ÷ 9 = 53.\n3) First share = 2 × 53 = 106.\n4) Second share = 7 × 53 = 371.\nFinal answer: $371$", "final_answer": "371", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e49c7687e0e540e6", "problem_id": "56e37d94f076f8f7", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Two friends split $477 in the ratio 2:7. How much does each receive?", "output": "Total ratio parts = 2+7 = 9.\nEach part = 477 ÷ 9 = 53.\nFirst share = 2 × 53 = 106.\nSecond share = 7 × 53 = 371.\nCheck: 106+371=477.\nFinal answer: $371$", "final_answer": "371", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "686a3b73534dbaa4", "problem_id": "548ef5bd85687b77", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_9=35$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_9=17.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=17 and k=9.\n4) Compute C(25,8) = 1081575.\nFinal answer: $1081575$", "final_answer": "1081575", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b26f88b6fa42a8be", "problem_id": "548ef5bd85687b77", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_9=35$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_9=17.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=17 and k=9.\nCompute C(25,8) = 1081575.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $1081575$", "final_answer": "1081575", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1735db9d0062d1cb", "problem_id": "e9bf59582f59abed", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$- 7 x - 8 y$ = $-76$\n$2 x + 6 y$ = $44$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $- 14 x - 16 y$ = $-152$.\n3) Multiply the second equation by $-7$: $- 14 x - 42 y$ = $-308$.\n4) Subtract to eliminate x: $26 y$ = $156$.\n5) Solve for y: y = $6$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, 6\\right)$", "final_answer": "(x,y)=\\left(4, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3af468d69f270f8a", "problem_id": "e9bf59582f59abed", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the linear system:\n$- 7 x - 8 y$ = $-76$\n$2 x + 6 y$ = $44$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $2$: $- 14 x - 16 y$ = $-152$.\nMultiply the second equation by $-7$: $- 14 x - 42 y$ = $-308$.\nFinal answer: $(x,y)=\\left(4, 6\\right)$", "final_answer": "(x,y)=\\left(4, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2f5d46517045dc21", "problem_id": "7f706ad765fa3bea", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A store sells a backpack for $940. It is discounted by 15%. You buy 3 backpacks. What is the total cost?", "output": "1) Discounted price = 940 × (1 − 15/100) = 940 × 85/100 = 799.\n2) Subtotal = 3 × 799 = 2397.\nFinal answer: $2397$", "final_answer": "2397", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1875da4c2dec8f62", "problem_id": "7f706ad765fa3bea", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A store sells a backpack for $940. It is discounted by 15%. You buy 3 backpacks. What is the total cost?", "output": "Discounted price = 940 × (1 − 15/100) = 940 × 85/100 = 799.\nSubtotal = 3 × 799 = 2397.\nCheck: recomputing the formula yields 2397.\nFinal answer: $2397$", "final_answer": "2397", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0b0ef6b2382a01ad", "problem_id": "9d6cc78c3c7ee641", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - x^{3} - x^{2} - 7 x - 31$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to - x^{3} - x^{2} - 8 x - 31.\n3) Then the limit is -143.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: -139.\nFinal answer: $-139$", "final_answer": "-139", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5d068abe4dff4ab7", "problem_id": "a28a666d7fab97bc", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 10 x + 8$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 10 x + 8$ = $2 \\left(x - 4\\right) \\left(x - 1\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $4$.\n5) From the second factor: x = $1$.\nFinal answer: $x \\in \\{4, 1\\}$", "final_answer": "x \\in \\{4, 1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "68e4fe527c562630", "problem_id": "5579be24c9519674", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A store sells a backpack for $141. It is discounted by 5%. You buy 11 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 141 × (1 − 5/100) = 141 × 95/100 = 2679/20.\n2) Subtotal = 11 × 2679/20 = 29469/20.\n3) Total with tax = 29469/20 × 105/100 = 618849/400.\nFinal answer: $\\frac{618849}{400}$", "final_answer": "\\frac{618849}{400}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "142cf276a25f330e", "problem_id": "0dc2f8488645f051", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{2} + 3 x + 5$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{2} + 2 x + 5.\n3) Then the limit is 17.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 19.\nFinal answer: $19$", "final_answer": "19", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7dbdd277fb141102", "problem_id": "76fca6e27bf4a02e", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 12 red balls and 13 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(25,8).\n3) Favorable samples: choose k reds and n−k blues: C(12,1)·C(13,7).\n4) Compute totals: C(25,8) = 1081575, and favorable = 20592.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{208}{10925}$", "final_answer": "\\frac{208}{10925}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "942c5c9d8c96acfc", "problem_id": "76fca6e27bf4a02e", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 12 red balls and 13 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(25,8).\nFavorable samples: choose k reds and n−k blues: C(12,1)·C(13,7).\nCompute totals: C(25,8) = 1081575, and favorable = 20592.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{208}{10925}$.\nFinal answer: $\\frac{208}{10925}$", "final_answer": "\\frac{208}{10925}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5bbecad5297a0d25", "problem_id": "96336b046e2b6dbc", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=16$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=16 and k=6.\n4) Compute C(21,5) = 20349.\nFinal answer: $20349$", "final_answer": "20349", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9907690a818c1981", "problem_id": "96336b046e2b6dbc", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=16$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=16 and k=6.\nFinal answer: $20349$", "final_answer": "20349", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0420b2e1fb3195b6", "problem_id": "0802f56675976727", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A store sells a backpack for $106. It is discounted by 5%. You buy 2 backpacks. What is the total cost?", "output": "1) Discounted price = 106 × (1 − 5/100) = 106 × 95/100 = 1007/10.\n2) Subtotal = 2 × 1007/10 = 1007/5.\nFinal answer: $\\frac{1007}{5}$", "final_answer": "\\frac{1007}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "994273d7d30e418f", "problem_id": "0802f56675976727", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "A store sells a backpack for $106. It is discounted by 5%. You buy 2 backpacks. What is the total cost?", "output": "$\\frac{1007}{5}$", "final_answer": "\\frac{1007}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7b1439c0eb0d3c9e", "problem_id": "08b38198e89cf982", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Worker A finishes a job in 9 hours and Worker B finishes it in 18 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/9 job/hour. B's rate = 1/18 job/hour.\n2) Combined rate = 1/9 + 1/18 = 1/6 job/hour.\n3) Time = 1 ÷ (combined rate) = 6 hours.\nFinal answer: $6$", "final_answer": "6", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "388680f86b8ad15a", "problem_id": "08b38198e89cf982", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Worker A finishes a job in 9 hours and Worker B finishes it in 18 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/9 job/hour. B's rate = 1/18 job/hour.\nCombined rate = 1/9 + 1/18 = 1/6 job/hour.\nTime = 1 ÷ (combined rate) = 6 hours.\nCheck: (combined rate)×(time) = 1/6×6 = 1.\nFinal answer: $6$", "final_answer": "6", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f6133e49537a8013", "problem_id": "38f74358b08cf593", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 18 times, what is the probability of exactly 12 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=18, k=12, p=$\\frac{1}{8}$ and simplify.\n4) Here C(18,12) = 18564.\nFinal answer: $\\frac{546009009}{4503599627370496}$", "final_answer": "\\frac{546009009}{4503599627370496}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b8f1830e67e17ccd", "problem_id": "38f74358b08cf593", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 18 times, what is the probability of exactly 12 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=18, k=12, p=$\\frac{1}{8}$ and simplify.\nHere C(18,12) = 18564.\nCheck: expression simplifies to $\\frac{546009009}{4503599627370496}$.\nFinal answer: $\\frac{546009009}{4503599627370496}$", "final_answer": "\\frac{546009009}{4503599627370496}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "97dbddff07a3f615", "problem_id": "23f171579c108a95", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the congruences: x≡0 (mod 3), x≡4 (mod 11). Give the solution modulo 33.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 33.\n3) Compute the CRT solution to get x ≡ 15 (mod 33).\nFinal answer: $x\\equiv 15\\pmod{33}$", "final_answer": "x\\equiv 15\\pmod{33}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e073e270ebb39ddb", "problem_id": "23f171579c108a95", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡0 (mod 3), x≡4 (mod 11). Give the solution modulo 33.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 33.\nCompute the CRT solution to get x ≡ 15 (mod 33).\nCheck: 15 mod 3=0 and mod 11=4.\nFinal answer: $x\\equiv 15\\pmod{33}$", "final_answer": "x\\equiv 15\\pmod{33}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cd25e58a260e7a78", "problem_id": "fc5ddcaff5299c40", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A store sells a backpack for $121. It is discounted by 5%. You buy 14 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 121 × (1 − 5/100) = 121 × 95/100 = 2299/20.\n2) Subtotal = 14 × 2299/20 = 16093/10.\n3) Total with tax = 16093/10 × 109/100 = 1754137/1000.\nFinal answer: $\\frac{1754137}{1000}$", "final_answer": "\\frac{1754137}{1000}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e42441fdaec64127", "problem_id": "fc5ddcaff5299c40", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A store sells a backpack for $121. It is discounted by 5%. You buy 14 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 121 × (1 − 5/100) = 121 × 95/100 = 2299/20.\nSubtotal = 14 × 2299/20 = 16093/10.\nTotal with tax = 16093/10 × 109/100 = 1754137/1000.\nFinal answer: $\\frac{1754137}{1000}$", "final_answer": "\\frac{1754137}{1000}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "70abbb5780deb698", "problem_id": "5d96134e70be7d0d", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A disease has prevalence $\\frac{1}{50}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{17}{250}$.\n3) Compute P(D|+) = $\\frac{19}{68}$.\nFinal answer: $\\frac{19}{68}$", "final_answer": "\\frac{19}{68}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "86c5dff6cd0e9bf5", "problem_id": "5d96134e70be7d0d", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A disease has prevalence $\\frac{1}{50}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{17}{250}$.\nCompute P(D|+) = $\\frac{19}{68}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{19}{68}$", "final_answer": "\\frac{19}{68}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "13bd2627920420c9", "problem_id": "47c0687fe4ea8f77", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.\nFinal answer: $x^{2}\\cdot \\cos{\\left(x \\right)} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + 3\\cdot x\\cdot \\cos{\\left(x \\right)} + 3\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - \\cos{\\left(x \\right)}$", "final_answer": "x^{2}\\cdot \\cos{\\left(x \\right)} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + 3\\cdot x\\cdot \\cos{\\left(x \\right)} + 3\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1af87b31fb34812e", "problem_id": "47c0687fe4ea8f77", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.\nFinal answer: $x^{2}\\cdot \\cos{\\left(x \\right)} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + 3\\cdot x\\cdot \\cos{\\left(x \\right)} + 3\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - \\cos{\\left(x \\right)}$", "final_answer": "x^{2}\\cdot \\cos{\\left(x \\right)} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + 3\\cdot x\\cdot \\cos{\\left(x \\right)} + 3\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1ab0b425ad261c49", "problem_id": "ca97d26d94a08616", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A cyclist rides at 86 miles per hour for 3/2 hours, then at 65 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 86 × 3/2 = 129 miles.\nSecond leg distance = 65 × 3 = 195 miles.\nTotal distance = 129 + 195 = 324 miles.\nCheck: adding both legs gives 324.\nFinal answer: $324$", "final_answer": "324", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e5ea0566ad86f9c3", "problem_id": "e87817263420430d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(x^{3} + 2 x^{2} + 2 x + 3\\right) \\cos{\\left(3 x - 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x e^{x}) = x e^{x}$.\n3) So $f'(x) = - 3 x^{3} \\sin{\\left(3 x - 2 \\right)} - 6 x^{2} \\sin{\\left(3 x - 2 \\right)} + 3 x^{2} \\cos{\\left(3 x - 2 \\right)} + x e^{x} - 6 x \\sin{\\left(3 x - 2 \\right)} + 4 x \\cos{\\left(3 x - 2 \\right)} + e^{x} - 9 \\sin{\\left(3 x - 2 \\right)} + 2 \\cos{\\left(3 x - 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x e^{x}) = x e^{x} + e^{x}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 1\\right) e^{x} - 3 \\left(x \\left(x \\left(x + 2\\right) + 2\\right) + 3\\right) \\sin{\\left(3 x - 2 \\right)} + \\left(3 x^{2} + 4 x + 2\\right) \\cos{\\left(3 x - 2 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 1\\right) e^{x} - 3 \\left(x \\left(x \\left(x + 2\\right) + 2\\right) + 3\\right) \\sin{\\left(3 x - 2 \\right)} + \\left(3 x^{2} + 4 x + 2\\right) \\cos{\\left(3 x - 2 \\right)} + e^{x}$", "final_answer": "\\left(x + 1\\right) e^{x} - 3 \\left(x \\left(x \\left(x + 2\\right) + 2\\right) + 3\\right) \\sin{\\left(3 x - 2 \\right)} + \\left(3 x^{2} + 4 x + 2\\right) \\cos{\\left(3 x - 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e4faa9887d40b040", "problem_id": "dc2ed41a8424457d", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 15 times, what is the probability of exactly 9 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=15, k=9, p=$\\frac{1}{4}$ and simplify.\nFinal answer: $\\frac{3648645}{1073741824}$", "final_answer": "\\frac{3648645}{1073741824}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6c997ce612aca06f", "problem_id": "dc2ed41a8424457d", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 15 times, what is the probability of exactly 9 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=15, k=9, p=$\\frac{1}{4}$ and simplify.\n4) Here C(15,9) = 5005.\nFinal answer: $\\frac{3648645}{1073741824}$", "final_answer": "\\frac{3648645}{1073741824}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c7f8e94f52fbeb25", "problem_id": "399c410826df04d1", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "You want 61 liters of a 35% salt solution. You have a 10% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 61−x liters is 40% solution.\n2) Salt equation: 10x + 40(61−x) = 35·61.\n3) (10-40)x = (35-40)·61 ⇒ x = 61/6.\n4) Then 61−x = 305/6.\nFinal answer: $61/6 L of 10% and 305/6 L of 40%$", "final_answer": "61/6 L of 10% and 305/6 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a3315f656183612", "problem_id": "399c410826df04d1", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "You want 61 liters of a 35% salt solution. You have a 10% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 61−x liters is 40% solution.\nSalt equation: 10x + 40(61−x) = 35·61.\n(10-40)x = (35-40)·61 ⇒ x = 61/6.\nThen 61−x = 305/6.\nCheck: 10·61/6 + 40·305/6 = 35·61.\nFinal answer: $61/6 L of 10% and 305/6 L of 40%$", "final_answer": "61/6 L of 10% and 305/6 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c63e43a7068790eb", "problem_id": "9897150815cee412", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A store sells a backpack for $314. It is discounted by 25%. You buy 6 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 314 × (1 − 25/100) = 314 × 75/100 = 471/2.\n2) Subtotal = 6 × 471/2 = 1413.\n3) Total with tax = 1413 × 105/100 = 29673/20.\nFinal answer: $\\frac{29673}{20}$", "final_answer": "\\frac{29673}{20}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a495c81ac9ae025f", "problem_id": "9897150815cee412", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A store sells a backpack for $314. It is discounted by 25%. You buy 6 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 314 × (1 − 25/100) = 314 × 75/100 = 471/2.\nSubtotal = 6 × 471/2 = 1413.\nTotal with tax = 1413 × 105/100 = 29673/20.\nFinal answer: $\\frac{29673}{20}$", "final_answer": "\\frac{29673}{20}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "45c2560006bdd456", "problem_id": "d00515e428185f98", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the linear system:\n$3 x + 4 y$ = $-17$\n$- x - 3 y$ = $14$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $- 3 x - 4 y$ = $17$.\n3) Multiply the second equation by $3$: $- 3 x - 9 y$ = $42$.\n4) Subtract to eliminate x: $5 y$ = $-25$.\n5) Solve for y: y = $-5$.\n6) Substitute back to find x: x = $1$.\nFinal answer: $(x,y)=\\left(1, -5\\right)$", "final_answer": "(x,y)=\\left(1, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7a142c64819aa559", "problem_id": "d00515e428185f98", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$3 x + 4 y$ = $-17$\n$- x - 3 y$ = $14$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $- 3 x - 4 y$ = $17$.\nMultiply the second equation by $3$: $- 3 x - 9 y$ = $42$.\nSubtract to eliminate x: $5 y$ = $-25$.\nSolve for y: y = $-5$.\nSubstitute back to find x: x = $1$.\nCheck: plug (x,y)=(1,-5) into both equations.\nFinal answer: $(x,y)=\\left(1, -5\\right)$", "final_answer": "(x,y)=\\left(1, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cbcd59730c8bc10a", "problem_id": "0d26cafeaff42269", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A store sells a backpack for $294. It is discounted by 15%. You buy 3 backpacks. What is the total cost?", "output": "1) Discounted price = 294 × (1 − 15/100) = 294 × 85/100 = 2499/10.\n2) Subtotal = 3 × 2499/10 = 7497/10.\nFinal answer: $\\frac{7497}{10}$", "final_answer": "\\frac{7497}{10}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2b59fb1a9617fd14", "problem_id": "0d26cafeaff42269", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $294. It is discounted by 15%. You buy 3 backpacks. What is the total cost?", "output": "Discounted price = 294 × (1 − 15/100) = 294 × 85/100 = 2499/10.\nSubtotal = 3 × 2499/10 = 7497/10.\nCheck: recomputing the formula yields 7497/10.\nFinal answer: $\\frac{7497}{10}$", "final_answer": "\\frac{7497}{10}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "482e143233ef839e", "problem_id": "1233771ba5f2ffa1", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - \\frac{43 x}{4} + 9$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$- \\frac{43}{4}$, c=$9$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{121}{16}$, so √Δ = $\\frac{11}{4}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{9}{4}$.\n7) x2 = $\\frac{4}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{9}{4}, \\frac{4}{3}\\}$", "final_answer": "x \\in \\{\\frac{9}{4}, \\frac{4}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "73016664ab153afe", "problem_id": "f3d3b4e037485f8c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(x^{2} - 3 x - 2\\right) \\cos{\\left(- 3 x^{2} + 3 x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\n3) So $f'(x) = 6 x^{3} \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} - 21 x^{2} \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} + x e^{x} - 3 x \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} + 2 x \\cos{\\left(\\left(- 3 x^{2} + 3 x\\right) + 3 \\right)} + 3 e^{x} + 6 \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} - 3 \\cos{\\left(\\left(- 3 x^{2} + 3 x\\right) + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = - 3 \\left(1 - 2 x\\right) \\left(x \\left(x - 3\\right) - 2\\right) \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} + \\left(x + 2\\right) e^{x} + \\left(2 x - 3\\right) \\cos{\\left(\\left(- 3 x^{2} + 3 x\\right) + 3 \\right)} + e^{x}$.\nFinal answer: $- 3 \\left(1 - 2 x\\right) \\left(x \\left(x - 3\\right) - 2\\right) \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} + \\left(x + 2\\right) e^{x} + \\left(2 x - 3\\right) \\cos{\\left(\\left(- 3 x^{2} + 3 x\\right) + 3 \\right)} + e^{x}$", "final_answer": "- 3 \\left(1 - 2 x\\right) \\left(x \\left(x - 3\\right) - 2\\right) \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} + \\left(x + 2\\right) e^{x} + \\left(2 x - 3\\right) \\cos{\\left(\\left(- 3 x^{2} + 3 x\\right) + 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "88c47e3cbc3bf74c", "problem_id": "8e95dac5e1ec9361", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$4 x - 6 y$ = $-24$\n$- 2 x$ = $12$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $- 8 x + 12 y$ = $48$.\n3) Multiply the second equation by $4$: $- 8 x$ = $48$.\n4) Subtract to eliminate x: $12 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $-6$.\nFinal answer: $(x,y)=\\left(-6, 0\\right)$", "final_answer": "(x,y)=\\left(-6, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "afe394fcef88be17", "problem_id": "8e95dac5e1ec9361", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the linear system:\n$4 x - 6 y$ = $-24$\n$- 2 x$ = $12$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $- 8 x + 12 y$ = $48$.\nMultiply the second equation by $4$: $- 8 x$ = $48$.\nFinal answer: $(x,y)=\\left(-6, 0\\right)$", "final_answer": "(x,y)=\\left(-6, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fa4be94026031524", "problem_id": "95fd6ea7eb92753e", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "You want 39 liters of a 30% salt solution. You have a 15% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 39−x liters is 40% solution.\n2) Salt equation: 15x + 40(39−x) = 30·39.\n3) (15-40)x = (30-40)·39 ⇒ x = 78/5.\n4) Then 39−x = 117/5.\nFinal answer: $78/5 L of 15% and 117/5 L of 40%$", "final_answer": "78/5 L of 15% and 117/5 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f6aa6482df157596", "problem_id": "95fd6ea7eb92753e", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 39 liters of a 30% salt solution. You have a 15% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 39−x liters is 40% solution.\nSalt equation: 15x + 40(39−x) = 30·39.\n(15-40)x = (30-40)·39 ⇒ x = 78/5.\nThen 39−x = 117/5.\nCheck: 15·78/5 + 40·117/5 = 30·39.\nFinal answer: $78/5 L of 15% and 117/5 L of 40%$", "final_answer": "78/5 L of 15% and 117/5 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "56a3b8ad72c7f809", "problem_id": "e744f1d2de978abc", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}$.\n3) Compute an antiderivative: F(x) = $\\frac{2 x^{3}}{3} + \\frac{3 x^{2}}{2} - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{2\\cdot x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{2\\cdot x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d09c9c02909eba98", "problem_id": "b6247d562fb05d69", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} - 4 x + 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(2 x^{2}) = 4 x^{2}$.\n3) So $f'(x) = 4 x^{2} - 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(2 x^{2}) = 4 x$.\nTherefore the correct derivative is $f'(x) = 4 \\left(x - 1\\right)$.\nFinal answer: $4 \\left(x - 1\\right)$", "final_answer": "4 \\left(x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ac464eb1f7d84064", "problem_id": "04ca0a85f58e9eaf", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "An urn has 13 red balls and 30 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(43,10).\n3) Favorable samples: choose k reds and n−k blues: C(13,2)·C(30,8).\n4) Compute totals: C(43,10) = 1917334783, and favorable = 456528150.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{35117550}{147487291}$", "final_answer": "\\frac{35117550}{147487291}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "59953ae4b6114696", "problem_id": "11024044ec7ec250", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{3} + 4 x^{2} + 9 x + 20$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} + 4 x^{2} + 8 x + 20.\n3) Then the limit is 68.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 70.\nFinal answer: $70$", "final_answer": "70", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ab70236e914e8a32", "problem_id": "f03f32ac5e3caa4d", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "You want 25 liters of a 40% salt solution. You have a 25% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 25−x liters is 60% solution.\nSalt equation: 25x + 60(25−x) = 40·25.\n(25-60)x = (40-60)·25 ⇒ x = 100/7.\nFinal answer: $100/7 L of 25% and 75/7 L of 60%$", "final_answer": "100/7 L of 25% and 75/7 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b05ba5d39482da31", "problem_id": "f03f32ac5e3caa4d", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "You want 25 liters of a 40% salt solution. You have a 25% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 25−x liters is 60% solution.\n2) Salt equation: 25x + 60(25−x) = 40·25.\n3) (25-60)x = (40-60)·25 ⇒ x = 100/7.\n4) Then 25−x = 75/7.\nFinal answer: $100/7 L of 25% and 75/7 L of 60%$", "final_answer": "100/7 L of 25% and 75/7 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "077c3fd4368f4d80", "problem_id": "011c1b281463aa14", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_9=23$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_9=5.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=5 and k=9.\n4) Compute C(13,8) = 1287.\nFinal answer: $1287$", "final_answer": "1287", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6cf17325d570e9c7", "problem_id": "011c1b281463aa14", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_9=23$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_9=5.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=5 and k=9.\nCompute C(13,8) = 1287.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $1287$", "final_answer": "1287", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "22ba294c2847fe3c", "problem_id": "84f9614e40e2a9fe", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the linear system:\n$2 x - 2 y$ = $-6$\n$x + y$ = $-5$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $2 x - 2 y$ = $-6$.\nMultiply the second equation by $2$: $2 x + 2 y$ = $-10$.\nFinal answer: $(x,y)=\\left(-4, -1\\right)$", "final_answer": "(x,y)=\\left(-4, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8966e69a467816fc", "problem_id": "84f9614e40e2a9fe", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$2 x - 2 y$ = $-6$\n$x + y$ = $-5$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $2 x - 2 y$ = $-6$.\n3) Multiply the second equation by $2$: $2 x + 2 y$ = $-10$.\n4) Subtract to eliminate x: $- 4 y$ = $4$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $-4$.\nFinal answer: $(x,y)=\\left(-4, -1\\right)$", "final_answer": "(x,y)=\\left(-4, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "85705027be5279fe", "problem_id": "1644eede615118dd", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many binary strings of length 8 contain exactly 2 ones with no two ones adjacent?", "output": "Reserve 1 mandatory zeros between the 2 ones.\nThis leaves n−k+1 = 7 valid slots for the ones.\nChoose the slots: C(7,2) = 21.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $21$", "final_answer": "21", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "de7febb1ac444653", "problem_id": "1644eede615118dd", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many binary strings of length 8 contain exactly 2 ones with no two ones adjacent?", "output": "1) Reserve 1 mandatory zeros between the 2 ones.\n2) This leaves n−k+1 = 7 valid slots for the ones.\n3) Choose the slots: C(7,2) = 21.\nFinal answer: $21$", "final_answer": "21", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8fd4873429419ce3", "problem_id": "bf4b04179aedb308", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A cyclist rides at 37 miles per hour for 1/2 hours, then at 35 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 37 × 1/2 = 37/2 miles.\n2) Second leg distance = 35 × 1 = 35 miles.\n3) Total distance = 37/2 + 35 = 107/2 miles.\nFinal answer: $\\frac{107}{2}$", "final_answer": "\\frac{107}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "885e04f2dc9f8248", "problem_id": "d2266df0b4a0f3d1", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(2 x^{3} - 3 x^{2} - 3 x\\right) \\cos{\\left(3 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 3 x^{2} \\cos{\\left(3 x \\right)}) = 3 x^{2} \\sin{\\left(3 x \\right)}$.\n3) So $f'(x) = - 6 x^{3} \\sin{\\left(3 x \\right)} + 3 x^{2} \\sin{\\left(3 x \\right)} + 6 x^{2} \\cos{\\left(3 x \\right)} + 2 x e^{x} + 9 x \\sin{\\left(3 x \\right)} - 4 e^{x} - 3 \\cos{\\left(3 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 3 x^{2} \\cos{\\left(3 x \\right)}) = 9 x^{2} \\sin{\\left(3 x \\right)} - 6 x \\cos{\\left(3 x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 3 x \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x \\right)} + 2 \\left(x - 3\\right) e^{x} + 3 \\left(2 x^{2} - 2 x - 1\\right) \\cos{\\left(3 x \\right)} + 2 e^{x}$.\nFinal answer: $- 3 x \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x \\right)} + 2 \\left(x - 3\\right) e^{x} + 3 \\left(2 x^{2} - 2 x - 1\\right) \\cos{\\left(3 x \\right)} + 2 e^{x}$", "final_answer": "- 3 x \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x \\right)} + 2 \\left(x - 3\\right) e^{x} + 3 \\left(2 x^{2} - 2 x - 1\\right) \\cos{\\left(3 x \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f8f98a2046a2e49f", "problem_id": "7794d743163db3f5", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 12 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=3, p=$\\frac{1}{10}$ and simplify.\n4) Here C(12,3) = 220.\nFinal answer: $\\frac{4261625379}{50000000000}$", "final_answer": "\\frac{4261625379}{50000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d4b11297651e3b45", "problem_id": "7794d743163db3f5", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 12 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=12, k=3, p=$\\frac{1}{10}$ and simplify.\nFinal answer: $\\frac{4261625379}{50000000000}$", "final_answer": "\\frac{4261625379}{50000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8ad3ad79d090e4aa", "problem_id": "834f1fae2a4b73be", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 2 x - 40$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 2 x - 40$ = $2 \\left(x - 5\\right) \\left(x + 4\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $5$.\n5) From the second factor: x = $-4$.\nFinal answer: $x \\in \\{5, -4\\}$", "final_answer": "x \\in \\{5, -4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3b6bf1cd97085d96", "problem_id": "834f1fae2a4b73be", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 2 x - 40$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} - 2 x - 40$ = $2 \\left(x - 5\\right) \\left(x + 4\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFinal answer: $x \\in \\{5, -4\\}$", "final_answer": "x \\in \\{5, -4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1a15de5715c9ab85", "problem_id": "197956f9cb864b21", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the congruences: x≡4 (mod 11), x≡1 (mod 10). Give the solution modulo 110.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 110.\n3) Compute the CRT solution to get x ≡ 81 (mod 110).\nFinal answer: $x\\equiv 81\\pmod{110}$", "final_answer": "x\\equiv 81\\pmod{110}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2c1c07dbf1074413", "problem_id": "197956f9cb864b21", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡4 (mod 11), x≡1 (mod 10). Give the solution modulo 110.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 110.\nCompute the CRT solution to get x ≡ 81 (mod 110).\nCheck: 81 mod 11=4 and mod 10=1.\nFinal answer: $x\\equiv 81\\pmod{110}$", "final_answer": "x\\equiv 81\\pmod{110}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c24a25bd1b69225a", "problem_id": "182f8d6a5f6e5cee", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} + 3 x^{2} - 2$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x^{3}) = 3 x^{3}$.\n3) So $f'(x) = 3 x^{3} + 6 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x^{3}) = 3 x^{2}$.\nTherefore the correct derivative is $f'(x) = 3 x \\left(x + 2\\right)$.\nFinal answer: $3 x \\left(x + 2\\right)$", "final_answer": "3 x \\left(x + 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2ab9fcaf1f02d8ce", "problem_id": "7c404bd03a7f7974", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡2 (mod 3), x≡3 (mod 7). Give the solution modulo 21.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 21.\n3) Compute the CRT solution to get x ≡ 17 (mod 21).\nFinal answer: $x\\equiv 17\\pmod{21}$", "final_answer": "x\\equiv 17\\pmod{21}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "84917941dc94b453", "problem_id": "91858649a0e7970a", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{3} + 2 x^{2} + 7 x + 15$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{4} - 2 x^{3} + 3 x^{2} + x + 1$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $8 x^{3} - 6 x^{2} + 6 x + 1$.\nEvaluate at a: p'(2) = $53$.\nCheck: the identity gives $53$.\nFinal answer: $53$", "final_answer": "53", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e45ee52b265058ee", "problem_id": "91858649a0e7970a", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{3} + 2 x^{2} + 7 x + 15$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $2 x^{4} - 2 x^{3} + 3 x^{2} + x + 1$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $8 x^{3} - 6 x^{2} + 6 x + 1$.\n5) Evaluate at a: p'(2) = $53$.\nFinal answer: $53$", "final_answer": "53", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "34593ceea0df44a1", "problem_id": "782934a1915a228b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} + 4 x - 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x^{2}) = - 4 x^{2}$.\n3) So $f'(x) = 4 - 4 x^{2}$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x^{2}) = - 4 x$.\nTherefore the correct derivative is $f'(x) = 4 \\left(1 - x\\right)$.\nFinal answer: $4 \\left(1 - x\\right)$", "final_answer": "4 \\left(1 - x\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7b2baeadf8ba70d1", "problem_id": "bbd90636f13742cb", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} x \\left(2 - 3 x\\right)$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} + x.\n3) Then the limit is -4.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: -5.\nFinal answer: $-5$", "final_answer": "-5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a488f458412087aa", "problem_id": "9f48e1a86bd96b81", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "You want 15 liters of a 20% salt solution. You have a 10% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 15−x liters is 50% solution.\n2) Salt equation: 10x + 50(15−x) = 20·15.\n3) (10-50)x = (20-50)·15 ⇒ x = 45/4.\n4) Then 15−x = 15/4.\nFinal answer: $45/4 L of 10% and 15/4 L of 50%$", "final_answer": "45/4 L of 10% and 15/4 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a02d68983a85d841", "problem_id": "9f48e1a86bd96b81", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "You want 15 liters of a 20% salt solution. You have a 10% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 15−x liters is 50% solution.\nSalt equation: 10x + 50(15−x) = 20·15.\n(10-50)x = (20-50)·15 ⇒ x = 45/4.\nThen 15−x = 15/4.\nCheck: 10·45/4 + 50·15/4 = 20·15.\nFinal answer: $45/4 L of 10% and 15/4 L of 50%$", "final_answer": "45/4 L of 10% and 15/4 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "63a1b6d7af5c6c75", "problem_id": "cf5e701a2bf6d0c7", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A store sells a backpack for $137. It is discounted by 5%. You buy 14 backpacks. What is the total cost?", "output": "1) Discounted price = 137 × (1 − 5/100) = 137 × 95/100 = 2603/20.\n2) Subtotal = 14 × 2603/20 = 18221/10.\nFinal answer: $\\frac{18221}{10}$", "final_answer": "\\frac{18221}{10}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9954aec2eb6c98b3", "problem_id": "cf5e701a2bf6d0c7", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A store sells a backpack for $137. It is discounted by 5%. You buy 14 backpacks. What is the total cost?", "output": "Discounted price = 137 × (1 − 5/100) = 137 × 95/100 = 2603/20.\nSubtotal = 14 × 2603/20 = 18221/10.\nCheck: recomputing the formula yields 18221/10.\nFinal answer: $\\frac{18221}{10}$", "final_answer": "\\frac{18221}{10}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2cdab5fb09e1b86c", "problem_id": "c64f293445e0c84b", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Worker A finishes a job in 21 hours and Worker B finishes it in 6 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/21 job/hour. B's rate = 1/6 job/hour.\n2) Combined rate = 1/21 + 1/6 = 3/14 job/hour.\n3) Time = 1 ÷ (combined rate) = 14/3 hours.\nFinal answer: $\\frac{14}{3}$", "final_answer": "\\frac{14}{3}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f2f7535dbf317b73", "problem_id": "c64f293445e0c84b", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Worker A finishes a job in 21 hours and Worker B finishes it in 6 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/21 job/hour. B's rate = 1/6 job/hour.\nCombined rate = 1/21 + 1/6 = 3/14 job/hour.\nTime = 1 ÷ (combined rate) = 14/3 hours.\nCheck: (combined rate)×(time) = 3/14×14/3 = 1.\nFinal answer: $\\frac{14}{3}$", "final_answer": "\\frac{14}{3}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2ee3ce24d226e62f", "problem_id": "827c89dd5205edd8", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many binary strings of length 9 contain exactly 3 ones with no two ones adjacent?", "output": "1) Reserve 2 mandatory zeros between the 3 ones.\n2) This leaves n−k+1 = 7 valid slots for the ones.\n3) Choose the slots: C(7,3) = 35.\nFinal answer: $35$", "final_answer": "35", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5821d3934f2c881e", "problem_id": "827c89dd5205edd8", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many binary strings of length 9 contain exactly 3 ones with no two ones adjacent?", "output": "Reserve 2 mandatory zeros between the 3 ones.\nThis leaves n−k+1 = 7 valid slots for the ones.\nChoose the slots: C(7,3) = 35.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $35$", "final_answer": "35", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff9f0a30621c4fd2", "problem_id": "c5464d40960f76aa", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 48 miles per hour for 2 hours, then at 41 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 48 × 2 = 96 miles.\nSecond leg distance = 41 × 3/2 = 123/2 miles.\nTotal distance = 96 + 123/2 = 315/2 miles.\nCheck: adding both legs gives 315/2.\nFinal answer: $\\frac{315}{2}$", "final_answer": "\\frac{315}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3cb2f7bd508710a3", "problem_id": "ee8ca69cc5ede5ab", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} x + 1$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 1.\n3) Then the limit is 1.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -1.\nFinal answer: $-1$", "final_answer": "-1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "cffa142115db891f", "problem_id": "d029a71d4961b0eb", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Two friends split $225 in the ratio 3:2. How much does each receive?", "output": "1) Total ratio parts = 3+2 = 5.\n2) Each part = 225 ÷ 5 = 45.\n3) First share = 3 × 45 = 135.\n4) Second share = 2 × 45 = 90.\nFinal answer: $90$", "final_answer": "90", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "51d06bf053dbb8ec", "problem_id": "5196868997c8b2a9", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{3} - x^{2} - 4$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 3 x^{3}) = - 9 x^{3}$.\n3) So $f'(x) = - 9 x^{3} - 2 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 3 x^{3}) = - 9 x^{2}$.\nTherefore the correct derivative is $f'(x) = x \\left(- 9 x - 2\\right)$.\nFinal answer: $x \\left(- 9 x - 2\\right)$", "final_answer": "x \\left(- 9 x - 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "dc75d7200859cdba", "problem_id": "1beaf11af9fb4d76", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A store sells a backpack for $192. It is discounted by 35%. You buy 5 backpacks. What is the total cost?", "output": "1) Discounted price = 192 × (1 − 35/100) = 192 × 65/100 = 624/5.\n2) Subtotal = 5 × 624/5 = 624.\nFinal answer: $624$", "final_answer": "624", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a2d798e2f95cb0d4", "problem_id": "1beaf11af9fb4d76", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "A store sells a backpack for $192. It is discounted by 35%. You buy 5 backpacks. What is the total cost?", "output": "Discounted price = 192 × (1 − 35/100) = 192 × 65/100 = 624/5.\nSubtotal = 5 × 624/5 = 624.\nFinal answer: $624$", "final_answer": "624", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f0b528dffe6dc8cc", "problem_id": "616f5eb2d277401e", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 54 x + 240$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} - 54 x + 240$ = $3 \\left(x - 10\\right) \\left(x - 8\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $10$.\n5) From the second factor: x = $8$.\nFinal answer: $x \\in \\{10, 8\\}$", "final_answer": "x \\in \\{10, 8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7504607f10b4faa2", "problem_id": "616f5eb2d277401e", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 54 x + 240$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} - 54 x + 240$ = $3 \\left(x - 10\\right) \\left(x - 8\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFinal answer: $x \\in \\{10, 8\\}$", "final_answer": "x \\in \\{10, 8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1aa8fc4efbe5eebd", "problem_id": "3d32b41d613a1764", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$- 5 x + 6 y$ = $19$\n$10 x + 3 y$ = $97$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $10$: $- 50 x + 60 y$ = $190$.\n3) Multiply the second equation by $-5$: $- 50 x - 15 y$ = $-485$.\n4) Subtract to eliminate x: $75 y$ = $675$.\n5) Solve for y: y = $9$.\n6) Substitute back to find x: x = $7$.\nFinal answer: $(x,y)=\\left(7, 9\\right)$", "final_answer": "(x,y)=\\left(7, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a1b1b4747bce1abd", "problem_id": "3d32b41d613a1764", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$- 5 x + 6 y$ = $19$\n$10 x + 3 y$ = $97$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $10$: $- 50 x + 60 y$ = $190$.\nMultiply the second equation by $-5$: $- 50 x - 15 y$ = $-485$.\nSubtract to eliminate x: $75 y$ = $675$.\nSolve for y: y = $9$.\nSubstitute back to find x: x = $7$.\nCheck: plug (x,y)=(7,9) into both equations.\nFinal answer: $(x,y)=\\left(7, 9\\right)$", "final_answer": "(x,y)=\\left(7, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "57558be6de72c756", "problem_id": "1caf36e71918de2f", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{2} - 4 x + 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(4 x^{2}) = 8 x^{2}$.\n3) So $f'(x) = 8 x^{2} - 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(4 x^{2}) = 8 x$.\nTherefore the correct derivative is $f'(x) = 4 \\left(2 x - 1\\right)$.\nFinal answer: $4 \\left(2 x - 1\\right)$", "final_answer": "4 \\left(2 x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8a44fc0581fa5377", "problem_id": "252b1908d4b3345b", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$3 x + 2 y$ = $7$\n$- 4 y$ = $28$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $3$: $- 12 y$ = $84$.\n4) Subtract to eliminate x: $12 y$ = $-84$.\n5) Solve for y: y = $-7$.\n6) Substitute back to find x: x = $7$.\nFinal answer: $(x,y)=\\left(7, -7\\right)$", "final_answer": "(x,y)=\\left(7, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3253d93eea102371", "problem_id": "2a4a1d1ab1d35f53", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A cyclist rides at 66 miles per hour for 1/2 hours, then at 69 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 66 × 1/2 = 33 miles.\n2) Second leg distance = 69 × 3/2 = 207/2 miles.\n3) Total distance = 33 + 207/2 = 273/2 miles.\nFinal answer: $\\frac{273}{2}$", "final_answer": "\\frac{273}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b471cf351e786430", "problem_id": "4166a418f2c3a4dd", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x \\sin{\\left(x \\right)}) = - x \\cos{\\left(x \\right)}$.\n3) So $f'(x) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x \\sin{\\left(x \\right)}) = - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$", "final_answer": "x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "452c98528961c18e", "problem_id": "388e859e1952f052", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Worker A finishes a job in 15 hours and Worker B finishes it in 7 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/15 job/hour. B's rate = 1/7 job/hour.\nCombined rate = 1/15 + 1/7 = 22/105 job/hour.\nTime = 1 ÷ (combined rate) = 105/22 hours.\nCheck: (combined rate)×(time) = 22/105×105/22 = 1.\nFinal answer: $\\frac{105}{22}$", "final_answer": "\\frac{105}{22}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "49c261f8b306a66c", "problem_id": "388e859e1952f052", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Worker A finishes a job in 15 hours and Worker B finishes it in 7 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/15 job/hour. B's rate = 1/7 job/hour.\n2) Combined rate = 1/15 + 1/7 = 22/105 job/hour.\n3) Time = 1 ÷ (combined rate) = 105/22 hours.\nFinal answer: $\\frac{105}{22}$", "final_answer": "\\frac{105}{22}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6ef98f2bd34debe1", "problem_id": "e747c5ef10b5dba4", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{7}$. If it is flipped 19 times, what is the probability of exactly 15 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=19, k=15, p=$\\frac{4}{7}$ and simplify.\nHere C(19,15) = 3876.\nCheck: expression simplifies to $\\frac{337107688095744}{11398895185373143}$.\nFinal answer: $\\frac{337107688095744}{11398895185373143}$", "final_answer": "\\frac{337107688095744}{11398895185373143}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0b0a180162ffbfb9", "problem_id": "e747c5ef10b5dba4", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{7}$. If it is flipped 19 times, what is the probability of exactly 15 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=19, k=15, p=$\\frac{4}{7}$ and simplify.\n4) Here C(19,15) = 3876.\nFinal answer: $\\frac{337107688095744}{11398895185373143}$", "final_answer": "\\frac{337107688095744}{11398895185373143}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a588efc8e74aa630", "problem_id": "fd58efc3b863da9a", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 16 red balls and 16 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(32,8).\n3) Favorable samples: choose k reds and n−k blues: C(16,2)·C(16,6).\n4) Compute totals: C(32,8) = 10518300, and favorable = 960960.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1232}{13485}$", "final_answer": "\\frac{1232}{13485}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0a383a07bfcd890f", "problem_id": "fd58efc3b863da9a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 16 red balls and 16 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(32,8).\nFavorable samples: choose k reds and n−k blues: C(16,2)·C(16,6).\nCompute totals: C(32,8) = 10518300, and favorable = 960960.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1232}{13485}$.\nFinal answer: $\\frac{1232}{13485}$", "final_answer": "\\frac{1232}{13485}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a75cc21488004f6", "problem_id": "633252c31c41d39c", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$x$ = $4$\n$- 3 y$ = $3$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $1$: $- 3 y$ = $3$.\nSubtract to eliminate x: $3 y$ = $-3$.\nSolve for y: y = $-1$.\nSubstitute back to find x: x = $4$.\nCheck: plug (x,y)=(4,-1) into both equations.\nFinal answer: $(x,y)=\\left(4, -1\\right)$", "final_answer": "(x,y)=\\left(4, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b52592c89b780750", "problem_id": "633252c31c41d39c", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$x$ = $4$\n$- 3 y$ = $3$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $1$: $- 3 y$ = $3$.\n4) Subtract to eliminate x: $3 y$ = $-3$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, -1\\right)$", "final_answer": "(x,y)=\\left(4, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "22e43ae64a94cc06", "problem_id": "139a9bc415d639ac", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - 2 x^{3} - 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{3} - 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{255 \\pi^{4}}{32} - \\frac{65 \\pi^{3}}{8} - \\frac{15 \\pi}{2} - 2 + \\frac{15 \\pi^{2}}{4}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{255\\cdot \\pi^{4}}{32} - \\frac{65\\cdot \\pi^{3}}{8} - \\frac{15\\cdot \\pi}{2} - 2 + \\frac{15\\cdot \\pi^{2}}{4}$", "final_answer": "- \\frac{255\\cdot \\pi^{4}}{32} - \\frac{65\\cdot \\pi^{3}}{8} - \\frac{15\\cdot \\pi}{2} - 2 + \\frac{15\\cdot \\pi^{2}}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5baf96041f5b6aff", "problem_id": "139a9bc415d639ac", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - 2 x^{3} - 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{3} - 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{255 \\pi^{4}}{32} - \\frac{65 \\pi^{3}}{8} - \\frac{15 \\pi}{2} - 2 + \\frac{15 \\pi^{2}}{4}$.\n4) Simplify the result.\nFinal answer: $- \\frac{255\\cdot \\pi^{4}}{32} - \\frac{65\\cdot \\pi^{3}}{8} - \\frac{15\\cdot \\pi}{2} - 2 + \\frac{15\\cdot \\pi^{2}}{4}$", "final_answer": "- \\frac{255\\cdot \\pi^{4}}{32} - \\frac{65\\cdot \\pi^{3}}{8} - \\frac{15\\cdot \\pi}{2} - 2 + \\frac{15\\cdot \\pi^{2}}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "862600a784dfa234", "problem_id": "3f20cbc98420e4f0", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$5 x - 4 y$ = $-30$\n$6 x + 4 y$ = $8$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $6$: $30 x - 24 y$ = $-180$.\n3) Multiply the second equation by $5$: $30 x + 20 y$ = $40$.\n4) Subtract to eliminate x: $- 44 y$ = $-220$.\n5) Solve for y: y = $5$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, 5\\right)$", "final_answer": "(x,y)=\\left(-2, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "422eed54e42fa410", "problem_id": "3f20cbc98420e4f0", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the linear system:\n$5 x - 4 y$ = $-30$\n$6 x + 4 y$ = $8$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $6$: $30 x - 24 y$ = $-180$.\nMultiply the second equation by $5$: $30 x + 20 y$ = $40$.\nFinal answer: $(x,y)=\\left(-2, 5\\right)$", "final_answer": "(x,y)=\\left(-2, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "66daca7179133575", "problem_id": "6247e5b9f89fb08c", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡0 (mod 7), x≡7 (mod 11). Give the solution modulo 77.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 77.\n3) Compute the CRT solution to get x ≡ 7 (mod 77).\nFinal answer: $x\\equiv 7\\pmod{77}$", "final_answer": "x\\equiv 7\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2e40c9040cbb500b", "problem_id": "6247e5b9f89fb08c", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡0 (mod 7), x≡7 (mod 11). Give the solution modulo 77.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 77.\nCompute the CRT solution to get x ≡ 7 (mod 77).\nCheck: 7 mod 7=0 and mod 11=7.\nFinal answer: $x\\equiv 7\\pmod{77}$", "final_answer": "x\\equiv 7\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e4f164b82284f3d2", "problem_id": "58694ba9233bb453", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Find the multiplicative inverse of 2 modulo 17.", "output": "1) We want x such that 2x ≡ 1 (mod 17).\n2) Use extended Euclid to find s,t with 2s + 17t = 1.\n3) One Bezout identity is: 2·(-8) + 17·(1) = 1.\n4) Reduce s modulo 17: x ≡ 9 (mod 17).\nFinal answer: $x\\equiv 9\\pmod{17}$", "final_answer": "x\\equiv 9\\pmod{17}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "046b7f973a60c231", "problem_id": "58694ba9233bb453", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the multiplicative inverse of 2 modulo 17.", "output": "We want x such that 2x ≡ 1 (mod 17).\nUse extended Euclid to find s,t with 2s + 17t = 1.\nOne Bezout identity is: 2·(-8) + 17·(1) = 1.\nReduce s modulo 17: x ≡ 9 (mod 17).\nCheck: 2·9 mod 17 = 1.\nFinal answer: $x\\equiv 9\\pmod{17}$", "final_answer": "x\\equiv 9\\pmod{17}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "414036ca480108f5", "problem_id": "0e4d2a2b261b5eab", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many binary strings of length 19 contain exactly 4 ones with no two ones adjacent?", "output": "1) Reserve 3 mandatory zeros between the 4 ones.\n2) This leaves n−k+1 = 16 valid slots for the ones.\n3) Choose the slots: C(16,4) = 1820.\nFinal answer: $1820$", "final_answer": "1820", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "df764e4d04dac3a6", "problem_id": "0e4d2a2b261b5eab", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many binary strings of length 19 contain exactly 4 ones with no two ones adjacent?", "output": "Reserve 3 mandatory zeros between the 4 ones.\nThis leaves n−k+1 = 16 valid slots for the ones.\nChoose the slots: C(16,4) = 1820.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $1820$", "final_answer": "1820", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7749f31b8a47dd45", "problem_id": "782934a1915a228b", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} + 4 x - 3$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-3) = 0.\n3) d/dx(- 2 x^{2}) = - 4 x.\n4) d/dx(4 x) = 4.\n5) Combine: f'(x) = 4 \\left(1 - x\\right).\nFinal answer: $4 - 4\\cdot x$", "final_answer": "4 - 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5330fd61b22526fa", "problem_id": "070b8d8e920c171e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- x^{2} + 3 x - 1\\right) \\cos{\\left(3 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\cos{\\left(3 x \\right)}) = \\sin{\\left(3 x \\right)}$.\n3) So $f'(x) = 3 x^{2} \\sin{\\left(3 x \\right)} + 2 x e^{x} - 9 x \\sin{\\left(3 x \\right)} - 2 x \\cos{\\left(3 x \\right)} + 4 e^{x} + \\sin{\\left(3 x \\right)} + 3 \\cos{\\left(3 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\cos{\\left(3 x \\right)}) = 3 \\sin{\\left(3 x \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(3 - 2 x\\right) \\cos{\\left(3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 3 \\left(x \\left(3 - x\\right) - 1\\right) \\sin{\\left(3 x \\right)} + 2 e^{x}$.\nFinal answer: $\\left(3 - 2 x\\right) \\cos{\\left(3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 3 \\left(x \\left(3 - x\\right) - 1\\right) \\sin{\\left(3 x \\right)} + 2 e^{x}$", "final_answer": "\\left(3 - 2 x\\right) \\cos{\\left(3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 3 \\left(x \\left(3 - x\\right) - 1\\right) \\sin{\\left(3 x \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "be97cd58bede9269", "problem_id": "5c59f82978da736e", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A cyclist rides at 23 miles per hour for 2 hours, then at 37 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 23 × 2 = 46 miles.\n2) Second leg distance = 37 × 1 = 37 miles.\n3) Total distance = 46 + 37 = 83 miles.\nFinal answer: $83$", "final_answer": "83", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "298bf4cd2402cc00", "problem_id": "fd2bcac6548591c0", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\left(x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)}\\right) + 3 \\sin{\\left(x \\right)}) = \\left(x^{2} + x + 3\\right) \\cos{\\left(x \\right)}$.\n3) So $f'(x) = x^{2} \\cos{\\left(x \\right)} + x \\cos{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\left(x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)}\\right) + 3 \\sin{\\left(x \\right)}) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} + 3 \\cos{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$", "final_answer": "x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "838d2cd55fb60aea", "problem_id": "ec011ccfb26f68dd", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 30 red balls and 17 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(47,7).\n3) Favorable samples: choose k reds and n−k blues: C(30,1)·C(17,6).\n4) Compute totals: C(47,7) = 62891499, and favorable = 371280.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{123760}{20963833}$", "final_answer": "\\frac{123760}{20963833}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1d79790f2c07af2f", "problem_id": "ec011ccfb26f68dd", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 30 red balls and 17 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(47,7).\nFavorable samples: choose k reds and n−k blues: C(30,1)·C(17,6).\nCompute totals: C(47,7) = 62891499, and favorable = 371280.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{123760}{20963833}$.\nFinal answer: $\\frac{123760}{20963833}$", "final_answer": "\\frac{123760}{20963833}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1537f9aa33300881", "problem_id": "dcfaa713fcca2e05", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A store sells a backpack for $158. It is discounted by 25%. You buy 8 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 158 × (1 − 25/100) = 158 × 75/100 = 237/2.\nSubtotal = 8 × 237/2 = 948.\nTotal with tax = 948 × 108/100 = 25596/25.\nCheck: recomputing the formula yields 25596/25.\nFinal answer: $\\frac{25596}{25}$", "final_answer": "\\frac{25596}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4838f2380b4a66f5", "problem_id": "dcfaa713fcca2e05", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A store sells a backpack for $158. It is discounted by 25%. You buy 8 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 158 × (1 − 25/100) = 158 × 75/100 = 237/2.\n2) Subtotal = 8 × 237/2 = 948.\n3) Total with tax = 948 × 108/100 = 25596/25.\nFinal answer: $\\frac{25596}{25}$", "final_answer": "\\frac{25596}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "79c8720f084f319e", "problem_id": "02f5ba068679179f", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(- x^{3} + 2 x^{2} - 2 x - 2\\right) \\cos{\\left(2 x - 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 2 \\cos{\\left(2 x - 2 \\right)}) = 2 \\sin{\\left(2 x - 2 \\right)}$.\n3) So $f'(x) = 2 x^{3} \\sin{\\left(2 x - 2 \\right)} - 4 x^{2} \\sin{\\left(2 x - 2 \\right)} - 3 x^{2} \\cos{\\left(2 x - 2 \\right)} + x e^{x} + 4 x \\sin{\\left(2 x - 2 \\right)} + 4 x \\cos{\\left(2 x - 2 \\right)} - 2 e^{x} + 2 \\sin{\\left(2 x - 2 \\right)} - 2 \\cos{\\left(2 x - 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 2 \\cos{\\left(2 x - 2 \\right)}) = 4 \\sin{\\left(2 x - 2 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x - 3\\right) e^{x} - 2 \\left(x \\left(x \\left(2 - x\\right) - 2\\right) - 2\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(- 3 x^{2} + 4 x - 2\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}$.\nFinal answer: $\\left(x - 3\\right) e^{x} - 2 \\left(x \\left(x \\left(2 - x\\right) - 2\\right) - 2\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(- 3 x^{2} + 4 x - 2\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}$", "final_answer": "\\left(x - 3\\right) e^{x} - 2 \\left(x \\left(x \\left(2 - x\\right) - 2\\right) - 2\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(- 3 x^{2} + 4 x - 2\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b1214f7223ffbe0a", "problem_id": "f73d1f84a7d9ae49", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Worker A finishes a job in 6 hours and Worker B finishes it in 4 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/6 job/hour. B's rate = 1/4 job/hour.\n2) Combined rate = 1/6 + 1/4 = 5/12 job/hour.\n3) Time = 1 ÷ (combined rate) = 12/5 hours.\nFinal answer: $\\frac{12}{5}$", "final_answer": "\\frac{12}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a91badc56c92b209", "problem_id": "0d937374803757a8", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Worker A finishes a job in 17 hours and Worker B finishes it in 9 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/17 job/hour. B's rate = 1/9 job/hour.\nCombined rate = 1/17 + 1/9 = 26/153 job/hour.\nTime = 1 ÷ (combined rate) = 153/26 hours.\nFinal answer: $\\frac{153}{26}$", "final_answer": "\\frac{153}{26}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5fb74369ea0dab1e", "problem_id": "0d937374803757a8", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Worker A finishes a job in 17 hours and Worker B finishes it in 9 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/17 job/hour. B's rate = 1/9 job/hour.\n2) Combined rate = 1/17 + 1/9 = 26/153 job/hour.\n3) Time = 1 ÷ (combined rate) = 153/26 hours.\nFinal answer: $\\frac{153}{26}$", "final_answer": "\\frac{153}{26}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a307bc9e41a0135c", "problem_id": "c511a5946a496db4", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Worker A finishes a job in 10 hours and Worker B finishes it in 13 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/10 job/hour. B's rate = 1/13 job/hour.\n2) Combined rate = 1/10 + 1/13 = 23/130 job/hour.\n3) Time = 1 ÷ (combined rate) = 130/23 hours.\nFinal answer: $\\frac{130}{23}$", "final_answer": "\\frac{130}{23}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dd07ecc0bc3c6691", "problem_id": "c511a5946a496db4", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Worker A finishes a job in 10 hours and Worker B finishes it in 13 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/10 job/hour. B's rate = 1/13 job/hour.\nCombined rate = 1/10 + 1/13 = 23/130 job/hour.\nTime = 1 ÷ (combined rate) = 130/23 hours.\nFinal answer: $\\frac{130}{23}$", "final_answer": "\\frac{130}{23}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "90f1100afc2d50a9", "problem_id": "56a6eae271e321fe", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_9=36$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_9=27.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=27 and k=9.\n4) Compute C(35,8) = 23535820.\nFinal answer: $23535820$", "final_answer": "23535820", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "badc1c91db2639a4", "problem_id": "56a6eae271e321fe", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_9=36$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_9=27.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=27 and k=9.\nFinal answer: $23535820$", "final_answer": "23535820", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "56c6a9ecf3634fc4", "problem_id": "be45aa58b2d0e88d", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 26 x + 72$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} + 26 x + 72$ = $2 \\left(x + 4\\right) \\left(x + 9\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-4$.\n5) From the second factor: x = $-9$.\nFinal answer: $x \\in \\{-4, -9\\}$", "final_answer": "x \\in \\{-4, -9\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c2452d349d2c64b3", "problem_id": "be45aa58b2d0e88d", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 26 x + 72$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} + 26 x + 72$ = $2 \\left(x + 4\\right) \\left(x + 9\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFinal answer: $x \\in \\{-4, -9\\}$", "final_answer": "x \\in \\{-4, -9\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "122fa0e41a4f22b1", "problem_id": "6b69c311273cb3f3", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A store sells a backpack for $88. It is discounted by 25%. You buy 6 backpacks. What is the total cost?", "output": "1) Discounted price = 88 × (1 − 25/100) = 88 × 75/100 = 66.\n2) Subtotal = 6 × 66 = 396.\nFinal answer: $396$", "final_answer": "396", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4dab0cfb02ccbb6d", "problem_id": "86ad5e3e1f9c98d9", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - x^{2} + 5 x - 10$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} + 4 x - 10.\n3) Then the limit is -22.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -24.\nFinal answer: $-24$", "final_answer": "-24", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "4118a4e4842b1de8", "problem_id": "0752aa2c0e5db963", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 6 x + 9$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} - 6 x + 9$ = $\\left(x - 3\\right)^{2}$.\nSet each factor equal to 0 and solve each linear equation.\nFinal answer: $x \\in \\{3, 3\\}$", "final_answer": "x \\in \\{3, 3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0c3d838200fa883b", "problem_id": "0752aa2c0e5db963", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 6 x + 9$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} - 6 x + 9$ = $\\left(x - 3\\right)^{2}$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $3$.\n5) From the second factor: x = $3$.\nFinal answer: $x \\in \\{3, 3\\}$", "final_answer": "x \\in \\{3, 3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b4b41b0bff11fa0c", "problem_id": "ef1f7a94876bf5cc", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 20 times, what is the probability of exactly 11 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=20, k=11, p=$\\frac{1}{8}$ and simplify.\n4) Here C(20,11) = 167960.\nFinal answer: $\\frac{847223978965}{144115188075855872}$", "final_answer": "\\frac{847223978965}{144115188075855872}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f25718bfe6317826", "problem_id": "ef1f7a94876bf5cc", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 20 times, what is the probability of exactly 11 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=20, k=11, p=$\\frac{1}{8}$ and simplify.\nHere C(20,11) = 167960.\nCheck: expression simplifies to $\\frac{847223978965}{144115188075855872}$.\nFinal answer: $\\frac{847223978965}{144115188075855872}$", "final_answer": "\\frac{847223978965}{144115188075855872}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "14c1492576c7944d", "problem_id": "6c65717c319a8098", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "You want 22 liters of a 45% salt solution. You have a 30% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 22−x liters is 60% solution.\n2) Salt equation: 30x + 60(22−x) = 45·22.\n3) (30-60)x = (45-60)·22 ⇒ x = 11.\n4) Then 22−x = 11.\nFinal answer: $11 L of 30% and 11 L of 60%$", "final_answer": "11 L of 30% and 11 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ddd2e708edf44369", "problem_id": "6c65717c319a8098", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 22 liters of a 45% salt solution. You have a 30% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 22−x liters is 60% solution.\nSalt equation: 30x + 60(22−x) = 45·22.\n(30-60)x = (45-60)·22 ⇒ x = 11.\nThen 22−x = 11.\nCheck: 30·11 + 60·11 = 45·22.\nFinal answer: $11 L of 30% and 11 L of 60%$", "final_answer": "11 L of 30% and 11 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ab09c4d010fb1f2d", "problem_id": "aa5e518fa990f1ce", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=19$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=19 and k=5.\n4) Compute C(23,4) = 8855.\nFinal answer: $8855$", "final_answer": "8855", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "37656e9f080e4674", "problem_id": "aa5e518fa990f1ce", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=19$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=19 and k=5.\nCompute C(23,4) = 8855.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $8855$", "final_answer": "8855", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9abc3cefefcec614", "problem_id": "4fab0ec2d4d51a41", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡1 (mod 11), x≡9 (mod 13). Give the solution modulo 143.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 143.\n3) Compute the CRT solution to get x ≡ 100 (mod 143).\nFinal answer: $x\\equiv 100\\pmod{143}$", "final_answer": "x\\equiv 100\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "00c9e380c92ef1b3", "problem_id": "4fab0ec2d4d51a41", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the congruences: x≡1 (mod 11), x≡9 (mod 13). Give the solution modulo 143.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 143.\nCompute the CRT solution to get x ≡ 100 (mod 143).\nFinal answer: $x\\equiv 100\\pmod{143}$", "final_answer": "x\\equiv 100\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "92f6dfb2d0ad4c2a", "problem_id": "086d6ba46e0c92e6", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$- 4 x - y$ = $-5$\n$10 x - 8 y$ = $-82$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $10$: $- 40 x - 10 y$ = $-50$.\n3) Multiply the second equation by $-4$: $- 40 x + 32 y$ = $328$.\n4) Subtract to eliminate x: $- 42 y$ = $-378$.\n5) Solve for y: y = $9$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, 9\\right)$", "final_answer": "(x,y)=\\left(-1, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "77b6745720e6c7a0", "problem_id": "086d6ba46e0c92e6", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the linear system:\n$- 4 x - y$ = $-5$\n$10 x - 8 y$ = $-82$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $10$: $- 40 x - 10 y$ = $-50$.\nMultiply the second equation by $-4$: $- 40 x + 32 y$ = $328$.\nFinal answer: $(x,y)=\\left(-1, 9\\right)$", "final_answer": "(x,y)=\\left(-1, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a03c88dfe1b4cdff", "problem_id": "526d1ac334f1c9ea", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$x + 2 y$ = $-8$\n$- x$ = $0$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $- x - 2 y$ = $8$.\nMultiply the second equation by $1$: $- x$ = $0$.\nSubtract to eliminate x: $- 2 y$ = $8$.\nSolve for y: y = $-4$.\nSubstitute back to find x: x = $0$.\nCheck: plug (x,y)=(0,-4) into both equations.\nFinal answer: $(x,y)=\\left(0, -4\\right)$", "final_answer": "(x,y)=\\left(0, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "79f6f44f5079bcd4", "problem_id": "526d1ac334f1c9ea", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$x + 2 y$ = $-8$\n$- x$ = $0$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $- x - 2 y$ = $8$.\n3) Multiply the second equation by $1$: $- x$ = $0$.\n4) Subtract to eliminate x: $- 2 y$ = $8$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, -4\\right)$", "final_answer": "(x,y)=\\left(0, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f29b03b8dc2b09b9", "problem_id": "94260815181c5ab5", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Worker A finishes a job in 16 hours and Worker B finishes it in 15 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/16 job/hour. B's rate = 1/15 job/hour.\n2) Combined rate = 1/16 + 1/15 = 31/240 job/hour.\n3) Time = 1 ÷ (combined rate) = 240/31 hours.\nFinal answer: $\\frac{240}{31}$", "final_answer": "\\frac{240}{31}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "860d2a5c3fa47e79", "problem_id": "94260815181c5ab5", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Worker A finishes a job in 16 hours and Worker B finishes it in 15 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/16 job/hour. B's rate = 1/15 job/hour.\nCombined rate = 1/16 + 1/15 = 31/240 job/hour.\nTime = 1 ÷ (combined rate) = 240/31 hours.\nFinal answer: $\\frac{240}{31}$", "final_answer": "\\frac{240}{31}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b4a8c0716b20d760", "problem_id": "a6a9f96a725a0478", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 x}{2} - 10$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{3}{2}$, c=$-10$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{169}{4}$, so √Δ = $\\frac{13}{2}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $4$.\n7) x2 = $- \\frac{5}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{4, - \\frac{5}{2}\\}$", "final_answer": "x \\in \\{4, - \\frac{5}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7d12d12a63c8c186", "problem_id": "884ffe89a210d9f4", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - x^{2} + 2 x - 7$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} + x - 7.\n3) Then the limit is -19.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -22.\nFinal answer: $-22$", "final_answer": "-22", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3b747cce14a962e1", "problem_id": "c6c6b3aa7d722e5c", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$3 x - 6 y$ = $9$\n$8 x + 8 y$ = $96$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $8$: $24 x - 48 y$ = $72$.\nMultiply the second equation by $3$: $24 x + 24 y$ = $288$.\nSubtract to eliminate x: $- 72 y$ = $-216$.\nSolve for y: y = $3$.\nSubstitute back to find x: x = $9$.\nCheck: plug (x,y)=(9,3) into both equations.\nFinal answer: $(x,y)=\\left(9, 3\\right)$", "final_answer": "(x,y)=\\left(9, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "37f8d53e148abd33", "problem_id": "c6c6b3aa7d722e5c", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$3 x - 6 y$ = $9$\n$8 x + 8 y$ = $96$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $8$: $24 x - 48 y$ = $72$.\n3) Multiply the second equation by $3$: $24 x + 24 y$ = $288$.\n4) Subtract to eliminate x: $- 72 y$ = $-216$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $9$.\nFinal answer: $(x,y)=\\left(9, 3\\right)$", "final_answer": "(x,y)=\\left(9, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "466ccb9d345aecef", "problem_id": "0b02fa7f2099f57b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - 2 x - 8$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to - 3 x - 8.\n3) Then the limit is -20.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: -16.\nFinal answer: $-16$", "final_answer": "-16", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6c64ccfd8ff0be6a", "problem_id": "6eb008fcd51aa945", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$12 x - 7 y$ = $-86$\n$8 x + 5 y$ = $-38$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $8$: $96 x - 56 y$ = $-688$.\nMultiply the second equation by $12$: $96 x + 60 y$ = $-456$.\nSubtract to eliminate x: $- 116 y$ = $-232$.\nSolve for y: y = $2$.\nSubstitute back to find x: x = $-6$.\nCheck: plug (x,y)=(-6,2) into both equations.\nFinal answer: $(x,y)=\\left(-6, 2\\right)$", "final_answer": "(x,y)=\\left(-6, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "acb40955a83e8a4f", "problem_id": "5adb0c43d90fe0dd", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the equation of the tangent line to $y=- 2 x^{2} - x + 3$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $- 4 x - 1$.\n2) Slope at x=-2: m = f'(-2) = $7$.\n3) Point on curve: (-2, f(-2)) = (-2, $-3$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -3 = 7(x − -2).\n6) Optionally expand: y = $7 x + 11$.\nFinal answer: $y = 7\\cdot x + 11$", "final_answer": "y = 7\\cdot x + 11", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c90ea018da0b5689", "problem_id": "5adb0c43d90fe0dd", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{2} - x + 3$ at $x=-2$.", "output": "Differentiate: f'(x) = $- 4 x - 1$.\nSlope at x=-2: m = f'(-2) = $7$.\nPoint on curve: (-2, f(-2)) = (-2, $-3$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -3 = 7(x − -2).\nOptionally expand: y = $7 x + 11$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 7\\cdot x + 11$", "final_answer": "y = 7\\cdot x + 11", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6442eda6451be2c4", "problem_id": "2266af0b0dafe10f", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} 2 x^{3} - 9 x^{2} + 33 x - 129$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $2 x^{4} - x^{3} - 3 x^{2} + 3 x + 1$ and a = -4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $8 x^{3} - 3 x^{2} - 6 x + 3$.\n5) Evaluate at a: p'(-4) = $-533$.\nFinal answer: $-533$", "final_answer": "-533", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "39abda1b7c833847", "problem_id": "2266af0b0dafe10f", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} 2 x^{3} - 9 x^{2} + 33 x - 129$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{4} - x^{3} - 3 x^{2} + 3 x + 1$ and a = -4.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-533$", "final_answer": "-533", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4e4a7a81fb72f452", "problem_id": "56520be7ec767feb", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_10=23$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_10=13.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=13 and k=10.\n4) Compute C(22,9) = 497420.\nFinal answer: $497420$", "final_answer": "497420", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cbd1773cefed30b5", "problem_id": "56520be7ec767feb", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_10=23$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_10=13.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=13 and k=10.\nCompute C(22,9) = 497420.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $497420$", "final_answer": "497420", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ed20417103a800d8", "problem_id": "14765753b5a4a573", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_5=34$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_5=24.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=24 and k=5.\n4) Compute C(28,4) = 20475.\nFinal answer: $20475$", "final_answer": "20475", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cabd2e9ee1f467ea", "problem_id": "14765753b5a4a573", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_5=34$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_5=24.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=24 and k=5.\nCompute C(28,4) = 20475.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $20475$", "final_answer": "20475", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "50db5714e0e22611", "problem_id": "3a51b3a525176194", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$- 2 x - 4 y$ = $-14$\n$4 x - 2 y$ = $18$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $- 8 x - 16 y$ = $-56$.\n3) Multiply the second equation by $-2$: $- 8 x + 4 y$ = $-36$.\n4) Subtract to eliminate x: $- 20 y$ = $-20$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, 1\\right)$", "final_answer": "(x,y)=\\left(5, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b0d53f6091b47038", "problem_id": "3a51b3a525176194", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$- 2 x - 4 y$ = $-14$\n$4 x - 2 y$ = $18$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $- 8 x - 16 y$ = $-56$.\nMultiply the second equation by $-2$: $- 8 x + 4 y$ = $-36$.\nSubtract to eliminate x: $- 20 y$ = $-20$.\nSolve for y: y = $1$.\nSubstitute back to find x: x = $5$.\nCheck: plug (x,y)=(5,1) into both equations.\nFinal answer: $(x,y)=\\left(5, 1\\right)$", "final_answer": "(x,y)=\\left(5, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "02c656571df9dcc2", "problem_id": "2c140e892c80df2d", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the multiplicative inverse of 9 modulo 11.", "output": "1) We want x such that 9x ≡ 1 (mod 11).\n2) Use extended Euclid to find s,t with 9s + 11t = 1.\n3) One Bezout identity is: 9·(5) + 11·(-4) = 1.\n4) Reduce s modulo 11: x ≡ 5 (mod 11).\nFinal answer: $x\\equiv 5\\pmod{11}$", "final_answer": "x\\equiv 5\\pmod{11}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dafcf3150f632e72", "problem_id": "2c140e892c80df2d", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the multiplicative inverse of 9 modulo 11.", "output": "We want x such that 9x ≡ 1 (mod 11).\nUse extended Euclid to find s,t with 9s + 11t = 1.\nOne Bezout identity is: 9·(5) + 11·(-4) = 1.\nReduce s modulo 11: x ≡ 5 (mod 11).\nCheck: 9·5 mod 11 = 1.\nFinal answer: $x\\equiv 5\\pmod{11}$", "final_answer": "x\\equiv 5\\pmod{11}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6e850d70885217b9", "problem_id": "2c94d488ddf51403", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$x + 2 y$ = $1$\n$4 x - 3 y$ = $26$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $4 x + 8 y$ = $4$.\nMultiply the second equation by $1$: $4 x - 3 y$ = $26$.\nSubtract to eliminate x: $11 y$ = $-22$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $5$.\nCheck: plug (x,y)=(5,-2) into both equations.\nFinal answer: $(x,y)=\\left(5, -2\\right)$", "final_answer": "(x,y)=\\left(5, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ec512e835867647", "problem_id": "2c94d488ddf51403", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the linear system:\n$x + 2 y$ = $1$\n$4 x - 3 y$ = $26$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $4 x + 8 y$ = $4$.\n3) Multiply the second equation by $1$: $4 x - 3 y$ = $26$.\n4) Subtract to eliminate x: $11 y$ = $-22$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, -2\\right)$", "final_answer": "(x,y)=\\left(5, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "75e431875da1a766", "problem_id": "f82642fa060c8187", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} 2 x^{4} + 3 x^{3} - x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{4} + 3 x^{3} - x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{65 \\pi^{3}}{24} - \\frac{15 \\pi^{2}}{4} - 1 + 10 \\pi + \\frac{765 \\pi^{4}}{64} + \\frac{205 \\pi^{5}}{16}$.\n4) Simplify the result.\nFinal answer: $- \\frac{65\\cdot \\pi^{3}}{24} - \\frac{15\\cdot \\pi^{2}}{4} - 1 + 10\\cdot \\pi + \\frac{765\\cdot \\pi^{4}}{64} + \\frac{205\\cdot \\pi^{5}}{16}$", "final_answer": "- \\frac{65\\cdot \\pi^{3}}{24} - \\frac{15\\cdot \\pi^{2}}{4} - 1 + 10\\cdot \\pi + \\frac{765\\cdot \\pi^{4}}{64} + \\frac{205\\cdot \\pi^{5}}{16}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3cebcfc8a192980a", "problem_id": "8becc4ad099fc928", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 3 x^{2} + 12 x - 35$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} + 11 x - 35.\n3) Then the limit is -95.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -98.\nFinal answer: $-98$", "final_answer": "-98", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3d8d358390d4dded", "problem_id": "612447c7c9b05f30", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A store sells a backpack for $68. It is discounted by 40%. You buy 9 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 68 × (1 − 40/100) = 68 × 60/100 = 204/5.\n2) Subtotal = 9 × 204/5 = 1836/5.\n3) Total with tax = 1836/5 × 109/100 = 50031/125.\nFinal answer: $\\frac{50031}{125}$", "final_answer": "\\frac{50031}{125}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fdd079f1598d7e35", "problem_id": "612447c7c9b05f30", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A store sells a backpack for $68. It is discounted by 40%. You buy 9 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 68 × (1 − 40/100) = 68 × 60/100 = 204/5.\nSubtotal = 9 × 204/5 = 1836/5.\nTotal with tax = 1836/5 × 109/100 = 50031/125.\nCheck: recomputing the formula yields 50031/125.\nFinal answer: $\\frac{50031}{125}$", "final_answer": "\\frac{50031}{125}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b99ae334c8c8ad85", "problem_id": "30b1fab2e37dc270", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{3} + 3 x^{2} - 3 x - 2$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-2) = 0.\n3) d/dx(- 3 x) = -3.\n4) d/dx(3 x^{2}) = 6 x.\n5) d/dx(3 x^{3}) = 9 x^{2}.\n6) Combine: f'(x) = 3 \\left(3 x^{2} + 2 x - 1\\right).\nFinal answer: $9\\cdot x^{2} + 6\\cdot x - 3$", "final_answer": "9\\cdot x^{2} + 6\\cdot x - 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a682fcbb3230904", "problem_id": "f018355b7ee08b26", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 3 x^{2} - 1$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} - x - 1.\n3) Then the limit is -3.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: -4.\nFinal answer: $-4$", "final_answer": "-4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c5d4ae6965af2cb1", "problem_id": "8ace5bf22ab7eae3", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{3} - x^{2} - x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1}{x^{2} + 2}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 2}) = \\frac{2 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 2}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 8 x^{2} + 8} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 8 x^{2} + 8} - \\frac{6 x^{4}}{2 x^{4} + 8 x^{2} + 8} + \\frac{4 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 8 x^{2} + 8} + \\frac{8 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 8 x^{2} + 8} - \\frac{34 x^{2}}{2 x^{4} + 8 x^{2} + 8} - \\frac{4 x}{2 x^{4} + 8 x^{2} + 8} + \\frac{4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 4 x^{2} + 4} + \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 8 x^{2} + 8} + \\frac{8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 8 x^{2} + 8} - \\frac{4}{2 x^{4} + 8 x^{2} + 8} + \\frac{2 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 2} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 2} - \\frac{2 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 2}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 2}) = - \\frac{4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 2\\right)^{2}} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 2} + \\frac{2 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 2}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 4 x \\left(- 3 x^{3} - x^{2} - x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) + \\left(x^{2} + 2\\right) \\left(- 18 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4 x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x^{2} + 2\\right)^{2}}$.\nFinal answer: $\\frac{- 4 x \\left(- 3 x^{3} - x^{2} - x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) + \\left(x^{2} + 2\\right) \\left(- 18 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4 x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x^{2} + 2\\right)^{2}}$", "final_answer": "\\frac{- 4 x \\left(- 3 x^{3} - x^{2} - x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) + \\left(x^{2} + 2\\right) \\left(- 18 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4 x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x^{2} + 2\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "98500848ebf0900d", "problem_id": "3251e5f0733774a3", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Worker A finishes a job in 20 hours and Worker B finishes it in 14 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/20 job/hour. B's rate = 1/14 job/hour.\n2) Combined rate = 1/20 + 1/14 = 17/140 job/hour.\n3) Time = 1 ÷ (combined rate) = 140/17 hours.\nFinal answer: $\\frac{140}{17}$", "final_answer": "\\frac{140}{17}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7e95b9655d5aa278", "problem_id": "bd45c202f38149d5", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 11 times, what is the probability of exactly 9 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=11, k=9, p=$\\frac{1}{4}$ and simplify.\nHere C(11,9) = 55.\nCheck: expression simplifies to $\\frac{495}{4194304}$.\nFinal answer: $\\frac{495}{4194304}$", "final_answer": "\\frac{495}{4194304}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aa5e185df47b91ae", "problem_id": "bd45c202f38149d5", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 11 times, what is the probability of exactly 9 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=11, k=9, p=$\\frac{1}{4}$ and simplify.\nFinal answer: $\\frac{495}{4194304}$", "final_answer": "\\frac{495}{4194304}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "745ef76d84eed7bd", "problem_id": "9078a693041d9ffc", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} x^{2} - 4 x + 7$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{3} - 2 x^{2} - x + 3$ and a = -2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $3 x^{2} - 4 x - 1$.\n5) Evaluate at a: p'(-2) = $19$.\nFinal answer: $19$", "final_answer": "19", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0232737cac553a12", "problem_id": "9078a693041d9ffc", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} x^{2} - 4 x + 7$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $x^{3} - 2 x^{2} - x + 3$ and a = -2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $19$", "final_answer": "19", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9d1193a590d3e5ca", "problem_id": "389ac5d92d4b30be", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} + 2 x^{2} - 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $6 x^{2} + 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 3$.\n2) Slope at x=-1: m = f'(-1) = $-1 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\n3) Point on curve: (-1, f(-1)) = (-1, $\\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 4$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 4 = -1 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}(x − -1).\n6) Optionally expand: y = $- x + \\frac{x}{2 e^{\\frac{1}{2}}} + 2 x \\sin{\\left(2 \\right)} + \\cos{\\left(2 \\right)} + \\frac{3}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)} + 3$.\nFinal answer: $y = - x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)} + 3$", "final_answer": "y = - x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)} + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "071ad90be65c7eac", "problem_id": "389ac5d92d4b30be", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} + 2 x^{2} - 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=-1$.", "output": "Differentiate: f'(x) = $6 x^{2} + 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 3$.\nSlope at x=-1: m = f'(-1) = $-1 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nPoint on curve: (-1, f(-1)) = (-1, $\\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 4$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 4 = -1 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}(x − -1).\nOptionally expand: y = $- x + \\frac{x}{2 e^{\\frac{1}{2}}} + 2 x \\sin{\\left(2 \\right)} + \\cos{\\left(2 \\right)} + \\frac{3}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)} + 3$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)} + 3$", "final_answer": "y = - x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)} + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aa14ecdc34d3613e", "problem_id": "16c513130a8576ae", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{8 x}{5} - \\frac{\\sqrt{3} x}{2} - \\frac{21}{5} - \\frac{7 \\sqrt{3}}{10}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{8}{5} - \\frac{\\sqrt{3}}{2}$, c=$- \\frac{21}{5} - \\frac{7 \\sqrt{3}}{10}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{22 \\sqrt{3}}{5} + \\frac{2011}{100}$, so √Δ = $\\sqrt{\\frac{22 \\sqrt{3}}{5} + \\frac{2011}{100}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{3}}{4} + \\frac{4}{5} + \\frac{\\sqrt{440 \\sqrt{3} + 2011}}{20}$.\n7) x2 = $- \\frac{\\sqrt{440 \\sqrt{3} + 2011}}{20} + \\frac{\\sqrt{3}}{4} + \\frac{4}{5}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{3}}{4} + \\frac{4}{5} + \\frac{\\sqrt{440\\cdot \\sqrt{3} + 2011}}{20}, - \\frac{\\sqrt{440\\cdot \\sqrt{3} + 2011}}{20} + \\frac{\\sqrt{3}}{4} + \\frac{4}{5}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{3}}{4} + \\frac{4}{5} + \\frac{\\sqrt{440\\cdot \\sqrt{3} + 2011}}{20}, - \\frac{\\sqrt{440\\cdot \\sqrt{3} + 2011}}{20} + \\frac{\\sqrt{3}}{4} + \\frac{4}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cfd36019f073aa1c", "problem_id": "16c513130a8576ae", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{8 x}{5} - \\frac{\\sqrt{3} x}{2} - \\frac{21}{5} - \\frac{7 \\sqrt{3}}{10}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{8}{5} - \\frac{\\sqrt{3}}{2}$, c=$- \\frac{21}{5} - \\frac{7 \\sqrt{3}}{10}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{22 \\sqrt{3}}{5} + \\frac{2011}{100}$, so √Δ = $\\sqrt{\\frac{22 \\sqrt{3}}{5} + \\frac{2011}{100}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{3}}{4} + \\frac{4}{5} + \\frac{\\sqrt{440 \\sqrt{3} + 2011}}{20}$.\nx2 = $- \\frac{\\sqrt{440 \\sqrt{3} + 2011}}{20} + \\frac{\\sqrt{3}}{4} + \\frac{4}{5}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{3}}{4} + \\frac{4}{5} + \\frac{\\sqrt{440\\cdot \\sqrt{3} + 2011}}{20}, - \\frac{\\sqrt{440\\cdot \\sqrt{3} + 2011}}{20} + \\frac{\\sqrt{3}}{4} + \\frac{4}{5}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{3}}{4} + \\frac{4}{5} + \\frac{\\sqrt{440\\cdot \\sqrt{3} + 2011}}{20}, - \\frac{\\sqrt{440\\cdot \\sqrt{3} + 2011}}{20} + \\frac{\\sqrt{3}}{4} + \\frac{4}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ffb177bb129ac80b", "problem_id": "a7f6c5cc3ebd3dde", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "You want 69 liters of a 50% salt solution. You have a 30% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 69−x liters is 70% solution.\n2) Salt equation: 30x + 70(69−x) = 50·69.\n3) (30-70)x = (50-70)·69 ⇒ x = 69/2.\n4) Then 69−x = 69/2.\nFinal answer: $69/2 L of 30% and 69/2 L of 70%$", "final_answer": "69/2 L of 30% and 69/2 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "232b2c21c2a35be0", "problem_id": "a7f6c5cc3ebd3dde", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 69 liters of a 50% salt solution. You have a 30% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 69−x liters is 70% solution.\nSalt equation: 30x + 70(69−x) = 50·69.\n(30-70)x = (50-70)·69 ⇒ x = 69/2.\nThen 69−x = 69/2.\nCheck: 30·69/2 + 70·69/2 = 50·69.\nFinal answer: $69/2 L of 30% and 69/2 L of 70%$", "final_answer": "69/2 L of 30% and 69/2 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fea5213dffc45d9d", "problem_id": "044d90fd5420d8aa", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Worker A finishes a job in 8 hours and Worker B finishes it in 10 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/8 job/hour. B's rate = 1/10 job/hour.\n2) Combined rate = 1/8 + 1/10 = 9/40 job/hour.\n3) Time = 1 ÷ (combined rate) = 40/9 hours.\nFinal answer: $\\frac{40}{9}$", "final_answer": "\\frac{40}{9}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "049af3128745dd1c", "problem_id": "044d90fd5420d8aa", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Worker A finishes a job in 8 hours and Worker B finishes it in 10 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/8 job/hour. B's rate = 1/10 job/hour.\nCombined rate = 1/8 + 1/10 = 9/40 job/hour.\nTime = 1 ÷ (combined rate) = 40/9 hours.\nCheck: (combined rate)×(time) = 9/40×40/9 = 1.\nFinal answer: $\\frac{40}{9}$", "final_answer": "\\frac{40}{9}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "414f10e7e299165f", "problem_id": "d566e60c988a3578", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} 3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3$.\nEvaluate at the bounds: F(\\pi) and F(0).\nSubtract: F(\\pi) − F(0) = $- \\frac{3 \\pi^{2}}{2} + 4 + 3 \\pi + \\pi^{3}$.\nFinal answer: $- \\frac{3\\cdot \\pi^{2}}{2} + 4 + 3\\cdot \\pi + \\pi^{3}$", "final_answer": "- \\frac{3\\cdot \\pi^{2}}{2} + 4 + 3\\cdot \\pi + \\pi^{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0700466d9aef60cf", "problem_id": "d566e60c988a3578", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} 3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3$.\n2) Evaluate at the bounds: F(\\pi) and F(0).\n3) Subtract: F(\\pi) − F(0) = $- \\frac{3 \\pi^{2}}{2} + 4 + 3 \\pi + \\pi^{3}$.\n4) Simplify the result.\nFinal answer: $- \\frac{3\\cdot \\pi^{2}}{2} + 4 + 3\\cdot \\pi + \\pi^{3}$", "final_answer": "- \\frac{3\\cdot \\pi^{2}}{2} + 4 + 3\\cdot \\pi + \\pi^{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a169346d9d1879b6", "problem_id": "ebabd495234d0b8c", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many binary strings of length 16 contain exactly 7 ones with no two ones adjacent?", "output": "1) Reserve 6 mandatory zeros between the 7 ones.\n2) This leaves n−k+1 = 10 valid slots for the ones.\n3) Choose the slots: C(10,7) = 120.\nFinal answer: $120$", "final_answer": "120", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2b8d6fd974c1e7a7", "problem_id": "ebabd495234d0b8c", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many binary strings of length 16 contain exactly 7 ones with no two ones adjacent?", "output": "Reserve 6 mandatory zeros between the 7 ones.\nThis leaves n−k+1 = 10 valid slots for the ones.\nChoose the slots: C(10,7) = 120.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $120$", "final_answer": "120", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "537596444c9a2965", "problem_id": "9d9fa85f48ed6995", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡3 (mod 4), x≡4 (mod 5). Give the solution modulo 20.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 20.\nCompute the CRT solution to get x ≡ 19 (mod 20).\nCheck: 19 mod 4=3 and mod 5=4.\nFinal answer: $x\\equiv 19\\pmod{20}$", "final_answer": "x\\equiv 19\\pmod{20}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c6589fc7f74abdd1", "problem_id": "9d9fa85f48ed6995", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡3 (mod 4), x≡4 (mod 5). Give the solution modulo 20.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 20.\n3) Compute the CRT solution to get x ≡ 19 (mod 20).\nFinal answer: $x\\equiv 19\\pmod{20}$", "final_answer": "x\\equiv 19\\pmod{20}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5839e4b4545cb8c6", "problem_id": "5f688ee064fec191", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the multiplicative inverse of 23 modulo 41.", "output": "We want x such that 23x ≡ 1 (mod 41).\nUse extended Euclid to find s,t with 23s + 41t = 1.\nOne Bezout identity is: 23·(-16) + 41·(9) = 1.\nReduce s modulo 41: x ≡ 25 (mod 41).\nCheck: 23·25 mod 41 = 1.\nFinal answer: $x\\equiv 25\\pmod{41}$", "final_answer": "x\\equiv 25\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "45f5e0ac83db0d46", "problem_id": "aaa146a38fe8ea5e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{2} + 2 x - 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(4 x^{2}) = 8 x^{2}$.\n3) So $f'(x) = 8 x^{2} + 2$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(4 x^{2}) = 8 x$.\nTherefore the correct derivative is $f'(x) = 2 \\left(4 x + 1\\right)$.\nFinal answer: $2 \\left(4 x + 1\\right)$", "final_answer": "2 \\left(4 x + 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "65f71a3bddd2adf0", "problem_id": "3f56429057ffb3b0", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many binary strings of length 12 contain exactly 6 ones with no two ones adjacent?", "output": "1) Reserve 5 mandatory zeros between the 6 ones.\n2) This leaves n−k+1 = 7 valid slots for the ones.\n3) Choose the slots: C(7,6) = 7.\nFinal answer: $7$", "final_answer": "7", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0b5ce5ce80937896", "problem_id": "23eafac488bc3cdd", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} x - 2$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to -2.\n3) Then the limit is -2.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -4.\nFinal answer: $-4$", "final_answer": "-4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "84a77d756832f4f5", "problem_id": "2f35f145c21aa30d", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 7 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=7, k=2, p=$\\frac{2}{5}$ and simplify.\n4) Here C(7,2) = 21.\nFinal answer: $\\frac{20412}{78125}$", "final_answer": "\\frac{20412}{78125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3193ca78ad6c9bc8", "problem_id": "2f35f145c21aa30d", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 7 times, what is the probability of exactly 2 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=7, k=2, p=$\\frac{2}{5}$ and simplify.\nHere C(7,2) = 21.\nCheck: expression simplifies to $\\frac{20412}{78125}$.\nFinal answer: $\\frac{20412}{78125}$", "final_answer": "\\frac{20412}{78125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4df682948a2a8893", "problem_id": "cd513dafeb41ece1", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the congruences: x≡0 (mod 9), x≡2 (mod 8). Give the solution modulo 72.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 72.\nCompute the CRT solution to get x ≡ 18 (mod 72).\nFinal answer: $x\\equiv 18\\pmod{72}$", "final_answer": "x\\equiv 18\\pmod{72}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6efb9c9fdd4dea18", "problem_id": "cd513dafeb41ece1", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡0 (mod 9), x≡2 (mod 8). Give the solution modulo 72.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 72.\n3) Compute the CRT solution to get x ≡ 18 (mod 72).\nFinal answer: $x\\equiv 18\\pmod{72}$", "final_answer": "x\\equiv 18\\pmod{72}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "01a20422c50bfe6c", "problem_id": "0a1b4ef049086406", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 15 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=15, k=6, p=$\\frac{1}{8}$ and simplify.\n4) Here C(15,6) = 5005.\nFinal answer: $\\frac{201969803035}{35184372088832}$", "final_answer": "\\frac{201969803035}{35184372088832}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e85e3274420e1ef9", "problem_id": "0a1b4ef049086406", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 15 times, what is the probability of exactly 6 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=15, k=6, p=$\\frac{1}{8}$ and simplify.\nFinal answer: $\\frac{201969803035}{35184372088832}$", "final_answer": "\\frac{201969803035}{35184372088832}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a8a5311896b94e9", "problem_id": "6307940c925eea62", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A disease has prevalence $\\frac{3}{100}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{561}{10000}$.\n3) Compute P(D|+) = $\\frac{90}{187}$.\nFinal answer: $\\frac{90}{187}$", "final_answer": "\\frac{90}{187}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0f594b031f0c7c00", "problem_id": "6307940c925eea62", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A disease has prevalence $\\frac{3}{100}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{561}{10000}$.\nCompute P(D|+) = $\\frac{90}{187}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{90}{187}$", "final_answer": "\\frac{90}{187}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a6ad19509f38c631", "problem_id": "b2e554943aa9cc0f", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Find the equation of the tangent line to $y=3 x^{2} + x + 2 \\sin{\\left(x \\right)} + 2$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $6 x + 2 \\cos{\\left(x \\right)} + 1$.\n2) Slope at x=-2: m = f'(-2) = $-11 + 2 \\cos{\\left(2 \\right)}$.\n3) Point on curve: (-2, f(-2)) = (-2, $12 - 2 \\sin{\\left(2 \\right)}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 12 - 2 \\sin{\\left(2 \\right)} = -11 + 2 \\cos{\\left(2 \\right)}(x − -2).\n6) Optionally expand: y = $- 11 x + 2 x \\cos{\\left(2 \\right)} - 10 - 2 \\sin{\\left(2 \\right)} + 4 \\cos{\\left(2 \\right)}$.\nFinal answer: $y = - 11\\cdot x + 2\\cdot x\\cdot \\cos{\\left(2 \\right)} - 10 - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)}$", "final_answer": "y = - 11\\cdot x + 2\\cdot x\\cdot \\cos{\\left(2 \\right)} - 10 - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "458ed11816a3bb43", "problem_id": "b2e554943aa9cc0f", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{2} + x + 2 \\sin{\\left(x \\right)} + 2$ at $x=-2$.", "output": "Differentiate: f'(x) = $6 x + 2 \\cos{\\left(x \\right)} + 1$.\nSlope at x=-2: m = f'(-2) = $-11 + 2 \\cos{\\left(2 \\right)}$.\nPoint on curve: (-2, f(-2)) = (-2, $12 - 2 \\sin{\\left(2 \\right)}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 12 - 2 \\sin{\\left(2 \\right)} = -11 + 2 \\cos{\\left(2 \\right)}(x − -2).\nOptionally expand: y = $- 11 x + 2 x \\cos{\\left(2 \\right)} - 10 - 2 \\sin{\\left(2 \\right)} + 4 \\cos{\\left(2 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 11\\cdot x + 2\\cdot x\\cdot \\cos{\\left(2 \\right)} - 10 - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)}$", "final_answer": "y = - 11\\cdot x + 2\\cdot x\\cdot \\cos{\\left(2 \\right)} - 10 - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1ab230f2a662d0f9", "problem_id": "eb6cf60360a9f1fd", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A cyclist rides at 31 miles per hour for 2 hours, then at 87 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 31 × 2 = 62 miles.\n2) Second leg distance = 87 × 1 = 87 miles.\n3) Total distance = 62 + 87 = 149 miles.\nFinal answer: $149$", "final_answer": "149", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5fed735e7f7c5e9b", "problem_id": "eb6cf60360a9f1fd", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "A cyclist rides at 31 miles per hour for 2 hours, then at 87 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 31 × 2 = 62 miles.\nSecond leg distance = 87 × 1 = 87 miles.\nTotal distance = 62 + 87 = 149 miles.\nFinal answer: $149$", "final_answer": "149", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6e103c3253ef4777", "problem_id": "7ad602b7d6879177", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- 3 x^{2} - 3 x - 3\\right) \\cos{\\left(2 x + 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\n3) So $f'(x) = 6 x^{2} \\sin{\\left(2 x + 1 \\right)} + 2 x e^{x} + 6 x \\sin{\\left(2 x + 1 \\right)} - 6 x \\cos{\\left(2 x + 1 \\right)} + 4 e^{x} + 6 \\sin{\\left(2 x + 1 \\right)} - 3 \\cos{\\left(2 x + 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = 3 \\left(- 2 x - 1\\right) \\cos{\\left(2 x + 1 \\right)} + 2 \\left(x + 1\\right) e^{x} - 6 \\left(x \\left(- x - 1\\right) - 1\\right) \\sin{\\left(2 x + 1 \\right)} + 2 e^{x}$.\nFinal answer: $3 \\left(- 2 x - 1\\right) \\cos{\\left(2 x + 1 \\right)} + 2 \\left(x + 1\\right) e^{x} - 6 \\left(x \\left(- x - 1\\right) - 1\\right) \\sin{\\left(2 x + 1 \\right)} + 2 e^{x}$", "final_answer": "3 \\left(- 2 x - 1\\right) \\cos{\\left(2 x + 1 \\right)} + 2 \\left(x + 1\\right) e^{x} - 6 \\left(x \\left(- x - 1\\right) - 1\\right) \\sin{\\left(2 x + 1 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f63c525a4a4371ab", "problem_id": "79c69936795afa7d", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 11 x + 8$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$11$, c=$8$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $25$, so √Δ = $5$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $-1$.\n7) x2 = $- \\frac{8}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{-1, - \\frac{8}{3}\\}$", "final_answer": "x \\in \\{-1, - \\frac{8}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eb976da54a5ea9f9", "problem_id": "18271610725b8939", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=22$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=22 and k=7.\n4) Compute C(28,6) = 376740.\nFinal answer: $376740$", "final_answer": "376740", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a52e69d6b72b4db6", "problem_id": "18271610725b8939", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=22$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=22 and k=7.\nCompute C(28,6) = 376740.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $376740$", "final_answer": "376740", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fa890318ab0f32c8", "problem_id": "450fa62c58e6212b", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 2 x^{3} - 2 x^{2} - x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 2 x^{3} - 2 x^{2} - x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 1$.\n3) Compute an antiderivative: F(x) = $- \\frac{x^{4}}{2} - \\frac{2 x^{3}}{3} - \\frac{x^{2}}{2} + x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{x^{4}}{2} - \\frac{2\\cdot x^{3}}{3} - \\frac{x^{2}}{2} + x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{x^{4}}{2} - \\frac{2\\cdot x^{3}}{3} - \\frac{x^{2}}{2} + x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "55579c4a7ce9225e", "problem_id": "450fa62c58e6212b", "style": "answer_only", "instruction": "Final answer only.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 2 x^{3} - 2 x^{2} - x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "$C - \\frac{x^{4}}{2} - \\frac{2\\cdot x^{3}}{3} - \\frac{x^{2}}{2} + x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{x^{4}}{2} - \\frac{2\\cdot x^{3}}{3} - \\frac{x^{2}}{2} + x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "669afaca2d23ba95", "problem_id": "5e11b29f1c50daaa", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$7 x - 5 y$ = $-28$\n$- 6 y$ = $42$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $7$: $- 42 y$ = $294$.\n4) Subtract to eliminate x: $42 y$ = $-294$.\n5) Solve for y: y = $-7$.\n6) Substitute back to find x: x = $-9$.\nFinal answer: $(x,y)=\\left(-9, -7\\right)$", "final_answer": "(x,y)=\\left(-9, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "07186f33f3ebf3a9", "problem_id": "5e11b29f1c50daaa", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the linear system:\n$7 x - 5 y$ = $-28$\n$- 6 y$ = $42$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $7$: $- 42 y$ = $294$.\nFinal answer: $(x,y)=\\left(-9, -7\\right)$", "final_answer": "(x,y)=\\left(-9, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d3389788892dc020", "problem_id": "9c5b941ccc698ffa", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡3 (mod 5), x≡4 (mod 11). Give the solution modulo 55.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 55.\n3) Compute the CRT solution to get x ≡ 48 (mod 55).\nFinal answer: $x\\equiv 48\\pmod{55}$", "final_answer": "x\\equiv 48\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "68b4e0ba4255c401", "problem_id": "130dadf694997c19", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- 2 x^{2} + 2 x + 1\\right) \\cos{\\left(x - 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x \\cos{\\left(x - 1 \\right)}) = - 2 x \\sin{\\left(x - 1 \\right)}$.\n3) So $f'(x) = 2 x^{2} \\sin{\\left(x - 1 \\right)} + 2 x e^{x} - 2 x \\sin{\\left(x - 1 \\right)} - 4 x \\cos{\\left(x - 1 \\right)} + 4 e^{x} - \\sin{\\left(x - 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x \\cos{\\left(x - 1 \\right)}) = - 2 x \\sin{\\left(x - 1 \\right)} + 2 \\cos{\\left(x - 1 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(1 - 2 x\\right) \\cos{\\left(x - 1 \\right)} + 2 \\left(x + 1\\right) e^{x} - \\left(2 x \\left(1 - x\\right) + 1\\right) \\sin{\\left(x - 1 \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(1 - 2 x\\right) \\cos{\\left(x - 1 \\right)} + 2 \\left(x + 1\\right) e^{x} - \\left(2 x \\left(1 - x\\right) + 1\\right) \\sin{\\left(x - 1 \\right)} + 2 e^{x}$", "final_answer": "2 \\left(1 - 2 x\\right) \\cos{\\left(x - 1 \\right)} + 2 \\left(x + 1\\right) e^{x} - \\left(2 x \\left(1 - x\\right) + 1\\right) \\sin{\\left(x - 1 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "161b3adce3d86066", "problem_id": "919e3d21b114cae2", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Two friends split $132 in the ratio 3:3. How much does each receive?", "output": "1) Total ratio parts = 3+3 = 6.\n2) Each part = 132 ÷ 6 = 22.\n3) First share = 3 × 22 = 66.\n4) Second share = 3 × 22 = 66.\nFinal answer: $66$", "final_answer": "66", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a0a9e7572b31c56f", "problem_id": "919e3d21b114cae2", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Two friends split $132 in the ratio 3:3. How much does each receive?", "output": "Total ratio parts = 3+3 = 6.\nEach part = 132 ÷ 6 = 22.\nFirst share = 3 × 22 = 66.\nFinal answer: $66$", "final_answer": "66", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a0e6db068a44e808", "problem_id": "182f8d6a5f6e5cee", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} + 3 x^{2} - 2$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-2) = 0.\n3) d/dx(x^{3}) = 3 x^{2}.\n4) d/dx(3 x^{2}) = 6 x.\n5) Combine: f'(x) = 3 x \\left(x + 2\\right).\nFinal answer: $3\\cdot x\\cdot \\left(x + 2\\right)$", "final_answer": "3\\cdot x\\cdot \\left(x + 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d6279bbc0c3e8b6c", "problem_id": "182f8d6a5f6e5cee", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} + 3 x^{2} - 2$.", "output": "Differentiate term-by-term:\nd/dx(-2) = 0.\nd/dx(x^{3}) = 3 x^{2}.\nFinal answer: $3\\cdot x\\cdot \\left(x + 2\\right)$", "final_answer": "3\\cdot x\\cdot \\left(x + 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d76e3dec50b30f94", "problem_id": "6f2ebf761e34206d", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - 3 x^{2} + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 3$ at $x=1$.", "output": "$y = - 15\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 12$", "final_answer": "y = - 15\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 12", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "15e5e17ee590112d", "problem_id": "6f2ebf761e34206d", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - 3 x^{2} + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 3$ at $x=1$.", "output": "1) Differentiate: f'(x) = $- 9 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)}$.\n2) Slope at x=1: m = f'(1) = $-15 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}$.\n3) Point on curve: (1, f(1)) = (1, $-3 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -3 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} = -15 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}(x − 1).\n6) Optionally expand: y = $- 15 x - 2 x \\sin{\\left(2 \\right)} + \\frac{x e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2 \\sin{\\left(2 \\right)} + 12$.\nFinal answer: $y = - 15\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 12$", "final_answer": "y = - 15\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 12", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aa83a6df497cb59c", "problem_id": "a6a5f30f451160d0", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A cyclist rides at 62 miles per hour for 1 hours, then at 21 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 62 × 1 = 62 miles.\n2) Second leg distance = 21 × 2 = 42 miles.\n3) Total distance = 62 + 42 = 104 miles.\nFinal answer: $104$", "final_answer": "104", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "32a13a4661ba0a9b", "problem_id": "a6a5f30f451160d0", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A cyclist rides at 62 miles per hour for 1 hours, then at 21 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 62 × 1 = 62 miles.\nSecond leg distance = 21 × 2 = 42 miles.\nTotal distance = 62 + 42 = 104 miles.\nFinal answer: $104$", "final_answer": "104", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cb1f84c8b6b01cdb", "problem_id": "64ab10787133ea82", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{6} x}{2} + \\frac{7 x}{2} - \\frac{9 \\sqrt{6}}{4} - \\frac{9}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{7}{2} - \\frac{\\sqrt{6}}{2}$, c=$- \\frac{9 \\sqrt{6}}{4} - \\frac{9}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{11 \\sqrt{6}}{2} + \\frac{127}{4}$, so √Δ = $\\sqrt{\\frac{11 \\sqrt{6}}{2} + \\frac{127}{4}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{7}{4} + \\frac{\\sqrt{6}}{4} + \\frac{\\sqrt{22 \\sqrt{6} + 127}}{4}$.\nx2 = $- \\frac{\\sqrt{22 \\sqrt{6} + 127}}{4} - \\frac{7}{4} + \\frac{\\sqrt{6}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{7}{4} + \\frac{\\sqrt{6}}{4} + \\frac{\\sqrt{22\\cdot \\sqrt{6} + 127}}{4}, - \\frac{\\sqrt{22\\cdot \\sqrt{6} + 127}}{4} - \\frac{7}{4} + \\frac{\\sqrt{6}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{7}{4} + \\frac{\\sqrt{6}}{4} + \\frac{\\sqrt{22\\cdot \\sqrt{6} + 127}}{4}, - \\frac{\\sqrt{22\\cdot \\sqrt{6} + 127}}{4} - \\frac{7}{4} + \\frac{\\sqrt{6}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4887ddb1bf07e0ed", "problem_id": "64ab10787133ea82", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{6} x}{2} + \\frac{7 x}{2} - \\frac{9 \\sqrt{6}}{4} - \\frac{9}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{7}{2} - \\frac{\\sqrt{6}}{2}$, c=$- \\frac{9 \\sqrt{6}}{4} - \\frac{9}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{11 \\sqrt{6}}{2} + \\frac{127}{4}$, so √Δ = $\\sqrt{\\frac{11 \\sqrt{6}}{2} + \\frac{127}{4}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{7}{4} + \\frac{\\sqrt{6}}{4} + \\frac{\\sqrt{22 \\sqrt{6} + 127}}{4}$.\n7) x2 = $- \\frac{\\sqrt{22 \\sqrt{6} + 127}}{4} - \\frac{7}{4} + \\frac{\\sqrt{6}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{7}{4} + \\frac{\\sqrt{6}}{4} + \\frac{\\sqrt{22\\cdot \\sqrt{6} + 127}}{4}, - \\frac{\\sqrt{22\\cdot \\sqrt{6} + 127}}{4} - \\frac{7}{4} + \\frac{\\sqrt{6}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{7}{4} + \\frac{\\sqrt{6}}{4} + \\frac{\\sqrt{22\\cdot \\sqrt{6} + 127}}{4}, - \\frac{\\sqrt{22\\cdot \\sqrt{6} + 127}}{4} - \\frac{7}{4} + \\frac{\\sqrt{6}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "122edea59dd5bd67", "problem_id": "a4898b82e7b4d45a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(- 2 x^{3} - 2 x + 3\\right) \\cos{\\left(2 x + 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x e^{x}) = 2 x e^{x}$.\n3) So $f'(x) = 4 x^{3} \\sin{\\left(2 x + 1 \\right)} - 6 x^{2} \\cos{\\left(2 x + 1 \\right)} + 2 x e^{x} + 4 x \\sin{\\left(2 x + 1 \\right)} + 4 e^{x} - 6 \\sin{\\left(2 x + 1 \\right)} - 2 \\cos{\\left(2 x + 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x e^{x}) = 2 x e^{x} + 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(\\left(x + 2\\right) e^{x} + \\left(- 3 x^{2} - 1\\right) \\cos{\\left(2 x + 1 \\right)} - \\left(2 x \\left(- x^{2} - 1\\right) + 3\\right) \\sin{\\left(2 x + 1 \\right)} + e^{x}\\right)$.\nFinal answer: $2 \\left(\\left(x + 2\\right) e^{x} + \\left(- 3 x^{2} - 1\\right) \\cos{\\left(2 x + 1 \\right)} - \\left(2 x \\left(- x^{2} - 1\\right) + 3\\right) \\sin{\\left(2 x + 1 \\right)} + e^{x}\\right)$", "final_answer": "2 \\left(\\left(x + 2\\right) e^{x} + \\left(- 3 x^{2} - 1\\right) \\cos{\\left(2 x + 1 \\right)} - \\left(2 x \\left(- x^{2} - 1\\right) + 3\\right) \\sin{\\left(2 x + 1 \\right)} + e^{x}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3649b2c2de85a979", "problem_id": "303d174720037e71", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$- 4 x + 2 y$ = $34$\n$- 5 x + 7 y$ = $74$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-5$: $20 x - 10 y$ = $-170$.\n3) Multiply the second equation by $-4$: $20 x - 28 y$ = $-296$.\n4) Subtract to eliminate x: $18 y$ = $126$.\n5) Solve for y: y = $7$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, 7\\right)$", "final_answer": "(x,y)=\\left(-5, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c0546c8a261301dc", "problem_id": "c57dfd9fa102b04b", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Two friends split $420 in the ratio 7:7. How much does each receive?", "output": "1) Total ratio parts = 7+7 = 14.\n2) Each part = 420 ÷ 14 = 30.\n3) First share = 7 × 30 = 210.\n4) Second share = 7 × 30 = 210.\nFinal answer: $210$", "final_answer": "210", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b438ec9140c137c", "problem_id": "c57dfd9fa102b04b", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $420 in the ratio 7:7. How much does each receive?", "output": "Total ratio parts = 7+7 = 14.\nEach part = 420 ÷ 14 = 30.\nFirst share = 7 × 30 = 210.\nSecond share = 7 × 30 = 210.\nCheck: 210+210=420.\nFinal answer: $210$", "final_answer": "210", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff54e0d3ed8ce052", "problem_id": "5c583ee1597f39b8", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int x^{3} - 3 x - 1\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $x^{3} - 3 x - 1$.\n3) Compute an antiderivative: F(x) = $\\frac{x^{4}}{4} - \\frac{3 x^{2}}{2} - x$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{x^{4}}{4} - \\frac{3\\cdot x^{2}}{2} - x$", "final_answer": "C + \\frac{x^{4}}{4} - \\frac{3\\cdot x^{2}}{2} - x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a6aecd7e3e778b5d", "problem_id": "5c583ee1597f39b8", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int x^{3} - 3 x - 1\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $x^{3} - 3 x - 1$.\nCompute an antiderivative: F(x) = $\\frac{x^{4}}{4} - \\frac{3 x^{2}}{2} - x$.\nFinal answer: $C + \\frac{x^{4}}{4} - \\frac{3\\cdot x^{2}}{2} - x$", "final_answer": "C + \\frac{x^{4}}{4} - \\frac{3\\cdot x^{2}}{2} - x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6180bab6133c4b51", "problem_id": "29ecfda3edcd32c8", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 20 red balls and 27 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(47,10).\nFavorable samples: choose k reds and n−k blues: C(20,1)·C(27,9).\nCompute totals: C(47,10) = 5178066751, and favorable = 93736500.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1500}{82861}$.\nFinal answer: $\\frac{1500}{82861}$", "final_answer": "\\frac{1500}{82861}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f475842386682bb1", "problem_id": "683f51bf6c7a9c8e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{4} - x^{3} - x^{2} - 2 x - 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x^{4}) = 4 x^{4}$.\n3) So $f'(x) = 4 x^{4} - 3 x^{2} - 2 x - 2$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x^{4}) = 4 x^{3}$.\nTherefore the correct derivative is $f'(x) = 4 x^{3} - 3 x^{2} - 2 x - 2$.\nFinal answer: $4 x^{3} - 3 x^{2} - 2 x - 2$", "final_answer": "4 x^{3} - 3 x^{2} - 2 x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6a933dba50e81054", "problem_id": "583f4b7b3e7a1071", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Worker A finishes a job in 15 hours and Worker B finishes it in 20 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/15 job/hour. B's rate = 1/20 job/hour.\nCombined rate = 1/15 + 1/20 = 7/60 job/hour.\nTime = 1 ÷ (combined rate) = 60/7 hours.\nCheck: (combined rate)×(time) = 7/60×60/7 = 1.\nFinal answer: $\\frac{60}{7}$", "final_answer": "\\frac{60}{7}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e2aa6edd472ada0b", "problem_id": "583f4b7b3e7a1071", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Worker A finishes a job in 15 hours and Worker B finishes it in 20 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/15 job/hour. B's rate = 1/20 job/hour.\n2) Combined rate = 1/15 + 1/20 = 7/60 job/hour.\n3) Time = 1 ÷ (combined rate) = 60/7 hours.\nFinal answer: $\\frac{60}{7}$", "final_answer": "\\frac{60}{7}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8d679113f2a32b9f", "problem_id": "1d8eab479cb44f15", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 10 x + 16$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} + 10 x + 16$ = $\\left(x + 2\\right) \\left(x + 8\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-2$.\n5) From the second factor: x = $-8$.\nFinal answer: $x \\in \\{-2, -8\\}$", "final_answer": "x \\in \\{-2, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f5da1b2adb5b72a6", "problem_id": "1d8eab479cb44f15", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 10 x + 16$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} + 10 x + 16$ = $\\left(x + 2\\right) \\left(x + 8\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $-2$.\nFrom the second factor: x = $-8$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-2, -8\\}$", "final_answer": "x \\in \\{-2, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e00c5eb6220ce6fa", "problem_id": "4b34743caf085b7c", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} x^{3} - 3 x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $x^{3} - 3 x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\pi).\n3) Subtract: F(\\pi) − F(- \\pi) = $2 \\pi \\left(- \\pi^{2} - 2\\right)$.\n4) Simplify the result.\nFinal answer: $2\\cdot \\pi\\cdot \\left(- \\pi^{2} - 2\\right)$", "final_answer": "2\\cdot \\pi\\cdot \\left(- \\pi^{2} - 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c93f1fd85aa1f11b", "problem_id": "4b34743caf085b7c", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} x^{3} - 3 x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $x^{3} - 3 x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2$.\nEvaluate at the bounds: F(\\pi) and F(- \\pi).\nSubtract: F(\\pi) − F(- \\pi) = $2 \\pi \\left(- \\pi^{2} - 2\\right)$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $2\\cdot \\pi\\cdot \\left(- \\pi^{2} - 2\\right)$", "final_answer": "2\\cdot \\pi\\cdot \\left(- \\pi^{2} - 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0ef215a995edb960", "problem_id": "fb6f6e625b39f624", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Two friends split $1352 in the ratio 5:8. How much does each receive?", "output": "Total ratio parts = 5+8 = 13.\nEach part = 1352 ÷ 13 = 104.\nFirst share = 5 × 104 = 520.\nSecond share = 8 × 104 = 832.\nCheck: 520+832=1352.\nFinal answer: $832$", "final_answer": "832", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e4027c8e1ddec51f", "problem_id": "fb6f6e625b39f624", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $1352 in the ratio 5:8. How much does each receive?", "output": "1) Total ratio parts = 5+8 = 13.\n2) Each part = 1352 ÷ 13 = 104.\n3) First share = 5 × 104 = 520.\n4) Second share = 8 × 104 = 832.\nFinal answer: $832$", "final_answer": "832", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eb2767e771acb03b", "problem_id": "3fd927a1f536626d", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 2 x^{3} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{3} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $-4 + 9 \\pi + \\frac{15 \\pi^{4}}{2}$.\n4) Simplify the result.\nFinal answer: $-4 + 9\\cdot \\pi + \\frac{15\\cdot \\pi^{4}}{2}$", "final_answer": "-4 + 9\\cdot \\pi + \\frac{15\\cdot \\pi^{4}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "882101bb836e177f", "problem_id": "3fd927a1f536626d", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 2 x^{3} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $2 x^{3} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $-4 + 9 \\pi + \\frac{15 \\pi^{4}}{2}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $-4 + 9\\cdot \\pi + \\frac{15\\cdot \\pi^{4}}{2}$", "final_answer": "-4 + 9\\cdot \\pi + \\frac{15\\cdot \\pi^{4}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4611d44068d7b8d1", "problem_id": "d7764e8b17ab7295", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A store sells a backpack for $132. It is discounted by 5%. You buy 7 backpacks. What is the total cost?", "output": "1) Discounted price = 132 × (1 − 5/100) = 132 × 95/100 = 627/5.\n2) Subtotal = 7 × 627/5 = 4389/5.\nFinal answer: $\\frac{4389}{5}$", "final_answer": "\\frac{4389}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b3ebbe000e3c2e92", "problem_id": "d7764e8b17ab7295", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "A store sells a backpack for $132. It is discounted by 5%. You buy 7 backpacks. What is the total cost?", "output": "Discounted price = 132 × (1 − 5/100) = 132 × 95/100 = 627/5.\nSubtotal = 7 × 627/5 = 4389/5.\nFinal answer: $\\frac{4389}{5}$", "final_answer": "\\frac{4389}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "20b69cea3f0672d4", "problem_id": "64c068779db76475", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Worker A finishes a job in 21 hours and Worker B finishes it in 18 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/21 job/hour. B's rate = 1/18 job/hour.\n2) Combined rate = 1/21 + 1/18 = 13/126 job/hour.\n3) Time = 1 ÷ (combined rate) = 126/13 hours.\nFinal answer: $\\frac{126}{13}$", "final_answer": "\\frac{126}{13}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b87bd4522c7bc45", "problem_id": "64c068779db76475", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Worker A finishes a job in 21 hours and Worker B finishes it in 18 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/21 job/hour. B's rate = 1/18 job/hour.\nCombined rate = 1/21 + 1/18 = 13/126 job/hour.\nTime = 1 ÷ (combined rate) = 126/13 hours.\nCheck: (combined rate)×(time) = 13/126×126/13 = 1.\nFinal answer: $\\frac{126}{13}$", "final_answer": "\\frac{126}{13}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5fdb8be27c885703", "problem_id": "deafe4a875199a22", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_8=25$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_8=1.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=1 and k=8.\n4) Compute C(8,7) = 8.\nFinal answer: $8$", "final_answer": "8", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d92126d751196bda", "problem_id": "deafe4a875199a22", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_8=25$?", "output": "Let y_i=x_i-3, then y_i≥0 and y_1+…+y_8=1.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=1 and k=8.\nCompute C(8,7) = 8.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $8$", "final_answer": "8", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c28dbda10a627298", "problem_id": "9c6d3a75df7704a5", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$- 8 x - 8 y$ = $-88$\n$x + 9 y$ = $75$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $- 8 x - 8 y$ = $-88$.\n3) Multiply the second equation by $-8$: $- 8 x - 72 y$ = $-600$.\n4) Subtract to eliminate x: $64 y$ = $512$.\n5) Solve for y: y = $8$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, 8\\right)$", "final_answer": "(x,y)=\\left(3, 8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cbe29881cc40ed0f", "problem_id": "d6334c18f690852d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} \\cos{\\left(x^{2} + 3 x + 1 \\right)} + 2 \\left(x + 1\\right) e^{x}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 x^{2} \\cos{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)}) = - 3 x^{2} \\sin{\\left(x^{2} + 3 x + 1 \\right)}$.\n3) So $f'(x) = - 3 x^{2} \\sin{\\left(x^{2} + 3 x + 1 \\right)} + 2 x e^{x} + 4 e^{x}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 x^{2} \\cos{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)}) = - 3 x^{2} \\left(2 x + 3\\right) \\sin{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 6 x \\cos{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)}$.\nTherefore the correct derivative is $f'(x) = - 3 x^{2} \\left(2 x + 3\\right) \\sin{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 6 x \\cos{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 2 \\left(x + 1\\right) e^{x} + 2 e^{x}$.\nFinal answer: $- 3 x^{2} \\left(2 x + 3\\right) \\sin{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 6 x \\cos{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 2 \\left(x + 1\\right) e^{x} + 2 e^{x}$", "final_answer": "- 3 x^{2} \\left(2 x + 3\\right) \\sin{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 6 x \\cos{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 2 \\left(x + 1\\right) e^{x} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f948bb49c64c9d64", "problem_id": "b86b4713e55e523f", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "A cyclist rides at 70 miles per hour for 2 hours, then at 49 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 70 × 2 = 140 miles.\nSecond leg distance = 49 × 3 = 147 miles.\nTotal distance = 140 + 147 = 287 miles.\nFinal answer: $287$", "final_answer": "287", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "34952b29ce6861da", "problem_id": "b86b4713e55e523f", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A cyclist rides at 70 miles per hour for 2 hours, then at 49 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 70 × 2 = 140 miles.\n2) Second leg distance = 49 × 3 = 147 miles.\n3) Total distance = 140 + 147 = 287 miles.\nFinal answer: $287$", "final_answer": "287", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6b704cca8f88aad6", "problem_id": "d555a4ca7959c29e", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A cyclist rides at 72 miles per hour for 2 hours, then at 31 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 72 × 2 = 144 miles.\n2) Second leg distance = 31 × 5/2 = 155/2 miles.\n3) Total distance = 144 + 155/2 = 443/2 miles.\nFinal answer: $\\frac{443}{2}$", "final_answer": "\\frac{443}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c98171af5b19253d", "problem_id": "d555a4ca7959c29e", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A cyclist rides at 72 miles per hour for 2 hours, then at 31 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 72 × 2 = 144 miles.\nSecond leg distance = 31 × 5/2 = 155/2 miles.\nTotal distance = 144 + 155/2 = 443/2 miles.\nCheck: adding both legs gives 443/2.\nFinal answer: $\\frac{443}{2}$", "final_answer": "\\frac{443}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b2811cabf577bb3", "problem_id": "399ece830e72a8be", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - 2 x^{3} - 5 x^{2} - 21 x - 83$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 2 x^{4} + 3 x^{3} - x^{2} + x + 3$ and a = 4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 8 x^{3} + 9 x^{2} - 2 x + 1$.\n5) Evaluate at a: p'(4) = $-375$.\nFinal answer: $-375$", "final_answer": "-375", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "df730d393f74ad98", "problem_id": "399ece830e72a8be", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - 2 x^{3} - 5 x^{2} - 21 x - 83$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 2 x^{4} + 3 x^{3} - x^{2} + x + 3$ and a = 4.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-375$", "final_answer": "-375", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0ac6021d1262634a", "problem_id": "4faed61ee731051e", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} + x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 9 x^{2} + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 1$.\n2) Slope at x=2: m = f'(2) = $-35 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\n3) Point on curve: (2, f(2)) = (2, $-21 + \\cos{\\left(4 \\right)} + e$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -21 + \\cos{\\left(4 \\right)} + e = -35 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\n6) Optionally expand: y = $- 35 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 49$.\nFinal answer: $y = - 35\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 49$", "final_answer": "y = - 35\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 49", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "565eab3ad8442b1b", "problem_id": "4faed61ee731051e", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} + x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=2$.", "output": "Differentiate: f'(x) = $- 9 x^{2} + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 1$.\nSlope at x=2: m = f'(2) = $-35 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\nPoint on curve: (2, f(2)) = (2, $-21 + \\cos{\\left(4 \\right)} + e$).\nFinal answer: $y = - 35\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 49$", "final_answer": "y = - 35\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 49", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "37b3ada6994106e2", "problem_id": "01397540f26e3b2c", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Worker A finishes a job in 7 hours and Worker B finishes it in 15 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/7 job/hour. B's rate = 1/15 job/hour.\n2) Combined rate = 1/7 + 1/15 = 22/105 job/hour.\n3) Time = 1 ÷ (combined rate) = 105/22 hours.\nFinal answer: $\\frac{105}{22}$", "final_answer": "\\frac{105}{22}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3fd57c497e0001f3", "problem_id": "01397540f26e3b2c", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Worker A finishes a job in 7 hours and Worker B finishes it in 15 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/7 job/hour. B's rate = 1/15 job/hour.\nCombined rate = 1/7 + 1/15 = 22/105 job/hour.\nTime = 1 ÷ (combined rate) = 105/22 hours.\nCheck: (combined rate)×(time) = 22/105×105/22 = 1.\nFinal answer: $\\frac{105}{22}$", "final_answer": "\\frac{105}{22}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "00977c599069c1bd", "problem_id": "0147fecc349262f3", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 18 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=18, k=3, p=$\\frac{4}{5}$ and simplify.\n4) Here C(18,3) = 816.\nFinal answer: $\\frac{52224}{3814697265625}$", "final_answer": "\\frac{52224}{3814697265625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f7828e7cf851d053", "problem_id": "0147fecc349262f3", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 18 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=18, k=3, p=$\\frac{4}{5}$ and simplify.\nHere C(18,3) = 816.\nCheck: expression simplifies to $\\frac{52224}{3814697265625}$.\nFinal answer: $\\frac{52224}{3814697265625}$", "final_answer": "\\frac{52224}{3814697265625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "945c437cc1a2d657", "problem_id": "8dfdaa94a2d1d93d", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Worker A finishes a job in 12 hours and Worker B finishes it in 14 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/12 job/hour. B's rate = 1/14 job/hour.\n2) Combined rate = 1/12 + 1/14 = 13/84 job/hour.\n3) Time = 1 ÷ (combined rate) = 84/13 hours.\nFinal answer: $\\frac{84}{13}$", "final_answer": "\\frac{84}{13}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dca344bea3c69b37", "problem_id": "8dfdaa94a2d1d93d", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Worker A finishes a job in 12 hours and Worker B finishes it in 14 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/12 job/hour. B's rate = 1/14 job/hour.\nCombined rate = 1/12 + 1/14 = 13/84 job/hour.\nTime = 1 ÷ (combined rate) = 84/13 hours.\nCheck: (combined rate)×(time) = 13/84×84/13 = 1.\nFinal answer: $\\frac{84}{13}$", "final_answer": "\\frac{84}{13}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d7e93a592da62ea2", "problem_id": "00ba6182f671895b", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- 8 x + 2 y$ = $-62$\n$x + 4 y$ = $-5$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $- 8 x + 2 y$ = $-62$.\nMultiply the second equation by $-8$: $- 8 x - 32 y$ = $40$.\nSubtract to eliminate x: $34 y$ = $-102$.\nSolve for y: y = $-3$.\nSubstitute back to find x: x = $7$.\nCheck: plug (x,y)=(7,-3) into both equations.\nFinal answer: $(x,y)=\\left(7, -3\\right)$", "final_answer": "(x,y)=\\left(7, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c27b1ee2f9488f39", "problem_id": "00ba6182f671895b", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$- 8 x + 2 y$ = $-62$\n$x + 4 y$ = $-5$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $- 8 x + 2 y$ = $-62$.\n3) Multiply the second equation by $-8$: $- 8 x - 32 y$ = $40$.\n4) Subtract to eliminate x: $34 y$ = $-102$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $7$.\nFinal answer: $(x,y)=\\left(7, -3\\right)$", "final_answer": "(x,y)=\\left(7, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e62a907cb97804c6", "problem_id": "fc6b40183423c61e", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{5 x}{2} - \\frac{\\sqrt{5} x}{3} + \\frac{\\sqrt{5}}{2} + \\frac{3}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{5}{2} - \\frac{\\sqrt{5}}{3}$, c=$\\frac{\\sqrt{5}}{2} + \\frac{3}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{29}{36} - \\frac{\\sqrt{5}}{3}$, so √Δ = $\\sqrt{\\frac{29}{36} - \\frac{\\sqrt{5}}{3}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{29 - 12 \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{6} + \\frac{5}{4}$.\n7) x2 = $- \\frac{\\sqrt{29 - 12 \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{6} + \\frac{5}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{29 - 12\\cdot \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{6} + \\frac{5}{4}, - \\frac{\\sqrt{29 - 12\\cdot \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{6} + \\frac{5}{4}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{29 - 12\\cdot \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{6} + \\frac{5}{4}, - \\frac{\\sqrt{29 - 12\\cdot \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{6} + \\frac{5}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fc91ad0cfa4d49ce", "problem_id": "fc6b40183423c61e", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{5 x}{2} - \\frac{\\sqrt{5} x}{3} + \\frac{\\sqrt{5}}{2} + \\frac{3}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{5}{2} - \\frac{\\sqrt{5}}{3}$, c=$\\frac{\\sqrt{5}}{2} + \\frac{3}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{29}{36} - \\frac{\\sqrt{5}}{3}$, so √Δ = $\\sqrt{\\frac{29}{36} - \\frac{\\sqrt{5}}{3}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{29 - 12 \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{6} + \\frac{5}{4}$.\nx2 = $- \\frac{\\sqrt{29 - 12 \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{6} + \\frac{5}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{29 - 12\\cdot \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{6} + \\frac{5}{4}, - \\frac{\\sqrt{29 - 12\\cdot \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{6} + \\frac{5}{4}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{29 - 12\\cdot \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{6} + \\frac{5}{4}, - \\frac{\\sqrt{29 - 12\\cdot \\sqrt{5}}}{12} + \\frac{\\sqrt{5}}{6} + \\frac{5}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ba539bbc75381ee8", "problem_id": "488f09ff375ad89f", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 2 \\sqrt{11} x + \\frac{27 x}{4} - \\frac{3 \\sqrt{11}}{2} + \\frac{9}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{27}{4} - 2 \\sqrt{11}$, c=$\\frac{9}{2} - \\frac{3 \\sqrt{11}}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{- \\frac{27}{8} + \\frac{\\sqrt{1145 - 336\\cdot \\sqrt{11}}}{8} + \\sqrt{11}, - \\frac{27}{8} - \\frac{\\sqrt{1145 - 336\\cdot \\sqrt{11}}}{8} + \\sqrt{11}\\}$", "final_answer": "x \\in \\{- \\frac{27}{8} + \\frac{\\sqrt{1145 - 336\\cdot \\sqrt{11}}}{8} + \\sqrt{11}, - \\frac{27}{8} - \\frac{\\sqrt{1145 - 336\\cdot \\sqrt{11}}}{8} + \\sqrt{11}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1b890b6a11fc05f0", "problem_id": "488f09ff375ad89f", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 2 \\sqrt{11} x + \\frac{27 x}{4} - \\frac{3 \\sqrt{11}}{2} + \\frac{9}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{27}{4} - 2 \\sqrt{11}$, c=$\\frac{9}{2} - \\frac{3 \\sqrt{11}}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{1145}{16} - 21 \\sqrt{11}$, so √Δ = $\\sqrt{\\frac{1145}{16} - 21 \\sqrt{11}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{27}{8} + \\frac{\\sqrt{1145 - 336 \\sqrt{11}}}{8} + \\sqrt{11}$.\n7) x2 = $- \\frac{27}{8} - \\frac{\\sqrt{1145 - 336 \\sqrt{11}}}{8} + \\sqrt{11}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{27}{8} + \\frac{\\sqrt{1145 - 336\\cdot \\sqrt{11}}}{8} + \\sqrt{11}, - \\frac{27}{8} - \\frac{\\sqrt{1145 - 336\\cdot \\sqrt{11}}}{8} + \\sqrt{11}\\}$", "final_answer": "x \\in \\{- \\frac{27}{8} + \\frac{\\sqrt{1145 - 336\\cdot \\sqrt{11}}}{8} + \\sqrt{11}, - \\frac{27}{8} - \\frac{\\sqrt{1145 - 336\\cdot \\sqrt{11}}}{8} + \\sqrt{11}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "989f96244041eb65", "problem_id": "da382adb8e63ac1a", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Two friends split $1358 in the ratio 5:9. How much does each receive?", "output": "1) Total ratio parts = 5+9 = 14.\n2) Each part = 1358 ÷ 14 = 97.\n3) First share = 5 × 97 = 485.\n4) Second share = 9 × 97 = 873.\nFinal answer: $873$", "final_answer": "873", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3460a5eb5a51335d", "problem_id": "da382adb8e63ac1a", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Two friends split $1358 in the ratio 5:9. How much does each receive?", "output": "Total ratio parts = 5+9 = 14.\nEach part = 1358 ÷ 14 = 97.\nFirst share = 5 × 97 = 485.\nFinal answer: $873$", "final_answer": "873", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a07e97b0ce75e2d", "problem_id": "92fe8011858adae9", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 3 x^{3} - 6 x^{2} + 21 x - 60$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{4} + 3 x^{3} + 3 x^{2} + 3 x + 2$ and a = -3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $12 x^{3} + 9 x^{2} + 6 x + 3$.\n5) Evaluate at a: p'(-3) = $-258$.\nFinal answer: $-258$", "final_answer": "-258", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ea849a2cdf0ded9", "problem_id": "92fe8011858adae9", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 3 x^{3} - 6 x^{2} + 21 x - 60$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $3 x^{4} + 3 x^{3} + 3 x^{2} + 3 x + 2$ and a = -3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-258$", "final_answer": "-258", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d757d0bc9aad7001", "problem_id": "ded2825e9d1f2dcf", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 2 x + 1$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $2 x^{2} + 3 x$ and a = -1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $4 x + 3$.\n5) Evaluate at a: p'(-1) = $-1$.\nFinal answer: $-1$", "final_answer": "-1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8db9688b9fcfba51", "problem_id": "ded2825e9d1f2dcf", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 2 x + 1$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{2} + 3 x$ and a = -1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $4 x + 3$.\nEvaluate at a: p'(-1) = $-1$.\nCheck: the identity gives $-1$.\nFinal answer: $-1$", "final_answer": "-1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d7e853078651d32c", "problem_id": "799f780909ba6a9e", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $700 in the ratio 7:7. How much does each receive?", "output": "1) Total ratio parts = 7+7 = 14.\n2) Each part = 700 ÷ 14 = 50.\n3) First share = 7 × 50 = 350.\n4) Second share = 7 × 50 = 350.\nFinal answer: $350$", "final_answer": "350", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec0b735eca0dc345", "problem_id": "799f780909ba6a9e", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Two friends split $700 in the ratio 7:7. How much does each receive?", "output": "Total ratio parts = 7+7 = 14.\nEach part = 700 ÷ 14 = 50.\nFirst share = 7 × 50 = 350.\nFinal answer: $350$", "final_answer": "350", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2ddb41b10cf4fa10", "problem_id": "3af26daa8fc506e7", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} 2 x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\pi).\n3) Subtract: F(\\pi) − F(- \\pi) = $\\frac{4 \\pi \\left(-3 + \\pi^{2}\\right)}{3}$.\n4) Simplify the result.\nFinal answer: $\\frac{4\\cdot \\pi\\cdot \\left(-3 + \\pi^{2}\\right)}{3}$", "final_answer": "\\frac{4\\cdot \\pi\\cdot \\left(-3 + \\pi^{2}\\right)}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f04b4b1a7a71e68f", "problem_id": "3af26daa8fc506e7", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} 2 x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $2 x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\nEvaluate at the bounds: F(\\pi) and F(- \\pi).\nSubtract: F(\\pi) − F(- \\pi) = $\\frac{4 \\pi \\left(-3 + \\pi^{2}\\right)}{3}$.\nFinal answer: $\\frac{4\\cdot \\pi\\cdot \\left(-3 + \\pi^{2}\\right)}{3}$", "final_answer": "\\frac{4\\cdot \\pi\\cdot \\left(-3 + \\pi^{2}\\right)}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cef59a5adc3907c7", "problem_id": "44c646db8cdf3fda", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡1 (mod 4), x≡3 (mod 13). Give the solution modulo 52.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 52.\n3) Compute the CRT solution to get x ≡ 29 (mod 52).\nFinal answer: $x\\equiv 29\\pmod{52}$", "final_answer": "x\\equiv 29\\pmod{52}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "81641cfde0ae191f", "problem_id": "44c646db8cdf3fda", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡1 (mod 4), x≡3 (mod 13). Give the solution modulo 52.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 52.\nCompute the CRT solution to get x ≡ 29 (mod 52).\nCheck: 29 mod 4=1 and mod 13=3.\nFinal answer: $x\\equiv 29\\pmod{52}$", "final_answer": "x\\equiv 29\\pmod{52}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c2310919de5d1dbe", "problem_id": "f2a9376c1c5de5ac", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "You want 14 liters of a 20% salt solution. You have a 15% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 14−x liters is 50% solution.\n2) Salt equation: 15x + 50(14−x) = 20·14.\n3) (15-50)x = (20-50)·14 ⇒ x = 12.\n4) Then 14−x = 2.\nFinal answer: $12 L of 15% and 2 L of 50%$", "final_answer": "12 L of 15% and 2 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "994ada5e4e071577", "problem_id": "f2a9376c1c5de5ac", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "You want 14 liters of a 20% salt solution. You have a 15% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 14−x liters is 50% solution.\nSalt equation: 15x + 50(14−x) = 20·14.\n(15-50)x = (20-50)·14 ⇒ x = 12.\nThen 14−x = 2.\nCheck: 15·12 + 50·2 = 20·14.\nFinal answer: $12 L of 15% and 2 L of 50%$", "final_answer": "12 L of 15% and 2 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a5e542e89c904063", "problem_id": "a2fb2cd1c9ccfc1e", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 14 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=14, k=8, p=$\\frac{1}{4}$ and simplify.\nHere C(14,8) = 3003.\nCheck: expression simplifies to $\\frac{2189187}{268435456}$.\nFinal answer: $\\frac{2189187}{268435456}$", "final_answer": "\\frac{2189187}{268435456}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "248f612aeb7cff0d", "problem_id": "a2fb2cd1c9ccfc1e", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 14 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=14, k=8, p=$\\frac{1}{4}$ and simplify.\nFinal answer: $\\frac{2189187}{268435456}$", "final_answer": "\\frac{2189187}{268435456}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a213704f06ea7ffc", "problem_id": "2b20b71e25c8ed33", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Two friends split $2772 in the ratio 9:9. How much does each receive?", "output": "Total ratio parts = 9+9 = 18.\nEach part = 2772 ÷ 18 = 154.\nFirst share = 9 × 154 = 1386.\nFinal answer: $1386$", "final_answer": "1386", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "338f1770bfd8ec60", "problem_id": "2b20b71e25c8ed33", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Two friends split $2772 in the ratio 9:9. How much does each receive?", "output": "Total ratio parts = 9+9 = 18.\nEach part = 2772 ÷ 18 = 154.\nFirst share = 9 × 154 = 1386.\nSecond share = 9 × 154 = 1386.\nCheck: 1386+1386=2772.\nFinal answer: $1386$", "final_answer": "1386", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "39670bad71454842", "problem_id": "e136974dc349f615", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "An urn has 11 red balls and 14 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(25,6).\n3) Favorable samples: choose k reds and n−k blues: C(11,1)·C(14,5).\n4) Compute totals: C(25,6) = 177100, and favorable = 22022.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{143}{1150}$", "final_answer": "\\frac{143}{1150}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "471c240033add77f", "problem_id": "a00373c18df48976", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 \\cos{\\left(2 x \\right)}) = - 3 \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - 3 \\sin{\\left(2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 \\cos{\\left(2 x \\right)}) = - 6 \\sin{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = 3 \\left(x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)}\\right)$.\nFinal answer: $3 \\left(x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)}\\right)$", "final_answer": "3 \\left(x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "dbcc8a96e9ea59ae", "problem_id": "9d168a1e7b8d4abf", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡7 (mod 8), x≡2 (mod 7). Give the solution modulo 56.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 56.\n3) Compute the CRT solution to get x ≡ 23 (mod 56).\nFinal answer: $x\\equiv 23\\pmod{56}$", "final_answer": "x\\equiv 23\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "41daa5bca2a2b445", "problem_id": "9d168a1e7b8d4abf", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡7 (mod 8), x≡2 (mod 7). Give the solution modulo 56.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 56.\nCompute the CRT solution to get x ≡ 23 (mod 56).\nCheck: 23 mod 8=7 and mod 7=2.\nFinal answer: $x\\equiv 23\\pmod{56}$", "final_answer": "x\\equiv 23\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7e26e2fdc30a77ae", "problem_id": "515993edd6160188", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A cyclist rides at 54 miles per hour for 1 hours, then at 74 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 54 × 1 = 54 miles.\n2) Second leg distance = 74 × 2 = 148 miles.\n3) Total distance = 54 + 148 = 202 miles.\nFinal answer: $202$", "final_answer": "202", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "de1b6c66f50bc8e6", "problem_id": "515993edd6160188", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A cyclist rides at 54 miles per hour for 1 hours, then at 74 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 54 × 1 = 54 miles.\nSecond leg distance = 74 × 2 = 148 miles.\nTotal distance = 54 + 148 = 202 miles.\nCheck: adding both legs gives 202.\nFinal answer: $202$", "final_answer": "202", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "71a2f4dc958f4773", "problem_id": "3e40764ad01a4af6", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$5 x - 8 y$ = $36$\n$7 x + 3 y$ = $-49$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $7$: $35 x - 56 y$ = $252$.\nMultiply the second equation by $5$: $35 x + 15 y$ = $-245$.\nSubtract to eliminate x: $- 71 y$ = $497$.\nSolve for y: y = $-7$.\nSubstitute back to find x: x = $-4$.\nCheck: plug (x,y)=(-4,-7) into both equations.\nFinal answer: $(x,y)=\\left(-4, -7\\right)$", "final_answer": "(x,y)=\\left(-4, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "04c53a1b1dbe2ad8", "problem_id": "3e40764ad01a4af6", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$5 x - 8 y$ = $36$\n$7 x + 3 y$ = $-49$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $7$: $35 x - 56 y$ = $252$.\n3) Multiply the second equation by $5$: $35 x + 15 y$ = $-245$.\n4) Subtract to eliminate x: $- 71 y$ = $497$.\n5) Solve for y: y = $-7$.\n6) Substitute back to find x: x = $-4$.\nFinal answer: $(x,y)=\\left(-4, -7\\right)$", "final_answer": "(x,y)=\\left(-4, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "054ba5851d650b1e", "problem_id": "394257105ba54bd4", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$x - 2 y$ = $5$\n$- 4 x - 4 y$ = $4$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $- 4 x + 8 y$ = $-20$.\nMultiply the second equation by $1$: $- 4 x - 4 y$ = $4$.\nSubtract to eliminate x: $12 y$ = $-24$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $1$.\nCheck: plug (x,y)=(1,-2) into both equations.\nFinal answer: $(x,y)=\\left(1, -2\\right)$", "final_answer": "(x,y)=\\left(1, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3693aa00dc1ebb80", "problem_id": "394257105ba54bd4", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$x - 2 y$ = $5$\n$- 4 x - 4 y$ = $4$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $- 4 x + 8 y$ = $-20$.\n3) Multiply the second equation by $1$: $- 4 x - 4 y$ = $4$.\n4) Subtract to eliminate x: $12 y$ = $-24$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $1$.\nFinal answer: $(x,y)=\\left(1, -2\\right)$", "final_answer": "(x,y)=\\left(1, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "60e5994d58d70656", "problem_id": "0ae59e1e003eed6b", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$- 4 x + 3 y$ = $-25$\n$x + 2 y$ = $-2$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $- 4 x + 3 y$ = $-25$.\nMultiply the second equation by $-4$: $- 4 x - 8 y$ = $8$.\nSubtract to eliminate x: $11 y$ = $-33$.\nSolve for y: y = $-3$.\nSubstitute back to find x: x = $4$.\nCheck: plug (x,y)=(4,-3) into both equations.\nFinal answer: $(x,y)=\\left(4, -3\\right)$", "final_answer": "(x,y)=\\left(4, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1259a08bf5bd8e05", "problem_id": "0ae59e1e003eed6b", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the linear system:\n$- 4 x + 3 y$ = $-25$\n$x + 2 y$ = $-2$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $- 4 x + 3 y$ = $-25$.\nMultiply the second equation by $-4$: $- 4 x - 8 y$ = $8$.\nFinal answer: $(x,y)=\\left(4, -3\\right)$", "final_answer": "(x,y)=\\left(4, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d9a40b37e08aac07", "problem_id": "5dae80afb45414ad", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$- 8 x + 8 y$ = $0$\n$2 x - 7 y$ = $20$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $- 16 x + 16 y$ = $0$.\n3) Multiply the second equation by $-8$: $- 16 x + 56 y$ = $-160$.\n4) Subtract to eliminate x: $- 40 y$ = $160$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $-4$.\nFinal answer: $(x,y)=\\left(-4, -4\\right)$", "final_answer": "(x,y)=\\left(-4, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ea6c4dcaf2c1afa5", "problem_id": "5dae80afb45414ad", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$- 8 x + 8 y$ = $0$\n$2 x - 7 y$ = $20$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $2$: $- 16 x + 16 y$ = $0$.\nMultiply the second equation by $-8$: $- 16 x + 56 y$ = $-160$.\nSubtract to eliminate x: $- 40 y$ = $160$.\nSolve for y: y = $-4$.\nSubstitute back to find x: x = $-4$.\nCheck: plug (x,y)=(-4,-4) into both equations.\nFinal answer: $(x,y)=\\left(-4, -4\\right)$", "final_answer": "(x,y)=\\left(-4, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3809032eb2cec32f", "problem_id": "aabec3341678fc57", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "You want 17 liters of a 25% salt solution. You have a 10% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 17−x liters is 60% solution.\n2) Salt equation: 10x + 60(17−x) = 25·17.\n3) (10-60)x = (25-60)·17 ⇒ x = 119/10.\n4) Then 17−x = 51/10.\nFinal answer: $119/10 L of 10% and 51/10 L of 60%$", "final_answer": "119/10 L of 10% and 51/10 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0da8d2283cc23cf8", "problem_id": "aabec3341678fc57", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "You want 17 liters of a 25% salt solution. You have a 10% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 17−x liters is 60% solution.\nSalt equation: 10x + 60(17−x) = 25·17.\n(10-60)x = (25-60)·17 ⇒ x = 119/10.\nThen 17−x = 51/10.\nCheck: 10·119/10 + 60·51/10 = 25·17.\nFinal answer: $119/10 L of 10% and 51/10 L of 60%$", "final_answer": "119/10 L of 10% and 51/10 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a32f8f2311020f2d", "problem_id": "c91479b18687894b", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $- \\frac{9 \\pi^{2}}{2} - 3 \\pi - 4 + 9 \\pi^{3}$.\n4) Simplify the result.\nFinal answer: $- \\frac{9\\cdot \\pi^{2}}{2} - 3\\cdot \\pi - 4 + 9\\cdot \\pi^{3}$", "final_answer": "- \\frac{9\\cdot \\pi^{2}}{2} - 3\\cdot \\pi - 4 + 9\\cdot \\pi^{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9b4c355747fa6028", "problem_id": "c91479b18687894b", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $- \\frac{9 \\pi^{2}}{2} - 3 \\pi - 4 + 9 \\pi^{3}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{9\\cdot \\pi^{2}}{2} - 3\\cdot \\pi - 4 + 9\\cdot \\pi^{3}$", "final_answer": "- \\frac{9\\cdot \\pi^{2}}{2} - 3\\cdot \\pi - 4 + 9\\cdot \\pi^{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ba2bec5dff53895d", "problem_id": "efb0d7abe57b49aa", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$2 x + 2 y$ = $-16$\n$- 4 x + 4 y$ = $-32$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $- 8 x - 8 y$ = $64$.\n3) Multiply the second equation by $2$: $- 8 x + 8 y$ = $-64$.\n4) Subtract to eliminate x: $- 16 y$ = $128$.\n5) Solve for y: y = $-8$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, -8\\right)$", "final_answer": "(x,y)=\\left(0, -8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1d504a393ef4a695", "problem_id": "efb0d7abe57b49aa", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$2 x + 2 y$ = $-16$\n$- 4 x + 4 y$ = $-32$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $- 8 x - 8 y$ = $64$.\nMultiply the second equation by $2$: $- 8 x + 8 y$ = $-64$.\nSubtract to eliminate x: $- 16 y$ = $128$.\nSolve for y: y = $-8$.\nSubstitute back to find x: x = $0$.\nCheck: plug (x,y)=(0,-8) into both equations.\nFinal answer: $(x,y)=\\left(0, -8\\right)$", "final_answer": "(x,y)=\\left(0, -8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d8520d23527a202d", "problem_id": "20757c0dc97578f8", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Worker A finishes a job in 7 hours and Worker B finishes it in 10 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/7 job/hour. B's rate = 1/10 job/hour.\nCombined rate = 1/7 + 1/10 = 17/70 job/hour.\nTime = 1 ÷ (combined rate) = 70/17 hours.\nFinal answer: $\\frac{70}{17}$", "final_answer": "\\frac{70}{17}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "69b5f33f2431516d", "problem_id": "20757c0dc97578f8", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Worker A finishes a job in 7 hours and Worker B finishes it in 10 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/7 job/hour. B's rate = 1/10 job/hour.\n2) Combined rate = 1/7 + 1/10 = 17/70 job/hour.\n3) Time = 1 ÷ (combined rate) = 70/17 hours.\nFinal answer: $\\frac{70}{17}$", "final_answer": "\\frac{70}{17}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1b6de21988380e55", "problem_id": "38d4e789b34078b9", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "An urn has 17 red balls and 23 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(40,12).\nFavorable samples: choose k reds and n−k blues: C(17,3)·C(23,9).\nFinal answer: $\\frac{43010}{432419}$", "final_answer": "\\frac{43010}{432419}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e04bb0a3c5545248", "problem_id": "38d4e789b34078b9", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "An urn has 17 red balls and 23 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(40,12).\n3) Favorable samples: choose k reds and n−k blues: C(17,3)·C(23,9).\n4) Compute totals: C(40,12) = 5586853480, and favorable = 555689200.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{43010}{432419}$", "final_answer": "\\frac{43010}{432419}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "750ed99094ef9777", "problem_id": "5c85f1ba3de55dc9", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$- 3 x - y$ = $7$\n$- 4 y$ = $16$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $-3$: $12 y$ = $-48$.\n4) Subtract to eliminate x: $- 12 y$ = $48$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, -4\\right)$", "final_answer": "(x,y)=\\left(-1, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a7fc3e019050644", "problem_id": "5c85f1ba3de55dc9", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$- 3 x - y$ = $7$\n$- 4 y$ = $16$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $-3$: $12 y$ = $-48$.\nSubtract to eliminate x: $- 12 y$ = $48$.\nSolve for y: y = $-4$.\nSubstitute back to find x: x = $-1$.\nCheck: plug (x,y)=(-1,-4) into both equations.\nFinal answer: $(x,y)=\\left(-1, -4\\right)$", "final_answer": "(x,y)=\\left(-1, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a67f919beacb6de2", "problem_id": "fd0999c8fea131c2", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 21 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=21, k=8, p=$\\frac{1}{5}$ and simplify.\nHere C(21,8) = 203490.\nCheck: expression simplifies to $\\frac{2731196547072}{95367431640625}$.\nFinal answer: $\\frac{2731196547072}{95367431640625}$", "final_answer": "\\frac{2731196547072}{95367431640625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4070b4495884bfe3", "problem_id": "fd0999c8fea131c2", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 21 times, what is the probability of exactly 8 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=21, k=8, p=$\\frac{1}{5}$ and simplify.\n4) Here C(21,8) = 203490.\nFinal answer: $\\frac{2731196547072}{95367431640625}$", "final_answer": "\\frac{2731196547072}{95367431640625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e9d2a73e84f72ac1", "problem_id": "787a419beec2efdf", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "An urn has 13 red balls and 21 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(34,9).\nFavorable samples: choose k reds and n−k blues: C(13,2)·C(21,7).\nFinal answer: $\\frac{1710}{9889}$", "final_answer": "\\frac{1710}{9889}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ad203ebd9f6158ca", "problem_id": "787a419beec2efdf", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 13 red balls and 21 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(34,9).\n3) Favorable samples: choose k reds and n−k blues: C(13,2)·C(21,7).\n4) Compute totals: C(34,9) = 52451256, and favorable = 9069840.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1710}{9889}$", "final_answer": "\\frac{1710}{9889}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "40f5483e1fafb9a0", "problem_id": "dfe5c5fabf42efe9", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 22 times, what is the probability of exactly 12 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=22, k=12, p=$\\frac{1}{5}$ and simplify.\n4) Here C(22,12) = 646646.\nFinal answer: $\\frac{678057476096}{2384185791015625}$", "final_answer": "\\frac{678057476096}{2384185791015625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e93fcfd30bf9d607", "problem_id": "dfe5c5fabf42efe9", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 22 times, what is the probability of exactly 12 heads?", "output": "$\\frac{678057476096}{2384185791015625}$", "final_answer": "\\frac{678057476096}{2384185791015625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "06a166272de5684b", "problem_id": "8c86134f5fef39f9", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡6 (mod 8), x≡6 (mod 11). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 6 (mod 88).\nFinal answer: $x\\equiv 6\\pmod{88}$", "final_answer": "x\\equiv 6\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b2ad95bbe92b2d8", "problem_id": "8c86134f5fef39f9", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the congruences: x≡6 (mod 8), x≡6 (mod 11). Give the solution modulo 88.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 88.\nCompute the CRT solution to get x ≡ 6 (mod 88).\nFinal answer: $x\\equiv 6\\pmod{88}$", "final_answer": "x\\equiv 6\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e458c19c3e11f4ed", "problem_id": "c59d3924de0772dd", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{3 x^{3} + 2 x^{2} + 2 x + \\left(x^{2} + 8\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{x^{2} + 8}$.", "output": "1) Let f(x) = $\\frac{3 x^{3} + 2 x^{2} + 2 x + \\left(x^{2} + 8\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{x^{2} + 8}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{- 4 x \\left(3 x^{3} + 2 x^{2} + 2 x + \\left(x^{2} + 8\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(x^{2} + 8\\right) \\left(18 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 x + \\left(x^{2} + 8\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 8\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x^{2} + 8\\right)^{2}}$.\nFinal answer: $\\frac{- 4\\cdot x\\cdot \\left(3\\cdot x^{3} + 2\\cdot x^{2} + 2\\cdot x + \\left(x^{2} + 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right) + \\left(x^{2} + 8\\right)\\cdot \\left(18\\cdot x^{2} + 4\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 8\\cdot x + \\left(x^{2} + 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 16\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\right)}{2\\cdot \\left(x^{2} + 8\\right)^{2}}$", "final_answer": "\\frac{- 4\\cdot x\\cdot \\left(3\\cdot x^{3} + 2\\cdot x^{2} + 2\\cdot x + \\left(x^{2} + 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right) + \\left(x^{2} + 8\\right)\\cdot \\left(18\\cdot x^{2} + 4\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 8\\cdot x + \\left(x^{2} + 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 16\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\right)}{2\\cdot \\left(x^{2} + 8\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d63fb74dadb2cbfd", "problem_id": "c59d3924de0772dd", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{3 x^{3} + 2 x^{2} + 2 x + \\left(x^{2} + 8\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{x^{2} + 8}$.", "output": "Let f(x) = $\\frac{3 x^{3} + 2 x^{2} + 2 x + \\left(x^{2} + 8\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{x^{2} + 8}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{- 4 x \\left(3 x^{3} + 2 x^{2} + 2 x + \\left(x^{2} + 8\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(x^{2} + 8\\right) \\left(18 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 x + \\left(x^{2} + 8\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 8\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x^{2} + 8\\right)^{2}}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\frac{- 4\\cdot x\\cdot \\left(3\\cdot x^{3} + 2\\cdot x^{2} + 2\\cdot x + \\left(x^{2} + 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right) + \\left(x^{2} + 8\\right)\\cdot \\left(18\\cdot x^{2} + 4\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 8\\cdot x + \\left(x^{2} + 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 16\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\right)}{2\\cdot \\left(x^{2} + 8\\right)^{2}}$", "final_answer": "\\frac{- 4\\cdot x\\cdot \\left(3\\cdot x^{3} + 2\\cdot x^{2} + 2\\cdot x + \\left(x^{2} + 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right) + \\left(x^{2} + 8\\right)\\cdot \\left(18\\cdot x^{2} + 4\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 8\\cdot x + \\left(x^{2} + 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 16\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\right)}{2\\cdot \\left(x^{2} + 8\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8aeee96c73ec5ab6", "problem_id": "80e582c094d1cd51", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$7 x - 8 y$ = $5$\n$- 5 y$ = $-10$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $7$: $- 35 y$ = $-70$.\n4) Subtract to eliminate x: $35 y$ = $70$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, 2\\right)$", "final_answer": "(x,y)=\\left(3, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3763acf108b7bcac", "problem_id": "80e582c094d1cd51", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$7 x - 8 y$ = $5$\n$- 5 y$ = $-10$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $7$: $- 35 y$ = $-70$.\nSubtract to eliminate x: $35 y$ = $70$.\nSolve for y: y = $2$.\nSubstitute back to find x: x = $3$.\nCheck: plug (x,y)=(3,2) into both equations.\nFinal answer: $(x,y)=\\left(3, 2\\right)$", "final_answer": "(x,y)=\\left(3, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a321a11b7c88322", "problem_id": "f62860f4189393f6", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 24 x + 48$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} + 24 x + 48$ = $3 \\left(x + 4\\right)^{2}$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-4$.\n5) From the second factor: x = $-4$.\nFinal answer: $x \\in \\{-4, -4\\}$", "final_answer": "x \\in \\{-4, -4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fb920ab1bb6a02f2", "problem_id": "f62860f4189393f6", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 24 x + 48$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} + 24 x + 48$ = $3 \\left(x + 4\\right)^{2}$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $-4$.\nFrom the second factor: x = $-4$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-4, -4\\}$", "final_answer": "x \\in \\{-4, -4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cd021f9f3e33c143", "problem_id": "9bc043feca913cbc", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A cyclist rides at 78 miles per hour for 3/2 hours, then at 89 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 78 × 3/2 = 117 miles.\n2) Second leg distance = 89 × 3/2 = 267/2 miles.\n3) Total distance = 117 + 267/2 = 501/2 miles.\nFinal answer: $\\frac{501}{2}$", "final_answer": "\\frac{501}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4fc725c466f55c26", "problem_id": "9bc043feca913cbc", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A cyclist rides at 78 miles per hour for 3/2 hours, then at 89 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 78 × 3/2 = 117 miles.\nSecond leg distance = 89 × 3/2 = 267/2 miles.\nTotal distance = 117 + 267/2 = 501/2 miles.\nCheck: adding both legs gives 501/2.\nFinal answer: $\\frac{501}{2}$", "final_answer": "\\frac{501}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "235ee470efa7e967", "problem_id": "2468a627b9dafa86", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Two friends split $975 in the ratio 8:7. How much does each receive?", "output": "Total ratio parts = 8+7 = 15.\nEach part = 975 ÷ 15 = 65.\nFirst share = 8 × 65 = 520.\nSecond share = 7 × 65 = 455.\nCheck: 520+455=975.\nFinal answer: $455$", "final_answer": "455", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4fc38aab8fbe3db9", "problem_id": "2468a627b9dafa86", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Two friends split $975 in the ratio 8:7. How much does each receive?", "output": "1) Total ratio parts = 8+7 = 15.\n2) Each part = 975 ÷ 15 = 65.\n3) First share = 8 × 65 = 520.\n4) Second share = 7 × 65 = 455.\nFinal answer: $455$", "final_answer": "455", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "40a308230f1b6db1", "problem_id": "362c2b411a0c5985", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=13$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=13 and k=5.\n4) Compute C(17,4) = 2380.\nFinal answer: $2380$", "final_answer": "2380", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d2cc8bf5f6159073", "problem_id": "5f1d52bcce33dad5", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\cos{\\left(2 x \\right)}) = - \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\cos{\\left(2 x \\right)}) = - 2 \\sin{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$", "final_answer": "3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "1bcf79010a9646ee", "problem_id": "93b58f8747b5969b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x^{3} + 5 x^{2} + 11 x + 25$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to x^{3} + 5 x^{2} + 10 x + 25.\n3) Then the limit is 73.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 75.\nFinal answer: $75$", "final_answer": "75", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "930187e7e19a9dde", "problem_id": "a3ca291d8d90c8a6", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Two friends split $333 in the ratio 2:7. How much does each receive?", "output": "1) Total ratio parts = 2+7 = 9.\n2) Each part = 333 ÷ 9 = 37.\n3) First share = 2 × 37 = 74.\n4) Second share = 7 × 37 = 259.\nFinal answer: $259$", "final_answer": "259", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e04f5fc33c0958ff", "problem_id": "7ef9cce63cd3044b", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A cyclist rides at 20 miles per hour for 1 hours, then at 64 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 20 × 1 = 20 miles.\n2) Second leg distance = 64 × 3 = 192 miles.\n3) Total distance = 20 + 192 = 212 miles.\nFinal answer: $212$", "final_answer": "212", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0afa5cd49c8da620", "problem_id": "f66393517368d768", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x^{2} - 2 x - 2$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{2} - 3 x - 2.\n3) Then the limit is -16.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -14.\nFinal answer: $-14$", "final_answer": "-14", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a97bfd773d98673b", "problem_id": "8ca33e415b9c53cf", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} + 3 x + 4$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x^{2}) = - 4 x^{2}$.\n3) So $f'(x) = 3 - 4 x^{2}$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x^{2}) = - 4 x$.\nTherefore the correct derivative is $f'(x) = 3 - 4 x$.\nFinal answer: $3 - 4 x$", "final_answer": "3 - 4 x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2be0fc8b960a5ec7", "problem_id": "5a6292abe694e73a", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$5 x - 3 y$ = $-22$\n$6 x - 6 y$ = $-36$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $6$: $30 x - 18 y$ = $-132$.\n3) Multiply the second equation by $5$: $30 x - 30 y$ = $-180$.\n4) Subtract to eliminate x: $12 y$ = $48$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, 4\\right)$", "final_answer": "(x,y)=\\left(-2, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "515728cde7a7b684", "problem_id": "432484413fd0c026", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 29 red balls and 21 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(50,8).\n3) Favorable samples: choose k reds and n−k blues: C(29,4)·C(21,4).\n4) Compute totals: C(50,8) = 536878650, and favorable = 142149735.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1353807}{5113130}$", "final_answer": "\\frac{1353807}{5113130}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "972ae0abbd720d66", "problem_id": "432484413fd0c026", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 29 red balls and 21 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(50,8).\nFavorable samples: choose k reds and n−k blues: C(29,4)·C(21,4).\nCompute totals: C(50,8) = 536878650, and favorable = 142149735.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1353807}{5113130}$.\nFinal answer: $\\frac{1353807}{5113130}$", "final_answer": "\\frac{1353807}{5113130}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "15cadb21bebbe5fc", "problem_id": "72dafc4636b0a9a9", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 2 x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $- 6 \\pi - 4 + 6 \\pi^{3}$.\n4) Simplify the result.\nFinal answer: $- 6\\cdot \\pi - 4 + 6\\cdot \\pi^{3}$", "final_answer": "- 6\\cdot \\pi - 4 + 6\\cdot \\pi^{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1749500a4acbefdd", "problem_id": "72dafc4636b0a9a9", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 2 x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $2 x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $- 6 \\pi - 4 + 6 \\pi^{3}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- 6\\cdot \\pi - 4 + 6\\cdot \\pi^{3}$", "final_answer": "- 6\\cdot \\pi - 4 + 6\\cdot \\pi^{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "486d4970feada99c", "problem_id": "5e72bb6d00b751f9", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the congruences: x≡2 (mod 4), x≡9 (mod 11). Give the solution modulo 44.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 44.\n3) Compute the CRT solution to get x ≡ 42 (mod 44).\nFinal answer: $x\\equiv 42\\pmod{44}$", "final_answer": "x\\equiv 42\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "15e20cd8cf1b3924", "problem_id": "5e72bb6d00b751f9", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Solve the congruences: x≡2 (mod 4), x≡9 (mod 11). Give the solution modulo 44.", "output": "$x\\equiv 42\\pmod{44}$", "final_answer": "x\\equiv 42\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a214d11bc908d215", "problem_id": "e14f503a2567a186", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{5} x + 4 x - 2 \\sqrt{5} + 4$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$4 - \\sqrt{5}$, c=$4 - 2 \\sqrt{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $5$, so √Δ = $\\sqrt{5}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $-2 + \\sqrt{5}$.\nx2 = $-2$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-2 + \\sqrt{5}, -2\\}$", "final_answer": "x \\in \\{-2 + \\sqrt{5}, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7f7ac2a6414d1fc6", "problem_id": "e87817263420430d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(x^{3} + 2 x^{2} + 2 x + 3\\right) \\cos{\\left(3 x - 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 \\cos{\\left(3 x - 2 \\right)}) = - 3 \\sin{\\left(3 x - 2 \\right)}$.\n3) So $f'(x) = - 3 x^{3} \\sin{\\left(3 x - 2 \\right)} - 6 x^{2} \\sin{\\left(3 x - 2 \\right)} + 3 x^{2} \\cos{\\left(3 x - 2 \\right)} + x e^{x} - 6 x \\sin{\\left(3 x - 2 \\right)} + 4 x \\cos{\\left(3 x - 2 \\right)} + 2 e^{x} - 3 \\sin{\\left(3 x - 2 \\right)} + 2 \\cos{\\left(3 x - 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 \\cos{\\left(3 x - 2 \\right)}) = - 9 \\sin{\\left(3 x - 2 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 1\\right) e^{x} - 3 \\left(x \\left(x \\left(x + 2\\right) + 2\\right) + 3\\right) \\sin{\\left(3 x - 2 \\right)} + \\left(3 x^{2} + 4 x + 2\\right) \\cos{\\left(3 x - 2 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 1\\right) e^{x} - 3 \\left(x \\left(x \\left(x + 2\\right) + 2\\right) + 3\\right) \\sin{\\left(3 x - 2 \\right)} + \\left(3 x^{2} + 4 x + 2\\right) \\cos{\\left(3 x - 2 \\right)} + e^{x}$", "final_answer": "\\left(x + 1\\right) e^{x} - 3 \\left(x \\left(x \\left(x + 2\\right) + 2\\right) + 3\\right) \\sin{\\left(3 x - 2 \\right)} + \\left(3 x^{2} + 4 x + 2\\right) \\cos{\\left(3 x - 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e75fea01d6f84167", "problem_id": "2aff2540673d084a", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 2 \\sqrt{7} x + x - \\sqrt{7}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$1 - 2 \\sqrt{7}$, c=$- \\sqrt{7}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $4 \\sqrt{7} + 29$, so √Δ = $\\sqrt{4 \\sqrt{7} + 29}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{1}{4} + \\frac{\\sqrt{7}}{2} + \\frac{\\sqrt{4 \\sqrt{7} + 29}}{4}$.\n7) x2 = $- \\frac{\\sqrt{4 \\sqrt{7} + 29}}{4} - \\frac{1}{4} + \\frac{\\sqrt{7}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{1}{4} + \\frac{\\sqrt{7}}{2} + \\frac{\\sqrt{4\\cdot \\sqrt{7} + 29}}{4}, - \\frac{\\sqrt{4\\cdot \\sqrt{7} + 29}}{4} - \\frac{1}{4} + \\frac{\\sqrt{7}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{1}{4} + \\frac{\\sqrt{7}}{2} + \\frac{\\sqrt{4\\cdot \\sqrt{7} + 29}}{4}, - \\frac{\\sqrt{4\\cdot \\sqrt{7} + 29}}{4} - \\frac{1}{4} + \\frac{\\sqrt{7}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c9b4b73fab1ec71a", "problem_id": "2aff2540673d084a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 2 \\sqrt{7} x + x - \\sqrt{7}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$1 - 2 \\sqrt{7}$, c=$- \\sqrt{7}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $4 \\sqrt{7} + 29$, so √Δ = $\\sqrt{4 \\sqrt{7} + 29}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{1}{4} + \\frac{\\sqrt{7}}{2} + \\frac{\\sqrt{4 \\sqrt{7} + 29}}{4}$.\nx2 = $- \\frac{\\sqrt{4 \\sqrt{7} + 29}}{4} - \\frac{1}{4} + \\frac{\\sqrt{7}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{1}{4} + \\frac{\\sqrt{7}}{2} + \\frac{\\sqrt{4\\cdot \\sqrt{7} + 29}}{4}, - \\frac{\\sqrt{4\\cdot \\sqrt{7} + 29}}{4} - \\frac{1}{4} + \\frac{\\sqrt{7}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{1}{4} + \\frac{\\sqrt{7}}{2} + \\frac{\\sqrt{4\\cdot \\sqrt{7} + 29}}{4}, - \\frac{\\sqrt{4\\cdot \\sqrt{7} + 29}}{4} - \\frac{1}{4} + \\frac{\\sqrt{7}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "deade287eca4ea27", "problem_id": "f22196f4a5bd5d97", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡3 (mod 9), x≡3 (mod 5). Give the solution modulo 45.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 45.\nCompute the CRT solution to get x ≡ 3 (mod 45).\nCheck: 3 mod 9=3 and mod 5=3.\nFinal answer: $x\\equiv 3\\pmod{45}$", "final_answer": "x\\equiv 3\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c43ee4422319fc62", "problem_id": "f22196f4a5bd5d97", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡3 (mod 9), x≡3 (mod 5). Give the solution modulo 45.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 45.\n3) Compute the CRT solution to get x ≡ 3 (mod 45).\nFinal answer: $x\\equiv 3\\pmod{45}$", "final_answer": "x\\equiv 3\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e8e882fd33fd2c25", "problem_id": "687294b07219eb0c", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A cyclist rides at 48 miles per hour for 1/2 hours, then at 43 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 48 × 1/2 = 24 miles.\n2) Second leg distance = 43 × 2 = 86 miles.\n3) Total distance = 24 + 86 = 110 miles.\nFinal answer: $110$", "final_answer": "110", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4f6c1f1aba1ac0cd", "problem_id": "687294b07219eb0c", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A cyclist rides at 48 miles per hour for 1/2 hours, then at 43 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 48 × 1/2 = 24 miles.\nSecond leg distance = 43 × 2 = 86 miles.\nTotal distance = 24 + 86 = 110 miles.\nCheck: adding both legs gives 110.\nFinal answer: $110$", "final_answer": "110", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6e5cad920e1fae70", "problem_id": "42c221a94af15f5a", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$- x + 8 y$ = $-54$\n$12 x + 8 y$ = $24$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $12$: $- 12 x + 96 y$ = $-648$.\n3) Multiply the second equation by $-1$: $- 12 x - 8 y$ = $-24$.\n4) Subtract to eliminate x: $104 y$ = $-624$.\n5) Solve for y: y = $-6$.\n6) Substitute back to find x: x = $6$.\nFinal answer: $(x,y)=\\left(6, -6\\right)$", "final_answer": "(x,y)=\\left(6, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e8ffa2c73319de8e", "problem_id": "cbfc2742c4c3ebdf", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- \\frac{x^{2}}{2} + x + \\left(x^{2} - x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\frac{1}{2}}{x^{2} - x + 1}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 1}) = - \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 1}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} - \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} - \\frac{4 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} + \\frac{3 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} + \\frac{6 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} - \\frac{x^{2}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} - \\frac{2 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 1} - \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} - \\frac{4 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} - \\frac{4 x}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} + \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 1} + \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 2 x + 2} - \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 1} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 1} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} + \\frac{2 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} + \\frac{3}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 1}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 1}) = - \\frac{x \\left(1 - 2 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(x^{2} - x\\right) + 1\\right)^{2}} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(x^{2} - x\\right) + 1\\right)} - \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 1} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 1}$.\nTherefore the correct derivative is $f'(x) = \\frac{\\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + 2 \\left(x \\left(x - 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(x \\left(x - 1\\right) + 1\\right) \\left(- 2 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{2 \\left(x \\left(x - 1\\right) + 1\\right)^{2}}$.\nFinal answer: $\\frac{\\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + 2 \\left(x \\left(x - 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(x \\left(x - 1\\right) + 1\\right) \\left(- 2 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{2 \\left(x \\left(x - 1\\right) + 1\\right)^{2}}$", "final_answer": "\\frac{\\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + 2 \\left(x \\left(x - 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(x \\left(x - 1\\right) + 1\\right) \\left(- 2 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{2 \\left(x \\left(x - 1\\right) + 1\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "578cd7094630c65d", "problem_id": "5cb99a50377f6d75", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$2 x + y$ = $11$\n$- 2 x + 4 y$ = $-46$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $- 4 x - 2 y$ = $-22$.\n3) Multiply the second equation by $2$: $- 4 x + 8 y$ = $-92$.\n4) Subtract to eliminate x: $- 10 y$ = $70$.\n5) Solve for y: y = $-7$.\n6) Substitute back to find x: x = $9$.\nFinal answer: $(x,y)=\\left(9, -7\\right)$", "final_answer": "(x,y)=\\left(9, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e08e234c1eea63fc", "problem_id": "5cb99a50377f6d75", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$2 x + y$ = $11$\n$- 2 x + 4 y$ = $-46$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $- 4 x - 2 y$ = $-22$.\nMultiply the second equation by $2$: $- 4 x + 8 y$ = $-92$.\nSubtract to eliminate x: $- 10 y$ = $70$.\nSolve for y: y = $-7$.\nSubstitute back to find x: x = $9$.\nCheck: plug (x,y)=(9,-7) into both equations.\nFinal answer: $(x,y)=\\left(9, -7\\right)$", "final_answer": "(x,y)=\\left(9, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f5bd431ff1319889", "problem_id": "7ce6d91b08125a8b", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 29 red balls and 25 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(54,8).\n3) Favorable samples: choose k reds and n−k blues: C(29,1)·C(25,7).\n4) Compute totals: C(54,8) = 1040465790, and favorable = 13940300.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1394030}{104046579}$", "final_answer": "\\frac{1394030}{104046579}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3928304b5267e568", "problem_id": "7ce6d91b08125a8b", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 29 red balls and 25 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(54,8).\nFavorable samples: choose k reds and n−k blues: C(29,1)·C(25,7).\nCompute totals: C(54,8) = 1040465790, and favorable = 13940300.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1394030}{104046579}$.\nFinal answer: $\\frac{1394030}{104046579}$", "final_answer": "\\frac{1394030}{104046579}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "699d99659aa5d3e4", "problem_id": "2911bbbbe3dea9c6", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} - 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\nEvaluate at the bounds: F(2 \\pi) and F(0).\nSubtract: F(2 \\pi) − F(0) = $2 \\pi \\left(- 4 \\pi^{2} + 2 + 3 \\pi\\right)$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $2\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} + 2 + 3\\cdot \\pi\\right)$", "final_answer": "2\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} + 2 + 3\\cdot \\pi\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f494446df04f487", "problem_id": "2911bbbbe3dea9c6", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} - 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(0).\n3) Subtract: F(2 \\pi) − F(0) = $2 \\pi \\left(- 4 \\pi^{2} + 2 + 3 \\pi\\right)$.\n4) Simplify the result.\nFinal answer: $2\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} + 2 + 3\\cdot \\pi\\right)$", "final_answer": "2\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} + 2 + 3\\cdot \\pi\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d2d4f96ff8c90e2c", "problem_id": "506fc5cf371de3f0", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 12 red balls and 23 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(35,8).\n3) Favorable samples: choose k reds and n−k blues: C(12,1)·C(23,7).\n4) Compute totals: C(35,8) = 23535820, and favorable = 2941884.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{3933}{31465}$", "final_answer": "\\frac{3933}{31465}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "68ddb490b2a34d2a", "problem_id": "506fc5cf371de3f0", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 12 red balls and 23 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(35,8).\nFavorable samples: choose k reds and n−k blues: C(12,1)·C(23,7).\nCompute totals: C(35,8) = 23535820, and favorable = 2941884.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{3933}{31465}$.\nFinal answer: $\\frac{3933}{31465}$", "final_answer": "\\frac{3933}{31465}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "43d9b2d3fc3ecf4e", "problem_id": "9bcb8922b8f78eba", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡7 (mod 9), x≡1 (mod 10). Give the solution modulo 90.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 90.\n3) Compute the CRT solution to get x ≡ 61 (mod 90).\nFinal answer: $x\\equiv 61\\pmod{90}$", "final_answer": "x\\equiv 61\\pmod{90}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7c8ddb31dcfaf7ae", "problem_id": "9bcb8922b8f78eba", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the congruences: x≡7 (mod 9), x≡1 (mod 10). Give the solution modulo 90.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 90.\nCompute the CRT solution to get x ≡ 61 (mod 90).\nFinal answer: $x\\equiv 61\\pmod{90}$", "final_answer": "x\\equiv 61\\pmod{90}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a19c9f738e8aba44", "problem_id": "cb14cde2e23a3e89", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the linear system:\n$- 7 x + 6 y$ = $80$\n$9 x + 4 y$ = $-56$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $9$: $- 63 x + 54 y$ = $720$.\nMultiply the second equation by $-7$: $- 63 x - 28 y$ = $392$.\nFinal answer: $(x,y)=\\left(-8, 4\\right)$", "final_answer": "(x,y)=\\left(-8, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1c1c092e9f138375", "problem_id": "cb14cde2e23a3e89", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$- 7 x + 6 y$ = $80$\n$9 x + 4 y$ = $-56$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $9$: $- 63 x + 54 y$ = $720$.\nMultiply the second equation by $-7$: $- 63 x - 28 y$ = $392$.\nSubtract to eliminate x: $82 y$ = $328$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $-8$.\nCheck: plug (x,y)=(-8,4) into both equations.\nFinal answer: $(x,y)=\\left(-8, 4\\right)$", "final_answer": "(x,y)=\\left(-8, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7f8360f2c063c351", "problem_id": "4484ccd46d0e1a64", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_7=31$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_7=24.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=24 and k=7.\n4) Compute C(30,6) = 593775.\nFinal answer: $593775$", "final_answer": "593775", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a48394fc8e8596f6", "problem_id": "4484ccd46d0e1a64", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_7=31$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_7=24.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=24 and k=7.\nCompute C(30,6) = 593775.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $593775$", "final_answer": "593775", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5146fff7eaec1ab8", "problem_id": "80f33aefb9285bde", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} 2 x^{3} - x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{3} - x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1$.\n2) Evaluate at the bounds: F(\\pi) and F(0).\n3) Subtract: F(\\pi) − F(0) = $- \\frac{\\pi^{3}}{3} + \\pi + 4 + \\frac{3 \\pi^{2}}{2} + \\frac{\\pi^{4}}{2}$.\n4) Simplify the result.\nFinal answer: $- \\frac{\\pi^{3}}{3} + \\pi + 4 + \\frac{3\\cdot \\pi^{2}}{2} + \\frac{\\pi^{4}}{2}$", "final_answer": "- \\frac{\\pi^{3}}{3} + \\pi + 4 + \\frac{3\\cdot \\pi^{2}}{2} + \\frac{\\pi^{4}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4297c18c0a6bbce5", "problem_id": "7f0d9dc95dbad082", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(- x^{2} - 3\\right) \\cos{\\left(- 3 x^{2} + 3 x + 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 6 e^{x}) = - 6 e^{x}$.\n3) So $f'(x) = - 6 x^{3} \\sin{\\left(- 3 x^{2} + 3 x + 2 \\right)} + 3 x^{2} \\sin{\\left(- 3 x^{2} + 3 x + 2 \\right)} + 2 x e^{x} - 18 x \\sin{\\left(- 3 x^{2} + 3 x + 2 \\right)} - 2 x \\cos{\\left(\\left(- 3 x^{2} + 3 x\\right) + 2 \\right)} - 4 e^{x} + 9 \\sin{\\left(- 3 x^{2} + 3 x + 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 6 e^{x}) = - 6 e^{x}$.\nTherefore the correct derivative is $f'(x) = - 2 x \\cos{\\left(\\left(- 3 x^{2} + 3 x\\right) + 2 \\right)} - 3 \\left(1 - 2 x\\right) \\left(- x^{2} - 3\\right) \\sin{\\left(- 3 x^{2} + 3 x + 2 \\right)} + 2 \\left(x - 3\\right) e^{x} + 2 e^{x}$.\nFinal answer: $- 2 x \\cos{\\left(\\left(- 3 x^{2} + 3 x\\right) + 2 \\right)} - 3 \\left(1 - 2 x\\right) \\left(- x^{2} - 3\\right) \\sin{\\left(- 3 x^{2} + 3 x + 2 \\right)} + 2 \\left(x - 3\\right) e^{x} + 2 e^{x}$", "final_answer": "- 2 x \\cos{\\left(\\left(- 3 x^{2} + 3 x\\right) + 2 \\right)} - 3 \\left(1 - 2 x\\right) \\left(- x^{2} - 3\\right) \\sin{\\left(- 3 x^{2} + 3 x + 2 \\right)} + 2 \\left(x - 3\\right) e^{x} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "cbcef925b03fc385", "problem_id": "5fd42154963faae4", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$3 x - 2 y$ = $-1$\n$x - y$ = $0$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $3 x - 2 y$ = $-1$.\nMultiply the second equation by $3$: $3 x - 3 y$ = $0$.\nSubtract to eliminate x: $y$ = $-1$.\nSolve for y: y = $-1$.\nSubstitute back to find x: x = $-1$.\nCheck: plug (x,y)=(-1,-1) into both equations.\nFinal answer: $(x,y)=\\left(-1, -1\\right)$", "final_answer": "(x,y)=\\left(-1, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a2cfc8b193ace77d", "problem_id": "5fd42154963faae4", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$3 x - 2 y$ = $-1$\n$x - y$ = $0$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $3 x - 2 y$ = $-1$.\n3) Multiply the second equation by $3$: $3 x - 3 y$ = $0$.\n4) Subtract to eliminate x: $y$ = $-1$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, -1\\right)$", "final_answer": "(x,y)=\\left(-1, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c070a04e418c5fd5", "problem_id": "e9d62f6b0797fcba", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "You want 32 liters of a 25% salt solution. You have a 15% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 32−x liters is 60% solution.\n2) Salt equation: 15x + 60(32−x) = 25·32.\n3) (15-60)x = (25-60)·32 ⇒ x = 224/9.\n4) Then 32−x = 64/9.\nFinal answer: $224/9 L of 15% and 64/9 L of 60%$", "final_answer": "224/9 L of 15% and 64/9 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d4bebb70534f3302", "problem_id": "e9d62f6b0797fcba", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "You want 32 liters of a 25% salt solution. You have a 15% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 32−x liters is 60% solution.\nSalt equation: 15x + 60(32−x) = 25·32.\n(15-60)x = (25-60)·32 ⇒ x = 224/9.\nThen 32−x = 64/9.\nCheck: 15·224/9 + 60·64/9 = 25·32.\nFinal answer: $224/9 L of 15% and 64/9 L of 60%$", "final_answer": "224/9 L of 15% and 64/9 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "91c69eb7cbca80ec", "problem_id": "36a5bffe0953f49e", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 6 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=6, k=2, p=$\\frac{1}{2}$ and simplify.\n4) Here C(6,2) = 15.\nFinal answer: $\\frac{15}{64}$", "final_answer": "\\frac{15}{64}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "737c7fe7adafd3c1", "problem_id": "36a5bffe0953f49e", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 6 times, what is the probability of exactly 2 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=6, k=2, p=$\\frac{1}{2}$ and simplify.\nHere C(6,2) = 15.\nCheck: expression simplifies to $\\frac{15}{64}$.\nFinal answer: $\\frac{15}{64}$", "final_answer": "\\frac{15}{64}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e3345ca7e6298db3", "problem_id": "aa8c80e4666e757c", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "An urn has 13 red balls and 19 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(32,8).\nFavorable samples: choose k reds and n−k blues: C(13,3)·C(19,5).\nFinal answer: $\\frac{7106}{22475}$", "final_answer": "\\frac{7106}{22475}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c0a0cb9f3023fb44", "problem_id": "aa8c80e4666e757c", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 13 red balls and 19 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(32,8).\n3) Favorable samples: choose k reds and n−k blues: C(13,3)·C(19,5).\n4) Compute totals: C(32,8) = 10518300, and favorable = 3325608.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{7106}{22475}$", "final_answer": "\\frac{7106}{22475}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "21eeed23d1cc5f1d", "problem_id": "f24ea93e51395c07", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} + 3 x$.", "output": "1) Differentiate term-by-term:\n2) d/dx(- 2 x^{2}) = - 4 x.\n3) d/dx(3 x) = 3.\n4) Combine: f'(x) = 3 - 4 x.\nFinal answer: $3 - 4\\cdot x$", "final_answer": "3 - 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7a4b72838c6593e2", "problem_id": "f24ea93e51395c07", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} + 3 x$.", "output": "Differentiate term-by-term:\nd/dx(- 2 x^{2}) = - 4 x.\nd/dx(3 x) = 3.\nCombine: f'(x) = 3 - 4 x.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $3 - 4\\cdot x$", "final_answer": "3 - 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b7714700639aaee4", "problem_id": "711f4b36b3c6da1e", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$7 x - 3 y$ = $-2$\n$- 6 x + 3 y$ = $0$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-6$: $- 42 x + 18 y$ = $12$.\n3) Multiply the second equation by $7$: $- 42 x + 21 y$ = $0$.\n4) Subtract to eliminate x: $- 3 y$ = $12$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, -4\\right)$", "final_answer": "(x,y)=\\left(-2, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ae859ff9392941ec", "problem_id": "711f4b36b3c6da1e", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the linear system:\n$7 x - 3 y$ = $-2$\n$- 6 x + 3 y$ = $0$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-6$: $- 42 x + 18 y$ = $12$.\nMultiply the second equation by $7$: $- 42 x + 21 y$ = $0$.\nFinal answer: $(x,y)=\\left(-2, -4\\right)$", "final_answer": "(x,y)=\\left(-2, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f21b506c44fca0e3", "problem_id": "8b03f3e762f03730", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $2 + \\frac{9 \\pi^{2}}{8} + \\frac{3 \\pi^{3}}{8}$.\n4) Simplify the result.\nFinal answer: $2 + \\frac{9\\cdot \\pi^{2}}{8} + \\frac{3\\cdot \\pi^{3}}{8}$", "final_answer": "2 + \\frac{9\\cdot \\pi^{2}}{8} + \\frac{3\\cdot \\pi^{3}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "981dcc14aaaf65fa", "problem_id": "8b03f3e762f03730", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\nEvaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $2 + \\frac{9 \\pi^{2}}{8} + \\frac{3 \\pi^{3}}{8}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $2 + \\frac{9\\cdot \\pi^{2}}{8} + \\frac{3\\cdot \\pi^{3}}{8}$", "final_answer": "2 + \\frac{9\\cdot \\pi^{2}}{8} + \\frac{3\\cdot \\pi^{3}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "089c01d5007ceaf5", "problem_id": "64c2db1961f7665e", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Two friends split $340 in the ratio 2:8. How much does each receive?", "output": "1) Total ratio parts = 2+8 = 10.\n2) Each part = 340 ÷ 10 = 34.\n3) First share = 2 × 34 = 68.\n4) Second share = 8 × 34 = 272.\nFinal answer: $272$", "final_answer": "272", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f644180e382b86dd", "problem_id": "64c2db1961f7665e", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Two friends split $340 in the ratio 2:8. How much does each receive?", "output": "Total ratio parts = 2+8 = 10.\nEach part = 340 ÷ 10 = 34.\nFirst share = 2 × 34 = 68.\nSecond share = 8 × 34 = 272.\nCheck: 68+272=340.\nFinal answer: $272$", "final_answer": "272", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ebe8e487110a3625", "problem_id": "ae5ebb9642461a0e", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} 3 x^{3} - 2 x - 2$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{4} - 3 x^{3} - 2 x^{2} - 2$ and a = 1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $12 x^{3} - 9 x^{2} - 4 x$.\n5) Evaluate at a: p'(1) = $-1$.\nFinal answer: $-1$", "final_answer": "-1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ebe3eab11ad69fb1", "problem_id": "ae5ebb9642461a0e", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} 3 x^{3} - 2 x - 2$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $3 x^{4} - 3 x^{3} - 2 x^{2} - 2$ and a = 1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-1$", "final_answer": "-1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a9210225483853bc", "problem_id": "4c2aea424516537d", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡0 (mod 4), x≡0 (mod 9). Give the solution modulo 36.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 36.\n3) Compute the CRT solution to get x ≡ 0 (mod 36).\nFinal answer: $x\\equiv 0\\pmod{36}$", "final_answer": "x\\equiv 0\\pmod{36}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ccaeeef3d71d1af8", "problem_id": "4c2aea424516537d", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡0 (mod 4), x≡0 (mod 9). Give the solution modulo 36.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 36.\nCompute the CRT solution to get x ≡ 0 (mod 36).\nCheck: 0 mod 4=0 and mod 9=0.\nFinal answer: $x\\equiv 0\\pmod{36}$", "final_answer": "x\\equiv 0\\pmod{36}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6d731528b6029e75", "problem_id": "f50329376ce2f04f", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 2 x^{3} - 4 x^{2} + 9 x - 16$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} - 4 x^{2} + 8 x - 16.\n3) Then the limit is -64.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -66.\nFinal answer: $-66$", "final_answer": "-66", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a8a1eda56bc5054c", "problem_id": "4dab2c74293157bd", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A cyclist rides at 78 miles per hour for 1/2 hours, then at 32 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 78 × 1/2 = 39 miles.\n2) Second leg distance = 32 × 3 = 96 miles.\n3) Total distance = 39 + 96 = 135 miles.\nFinal answer: $135$", "final_answer": "135", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0996a79b288ce489", "problem_id": "92f3e3646a2b6ef2", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 21 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=21, k=5, p=$\\frac{4}{5}$ and simplify.\n4) Here C(21,5) = 20349.\nFinal answer: $\\frac{20837376}{476837158203125}$", "final_answer": "\\frac{20837376}{476837158203125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db6c99c2108fca1a", "problem_id": "92f3e3646a2b6ef2", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 21 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=21, k=5, p=$\\frac{4}{5}$ and simplify.\nHere C(21,5) = 20349.\nCheck: expression simplifies to $\\frac{20837376}{476837158203125}$.\nFinal answer: $\\frac{20837376}{476837158203125}$", "final_answer": "\\frac{20837376}{476837158203125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ef84745c1264a521", "problem_id": "413d95a20cf6115a", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "You want 14 liters of a 25% salt solution. You have a 10% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 14−x liters is 70% solution.\n2) Salt equation: 10x + 70(14−x) = 25·14.\n3) (10-70)x = (25-70)·14 ⇒ x = 21/2.\n4) Then 14−x = 7/2.\nFinal answer: $21/2 L of 10% and 7/2 L of 70%$", "final_answer": "21/2 L of 10% and 7/2 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "35883a186d0a8ef3", "problem_id": "413d95a20cf6115a", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "You want 14 liters of a 25% salt solution. You have a 10% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 14−x liters is 70% solution.\nSalt equation: 10x + 70(14−x) = 25·14.\n(10-70)x = (25-70)·14 ⇒ x = 21/2.\nFinal answer: $21/2 L of 10% and 7/2 L of 70%$", "final_answer": "21/2 L of 10% and 7/2 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72138a9dd05b6e1c", "problem_id": "399ece830e72a8be", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - 2 x^{3} - 5 x^{2} - 21 x - 83$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{3} - 5 x^{2} - 22 x - 83.\n3) Then the limit is -379.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: -375.\nFinal answer: $-375$", "final_answer": "-375", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "05de5bb0dd6251a8", "problem_id": "7e7f574a971adbf6", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{7 x}{2} + \\frac{5}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$- \\frac{7}{2}$, c=$\\frac{5}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{9}{4}$, so √Δ = $\\frac{3}{2}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{5}{4}$.\n7) x2 = $\\frac{1}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{5}{4}, \\frac{1}{2}\\}$", "final_answer": "x \\in \\{\\frac{5}{4}, \\frac{1}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6b43759b1b0cb62c", "problem_id": "0b6ab783ce0ab756", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$3 x - 3 y$ = $-9$\n$- 4 x + y$ = $9$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $- 12 x + 12 y$ = $36$.\nMultiply the second equation by $3$: $- 12 x + 3 y$ = $27$.\nSubtract to eliminate x: $9 y$ = $9$.\nSolve for y: y = $1$.\nSubstitute back to find x: x = $-2$.\nCheck: plug (x,y)=(-2,1) into both equations.\nFinal answer: $(x,y)=\\left(-2, 1\\right)$", "final_answer": "(x,y)=\\left(-2, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "25d65fbc7eb0c99c", "problem_id": "0b6ab783ce0ab756", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$3 x - 3 y$ = $-9$\n$- 4 x + y$ = $9$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $- 12 x + 12 y$ = $36$.\n3) Multiply the second equation by $3$: $- 12 x + 3 y$ = $27$.\n4) Subtract to eliminate x: $9 y$ = $9$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, 1\\right)$", "final_answer": "(x,y)=\\left(-2, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1d78b538a5c72cfe", "problem_id": "4da1a9fed0c66e75", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Worker A finishes a job in 9 hours and Worker B finishes it in 17 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/9 job/hour. B's rate = 1/17 job/hour.\n2) Combined rate = 1/9 + 1/17 = 26/153 job/hour.\n3) Time = 1 ÷ (combined rate) = 153/26 hours.\nFinal answer: $\\frac{153}{26}$", "final_answer": "\\frac{153}{26}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b62ba06df8c75776", "problem_id": "4da1a9fed0c66e75", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Worker A finishes a job in 9 hours and Worker B finishes it in 17 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/9 job/hour. B's rate = 1/17 job/hour.\nCombined rate = 1/9 + 1/17 = 26/153 job/hour.\nTime = 1 ÷ (combined rate) = 153/26 hours.\nFinal answer: $\\frac{153}{26}$", "final_answer": "\\frac{153}{26}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8320d64655441913", "problem_id": "e13bf91b28d92e88", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "You want 50 liters of a 40% salt solution. You have a 25% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 50−x liters is 60% solution.\n2) Salt equation: 25x + 60(50−x) = 40·50.\n3) (25-60)x = (40-60)·50 ⇒ x = 200/7.\n4) Then 50−x = 150/7.\nFinal answer: $200/7 L of 25% and 150/7 L of 60%$", "final_answer": "200/7 L of 25% and 150/7 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ca1777088cb1c543", "problem_id": "e13bf91b28d92e88", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "You want 50 liters of a 40% salt solution. You have a 25% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 50−x liters is 60% solution.\nSalt equation: 25x + 60(50−x) = 40·50.\n(25-60)x = (40-60)·50 ⇒ x = 200/7.\nThen 50−x = 150/7.\nCheck: 25·200/7 + 60·150/7 = 40·50.\nFinal answer: $200/7 L of 25% and 150/7 L of 60%$", "final_answer": "200/7 L of 25% and 150/7 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2657feeafe0c861a", "problem_id": "592f17c7cce1a436", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} x^{3} + 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $x^{3} + 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{9 \\pi}{2} + 2 + \\frac{9 \\pi^{2}}{8} + \\frac{15 \\pi^{4}}{64} + \\frac{9 \\pi^{3}}{8}$.\n4) Simplify the result.\nFinal answer: $- \\frac{9\\cdot \\pi}{2} + 2 + \\frac{9\\cdot \\pi^{2}}{8} + \\frac{15\\cdot \\pi^{4}}{64} + \\frac{9\\cdot \\pi^{3}}{8}$", "final_answer": "- \\frac{9\\cdot \\pi}{2} + 2 + \\frac{9\\cdot \\pi^{2}}{8} + \\frac{15\\cdot \\pi^{4}}{64} + \\frac{9\\cdot \\pi^{3}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "18f3aaa251455d57", "problem_id": "1973eb501f26a2f2", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} + 2 x + \\left(x^{2} - x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{x^{2} - x + 7}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{7 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7}) = \\frac{7 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 7}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{4 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{15 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{30 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{2 x^{2}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{14 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{28 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{40 x}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{14 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 15 x^{2} - 14 x + 49} + \\frac{49 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{98 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{34}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{7 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 15 x^{2} - 14 x + 49} - \\frac{7 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 2 x + 14} + \\frac{7 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 7} - \\frac{7 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{7 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7}) = \\frac{7 \\left(1 - 2 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(x^{2} - x\\right) + 7\\right)^{2}} + \\frac{7 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(x^{2} - x\\right) + 7\\right)} + \\frac{7 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(x - 1\\right) + 7\\right) \\left(- 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x - 1\\right) + 7\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(x - 1\\right) + 7\\right) \\left(- 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x - 1\\right) + 7\\right)^{2}}$", "final_answer": "\\frac{2 \\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(x - 1\\right) + 7\\right) \\left(- 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x - 1\\right) + 7\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "fc0fda47740bc019", "problem_id": "e83dffe1871f2b96", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 23 red balls and 17 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(40,8).\n3) Favorable samples: choose k reds and n−k blues: C(23,2)·C(17,6).\n4) Compute totals: C(40,8) = 76904685, and favorable = 3131128.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1288}{31635}$", "final_answer": "\\frac{1288}{31635}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a3993d3377830b35", "problem_id": "e83dffe1871f2b96", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "An urn has 23 red balls and 17 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(40,8).\nFavorable samples: choose k reds and n−k blues: C(23,2)·C(17,6).\nFinal answer: $\\frac{1288}{31635}$", "final_answer": "\\frac{1288}{31635}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9e6edfce8b0d2135", "problem_id": "3a529c19c1bcb49b", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A store sells a backpack for $855. It is discounted by 25%. You buy 9 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 855 × (1 − 25/100) = 855 × 75/100 = 2565/4.\n2) Subtotal = 9 × 2565/4 = 23085/4.\n3) Total with tax = 23085/4 × 110/100 = 50787/8.\nFinal answer: $\\frac{50787}{8}$", "final_answer": "\\frac{50787}{8}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "54ce4c4135112666", "problem_id": "4cf4c136e5c984ce", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x \\sin{\\left(x \\right)}) = 2 x \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x \\sin{\\left(x \\right)}) = 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.\nFinal answer: $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$", "final_answer": "- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "15fea4e4b9214eba", "problem_id": "532e583a50559641", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the equation of the tangent line to $y=- x^{2} - 4 x - 4$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 2 x - 4$.\n2) Slope at x=2: m = f'(2) = $-8$.\n3) Point on curve: (2, f(2)) = (2, $-16$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -16 = -8(x − 2).\n6) Optionally expand: y = $- 8 x$.\nFinal answer: $y = - 8\\cdot x$", "final_answer": "y = - 8\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dd2f5a068926d1d0", "problem_id": "532e583a50559641", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the equation of the tangent line to $y=- x^{2} - 4 x - 4$ at $x=2$.", "output": "Differentiate: f'(x) = $- 2 x - 4$.\nSlope at x=2: m = f'(2) = $-8$.\nPoint on curve: (2, f(2)) = (2, $-16$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -16 = -8(x − 2).\nOptionally expand: y = $- 8 x$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 8\\cdot x$", "final_answer": "y = - 8\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "212dfc17694d7d5a", "problem_id": "d4e24da9067a8024", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "You want 25 liters of a 30% salt solution. You have a 20% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 25−x liters is 40% solution.\n2) Salt equation: 20x + 40(25−x) = 30·25.\n3) (20-40)x = (30-40)·25 ⇒ x = 25/2.\n4) Then 25−x = 25/2.\nFinal answer: $25/2 L of 20% and 25/2 L of 40%$", "final_answer": "25/2 L of 20% and 25/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2e7efcb0a5285394", "problem_id": "d4e24da9067a8024", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 25 liters of a 30% salt solution. You have a 20% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 25−x liters is 40% solution.\nSalt equation: 20x + 40(25−x) = 30·25.\n(20-40)x = (30-40)·25 ⇒ x = 25/2.\nThen 25−x = 25/2.\nCheck: 20·25/2 + 40·25/2 = 30·25.\nFinal answer: $25/2 L of 20% and 25/2 L of 40%$", "final_answer": "25/2 L of 20% and 25/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "455d8fbf57a4437e", "problem_id": "d7ef12a947ab6dc5", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the multiplicative inverse of 6 modulo 43.", "output": "1) We want x such that 6x ≡ 1 (mod 43).\n2) Use extended Euclid to find s,t with 6s + 43t = 1.\n3) One Bezout identity is: 6·(-7) + 43·(1) = 1.\n4) Reduce s modulo 43: x ≡ 36 (mod 43).\nFinal answer: $x\\equiv 36\\pmod{43}$", "final_answer": "x\\equiv 36\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f28a07b785982a85", "problem_id": "d7ef12a947ab6dc5", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Find the multiplicative inverse of 6 modulo 43.", "output": "We want x such that 6x ≡ 1 (mod 43).\nUse extended Euclid to find s,t with 6s + 43t = 1.\nOne Bezout identity is: 6·(-7) + 43·(1) = 1.\nFinal answer: $x\\equiv 36\\pmod{43}$", "final_answer": "x\\equiv 36\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "60cfafb17a341f89", "problem_id": "63bed2671c6cdb57", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x \\left(- x^{2} - x + 1\\right)$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - x^{3} - x^{2}.\n3) Then the limit is -12.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -10.\nFinal answer: $-10$", "final_answer": "-10", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "59a377dae800610c", "problem_id": "8d85321ac9a95c8f", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_6=35$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_6=17.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=17 and k=6.\n4) Compute C(22,5) = 26334.\nFinal answer: $26334$", "final_answer": "26334", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e2f4fd6e53dd3a0e", "problem_id": "72a1763b38dfa521", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(- x^{3} - x^{2}\\right) \\cos{\\left(x^{2} + 2 x - 2 \\right)}$.", "output": "Let f(x) = $\\left(x - 3\\right) e^{x} + \\left(- x^{3} - x^{2}\\right) \\cos{\\left(x^{2} + 2 x - 2 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- 2 x^{2} \\left(- x - 1\\right) \\left(x + 1\\right) \\sin{\\left(x^{2} + 2 x - 2 \\right)} + x \\left(- 3 x - 2\\right) \\cos{\\left(x^{2} + 2 x - 2 \\right)} + \\left(x - 3\\right) e^{x} + e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 2\\cdot x^{2}\\cdot \\left(- x - 1\\right)\\cdot \\left(x + 1\\right)\\cdot \\sin{\\left(x^{2} + 2\\cdot x - 2 \\right)} + x\\cdot \\left(- 3\\cdot x - 2\\right)\\cdot \\cos{\\left(x^{2} + 2\\cdot x - 2 \\right)} + \\left(x - 3\\right)\\cdot e^{x} + e^{x}$", "final_answer": "- 2\\cdot x^{2}\\cdot \\left(- x - 1\\right)\\cdot \\left(x + 1\\right)\\cdot \\sin{\\left(x^{2} + 2\\cdot x - 2 \\right)} + x\\cdot \\left(- 3\\cdot x - 2\\right)\\cdot \\cos{\\left(x^{2} + 2\\cdot x - 2 \\right)} + \\left(x - 3\\right)\\cdot e^{x} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6c14732efa81f959", "problem_id": "72a1763b38dfa521", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(- x^{3} - x^{2}\\right) \\cos{\\left(x^{2} + 2 x - 2 \\right)}$.", "output": "1) Let f(x) = $\\left(x - 3\\right) e^{x} + \\left(- x^{3} - x^{2}\\right) \\cos{\\left(x^{2} + 2 x - 2 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- 2 x^{2} \\left(- x - 1\\right) \\left(x + 1\\right) \\sin{\\left(x^{2} + 2 x - 2 \\right)} + x \\left(- 3 x - 2\\right) \\cos{\\left(x^{2} + 2 x - 2 \\right)} + \\left(x - 3\\right) e^{x} + e^{x}$.\nFinal answer: $- 2\\cdot x^{2}\\cdot \\left(- x - 1\\right)\\cdot \\left(x + 1\\right)\\cdot \\sin{\\left(x^{2} + 2\\cdot x - 2 \\right)} + x\\cdot \\left(- 3\\cdot x - 2\\right)\\cdot \\cos{\\left(x^{2} + 2\\cdot x - 2 \\right)} + \\left(x - 3\\right)\\cdot e^{x} + e^{x}$", "final_answer": "- 2\\cdot x^{2}\\cdot \\left(- x - 1\\right)\\cdot \\left(x + 1\\right)\\cdot \\sin{\\left(x^{2} + 2\\cdot x - 2 \\right)} + x\\cdot \\left(- 3\\cdot x - 2\\right)\\cdot \\cos{\\left(x^{2} + 2\\cdot x - 2 \\right)} + \\left(x - 3\\right)\\cdot e^{x} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6412e780c2421a99", "problem_id": "94919c0ae6a6515b", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Worker A finishes a job in 4 hours and Worker B finishes it in 16 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/4 job/hour. B's rate = 1/16 job/hour.\n2) Combined rate = 1/4 + 1/16 = 5/16 job/hour.\n3) Time = 1 ÷ (combined rate) = 16/5 hours.\nFinal answer: $\\frac{16}{5}$", "final_answer": "\\frac{16}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2e261477ef22ca80", "problem_id": "94919c0ae6a6515b", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Worker A finishes a job in 4 hours and Worker B finishes it in 16 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/4 job/hour. B's rate = 1/16 job/hour.\nCombined rate = 1/4 + 1/16 = 5/16 job/hour.\nTime = 1 ÷ (combined rate) = 16/5 hours.\nCheck: (combined rate)×(time) = 5/16×16/5 = 1.\nFinal answer: $\\frac{16}{5}$", "final_answer": "\\frac{16}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "09489857f2827f4b", "problem_id": "5bbfa5bfc6953dc3", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{7}$. If it is flipped 11 times, what is the probability of exactly 8 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=8, p=$\\frac{1}{7}$ and simplify.\n4) Here C(11,8) = 165.\nFinal answer: $\\frac{35640}{1977326743}$", "final_answer": "\\frac{35640}{1977326743}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e3da46c87b016422", "problem_id": "5bbfa5bfc6953dc3", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{7}$. If it is flipped 11 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=11, k=8, p=$\\frac{1}{7}$ and simplify.\nHere C(11,8) = 165.\nCheck: expression simplifies to $\\frac{35640}{1977326743}$.\nFinal answer: $\\frac{35640}{1977326743}$", "final_answer": "\\frac{35640}{1977326743}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "73e0d21a8f359e06", "problem_id": "00d9f80be60b1a29", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$- 4 x + 7 y$ = $43$\n$- 5 x - 5 y$ = $-70$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-5$: $20 x - 35 y$ = $-215$.\n3) Multiply the second equation by $-4$: $20 x + 20 y$ = $280$.\n4) Subtract to eliminate x: $- 55 y$ = $-495$.\n5) Solve for y: y = $9$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, 9\\right)$", "final_answer": "(x,y)=\\left(5, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b4b4febd28f8cdc5", "problem_id": "00d9f80be60b1a29", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- 4 x + 7 y$ = $43$\n$- 5 x - 5 y$ = $-70$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-5$: $20 x - 35 y$ = $-215$.\nMultiply the second equation by $-4$: $20 x + 20 y$ = $280$.\nSubtract to eliminate x: $- 55 y$ = $-495$.\nSolve for y: y = $9$.\nSubstitute back to find x: x = $5$.\nCheck: plug (x,y)=(5,9) into both equations.\nFinal answer: $(x,y)=\\left(5, 9\\right)$", "final_answer": "(x,y)=\\left(5, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "75f7ab44be9583ce", "problem_id": "954dc9b1336df2a1", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(2 x^{2} - 1\\right) \\cos{\\left(3 x^{2} - 3 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\n3) So $f'(x) = - 12 x^{3} \\sin{\\left(3 x^{2} - 3 x \\right)} + 6 x^{2} \\sin{\\left(3 x^{2} - 3 x \\right)} + x e^{x} + 6 x \\sin{\\left(3 x^{2} - 3 x \\right)} + 4 x \\cos{\\left(3 x^{2} - 3 x \\right)} + 3 e^{x} - 3 \\sin{\\left(3 x^{2} - 3 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = 4 x \\cos{\\left(3 x^{2} - 3 x \\right)} + \\left(x + 2\\right) e^{x} - 3 \\left(2 x - 1\\right) \\left(2 x^{2} - 1\\right) \\sin{\\left(3 x^{2} - 3 x \\right)} + e^{x}$.\nFinal answer: $4 x \\cos{\\left(3 x^{2} - 3 x \\right)} + \\left(x + 2\\right) e^{x} - 3 \\left(2 x - 1\\right) \\left(2 x^{2} - 1\\right) \\sin{\\left(3 x^{2} - 3 x \\right)} + e^{x}$", "final_answer": "4 x \\cos{\\left(3 x^{2} - 3 x \\right)} + \\left(x + 2\\right) e^{x} - 3 \\left(2 x - 1\\right) \\left(2 x^{2} - 1\\right) \\sin{\\left(3 x^{2} - 3 x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a1ffef194061bb39", "problem_id": "0136816a6f3adaed", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A store sells a backpack for $142. It is discounted by 5%. You buy 8 backpacks. What is the total cost?", "output": "1) Discounted price = 142 × (1 − 5/100) = 142 × 95/100 = 1349/10.\n2) Subtotal = 8 × 1349/10 = 5396/5.\nFinal answer: $\\frac{5396}{5}$", "final_answer": "\\frac{5396}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1ffc98e603ba97b5", "problem_id": "cb6b6d0eef6fb94f", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡2 (mod 3), x≡4 (mod 7). Give the solution modulo 21.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 21.\n3) Compute the CRT solution to get x ≡ 11 (mod 21).\nFinal answer: $x\\equiv 11\\pmod{21}$", "final_answer": "x\\equiv 11\\pmod{21}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0b919b2a0dfeb677", "problem_id": "cb6b6d0eef6fb94f", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 3), x≡4 (mod 7). Give the solution modulo 21.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 21.\nCompute the CRT solution to get x ≡ 11 (mod 21).\nCheck: 11 mod 3=2 and mod 7=4.\nFinal answer: $x\\equiv 11\\pmod{21}$", "final_answer": "x\\equiv 11\\pmod{21}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d4545cf464451d40", "problem_id": "166aa29a60f1b9d0", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 12 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=4, p=$\\frac{3}{8}$ and simplify.\n4) Here C(12,4) = 495.\nFinal answer: $\\frac{15662109375}{68719476736}$", "final_answer": "\\frac{15662109375}{68719476736}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "724335e8c600a835", "problem_id": "166aa29a60f1b9d0", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 12 times, what is the probability of exactly 4 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=12, k=4, p=$\\frac{3}{8}$ and simplify.\nHere C(12,4) = 495.\nCheck: expression simplifies to $\\frac{15662109375}{68719476736}$.\nFinal answer: $\\frac{15662109375}{68719476736}$", "final_answer": "\\frac{15662109375}{68719476736}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a6b9400d2efb37cc", "problem_id": "0b666892336485a9", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A disease has prevalence $\\frac{3}{100}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{49}{400}$.\nCompute P(D|+) = $\\frac{51}{245}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{51}{245}$", "final_answer": "\\frac{51}{245}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e21ac0ea201d255c", "problem_id": "0b666892336485a9", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A disease has prevalence $\\frac{3}{100}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{49}{400}$.\n3) Compute P(D|+) = $\\frac{51}{245}$.\nFinal answer: $\\frac{51}{245}$", "final_answer": "\\frac{51}{245}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "908b5378c0604986", "problem_id": "7ac046ff91d962ef", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡5 (mod 8), x≡5 (mod 7). Give the solution modulo 56.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 56.\n3) Compute the CRT solution to get x ≡ 5 (mod 56).\nFinal answer: $x\\equiv 5\\pmod{56}$", "final_answer": "x\\equiv 5\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "71096ac8a4bafcf6", "problem_id": "8b1a348fa710b7f2", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} - 2 x + \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{2 x - 5}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 5}) = - \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 5}$.\n3) So $f'(x) = \\frac{4 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{2} - 40 x + 50} + \\frac{8 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{2} - 40 x + 50} - \\frac{4 x^{2}}{8 x^{2} - 40 x + 50} - \\frac{20 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{2} - 40 x + 50} - \\frac{40 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{2} - 40 x + 50} + \\frac{20 x}{8 x^{2} - 40 x + 50} + \\frac{25 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{2} - 40 x + 50} + \\frac{50 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{2} - 40 x + 50} + \\frac{12}{8 x^{2} - 40 x + 50} - \\frac{10 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{4 x^{2} - 20 x + 25} + \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{4 x - 10} + \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 5} - \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 5}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 5}) = - \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(2 x - 5\\right)} - \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 5} + \\frac{10 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x - 5\\right)^{2}}$.\nTherefore the correct derivative is $f'(x) = \\frac{4 x^{2} + 8 x + \\left(2 x - 5\\right) \\left(- 4 x + \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4\\right) - 4 \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8}{2 \\left(2 x - 5\\right)^{2}}$.\nFinal answer: $\\frac{4 x^{2} + 8 x + \\left(2 x - 5\\right) \\left(- 4 x + \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4\\right) - 4 \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8}{2 \\left(2 x - 5\\right)^{2}}$", "final_answer": "\\frac{4 x^{2} + 8 x + \\left(2 x - 5\\right) \\left(- 4 x + \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4\\right) - 4 \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8}{2 \\left(2 x - 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0c800565e7bc0e42", "problem_id": "bbb92d67c0e97e7e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(3 x^{2} + 3\\right) \\cos{\\left(x^{2} - 3 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x e^{x}) = x e^{x}$.\n3) So $f'(x) = - 6 x^{3} \\sin{\\left(x^{2} - 3 x \\right)} + 9 x^{2} \\sin{\\left(x^{2} - 3 x \\right)} + x e^{x} - 6 x \\sin{\\left(x^{2} - 3 x \\right)} + 6 x \\cos{\\left(x^{2} - 3 x \\right)} + e^{x} + 9 \\sin{\\left(x^{2} - 3 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x e^{x}) = x e^{x} + e^{x}$.\nTherefore the correct derivative is $f'(x) = 6 x \\cos{\\left(x^{2} - 3 x \\right)} + \\left(x + 1\\right) e^{x} - 3 \\left(2 x - 3\\right) \\left(x^{2} + 1\\right) \\sin{\\left(x^{2} - 3 x \\right)} + e^{x}$.\nFinal answer: $6 x \\cos{\\left(x^{2} - 3 x \\right)} + \\left(x + 1\\right) e^{x} - 3 \\left(2 x - 3\\right) \\left(x^{2} + 1\\right) \\sin{\\left(x^{2} - 3 x \\right)} + e^{x}$", "final_answer": "6 x \\cos{\\left(x^{2} - 3 x \\right)} + \\left(x + 1\\right) e^{x} - 3 \\left(2 x - 3\\right) \\left(x^{2} + 1\\right) \\sin{\\left(x^{2} - 3 x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ec65ee8445a54c0c", "problem_id": "8ea3f5cbd8ebdd6e", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- 7 x + 8 y$ = $-103$\n$- 6 x - 8 y$ = $-14$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-6$: $42 x - 48 y$ = $618$.\nMultiply the second equation by $-7$: $42 x + 56 y$ = $98$.\nSubtract to eliminate x: $- 104 y$ = $520$.\nSolve for y: y = $-5$.\nSubstitute back to find x: x = $9$.\nCheck: plug (x,y)=(9,-5) into both equations.\nFinal answer: $(x,y)=\\left(9, -5\\right)$", "final_answer": "(x,y)=\\left(9, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ee13a51009ebc130", "problem_id": "de50f32c5c34c473", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$- 8 x - 5 y$ = $97$\n$2 x - y$ = $-13$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $- 16 x - 10 y$ = $194$.\n3) Multiply the second equation by $-8$: $- 16 x + 8 y$ = $104$.\n4) Subtract to eliminate x: $- 18 y$ = $90$.\n5) Solve for y: y = $-5$.\n6) Substitute back to find x: x = $-9$.\nFinal answer: $(x,y)=\\left(-9, -5\\right)$", "final_answer": "(x,y)=\\left(-9, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "08a7fa6bc5d5b7f1", "problem_id": "0fc8045e26442222", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{4} - 4 x^{3} - x^{2} - x + 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- x) = - x$.\n3) So $f'(x) = - 8 x^{3} - 12 x^{2} - 3 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- x) = -1$.\nTherefore the correct derivative is $f'(x) = - 8 x^{3} - 12 x^{2} - 2 x - 1$.\nFinal answer: $- 8 x^{3} - 12 x^{2} - 2 x - 1$", "final_answer": "- 8 x^{3} - 12 x^{2} - 2 x - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a4122d94fe457a6b", "problem_id": "36530a3aac195086", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A cyclist rides at 24 miles per hour for 1/2 hours, then at 49 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 24 × 1/2 = 12 miles.\n2) Second leg distance = 49 × 3/2 = 147/2 miles.\n3) Total distance = 12 + 147/2 = 171/2 miles.\nFinal answer: $\\frac{171}{2}$", "final_answer": "\\frac{171}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e4c54a21ad9a2bde", "problem_id": "36530a3aac195086", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A cyclist rides at 24 miles per hour for 1/2 hours, then at 49 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 24 × 1/2 = 12 miles.\nSecond leg distance = 49 × 3/2 = 147/2 miles.\nTotal distance = 12 + 147/2 = 171/2 miles.\nCheck: adding both legs gives 171/2.\nFinal answer: $\\frac{171}{2}$", "final_answer": "\\frac{171}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "58ac2f8c1854062b", "problem_id": "9d589512bbcc7591", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡6 (mod 11), x≡6 (mod 7). Give the solution modulo 77.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 77.\n3) Compute the CRT solution to get x ≡ 6 (mod 77).\nFinal answer: $x\\equiv 6\\pmod{77}$", "final_answer": "x\\equiv 6\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f76fdaacdc9c5ee3", "problem_id": "9d589512bbcc7591", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡6 (mod 11), x≡6 (mod 7). Give the solution modulo 77.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 77.\nCompute the CRT solution to get x ≡ 6 (mod 77).\nCheck: 6 mod 11=6 and mod 7=6.\nFinal answer: $x\\equiv 6\\pmod{77}$", "final_answer": "x\\equiv 6\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8aa6c6826ffe1787", "problem_id": "4ebcb8469277b299", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "You want 59 liters of a 35% salt solution. You have a 15% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 59−x liters is 60% solution.\n2) Salt equation: 15x + 60(59−x) = 35·59.\n3) (15-60)x = (35-60)·59 ⇒ x = 295/9.\n4) Then 59−x = 236/9.\nFinal answer: $295/9 L of 15% and 236/9 L of 60%$", "final_answer": "295/9 L of 15% and 236/9 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "772aa58ff968907c", "problem_id": "4ebcb8469277b299", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "You want 59 liters of a 35% salt solution. You have a 15% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 59−x liters is 60% solution.\nSalt equation: 15x + 60(59−x) = 35·59.\n(15-60)x = (35-60)·59 ⇒ x = 295/9.\nThen 59−x = 236/9.\nCheck: 15·295/9 + 60·236/9 = 35·59.\nFinal answer: $295/9 L of 15% and 236/9 L of 60%$", "final_answer": "295/9 L of 15% and 236/9 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "09d4ef751bdc0efb", "problem_id": "fc4f074049e344f4", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A store sells a backpack for $170. It is discounted by 20%. You buy 10 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 170 × (1 − 20/100) = 170 × 80/100 = 136.\n2) Subtotal = 10 × 136 = 1360.\n3) Total with tax = 1360 × 108/100 = 7344/5.\nFinal answer: $\\frac{7344}{5}$", "final_answer": "\\frac{7344}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f83febcc68abcb7c", "problem_id": "f0cace33ed62a217", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 2 x^{3} - 3 x^{2} + 3 x - 5$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} - 3 x^{2} + 2 x - 5.\n3) Then the limit is -12.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: -13.\nFinal answer: $-13$", "final_answer": "-13", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "69a1ab35f7b2988f", "problem_id": "28c06b4be7e207e3", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the multiplicative inverse of 5 modulo 11.", "output": "1) We want x such that 5x ≡ 1 (mod 11).\n2) Use extended Euclid to find s,t with 5s + 11t = 1.\n3) One Bezout identity is: 5·(-2) + 11·(1) = 1.\n4) Reduce s modulo 11: x ≡ 9 (mod 11).\nFinal answer: $x\\equiv 9\\pmod{11}$", "final_answer": "x\\equiv 9\\pmod{11}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "841c0a2e88dd6e5a", "problem_id": "28c06b4be7e207e3", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 5 modulo 11.", "output": "We want x such that 5x ≡ 1 (mod 11).\nUse extended Euclid to find s,t with 5s + 11t = 1.\nOne Bezout identity is: 5·(-2) + 11·(1) = 1.\nReduce s modulo 11: x ≡ 9 (mod 11).\nCheck: 5·9 mod 11 = 1.\nFinal answer: $x\\equiv 9\\pmod{11}$", "final_answer": "x\\equiv 9\\pmod{11}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1c43d8b83f0aabd7", "problem_id": "76ae89ddee64322a", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=25$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=25 and k=6.\n4) Compute C(30,5) = 142506.\nFinal answer: $142506$", "final_answer": "142506", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7b439081da7499d0", "problem_id": "76ae89ddee64322a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=25$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=25 and k=6.\nCompute C(30,5) = 142506.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $142506$", "final_answer": "142506", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dffd06844dc506e9", "problem_id": "8dca75f2954332d2", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{7 x}{4} + \\frac{3}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{7}{4}$, c=$\\frac{3}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{1}{16}$, so √Δ = $\\frac{1}{4}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $1$.\n7) x2 = $\\frac{3}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{1, \\frac{3}{4}\\}$", "final_answer": "x \\in \\{1, \\frac{3}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6aa89e5250b21be3", "problem_id": "8dca75f2954332d2", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{7 x}{4} + \\frac{3}{4}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{7}{4}$, c=$\\frac{3}{4}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{1}{16}$, so √Δ = $\\frac{1}{4}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $1$.\nx2 = $\\frac{3}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{1, \\frac{3}{4}\\}$", "final_answer": "x \\in \\{1, \\frac{3}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f69db378987559ec", "problem_id": "87fa2a5d0099e159", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many binary strings of length 23 contain exactly 4 ones with no two ones adjacent?", "output": "Reserve 3 mandatory zeros between the 4 ones.\nThis leaves n−k+1 = 20 valid slots for the ones.\nChoose the slots: C(20,4) = 4845.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $4845$", "final_answer": "4845", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a0bc58c120252f95", "problem_id": "87fa2a5d0099e159", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "How many binary strings of length 23 contain exactly 4 ones with no two ones adjacent?", "output": "1) Reserve 3 mandatory zeros between the 4 ones.\n2) This leaves n−k+1 = 20 valid slots for the ones.\n3) Choose the slots: C(20,4) = 4845.\nFinal answer: $4845$", "final_answer": "4845", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f637a06f88d1890c", "problem_id": "466cf59764f18346", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A cyclist rides at 42 miles per hour for 1 hours, then at 58 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 42 × 1 = 42 miles.\n2) Second leg distance = 58 × 2 = 116 miles.\n3) Total distance = 42 + 116 = 158 miles.\nFinal answer: $158$", "final_answer": "158", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72ae6fb69925517d", "problem_id": "466cf59764f18346", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A cyclist rides at 42 miles per hour for 1 hours, then at 58 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 42 × 1 = 42 miles.\nSecond leg distance = 58 × 2 = 116 miles.\nTotal distance = 42 + 116 = 158 miles.\nFinal answer: $158$", "final_answer": "158", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "481865d130da9772", "problem_id": "0efc82114dc4260f", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A disease has prevalence $\\frac{1}{100}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{27}{250}$.\n3) Compute P(D|+) = $\\frac{1}{12}$.\nFinal answer: $\\frac{1}{12}$", "final_answer": "\\frac{1}{12}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fd08730cec0b6ceb", "problem_id": "0efc82114dc4260f", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "A disease has prevalence $\\frac{1}{100}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{27}{250}$.\nCompute P(D|+) = $\\frac{1}{12}$.\nFinal answer: $\\frac{1}{12}$", "final_answer": "\\frac{1}{12}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1da0914c9df7c6dc", "problem_id": "2f9d514502064ac7", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A store sells a backpack for $130. It is discounted by 10%. You buy 3 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 130 × (1 − 10/100) = 130 × 90/100 = 117.\n2) Subtotal = 3 × 117 = 351.\n3) Total with tax = 351 × 109/100 = 38259/100.\nFinal answer: $\\frac{38259}{100}$", "final_answer": "\\frac{38259}{100}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "35a173755c39c550", "problem_id": "2f9d514502064ac7", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $130. It is discounted by 10%. You buy 3 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 130 × (1 − 10/100) = 130 × 90/100 = 117.\nSubtotal = 3 × 117 = 351.\nTotal with tax = 351 × 109/100 = 38259/100.\nCheck: recomputing the formula yields 38259/100.\nFinal answer: $\\frac{38259}{100}$", "final_answer": "\\frac{38259}{100}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d1298d566b078c60", "problem_id": "3b322962bbbd8428", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A store sells a backpack for $97. It is discounted by 5%. You buy 10 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 97 × (1 − 5/100) = 97 × 95/100 = 1843/20.\n2) Subtotal = 10 × 1843/20 = 1843/2.\n3) Total with tax = 1843/2 × 107/100 = 197201/200.\nFinal answer: $\\frac{197201}{200}$", "final_answer": "\\frac{197201}{200}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fcb5255eecbef2a6", "problem_id": "3b322962bbbd8428", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A store sells a backpack for $97. It is discounted by 5%. You buy 10 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 97 × (1 − 5/100) = 97 × 95/100 = 1843/20.\nSubtotal = 10 × 1843/20 = 1843/2.\nTotal with tax = 1843/2 × 107/100 = 197201/200.\nFinal answer: $\\frac{197201}{200}$", "final_answer": "\\frac{197201}{200}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0085206be2ac33f9", "problem_id": "c5067f215e11a643", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - 4 x^{3} + x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 4 x^{3} + x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{255 \\pi^{4}}{16} - 10 \\pi - 1 + \\frac{65 \\pi^{3}}{24}$.\n4) Simplify the result.\nFinal answer: $- \\frac{255\\cdot \\pi^{4}}{16} - 10\\cdot \\pi - 1 + \\frac{65\\cdot \\pi^{3}}{24}$", "final_answer": "- \\frac{255\\cdot \\pi^{4}}{16} - 10\\cdot \\pi - 1 + \\frac{65\\cdot \\pi^{3}}{24}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d016044122e6caf5", "problem_id": "c5067f215e11a643", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - 4 x^{3} + x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 4 x^{3} + x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{255 \\pi^{4}}{16} - 10 \\pi - 1 + \\frac{65 \\pi^{3}}{24}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{255\\cdot \\pi^{4}}{16} - 10\\cdot \\pi - 1 + \\frac{65\\cdot \\pi^{3}}{24}$", "final_answer": "- \\frac{255\\cdot \\pi^{4}}{16} - 10\\cdot \\pi - 1 + \\frac{65\\cdot \\pi^{3}}{24}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7dd8b9233f5af537", "problem_id": "45a7ab422a5a8f52", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{13 x}{6} - \\frac{5}{6}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{13}{6}$, c=$- \\frac{5}{6}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{289}{36}$, so √Δ = $\\frac{17}{6}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{5}{2}$.\n7) x2 = $- \\frac{1}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{5}{2}, - \\frac{1}{3}\\}$", "final_answer": "x \\in \\{\\frac{5}{2}, - \\frac{1}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "07ff113292aaa9a0", "problem_id": "45a7ab422a5a8f52", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{13 x}{6} - \\frac{5}{6}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{13}{6}$, c=$- \\frac{5}{6}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{5}{2}, - \\frac{1}{3}\\}$", "final_answer": "x \\in \\{\\frac{5}{2}, - \\frac{1}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ffa568875e0cbfbd", "problem_id": "e53812493d9cf08a", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $329 in the ratio 5:2. How much does each receive?", "output": "1) Total ratio parts = 5+2 = 7.\n2) Each part = 329 ÷ 7 = 47.\n3) First share = 5 × 47 = 235.\n4) Second share = 2 × 47 = 94.\nFinal answer: $94$", "final_answer": "94", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "977feeccc1438e63", "problem_id": "e53812493d9cf08a", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Two friends split $329 in the ratio 5:2. How much does each receive?", "output": "Total ratio parts = 5+2 = 7.\nEach part = 329 ÷ 7 = 47.\nFirst share = 5 × 47 = 235.\nSecond share = 2 × 47 = 94.\nCheck: 235+94=329.\nFinal answer: $94$", "final_answer": "94", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2fbfcfc8d2af923b", "problem_id": "0ac391262fd551de", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} - 3 x^{2} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 3$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $- 3 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\n2) Slope at x=-2: m = f'(-2) = $-2 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}$.\n3) Point on curve: (-2, f(-2)) = (-2, $\\cos{\\left(4 \\right)} + e^{-1} + 3$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(4 \\right)} + e^{-1} + 3 = -2 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}(x − -2).\n6) Optionally expand: y = $- 2 x + 2 x \\sin{\\left(4 \\right)} + \\frac{x}{2 e} + 4 \\sin{\\left(4 \\right)} - 1 + \\cos{\\left(4 \\right)} + \\frac{2}{e}$.\nFinal answer: $y = - 2\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 4\\cdot \\sin{\\left(4 \\right)} - 1 + \\cos{\\left(4 \\right)} + \\frac{2}{e}$", "final_answer": "y = - 2\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 4\\cdot \\sin{\\left(4 \\right)} - 1 + \\cos{\\left(4 \\right)} + \\frac{2}{e}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f15af99c34465469", "problem_id": "0ac391262fd551de", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} - 3 x^{2} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 3$ at $x=-2$.", "output": "Differentiate: f'(x) = $- 3 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\nSlope at x=-2: m = f'(-2) = $-2 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}$.\nPoint on curve: (-2, f(-2)) = (-2, $\\cos{\\left(4 \\right)} + e^{-1} + 3$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − \\cos{\\left(4 \\right)} + e^{-1} + 3 = -2 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}(x − -2).\nOptionally expand: y = $- 2 x + 2 x \\sin{\\left(4 \\right)} + \\frac{x}{2 e} + 4 \\sin{\\left(4 \\right)} - 1 + \\cos{\\left(4 \\right)} + \\frac{2}{e}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 2\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 4\\cdot \\sin{\\left(4 \\right)} - 1 + \\cos{\\left(4 \\right)} + \\frac{2}{e}$", "final_answer": "y = - 2\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 4\\cdot \\sin{\\left(4 \\right)} - 1 + \\cos{\\left(4 \\right)} + \\frac{2}{e}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4983fc77d70b9f0a", "problem_id": "cb05d8c81c756ed5", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} x^{3} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $x^{3} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\nSubtract: F(\\frac{\\pi}{2}) − F(0) = $\\frac{\\pi^{4}}{64} + 2 + \\pi$.\nFinal answer: $\\frac{\\pi^{4}}{64} + 2 + \\pi$", "final_answer": "\\frac{\\pi^{4}}{64} + 2 + \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6356c72def1f154b", "problem_id": "cb05d8c81c756ed5", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} x^{3} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $x^{3} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\nSubtract: F(\\frac{\\pi}{2}) − F(0) = $\\frac{\\pi^{4}}{64} + 2 + \\pi$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{\\pi^{4}}{64} + 2 + \\pi$", "final_answer": "\\frac{\\pi^{4}}{64} + 2 + \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2900732033aea588", "problem_id": "2412cdcdf3201af4", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x^{3} - 5 x^{2} - 12 x - 22$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 2 x^{4} - x^{3} - 2 x^{2} + 2 x - 3$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 8 x^{3} - 3 x^{2} - 4 x + 2$.\n5) Evaluate at a: p'(2) = $-82$.\nFinal answer: $-82$", "final_answer": "-82", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6c5f315e4cde9ace", "problem_id": "2412cdcdf3201af4", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x^{3} - 5 x^{2} - 12 x - 22$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 2 x^{4} - x^{3} - 2 x^{2} + 2 x - 3$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 8 x^{3} - 3 x^{2} - 4 x + 2$.\nEvaluate at a: p'(2) = $-82$.\nCheck: the identity gives $-82$.\nFinal answer: $-82$", "final_answer": "-82", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f85be1ac83fc360c", "problem_id": "857cfecb6ae38c5c", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A store sells a backpack for $99. It is discounted by 5%. You buy 14 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 99 × (1 − 5/100) = 99 × 95/100 = 1881/20.\n2) Subtotal = 14 × 1881/20 = 13167/10.\n3) Total with tax = 13167/10 × 110/100 = 144837/100.\nFinal answer: $\\frac{144837}{100}$", "final_answer": "\\frac{144837}{100}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dee09267aa09ffac", "problem_id": "857cfecb6ae38c5c", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A store sells a backpack for $99. It is discounted by 5%. You buy 14 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 99 × (1 − 5/100) = 99 × 95/100 = 1881/20.\nSubtotal = 14 × 1881/20 = 13167/10.\nTotal with tax = 13167/10 × 110/100 = 144837/100.\nCheck: recomputing the formula yields 144837/100.\nFinal answer: $\\frac{144837}{100}$", "final_answer": "\\frac{144837}{100}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6d966f63655ed530", "problem_id": "49003bcb9bf637da", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A store sells a backpack for $680. It is discounted by 30%. You buy 6 backpacks. What is the total cost?", "output": "1) Discounted price = 680 × (1 − 30/100) = 680 × 70/100 = 476.\n2) Subtotal = 6 × 476 = 2856.\nFinal answer: $2856$", "final_answer": "2856", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5ed6c93dad878239", "problem_id": "49003bcb9bf637da", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A store sells a backpack for $680. It is discounted by 30%. You buy 6 backpacks. What is the total cost?", "output": "Discounted price = 680 × (1 − 30/100) = 680 × 70/100 = 476.\nSubtotal = 6 × 476 = 2856.\nFinal answer: $2856$", "final_answer": "2856", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "11f1887f672d48b3", "problem_id": "6e70c2669c322bbe", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many binary strings of length 26 contain exactly 3 ones with no two ones adjacent?", "output": "1) Reserve 2 mandatory zeros between the 3 ones.\n2) This leaves n−k+1 = 24 valid slots for the ones.\n3) Choose the slots: C(24,3) = 2024.\nFinal answer: $2024$", "final_answer": "2024", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7b2baeadf8ba70d1", "problem_id": "bbd90636f13742cb", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} x \\left(2 - 3 x\\right)$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} + x.\n3) Then the limit is -4.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: -5.\nFinal answer: $-5$", "final_answer": "-5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a500b385505abb40", "problem_id": "cdcce947cf49d4d0", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A cyclist rides at 67 miles per hour for 1 hours, then at 39 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 67 × 1 = 67 miles.\n2) Second leg distance = 39 × 1 = 39 miles.\n3) Total distance = 67 + 39 = 106 miles.\nFinal answer: $106$", "final_answer": "106", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "478a7a2c6681d1d1", "problem_id": "cdcce947cf49d4d0", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 67 miles per hour for 1 hours, then at 39 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 67 × 1 = 67 miles.\nSecond leg distance = 39 × 1 = 39 miles.\nTotal distance = 67 + 39 = 106 miles.\nCheck: adding both legs gives 106.\nFinal answer: $106$", "final_answer": "106", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cbf9e0e566492cf6", "problem_id": "7d8f079659c10f7a", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 4$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} - 4$ = $\\left(x - 2\\right) \\left(x + 2\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $2$.\nFrom the second factor: x = $-2$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{2, -2\\}$", "final_answer": "x \\in \\{2, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dd3deff840a56e28", "problem_id": "7d8f079659c10f7a", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 4$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} - 4$ = $\\left(x - 2\\right) \\left(x + 2\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $2$.\n5) From the second factor: x = $-2$.\nFinal answer: $x \\in \\{2, -2\\}$", "final_answer": "x \\in \\{2, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f5fe88dcb2d4e1f3", "problem_id": "54482718efedf40b", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A cyclist rides at 58 miles per hour for 1/2 hours, then at 88 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 58 × 1/2 = 29 miles.\nSecond leg distance = 88 × 5/2 = 220 miles.\nTotal distance = 29 + 220 = 249 miles.\nFinal answer: $249$", "final_answer": "249", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "021c4f96f2f88896", "problem_id": "54482718efedf40b", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A cyclist rides at 58 miles per hour for 1/2 hours, then at 88 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 58 × 1/2 = 29 miles.\n2) Second leg distance = 88 × 5/2 = 220 miles.\n3) Total distance = 29 + 220 = 249 miles.\nFinal answer: $249$", "final_answer": "249", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8231492b6f945ab5", "problem_id": "5413fbd435308408", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} 3 x^{4} + x^{3} + 2 x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{4} + x^{3} + 2 x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{15 \\pi^{4}}{64} + 5 + \\frac{3 \\pi^{2}}{2} + \\frac{3 \\pi^{3}}{4} + \\frac{99 \\pi^{5}}{160}$.\n4) Simplify the result.\nFinal answer: $- \\frac{15\\cdot \\pi^{4}}{64} + 5 + \\frac{3\\cdot \\pi^{2}}{2} + \\frac{3\\cdot \\pi^{3}}{4} + \\frac{99\\cdot \\pi^{5}}{160}$", "final_answer": "- \\frac{15\\cdot \\pi^{4}}{64} + 5 + \\frac{3\\cdot \\pi^{2}}{2} + \\frac{3\\cdot \\pi^{3}}{4} + \\frac{99\\cdot \\pi^{5}}{160}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e5d2d4376f45506e", "problem_id": "5413fbd435308408", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} 3 x^{4} + x^{3} + 2 x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{4} + x^{3} + 2 x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{15 \\pi^{4}}{64} + 5 + \\frac{3 \\pi^{2}}{2} + \\frac{3 \\pi^{3}}{4} + \\frac{99 \\pi^{5}}{160}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{15\\cdot \\pi^{4}}{64} + 5 + \\frac{3\\cdot \\pi^{2}}{2} + \\frac{3\\cdot \\pi^{3}}{4} + \\frac{99\\cdot \\pi^{5}}{160}$", "final_answer": "- \\frac{15\\cdot \\pi^{4}}{64} + 5 + \\frac{3\\cdot \\pi^{2}}{2} + \\frac{3\\cdot \\pi^{3}}{4} + \\frac{99\\cdot \\pi^{5}}{160}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "775462474904694a", "problem_id": "f62b1af8963ff7a5", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡3 (mod 8), x≡5 (mod 9). Give the solution modulo 72.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 72.\n3) Compute the CRT solution to get x ≡ 59 (mod 72).\nFinal answer: $x\\equiv 59\\pmod{72}$", "final_answer": "x\\equiv 59\\pmod{72}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a98516da662f83c9", "problem_id": "f62b1af8963ff7a5", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡3 (mod 8), x≡5 (mod 9). Give the solution modulo 72.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 72.\nCompute the CRT solution to get x ≡ 59 (mod 72).\nCheck: 59 mod 8=3 and mod 9=5.\nFinal answer: $x\\equiv 59\\pmod{72}$", "final_answer": "x\\equiv 59\\pmod{72}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8df39bc219af073d", "problem_id": "abbf9262a3053175", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=20$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=20 and k=7.\n4) Compute C(26,6) = 230230.\nFinal answer: $230230$", "final_answer": "230230", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "570cb3048aaf3de1", "problem_id": "abbf9262a3053175", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=20$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=20 and k=7.\nCompute C(26,6) = 230230.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $230230$", "final_answer": "230230", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b5a666123d11df47", "problem_id": "f4bd5b17dcd256be", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Two friends split $42 in the ratio 1:5. How much does each receive?", "output": "1) Total ratio parts = 1+5 = 6.\n2) Each part = 42 ÷ 6 = 7.\n3) First share = 1 × 7 = 7.\n4) Second share = 5 × 7 = 35.\nFinal answer: $35$", "final_answer": "35", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a0d1f6176428e8eb", "problem_id": "f4bd5b17dcd256be", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Two friends split $42 in the ratio 1:5. How much does each receive?", "output": "Total ratio parts = 1+5 = 6.\nEach part = 42 ÷ 6 = 7.\nFirst share = 1 × 7 = 7.\nSecond share = 5 × 7 = 35.\nCheck: 7+35=42.\nFinal answer: $35$", "final_answer": "35", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "09041502fa76c694", "problem_id": "d2c6ae5adcc3ea2c", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Worker A finishes a job in 19 hours and Worker B finishes it in 6 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/19 job/hour. B's rate = 1/6 job/hour.\n2) Combined rate = 1/19 + 1/6 = 25/114 job/hour.\n3) Time = 1 ÷ (combined rate) = 114/25 hours.\nFinal answer: $\\frac{114}{25}$", "final_answer": "\\frac{114}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4815b09a49fc72bd", "problem_id": "d2c6ae5adcc3ea2c", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Worker A finishes a job in 19 hours and Worker B finishes it in 6 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/19 job/hour. B's rate = 1/6 job/hour.\nCombined rate = 1/19 + 1/6 = 25/114 job/hour.\nTime = 1 ÷ (combined rate) = 114/25 hours.\nFinal answer: $\\frac{114}{25}$", "final_answer": "\\frac{114}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a29ce0bcec77f4d", "problem_id": "d691a5015f8b3792", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$- 3 x + 4 y$ = $-28$\n$- x - 4 y$ = $12$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $3 x - 4 y$ = $28$.\n3) Multiply the second equation by $-3$: $3 x + 12 y$ = $-36$.\n4) Subtract to eliminate x: $- 16 y$ = $64$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, -4\\right)$", "final_answer": "(x,y)=\\left(4, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7e19c892373cdba7", "problem_id": "af7b0f60f2a2007c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{3} - 3 x^{2} + x - 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x) = x$.\n3) So $f'(x) = 12 x^{2} - 5 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x) = 1$.\nTherefore the correct derivative is $f'(x) = 12 x^{2} - 6 x + 1$.\nFinal answer: $12 x^{2} - 6 x + 1$", "final_answer": "12 x^{2} - 6 x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b231ecf58776f076", "problem_id": "cc541c991b0033ff", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 3 x + 1$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$-3$, c=$1$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $1$, so √Δ = $1$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $1$.\n7) x2 = $\\frac{1}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{1, \\frac{1}{2}\\}$", "final_answer": "x \\in \\{1, \\frac{1}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b548fbd1f0037023", "problem_id": "cc541c991b0033ff", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 3 x + 1$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$-3$, c=$1$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $1$, so √Δ = $1$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $1$.\nx2 = $\\frac{1}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{1, \\frac{1}{2}\\}$", "final_answer": "x \\in \\{1, \\frac{1}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d8e15e42d880f08d", "problem_id": "74ed30bc05f8082b", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 2 x^{4} + x^{3} + 2 x + 3\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 2 x^{4} + x^{3} + 2 x + 3$.\nCompute an antiderivative: F(x) = $- \\frac{2 x^{5}}{5} + \\frac{x^{4}}{4} + x^{2} + 3 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - \\frac{2\\cdot x^{5}}{5} + \\frac{x^{4}}{4} + x^{2} + 3\\cdot x$", "final_answer": "C - \\frac{2\\cdot x^{5}}{5} + \\frac{x^{4}}{4} + x^{2} + 3\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d7ba2064cc3d4974", "problem_id": "74ed30bc05f8082b", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 2 x^{4} + x^{3} + 2 x + 3\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 2 x^{4} + x^{3} + 2 x + 3$.\n3) Compute an antiderivative: F(x) = $- \\frac{2 x^{5}}{5} + \\frac{x^{4}}{4} + x^{2} + 3 x$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{2\\cdot x^{5}}{5} + \\frac{x^{4}}{4} + x^{2} + 3\\cdot x$", "final_answer": "C - \\frac{2\\cdot x^{5}}{5} + \\frac{x^{4}}{4} + x^{2} + 3\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b11629037b797778", "problem_id": "17073a40d495b172", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A cyclist rides at 24 miles per hour for 2 hours, then at 51 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 24 × 2 = 48 miles.\n2) Second leg distance = 51 × 2 = 102 miles.\n3) Total distance = 48 + 102 = 150 miles.\nFinal answer: $150$", "final_answer": "150", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dd66352382f21d1a", "problem_id": "17073a40d495b172", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A cyclist rides at 24 miles per hour for 2 hours, then at 51 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 24 × 2 = 48 miles.\nSecond leg distance = 51 × 2 = 102 miles.\nTotal distance = 48 + 102 = 150 miles.\nCheck: adding both legs gives 150.\nFinal answer: $150$", "final_answer": "150", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f784f152a0efc63", "problem_id": "d020904e21642e8a", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 17 red balls and 20 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(37,10).\nFavorable samples: choose k reds and n−k blues: C(17,1)·C(20,9).\nCompute totals: C(37,10) = 348330136, and favorable = 2855320.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{20995}{2561251}$.\nFinal answer: $\\frac{20995}{2561251}$", "final_answer": "\\frac{20995}{2561251}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c9672c0d6ef9eaa3", "problem_id": "d020904e21642e8a", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 17 red balls and 20 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(37,10).\n3) Favorable samples: choose k reds and n−k blues: C(17,1)·C(20,9).\n4) Compute totals: C(37,10) = 348330136, and favorable = 2855320.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{20995}{2561251}$", "final_answer": "\\frac{20995}{2561251}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a1dab6cc21c621eb", "problem_id": "e39e8c6c3f41bce0", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} + 3 x^{2} + 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=1$.", "output": "1) Differentiate: f'(x) = $6 x^{2} + 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 2$.\n2) Slope at x=1: m = f'(1) = $- 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 14$.\n3) Point on curve: (1, f(1)) = (1, $\\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} + 6$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} + 6 = - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 14(x − 1).\n6) Optionally expand: y = $- 2 x \\sin{\\left(2 \\right)} + \\frac{x e^{\\frac{1}{2}}}{2} + 14 x - 8 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 14\\cdot x - 8 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 14\\cdot x - 8 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "37e0e75acc6323a1", "problem_id": "e39e8c6c3f41bce0", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} + 3 x^{2} + 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=1$.", "output": "Differentiate: f'(x) = $6 x^{2} + 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 2$.\nSlope at x=1: m = f'(1) = $- 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 14$.\nPoint on curve: (1, f(1)) = (1, $\\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} + 6$).\nFinal answer: $y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 14\\cdot x - 8 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 14\\cdot x - 8 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "812531e2920fd4ce", "problem_id": "1d849ea1f8d9b4a1", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 6 x - 8$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 6 x - 8$ = $2 \\left(x - 4\\right) \\left(x + 1\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $4$.\n5) From the second factor: x = $-1$.\nFinal answer: $x \\in \\{4, -1\\}$", "final_answer": "x \\in \\{4, -1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "34ee28228597a904", "problem_id": "1d849ea1f8d9b4a1", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 6 x - 8$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} - 6 x - 8$ = $2 \\left(x - 4\\right) \\left(x + 1\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $4$.\nFrom the second factor: x = $-1$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{4, -1\\}$", "final_answer": "x \\in \\{4, -1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "652962104bc95172", "problem_id": "db78d67e139f1c64", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} x^{2} + x + 6$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to x^{2} + 6.\n3) Then the limit is 15.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: 18.\nFinal answer: $18$", "final_answer": "18", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "527a4ba5dadc0675", "problem_id": "e7f348aad6fbaf3d", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A cyclist rides at 82 miles per hour for 3/2 hours, then at 77 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 82 × 3/2 = 123 miles.\n2) Second leg distance = 77 × 3 = 231 miles.\n3) Total distance = 123 + 231 = 354 miles.\nFinal answer: $354$", "final_answer": "354", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6f693176ea5ebf7d", "problem_id": "af6f6e2247fcdce0", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{5} x + \\frac{7 x}{2} - 15 + \\frac{15 \\sqrt{5}}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{7}{2} - 3 \\sqrt{5}$, c=$-15 + \\frac{15 \\sqrt{5}}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{469}{4} - 51 \\sqrt{5}$, so √Δ = $\\sqrt{\\frac{469}{4} - 51 \\sqrt{5}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{7}{4} + \\frac{\\sqrt{469 - 204 \\sqrt{5}}}{4} + \\frac{3 \\sqrt{5}}{2}$.\n7) x2 = $- \\frac{7}{4} - \\frac{\\sqrt{469 - 204 \\sqrt{5}}}{4} + \\frac{3 \\sqrt{5}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{7}{4} + \\frac{\\sqrt{469 - 204\\cdot \\sqrt{5}}}{4} + \\frac{3\\cdot \\sqrt{5}}{2}, - \\frac{7}{4} - \\frac{\\sqrt{469 - 204\\cdot \\sqrt{5}}}{4} + \\frac{3\\cdot \\sqrt{5}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{7}{4} + \\frac{\\sqrt{469 - 204\\cdot \\sqrt{5}}}{4} + \\frac{3\\cdot \\sqrt{5}}{2}, - \\frac{7}{4} - \\frac{\\sqrt{469 - 204\\cdot \\sqrt{5}}}{4} + \\frac{3\\cdot \\sqrt{5}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d92853f7f65c48cb", "problem_id": "af6f6e2247fcdce0", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{5} x + \\frac{7 x}{2} - 15 + \\frac{15 \\sqrt{5}}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{7}{2} - 3 \\sqrt{5}$, c=$-15 + \\frac{15 \\sqrt{5}}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{469}{4} - 51 \\sqrt{5}$, so √Δ = $\\sqrt{\\frac{469}{4} - 51 \\sqrt{5}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{7}{4} + \\frac{\\sqrt{469 - 204 \\sqrt{5}}}{4} + \\frac{3 \\sqrt{5}}{2}$.\nx2 = $- \\frac{7}{4} - \\frac{\\sqrt{469 - 204 \\sqrt{5}}}{4} + \\frac{3 \\sqrt{5}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{7}{4} + \\frac{\\sqrt{469 - 204\\cdot \\sqrt{5}}}{4} + \\frac{3\\cdot \\sqrt{5}}{2}, - \\frac{7}{4} - \\frac{\\sqrt{469 - 204\\cdot \\sqrt{5}}}{4} + \\frac{3\\cdot \\sqrt{5}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{7}{4} + \\frac{\\sqrt{469 - 204\\cdot \\sqrt{5}}}{4} + \\frac{3\\cdot \\sqrt{5}}{2}, - \\frac{7}{4} - \\frac{\\sqrt{469 - 204\\cdot \\sqrt{5}}}{4} + \\frac{3\\cdot \\sqrt{5}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ceba93d779b9cec8", "problem_id": "f063a65fe15b6ef3", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} - 2 x^{3} - 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{3} - 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\nSubtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{\\pi^{4}}{32} - \\frac{\\pi^{3}}{12} + 2 + \\frac{3 \\pi^{2}}{8}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{\\pi^{4}}{32} - \\frac{\\pi^{3}}{12} + 2 + \\frac{3\\cdot \\pi^{2}}{8}$", "final_answer": "- \\frac{\\pi^{4}}{32} - \\frac{\\pi^{3}}{12} + 2 + \\frac{3\\cdot \\pi^{2}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "74b76fc876392d70", "problem_id": "f063a65fe15b6ef3", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} - 2 x^{3} - 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{3} - 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\n3) Subtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{\\pi^{4}}{32} - \\frac{\\pi^{3}}{12} + 2 + \\frac{3 \\pi^{2}}{8}$.\n4) Simplify the result.\nFinal answer: $- \\frac{\\pi^{4}}{32} - \\frac{\\pi^{3}}{12} + 2 + \\frac{3\\cdot \\pi^{2}}{8}$", "final_answer": "- \\frac{\\pi^{4}}{32} - \\frac{\\pi^{3}}{12} + 2 + \\frac{3\\cdot \\pi^{2}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "86692945bc23d8c7", "problem_id": "0f2842e067f8497e", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} 3 x^{3} + 4 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{3} + 4 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- 6 \\pi - \\frac{9 \\pi^{2}}{8} - 1 + \\frac{3 \\pi^{3}}{2} + \\frac{45 \\pi^{4}}{64}$.\n4) Simplify the result.\nFinal answer: $- 6\\cdot \\pi - \\frac{9\\cdot \\pi^{2}}{8} - 1 + \\frac{3\\cdot \\pi^{3}}{2} + \\frac{45\\cdot \\pi^{4}}{64}$", "final_answer": "- 6\\cdot \\pi - \\frac{9\\cdot \\pi^{2}}{8} - 1 + \\frac{3\\cdot \\pi^{3}}{2} + \\frac{45\\cdot \\pi^{4}}{64}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1233b726f467ea68", "problem_id": "0f2842e067f8497e", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} 3 x^{3} + 4 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{3} + 4 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4$.\nEvaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- 6 \\pi - \\frac{9 \\pi^{2}}{8} - 1 + \\frac{3 \\pi^{3}}{2} + \\frac{45 \\pi^{4}}{64}$.\nFinal answer: $- 6\\cdot \\pi - \\frac{9\\cdot \\pi^{2}}{8} - 1 + \\frac{3\\cdot \\pi^{3}}{2} + \\frac{45\\cdot \\pi^{4}}{64}$", "final_answer": "- 6\\cdot \\pi - \\frac{9\\cdot \\pi^{2}}{8} - 1 + \\frac{3\\cdot \\pi^{3}}{2} + \\frac{45\\cdot \\pi^{4}}{64}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f0855a33f941112", "problem_id": "0308f71b40ff3074", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Two friends split $702 in the ratio 8:5. How much does each receive?", "output": "1) Total ratio parts = 8+5 = 13.\n2) Each part = 702 ÷ 13 = 54.\n3) First share = 8 × 54 = 432.\n4) Second share = 5 × 54 = 270.\nFinal answer: $270$", "final_answer": "270", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2de72af05bd65c7b", "problem_id": "0308f71b40ff3074", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $702 in the ratio 8:5. How much does each receive?", "output": "Total ratio parts = 8+5 = 13.\nEach part = 702 ÷ 13 = 54.\nFirst share = 8 × 54 = 432.\nSecond share = 5 × 54 = 270.\nCheck: 432+270=702.\nFinal answer: $270$", "final_answer": "270", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3d44e551143f3e97", "problem_id": "092d17eded2b9cf7", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 2 \\sqrt{7} x - \\frac{3 x}{2} + \\frac{1}{2} + \\sqrt{7}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- 2 \\sqrt{7} - \\frac{3}{2}$, c=$\\frac{1}{2} + \\sqrt{7}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $2 \\sqrt{7} + \\frac{113}{4}$, so √Δ = $\\sqrt{2 \\sqrt{7} + \\frac{113}{4}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{3}{4} + \\sqrt{7} + \\frac{\\sqrt{8 \\sqrt{7} + 113}}{4}$.\n7) x2 = $- \\frac{\\sqrt{8 \\sqrt{7} + 113}}{4} + \\frac{3}{4} + \\sqrt{7}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{3}{4} + \\sqrt{7} + \\frac{\\sqrt{8\\cdot \\sqrt{7} + 113}}{4}, - \\frac{\\sqrt{8\\cdot \\sqrt{7} + 113}}{4} + \\frac{3}{4} + \\sqrt{7}\\}$", "final_answer": "x \\in \\{\\frac{3}{4} + \\sqrt{7} + \\frac{\\sqrt{8\\cdot \\sqrt{7} + 113}}{4}, - \\frac{\\sqrt{8\\cdot \\sqrt{7} + 113}}{4} + \\frac{3}{4} + \\sqrt{7}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c9acb7eb29cf9d93", "problem_id": "93e08bc7a6c9189e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{2} + 2 x + \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{3 x + 7}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{7 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 7}) = \\frac{7 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 7}$.\n3) So $f'(x) = \\frac{9 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} + 84 x + 98} + \\frac{18 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} + 84 x + 98} + \\frac{6 x^{2}}{18 x^{2} + 84 x + 98} + \\frac{42 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} + 84 x + 98} + \\frac{84 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} + 84 x + 98} + \\frac{28 x}{18 x^{2} + 84 x + 98} + \\frac{49 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} + 84 x + 98} + \\frac{98 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} + 84 x + 98} + \\frac{46}{18 x^{2} + 84 x + 98} + \\frac{21 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{2} + 42 x + 49} - \\frac{7 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x + 14} + \\frac{7 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 7} - \\frac{7 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 7}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{7 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 7}) = \\frac{7 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(3 x + 7\\right)} + \\frac{7 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 7} - \\frac{21 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(3 x + 7\\right)^{2}}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 6 x^{2} - 12 x + \\left(3 x + 7\\right) \\left(4 x + \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 6 \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 18}{2 \\left(3 x + 7\\right)^{2}}$.\nFinal answer: $\\frac{- 6 x^{2} - 12 x + \\left(3 x + 7\\right) \\left(4 x + \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 6 \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 18}{2 \\left(3 x + 7\\right)^{2}}$", "final_answer": "\\frac{- 6 x^{2} - 12 x + \\left(3 x + 7\\right) \\left(4 x + \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 6 \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 18}{2 \\left(3 x + 7\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "cb442487e139a8c1", "problem_id": "e4bb9307120a81c1", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- 5 y$ = $35$\n$- 8 x + 4 y$ = $-20$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-8$: $40 y$ = $-280$.\nMultiply the second equation by $0$: $0$ = $0$.\nSubtract to eliminate x: $40 y$ = $-280$.\nSolve for y: y = $-7$.\nSubstitute back to find x: x = $-1$.\nCheck: plug (x,y)=(-1,-7) into both equations.\nFinal answer: $(x,y)=\\left(-1, -7\\right)$", "final_answer": "(x,y)=\\left(-1, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b3f131d18a3b650d", "problem_id": "2458bd55b3c9d678", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 2 x^{2} - 5 x + 16$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{2} - 6 x + 16.\n3) Then the limit is 52.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: 49.\nFinal answer: $49$", "final_answer": "49", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "956040feb2351d01", "problem_id": "48732515d6e01900", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many binary strings of length 17 contain exactly 6 ones with no two ones adjacent?", "output": "1) Reserve 5 mandatory zeros between the 6 ones.\n2) This leaves n−k+1 = 12 valid slots for the ones.\n3) Choose the slots: C(12,6) = 924.\nFinal answer: $924$", "final_answer": "924", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5919a09d2bd50259", "problem_id": "48732515d6e01900", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "How many binary strings of length 17 contain exactly 6 ones with no two ones adjacent?", "output": "Reserve 5 mandatory zeros between the 6 ones.\nThis leaves n−k+1 = 12 valid slots for the ones.\nChoose the slots: C(12,6) = 924.\nFinal answer: $924$", "final_answer": "924", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bc3abe4b2af31786", "problem_id": "7bbb79ffa6d514a4", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$4 y$ = $16$\n$2 x - 3 y$ = $-10$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $8 y$ = $32$.\n3) Multiply the second equation by $0$: $0$ = $0$.\n4) Subtract to eliminate x: $8 y$ = $32$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $1$.\nFinal answer: $(x,y)=\\left(1, 4\\right)$", "final_answer": "(x,y)=\\left(1, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "129d31f93864d223", "problem_id": "7bbb79ffa6d514a4", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$4 y$ = $16$\n$2 x - 3 y$ = $-10$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $2$: $8 y$ = $32$.\nMultiply the second equation by $0$: $0$ = $0$.\nSubtract to eliminate x: $8 y$ = $32$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $1$.\nCheck: plug (x,y)=(1,4) into both equations.\nFinal answer: $(x,y)=\\left(1, 4\\right)$", "final_answer": "(x,y)=\\left(1, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1808ae1b83b9da41", "problem_id": "24ade34e3ea980c1", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{2} x + \\frac{44 x}{15} - \\frac{8 \\sqrt{2}}{5} + \\frac{32}{15}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{44}{15} - \\sqrt{2}$, c=$\\frac{32}{15} - \\frac{8 \\sqrt{2}}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{8 \\sqrt{2}}{15} + \\frac{466}{225}$, so √Δ = $\\sqrt{\\frac{8 \\sqrt{2}}{15} + \\frac{466}{225}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{22}{15} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{120 \\sqrt{2} + 466}}{30}$.\nx2 = $- \\frac{22}{15} - \\frac{\\sqrt{120 \\sqrt{2} + 466}}{30} + \\frac{\\sqrt{2}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{22}{15} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{120\\cdot \\sqrt{2} + 466}}{30}, - \\frac{22}{15} - \\frac{\\sqrt{120\\cdot \\sqrt{2} + 466}}{30} + \\frac{\\sqrt{2}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{22}{15} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{120\\cdot \\sqrt{2} + 466}}{30}, - \\frac{22}{15} - \\frac{\\sqrt{120\\cdot \\sqrt{2} + 466}}{30} + \\frac{\\sqrt{2}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff06c16e7fccce64", "problem_id": "24ade34e3ea980c1", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{2} x + \\frac{44 x}{15} - \\frac{8 \\sqrt{2}}{5} + \\frac{32}{15}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{44}{15} - \\sqrt{2}$, c=$\\frac{32}{15} - \\frac{8 \\sqrt{2}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{8 \\sqrt{2}}{15} + \\frac{466}{225}$, so √Δ = $\\sqrt{\\frac{8 \\sqrt{2}}{15} + \\frac{466}{225}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{22}{15} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{120 \\sqrt{2} + 466}}{30}$.\n7) x2 = $- \\frac{22}{15} - \\frac{\\sqrt{120 \\sqrt{2} + 466}}{30} + \\frac{\\sqrt{2}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{22}{15} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{120\\cdot \\sqrt{2} + 466}}{30}, - \\frac{22}{15} - \\frac{\\sqrt{120\\cdot \\sqrt{2} + 466}}{30} + \\frac{\\sqrt{2}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{22}{15} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{120\\cdot \\sqrt{2} + 466}}{30}, - \\frac{22}{15} - \\frac{\\sqrt{120\\cdot \\sqrt{2} + 466}}{30} + \\frac{\\sqrt{2}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "44a58d702928822d", "problem_id": "89ad2c49c75600e6", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Find the multiplicative inverse of 38 modulo 47.", "output": "We want x such that 38x ≡ 1 (mod 47).\nUse extended Euclid to find s,t with 38s + 47t = 1.\nOne Bezout identity is: 38·(-21) + 47·(17) = 1.\nFinal answer: $x\\equiv 26\\pmod{47}$", "final_answer": "x\\equiv 26\\pmod{47}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "237cca119fc1e607", "problem_id": "89ad2c49c75600e6", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the multiplicative inverse of 38 modulo 47.", "output": "1) We want x such that 38x ≡ 1 (mod 47).\n2) Use extended Euclid to find s,t with 38s + 47t = 1.\n3) One Bezout identity is: 38·(-21) + 47·(17) = 1.\n4) Reduce s modulo 47: x ≡ 26 (mod 47).\nFinal answer: $x\\equiv 26\\pmod{47}$", "final_answer": "x\\equiv 26\\pmod{47}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "177bb5b949d11c35", "problem_id": "0092d0af8dec0cdb", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} - x^{3} - 2 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{3} - 2 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $\\frac{\\pi \\left(-18 - \\pi^{2}\\right)}{6}$.\n4) Simplify the result.\nFinal answer: $\\frac{\\pi\\cdot \\left(-18 - \\pi^{2}\\right)}{6}$", "final_answer": "\\frac{\\pi\\cdot \\left(-18 - \\pi^{2}\\right)}{6}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "04aa92113aaf2b27", "problem_id": "0092d0af8dec0cdb", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} - x^{3} - 2 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{3} - 2 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $\\frac{\\pi \\left(-18 - \\pi^{2}\\right)}{6}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{\\pi\\cdot \\left(-18 - \\pi^{2}\\right)}{6}$", "final_answer": "\\frac{\\pi\\cdot \\left(-18 - \\pi^{2}\\right)}{6}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "93bb0fb54f5bd9be", "problem_id": "7e196ced5da430b9", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 12 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=2, p=$\\frac{2}{7}$ and simplify.\n4) Here C(12,2) = 66.\nFinal answer: $\\frac{2578125000}{13841287201}$", "final_answer": "\\frac{2578125000}{13841287201}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f19461620c2303a9", "problem_id": "7e196ced5da430b9", "style": "answer_only", "instruction": "Final answer only.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 12 times, what is the probability of exactly 2 heads?", "output": "$\\frac{2578125000}{13841287201}$", "final_answer": "\\frac{2578125000}{13841287201}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "13b30dc9c387ba0b", "problem_id": "336b4955194c3cad", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "An urn has 14 red balls and 17 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(31,10).\n3) Favorable samples: choose k reds and n−k blues: C(14,1)·C(17,9).\n4) Compute totals: C(31,10) = 44352165, and favorable = 340340.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{476}{62031}$", "final_answer": "\\frac{476}{62031}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bdda144f53b0448b", "problem_id": "336b4955194c3cad", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 14 red balls and 17 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(31,10).\nFavorable samples: choose k reds and n−k blues: C(14,1)·C(17,9).\nCompute totals: C(31,10) = 44352165, and favorable = 340340.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{476}{62031}$.\nFinal answer: $\\frac{476}{62031}$", "final_answer": "\\frac{476}{62031}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3a6dc0abe876d317", "problem_id": "c3f6f94e5971859e", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the multiplicative inverse of 9 modulo 19.", "output": "We want x such that 9x ≡ 1 (mod 19).\nUse extended Euclid to find s,t with 9s + 19t = 1.\nOne Bezout identity is: 9·(-2) + 19·(1) = 1.\nReduce s modulo 19: x ≡ 17 (mod 19).\nCheck: 9·17 mod 19 = 1.\nFinal answer: $x\\equiv 17\\pmod{19}$", "final_answer": "x\\equiv 17\\pmod{19}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a926c5a026f682c", "problem_id": "c3f6f94e5971859e", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the multiplicative inverse of 9 modulo 19.", "output": "1) We want x such that 9x ≡ 1 (mod 19).\n2) Use extended Euclid to find s,t with 9s + 19t = 1.\n3) One Bezout identity is: 9·(-2) + 19·(1) = 1.\n4) Reduce s modulo 19: x ≡ 17 (mod 19).\nFinal answer: $x\\equiv 17\\pmod{19}$", "final_answer": "x\\equiv 17\\pmod{19}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fd74718b11579a6f", "problem_id": "afee69877a4048d7", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A store sells a backpack for $100. It is discounted by 35%. You buy 3 backpacks. What is the total cost?", "output": "Discounted price = 100 × (1 − 35/100) = 100 × 65/100 = 65.\nSubtotal = 3 × 65 = 195.\nFinal answer: $195$", "final_answer": "195", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "19f27663287921e3", "problem_id": "afee69877a4048d7", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A store sells a backpack for $100. It is discounted by 35%. You buy 3 backpacks. What is the total cost?", "output": "Discounted price = 100 × (1 − 35/100) = 100 × 65/100 = 65.\nSubtotal = 3 × 65 = 195.\nCheck: recomputing the formula yields 195.\nFinal answer: $195$", "final_answer": "195", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dabb60dfbd86baf4", "problem_id": "2d4dbf5149de7f81", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡3 (mod 4), x≡2 (mod 7). Give the solution modulo 28.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 28.\nCompute the CRT solution to get x ≡ 23 (mod 28).\nCheck: 23 mod 4=3 and mod 7=2.\nFinal answer: $x\\equiv 23\\pmod{28}$", "final_answer": "x\\equiv 23\\pmod{28}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a915ae1580815af8", "problem_id": "2d4dbf5149de7f81", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡3 (mod 4), x≡2 (mod 7). Give the solution modulo 28.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 28.\n3) Compute the CRT solution to get x ≡ 23 (mod 28).\nFinal answer: $x\\equiv 23\\pmod{28}$", "final_answer": "x\\equiv 23\\pmod{28}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9dea1093666c39fb", "problem_id": "288ddd022581ce76", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the multiplicative inverse of 15 modulo 23.", "output": "1) We want x such that 15x ≡ 1 (mod 23).\n2) Use extended Euclid to find s,t with 15s + 23t = 1.\n3) One Bezout identity is: 15·(-3) + 23·(2) = 1.\n4) Reduce s modulo 23: x ≡ 20 (mod 23).\nFinal answer: $x\\equiv 20\\pmod{23}$", "final_answer": "x\\equiv 20\\pmod{23}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c4e5a800037351b1", "problem_id": "288ddd022581ce76", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the multiplicative inverse of 15 modulo 23.", "output": "We want x such that 15x ≡ 1 (mod 23).\nUse extended Euclid to find s,t with 15s + 23t = 1.\nOne Bezout identity is: 15·(-3) + 23·(2) = 1.\nReduce s modulo 23: x ≡ 20 (mod 23).\nCheck: 15·20 mod 23 = 1.\nFinal answer: $x\\equiv 20\\pmod{23}$", "final_answer": "x\\equiv 20\\pmod{23}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4de731e321bba272", "problem_id": "62c7d406f4418a11", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A store sells a backpack for $126. It is discounted by 30%. You buy 9 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 126 × (1 − 30/100) = 126 × 70/100 = 441/5.\n2) Subtotal = 9 × 441/5 = 3969/5.\n3) Total with tax = 3969/5 × 108/100 = 107163/125.\nFinal answer: $\\frac{107163}{125}$", "final_answer": "\\frac{107163}{125}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9318d138c0d4ab06", "problem_id": "62c7d406f4418a11", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "A store sells a backpack for $126. It is discounted by 30%. You buy 9 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 126 × (1 − 30/100) = 126 × 70/100 = 441/5.\nSubtotal = 9 × 441/5 = 3969/5.\nTotal with tax = 3969/5 × 108/100 = 107163/125.\nFinal answer: $\\frac{107163}{125}$", "final_answer": "\\frac{107163}{125}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "da9e2923ec39a489", "problem_id": "49ed616b9b11bb44", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 x - \\frac{3 \\sqrt{3} x}{2} + 2 + 3 \\sqrt{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$-3 - \\frac{3 \\sqrt{3}}{2}$, c=$2 + 3 \\sqrt{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{31}{4} - 3 \\sqrt{3}$, so √Δ = $\\sqrt{\\frac{31}{4} - 3 \\sqrt{3}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{31 - 12 \\sqrt{3}}}{4} + \\frac{3 \\sqrt{3}}{4} + \\frac{3}{2}$.\n7) x2 = $- \\frac{\\sqrt{31 - 12 \\sqrt{3}}}{4} + \\frac{3 \\sqrt{3}}{4} + \\frac{3}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{31 - 12\\cdot \\sqrt{3}}}{4} + \\frac{3\\cdot \\sqrt{3}}{4} + \\frac{3}{2}, - \\frac{\\sqrt{31 - 12\\cdot \\sqrt{3}}}{4} + \\frac{3\\cdot \\sqrt{3}}{4} + \\frac{3}{2}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{31 - 12\\cdot \\sqrt{3}}}{4} + \\frac{3\\cdot \\sqrt{3}}{4} + \\frac{3}{2}, - \\frac{\\sqrt{31 - 12\\cdot \\sqrt{3}}}{4} + \\frac{3\\cdot \\sqrt{3}}{4} + \\frac{3}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6f958d4345d415d4", "problem_id": "fe88d3ab437c78a9", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} - x^{2} - 3 x - 3$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} - 4 x - 3.\n3) Then the limit is -8.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: -7.\nFinal answer: $-7$", "final_answer": "-7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "38510ae9a29c6cb8", "problem_id": "dae7790aacc8480d", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A store sells a backpack for $140. It is discounted by 30%. You buy 4 backpacks. What is the total cost?", "output": "1) Discounted price = 140 × (1 − 30/100) = 140 × 70/100 = 98.\n2) Subtotal = 4 × 98 = 392.\nFinal answer: $392$", "final_answer": "392", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2732bff5d9f6076c", "problem_id": "dae7790aacc8480d", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A store sells a backpack for $140. It is discounted by 30%. You buy 4 backpacks. What is the total cost?", "output": "Discounted price = 140 × (1 − 30/100) = 140 × 70/100 = 98.\nSubtotal = 4 × 98 = 392.\nFinal answer: $392$", "final_answer": "392", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e1687b4eff304a1c", "problem_id": "5ccc4a81ebce7c05", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{4} - x^{3} + 2 x^{2} + x\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 3 x^{4} - x^{3} + 2 x^{2} + x$.\n3) Compute an antiderivative: F(x) = $- \\frac{3 x^{5}}{5} - \\frac{x^{4}}{4} + \\frac{2 x^{3}}{3} + \\frac{x^{2}}{2}$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{3\\cdot x^{5}}{5} - \\frac{x^{4}}{4} + \\frac{2\\cdot x^{3}}{3} + \\frac{x^{2}}{2}$", "final_answer": "C - \\frac{3\\cdot x^{5}}{5} - \\frac{x^{4}}{4} + \\frac{2\\cdot x^{3}}{3} + \\frac{x^{2}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "283b59eaee7be44d", "problem_id": "5ccc4a81ebce7c05", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{4} - x^{3} + 2 x^{2} + x\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 3 x^{4} - x^{3} + 2 x^{2} + x$.\nCompute an antiderivative: F(x) = $- \\frac{3 x^{5}}{5} - \\frac{x^{4}}{4} + \\frac{2 x^{3}}{3} + \\frac{x^{2}}{2}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - \\frac{3\\cdot x^{5}}{5} - \\frac{x^{4}}{4} + \\frac{2\\cdot x^{3}}{3} + \\frac{x^{2}}{2}$", "final_answer": "C - \\frac{3\\cdot x^{5}}{5} - \\frac{x^{4}}{4} + \\frac{2\\cdot x^{3}}{3} + \\frac{x^{2}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9d8eb46c04788245", "problem_id": "f4d82191ee232a8d", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 21 red balls and 22 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(43,12).\n3) Favorable samples: choose k reds and n−k blues: C(21,3)·C(22,9).\n4) Compute totals: C(43,12) = 15338678264, and favorable = 661568600.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{36575}{848003}$", "final_answer": "\\frac{36575}{848003}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3ba3153236168c71", "problem_id": "f4d82191ee232a8d", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 21 red balls and 22 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(43,12).\nFavorable samples: choose k reds and n−k blues: C(21,3)·C(22,9).\nCompute totals: C(43,12) = 15338678264, and favorable = 661568600.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{36575}{848003}$.\nFinal answer: $\\frac{36575}{848003}$", "final_answer": "\\frac{36575}{848003}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ade2755283014100", "problem_id": "3875417b29c129cd", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A store sells a backpack for $30. It is discounted by 35%. You buy 3 backpacks. What is the total cost?", "output": "1) Discounted price = 30 × (1 − 35/100) = 30 × 65/100 = 39/2.\n2) Subtotal = 3 × 39/2 = 117/2.\nFinal answer: $\\frac{117}{2}$", "final_answer": "\\frac{117}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3205199c546e9cb2", "problem_id": "4f060fc8a978793b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(3 x^{2} + 2 x + 2\\right) \\cos{\\left(x - 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 \\cos{\\left(x - 3 \\right)}) = - 2 \\sin{\\left(x - 3 \\right)}$.\n3) So $f'(x) = - 3 x^{2} \\sin{\\left(x - 3 \\right)} + x e^{x} - 2 x \\sin{\\left(x - 3 \\right)} + 6 x \\cos{\\left(x - 3 \\right)} + 3 e^{x} - 2 \\sin{\\left(x - 3 \\right)} + 2 \\cos{\\left(x - 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 \\cos{\\left(x - 3 \\right)}) = - 2 \\sin{\\left(x - 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 2\\right) e^{x} + 2 \\left(3 x + 1\\right) \\cos{\\left(x - 3 \\right)} - \\left(x \\left(3 x + 2\\right) + 2\\right) \\sin{\\left(x - 3 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 2\\right) e^{x} + 2 \\left(3 x + 1\\right) \\cos{\\left(x - 3 \\right)} - \\left(x \\left(3 x + 2\\right) + 2\\right) \\sin{\\left(x - 3 \\right)} + e^{x}$", "final_answer": "\\left(x + 2\\right) e^{x} + 2 \\left(3 x + 1\\right) \\cos{\\left(x - 3 \\right)} - \\left(x \\left(3 x + 2\\right) + 2\\right) \\sin{\\left(x - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "73d8bdd492d3e02e", "problem_id": "661a25de9ebf8cc3", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A store sells a backpack for $395. It is discounted by 30%. You buy 9 backpacks. What is the total cost?", "output": "1) Discounted price = 395 × (1 − 30/100) = 395 × 70/100 = 553/2.\n2) Subtotal = 9 × 553/2 = 4977/2.\nFinal answer: $\\frac{4977}{2}$", "final_answer": "\\frac{4977}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f88994e49a6c253", "problem_id": "661a25de9ebf8cc3", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "A store sells a backpack for $395. It is discounted by 30%. You buy 9 backpacks. What is the total cost?", "output": "$\\frac{4977}{2}$", "final_answer": "\\frac{4977}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "981f1616ac686ebb", "problem_id": "8350be13cf79e6a0", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{24 x}{5} + \\frac{14}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$\\frac{24}{5}$, c=$\\frac{14}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{16}{25}$, so √Δ = $\\frac{4}{5}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $-1$.\n7) x2 = $- \\frac{7}{5}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{-1, - \\frac{7}{5}\\}$", "final_answer": "x \\in \\{-1, - \\frac{7}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db387bc3e268ce04", "problem_id": "8350be13cf79e6a0", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{24 x}{5} + \\frac{14}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$\\frac{24}{5}$, c=$\\frac{14}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{16}{25}$, so √Δ = $\\frac{4}{5}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $-1$.\nx2 = $- \\frac{7}{5}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-1, - \\frac{7}{5}\\}$", "final_answer": "x \\in \\{-1, - \\frac{7}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6d926e64cafa445b", "problem_id": "94cffce7e5cccf76", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 2 x - 3$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $2 x^{2} + x - 1$ and a = -2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $4 x + 1$.\n5) Evaluate at a: p'(-2) = $-7$.\nFinal answer: $-7$", "final_answer": "-7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f11382da586d824", "problem_id": "94cffce7e5cccf76", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 2 x - 3$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{2} + x - 1$ and a = -2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $4 x + 1$.\nEvaluate at a: p'(-2) = $-7$.\nCheck: the identity gives $-7$.\nFinal answer: $-7$", "final_answer": "-7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b7a90348edfb125", "problem_id": "e8ea7e17163d28ac", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(3 x^{3} + 2 x^{2} + 2 x - 3\\right) \\cos{\\left(3 x^{2} + x \\right)}$.", "output": "1) Let f(x) = $2 \\left(x + 1\\right) e^{x} + \\left(3 x^{3} + 2 x^{2} + 2 x - 3\\right) \\cos{\\left(3 x^{2} + x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $2 \\left(x + 1\\right) e^{x} - \\left(6 x + 1\\right) \\left(3 x^{3} + 2 x^{2} + 2 x - 3\\right) \\sin{\\left(3 x^{2} + x \\right)} + \\left(9 x^{2} + 4 x + 2\\right) \\cos{\\left(3 x^{2} + x \\right)} + 2 e^{x}$.\nFinal answer: $\\left(2\\cdot x + 2\\right)\\cdot e^{x} - \\left(6\\cdot x + 1\\right)\\cdot \\left(3\\cdot x^{3} + 2\\cdot x^{2} + 2\\cdot x - 3\\right)\\cdot \\sin{\\left(3\\cdot x^{2} + x \\right)} + \\left(9\\cdot x^{2} + 4\\cdot x + 2\\right)\\cdot \\cos{\\left(3\\cdot x^{2} + x \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x + 2\\right)\\cdot e^{x} - \\left(6\\cdot x + 1\\right)\\cdot \\left(3\\cdot x^{3} + 2\\cdot x^{2} + 2\\cdot x - 3\\right)\\cdot \\sin{\\left(3\\cdot x^{2} + x \\right)} + \\left(9\\cdot x^{2} + 4\\cdot x + 2\\right)\\cdot \\cos{\\left(3\\cdot x^{2} + x \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2dc5f411881ac26b", "problem_id": "e8ea7e17163d28ac", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(3 x^{3} + 2 x^{2} + 2 x - 3\\right) \\cos{\\left(3 x^{2} + x \\right)}$.", "output": "Let f(x) = $2 \\left(x + 1\\right) e^{x} + \\left(3 x^{3} + 2 x^{2} + 2 x - 3\\right) \\cos{\\left(3 x^{2} + x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $2 \\left(x + 1\\right) e^{x} - \\left(6 x + 1\\right) \\left(3 x^{3} + 2 x^{2} + 2 x - 3\\right) \\sin{\\left(3 x^{2} + x \\right)} + \\left(9 x^{2} + 4 x + 2\\right) \\cos{\\left(3 x^{2} + x \\right)} + 2 e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\left(2\\cdot x + 2\\right)\\cdot e^{x} - \\left(6\\cdot x + 1\\right)\\cdot \\left(3\\cdot x^{3} + 2\\cdot x^{2} + 2\\cdot x - 3\\right)\\cdot \\sin{\\left(3\\cdot x^{2} + x \\right)} + \\left(9\\cdot x^{2} + 4\\cdot x + 2\\right)\\cdot \\cos{\\left(3\\cdot x^{2} + x \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x + 2\\right)\\cdot e^{x} - \\left(6\\cdot x + 1\\right)\\cdot \\left(3\\cdot x^{3} + 2\\cdot x^{2} + 2\\cdot x - 3\\right)\\cdot \\sin{\\left(3\\cdot x^{2} + x \\right)} + \\left(9\\cdot x^{2} + 4\\cdot x + 2\\right)\\cdot \\cos{\\left(3\\cdot x^{2} + x \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b063d4d1381cb837", "problem_id": "c19a58d418fe2b7f", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 x - 18$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} - 3 x - 18$ = $\\left(x - 6\\right) \\left(x + 3\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $6$.\n5) From the second factor: x = $-3$.\nFinal answer: $x \\in \\{6, -3\\}$", "final_answer": "x \\in \\{6, -3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2c3efe7366c4b03b", "problem_id": "c19a58d418fe2b7f", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 x - 18$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} - 3 x - 18$ = $\\left(x - 6\\right) \\left(x + 3\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $6$.\nFrom the second factor: x = $-3$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{6, -3\\}$", "final_answer": "x \\in \\{6, -3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b9fa3810660aa85b", "problem_id": "7773e2af1e2cad87", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡5 (mod 8), x≡2 (mod 7). Give the solution modulo 56.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 56.\n3) Compute the CRT solution to get x ≡ 37 (mod 56).\nFinal answer: $x\\equiv 37\\pmod{56}$", "final_answer": "x\\equiv 37\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "464e6ae5a47e142f", "problem_id": "7773e2af1e2cad87", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡5 (mod 8), x≡2 (mod 7). Give the solution modulo 56.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 56.\nCompute the CRT solution to get x ≡ 37 (mod 56).\nCheck: 37 mod 8=5 and mod 7=2.\nFinal answer: $x\\equiv 37\\pmod{56}$", "final_answer": "x\\equiv 37\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a86fd73e1ad32f0d", "problem_id": "c4af0e4fb58b299b", "style": "answer_only", "instruction": "Final answer only.", "input": "An urn has 19 red balls and 12 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "$\\frac{14212}{58435}$", "final_answer": "\\frac{14212}{58435}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6e795cc01d21f061", "problem_id": "c4af0e4fb58b299b", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 19 red balls and 12 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(31,8).\n3) Favorable samples: choose k reds and n−k blues: C(19,4)·C(12,4).\n4) Compute totals: C(31,8) = 7888725, and favorable = 1918620.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{14212}{58435}$", "final_answer": "\\frac{14212}{58435}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "352398c9006bdab8", "problem_id": "96eefa8fbfdd841d", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Two friends split $1430 in the ratio 8:2. How much does each receive?", "output": "1) Total ratio parts = 8+2 = 10.\n2) Each part = 1430 ÷ 10 = 143.\n3) First share = 8 × 143 = 1144.\n4) Second share = 2 × 143 = 286.\nFinal answer: $286$", "final_answer": "286", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6fc7279978d690d5", "problem_id": "96eefa8fbfdd841d", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Two friends split $1430 in the ratio 8:2. How much does each receive?", "output": "Total ratio parts = 8+2 = 10.\nEach part = 1430 ÷ 10 = 143.\nFirst share = 8 × 143 = 1144.\nFinal answer: $286$", "final_answer": "286", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d24bb0ddee31c625", "problem_id": "395da71e080223d7", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "An urn has 19 red balls and 10 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(29,12).\nFavorable samples: choose k reds and n−k blues: C(19,5)·C(10,7).\nFinal answer: $\\frac{1632}{60697}$", "final_answer": "\\frac{1632}{60697}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b974e32d76faa21b", "problem_id": "395da71e080223d7", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 19 red balls and 10 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(29,12).\nFavorable samples: choose k reds and n−k blues: C(19,5)·C(10,7).\nCompute totals: C(29,12) = 51895935, and favorable = 1395360.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1632}{60697}$.\nFinal answer: $\\frac{1632}{60697}$", "final_answer": "\\frac{1632}{60697}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a784242c6f6e9ea", "problem_id": "cb5b777feb33973d", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A store sells a backpack for $62. It is discounted by 35%. You buy 5 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 62 × (1 − 35/100) = 62 × 65/100 = 403/10.\n2) Subtotal = 5 × 403/10 = 403/2.\n3) Total with tax = 403/2 × 108/100 = 10881/50.\nFinal answer: $\\frac{10881}{50}$", "final_answer": "\\frac{10881}{50}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6286495c26b1df60", "problem_id": "cb5b777feb33973d", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A store sells a backpack for $62. It is discounted by 35%. You buy 5 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 62 × (1 − 35/100) = 62 × 65/100 = 403/10.\nSubtotal = 5 × 403/10 = 403/2.\nTotal with tax = 403/2 × 108/100 = 10881/50.\nCheck: recomputing the formula yields 10881/50.\nFinal answer: $\\frac{10881}{50}$", "final_answer": "\\frac{10881}{50}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "77c48e3664789850", "problem_id": "9d3d72eaf058d2a7", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} 3 x^{3} + 6 x^{2} + 15 x + 48$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{4} - 3 x^{3} - 3 x^{2} + 3 x + 2$ and a = 3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $12 x^{3} - 9 x^{2} - 6 x + 3$.\n5) Evaluate at a: p'(3) = $228$.\nFinal answer: $228$", "final_answer": "228", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "facdefb2735f2b4e", "problem_id": "9d3d72eaf058d2a7", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} 3 x^{3} + 6 x^{2} + 15 x + 48$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $3 x^{4} - 3 x^{3} - 3 x^{2} + 3 x + 2$ and a = 3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $12 x^{3} - 9 x^{2} - 6 x + 3$.\nEvaluate at a: p'(3) = $228$.\nCheck: the identity gives $228$.\nFinal answer: $228$", "final_answer": "228", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0310c8076ac0e1c1", "problem_id": "2c6c3e7fba61902d", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$- x - y$ = $-2$\n$- 2 y$ = $0$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $-1$: $2 y$ = $0$.\n4) Subtract to eliminate x: $- 2 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $2$.\nFinal answer: $(x,y)=\\left(2, 0\\right)$", "final_answer": "(x,y)=\\left(2, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2f865d8c8301642d", "problem_id": "2c6c3e7fba61902d", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the linear system:\n$- x - y$ = $-2$\n$- 2 y$ = $0$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $-1$: $2 y$ = $0$.\nFinal answer: $(x,y)=\\left(2, 0\\right)$", "final_answer": "(x,y)=\\left(2, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eae952d9ffd820c8", "problem_id": "6814324ada013905", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 1$ at $x=1$.", "output": "1) Differentiate: f'(x) = $6 x + 2 \\cos{\\left(x \\right)} + 2$.\n2) Slope at x=1: m = f'(1) = $2 \\cos{\\left(1 \\right)} + 8$.\n3) Point on curve: (1, f(1)) = (1, $2 \\sin{\\left(1 \\right)} + 4$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 2 \\sin{\\left(1 \\right)} + 4 = 2 \\cos{\\left(1 \\right)} + 8(x − 1).\n6) Optionally expand: y = $2 x \\cos{\\left(1 \\right)} + 8 x - 4 - 2 \\cos{\\left(1 \\right)} + 2 \\sin{\\left(1 \\right)}$.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 8\\cdot x - 4 - 2\\cdot \\cos{\\left(1 \\right)} + 2\\cdot \\sin{\\left(1 \\right)}$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 8\\cdot x - 4 - 2\\cdot \\cos{\\left(1 \\right)} + 2\\cdot \\sin{\\left(1 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "460d5c626cdc8a2f", "problem_id": "6814324ada013905", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 1$ at $x=1$.", "output": "Differentiate: f'(x) = $6 x + 2 \\cos{\\left(x \\right)} + 2$.\nSlope at x=1: m = f'(1) = $2 \\cos{\\left(1 \\right)} + 8$.\nPoint on curve: (1, f(1)) = (1, $2 \\sin{\\left(1 \\right)} + 4$).\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 8\\cdot x - 4 - 2\\cdot \\cos{\\left(1 \\right)} + 2\\cdot \\sin{\\left(1 \\right)}$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 8\\cdot x - 4 - 2\\cdot \\cos{\\left(1 \\right)} + 2\\cdot \\sin{\\left(1 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c7894cb37706c63f", "problem_id": "97bd8a26c70084ec", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A cyclist rides at 60 miles per hour for 1 hours, then at 23 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 60 × 1 = 60 miles.\n2) Second leg distance = 23 × 3 = 69 miles.\n3) Total distance = 60 + 69 = 129 miles.\nFinal answer: $129$", "final_answer": "129", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5f460837b17094a4", "problem_id": "97bd8a26c70084ec", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "A cyclist rides at 60 miles per hour for 1 hours, then at 23 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 60 × 1 = 60 miles.\nSecond leg distance = 23 × 3 = 69 miles.\nTotal distance = 60 + 69 = 129 miles.\nFinal answer: $129$", "final_answer": "129", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9a2d6658fb5298c8", "problem_id": "289323fa25e5ee4c", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} - 3 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\n2) Evaluate at the bounds: F(\\pi) and F(0).\n3) Subtract: F(\\pi) − F(0) = $- \\pi^{3} - 2 \\pi - \\frac{\\pi^{2}}{2} + 4$.\n4) Simplify the result.\nFinal answer: $- \\pi^{3} - 2\\cdot \\pi - \\frac{\\pi^{2}}{2} + 4$", "final_answer": "- \\pi^{3} - 2\\cdot \\pi - \\frac{\\pi^{2}}{2} + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1d57bbe9be13433f", "problem_id": "289323fa25e5ee4c", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} - 3 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\nEvaluate at the bounds: F(\\pi) and F(0).\nSubtract: F(\\pi) − F(0) = $- \\pi^{3} - 2 \\pi - \\frac{\\pi^{2}}{2} + 4$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\pi^{3} - 2\\cdot \\pi - \\frac{\\pi^{2}}{2} + 4$", "final_answer": "- \\pi^{3} - 2\\cdot \\pi - \\frac{\\pi^{2}}{2} + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "205690f08bd1d5c3", "problem_id": "b0c9a55baa2929e5", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Worker A finishes a job in 10 hours and Worker B finishes it in 18 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/10 job/hour. B's rate = 1/18 job/hour.\n2) Combined rate = 1/10 + 1/18 = 7/45 job/hour.\n3) Time = 1 ÷ (combined rate) = 45/7 hours.\nFinal answer: $\\frac{45}{7}$", "final_answer": "\\frac{45}{7}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4b6214079af98a74", "problem_id": "08dad0ffebbc8c3d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x - 4$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 2 x - 4.\n3) Then the limit is -8.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -6.\nFinal answer: $-6$", "final_answer": "-6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "fa53459510b38c70", "problem_id": "e098da9a43c81b68", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many binary strings of length 20 contain exactly 5 ones with no two ones adjacent?", "output": "1) Reserve 4 mandatory zeros between the 5 ones.\n2) This leaves n−k+1 = 16 valid slots for the ones.\n3) Choose the slots: C(16,5) = 4368.\nFinal answer: $4368$", "final_answer": "4368", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5461482f44253ee5", "problem_id": "e098da9a43c81b68", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many binary strings of length 20 contain exactly 5 ones with no two ones adjacent?", "output": "Reserve 4 mandatory zeros between the 5 ones.\nThis leaves n−k+1 = 16 valid slots for the ones.\nChoose the slots: C(16,5) = 4368.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $4368$", "final_answer": "4368", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d18a310fcf7ea042", "problem_id": "d5161946143d352a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{2 x^{2} + 3 x + \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{x + 2}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 2}) = \\frac{2 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 2}$.\n3) So $f'(x) = \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 8 x + 8} + \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + 8 x + 8} + \\frac{4 x^{2}}{2 x^{2} + 8 x + 8} + \\frac{4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 8 x + 8} + \\frac{8 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + 8 x + 8} + \\frac{16 x}{2 x^{2} + 8 x + 8} + \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 8 x + 8} + \\frac{8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + 8 x + 8} + \\frac{8}{2 x^{2} + 8 x + 8} + \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 4 x + 4} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 2} - \\frac{2 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 2} + \\frac{2 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 2}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 2}) = \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 2} + \\frac{2 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 2} - \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x + 2\\right)^{2}}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 4 x^{2} - 6 x + \\left(x + 2\\right) \\left(8 x + \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4}{2 \\left(x + 2\\right)^{2}}$.\nFinal answer: $\\frac{- 4 x^{2} - 6 x + \\left(x + 2\\right) \\left(8 x + \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4}{2 \\left(x + 2\\right)^{2}}$", "final_answer": "\\frac{- 4 x^{2} - 6 x + \\left(x + 2\\right) \\left(8 x + \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4}{2 \\left(x + 2\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "60419e9098b6870f", "problem_id": "0855e8fa430a5fb9", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡1 (mod 7), x≡8 (mod 11). Give the solution modulo 77.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 77.\n3) Compute the CRT solution to get x ≡ 8 (mod 77).\nFinal answer: $x\\equiv 8\\pmod{77}$", "final_answer": "x\\equiv 8\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e1b9d0a90dd389c5", "problem_id": "d3c672e2a47d0653", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the multiplicative inverse of 8 modulo 43.", "output": "1) We want x such that 8x ≡ 1 (mod 43).\n2) Use extended Euclid to find s,t with 8s + 43t = 1.\n3) One Bezout identity is: 8·(-16) + 43·(3) = 1.\n4) Reduce s modulo 43: x ≡ 27 (mod 43).\nFinal answer: $x\\equiv 27\\pmod{43}$", "final_answer": "x\\equiv 27\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4e6931ad2e9d6e58", "problem_id": "d3c672e2a47d0653", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the multiplicative inverse of 8 modulo 43.", "output": "We want x such that 8x ≡ 1 (mod 43).\nUse extended Euclid to find s,t with 8s + 43t = 1.\nOne Bezout identity is: 8·(-16) + 43·(3) = 1.\nReduce s modulo 43: x ≡ 27 (mod 43).\nCheck: 8·27 mod 43 = 1.\nFinal answer: $x\\equiv 27\\pmod{43}$", "final_answer": "x\\equiv 27\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "253fed3d945bc175", "problem_id": "60f6e06325e659f5", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 12 red balls and 13 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(25,4).\n3) Favorable samples: choose k reds and n−k blues: C(12,2)·C(13,2).\n4) Compute totals: C(25,4) = 12650, and favorable = 5148.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{234}{575}$", "final_answer": "\\frac{234}{575}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3c0d477a2ae7f5ab", "problem_id": "60f6e06325e659f5", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 12 red balls and 13 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(25,4).\nFavorable samples: choose k reds and n−k blues: C(12,2)·C(13,2).\nCompute totals: C(25,4) = 12650, and favorable = 5148.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{234}{575}$.\nFinal answer: $\\frac{234}{575}$", "final_answer": "\\frac{234}{575}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d95a02340edbe8c7", "problem_id": "e04dd85641fb8bac", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "You want 35 liters of a 30% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 35−x liters is 50% solution.\n2) Salt equation: 25x + 50(35−x) = 30·35.\n3) (25-50)x = (30-50)·35 ⇒ x = 28.\n4) Then 35−x = 7.\nFinal answer: $28 L of 25% and 7 L of 50%$", "final_answer": "28 L of 25% and 7 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eeed3c2d9323efb2", "problem_id": "e04dd85641fb8bac", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "You want 35 liters of a 30% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 35−x liters is 50% solution.\nSalt equation: 25x + 50(35−x) = 30·35.\n(25-50)x = (30-50)·35 ⇒ x = 28.\nThen 35−x = 7.\nCheck: 25·28 + 50·7 = 30·35.\nFinal answer: $28 L of 25% and 7 L of 50%$", "final_answer": "28 L of 25% and 7 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6433f655db488893", "problem_id": "efb79af1b7251516", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_7=23$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_7=16.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=16 and k=7.\nCompute C(22,6) = 74613.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $74613$", "final_answer": "74613", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a13f0ee9d7621514", "problem_id": "d903d488291fd67c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 2 x^{2} + 2 x + \\left(x^{2} + 3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{x^{2} + 3 x - 5}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 5}) = \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3 x - 5}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{6 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{12 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{16 x^{2}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{6 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 6 x^{3} - x^{2} - 30 x + 25} - \\frac{30 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{60 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{48 x}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{9 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 6 x^{3} - x^{2} - 30 x + 25} - \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 6 x - 10} + \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3 x - 5} - \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 5} + \\frac{25 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{50 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{8}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 5}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 5}) = \\frac{3 x \\left(- 2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(x^{2} + 3 x\\right) - 5\\right)^{2}} + \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(x^{2} + 3 x\\right) - 5\\right)} + \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 5} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 5}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(- 2 x - 3\\right) \\left(- 2 x^{2} + 2 x + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(x \\left(x + 3\\right) - 5\\right) \\left(- 8 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x + 3\\right) - 5\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(- 2 x - 3\\right) \\left(- 2 x^{2} + 2 x + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(x \\left(x + 3\\right) - 5\\right) \\left(- 8 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x + 3\\right) - 5\\right)^{2}}$", "final_answer": "\\frac{2 \\left(- 2 x - 3\\right) \\left(- 2 x^{2} + 2 x + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(x \\left(x + 3\\right) - 5\\right) \\left(- 8 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x + 3\\right) - 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b9ba815ca4e46c73", "problem_id": "1ef53eb30eac9167", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the equation of the tangent line to $y=x^{2} + 3$ at $x=0$.", "output": "1) Differentiate: f'(x) = $2 x$.\n2) Slope at x=0: m = f'(0) = $0$.\n3) Point on curve: (0, f(0)) = (0, $3$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 3 = 0(x − 0).\n6) Optionally expand: y = $3$.\nFinal answer: $y = 3$", "final_answer": "y = 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "40f96c2700ff0257", "problem_id": "1ef53eb30eac9167", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the equation of the tangent line to $y=x^{2} + 3$ at $x=0$.", "output": "Differentiate: f'(x) = $2 x$.\nSlope at x=0: m = f'(0) = $0$.\nPoint on curve: (0, f(0)) = (0, $3$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 3 = 0(x − 0).\nOptionally expand: y = $3$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 3$", "final_answer": "y = 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "36832f18acbc3d4f", "problem_id": "bd153da715513b1e", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 8 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=8, k=3, p=$\\frac{3}{8}$ and simplify.\n4) Here C(8,3) = 56.\nFinal answer: $\\frac{590625}{2097152}$", "final_answer": "\\frac{590625}{2097152}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3571ca31696a06b4", "problem_id": "bd153da715513b1e", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 8 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=8, k=3, p=$\\frac{3}{8}$ and simplify.\nFinal answer: $\\frac{590625}{2097152}$", "final_answer": "\\frac{590625}{2097152}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6cb9134969234be9", "problem_id": "7d78fe49f8c1878b", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many binary strings of length 25 contain exactly 7 ones with no two ones adjacent?", "output": "1) Reserve 6 mandatory zeros between the 7 ones.\n2) This leaves n−k+1 = 19 valid slots for the ones.\n3) Choose the slots: C(19,7) = 50388.\nFinal answer: $50388$", "final_answer": "50388", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "605e29252878cfca", "problem_id": "7d78fe49f8c1878b", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many binary strings of length 25 contain exactly 7 ones with no two ones adjacent?", "output": "Reserve 6 mandatory zeros between the 7 ones.\nThis leaves n−k+1 = 19 valid slots for the ones.\nChoose the slots: C(19,7) = 50388.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $50388$", "final_answer": "50388", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f2e47ab320bdf035", "problem_id": "a45dba5993de25e1", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A store sells a backpack for $770. It is discounted by 40%. You buy 7 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 770 × (1 − 40/100) = 770 × 60/100 = 462.\n2) Subtotal = 7 × 462 = 3234.\n3) Total with tax = 3234 × 108/100 = 87318/25.\nFinal answer: $\\frac{87318}{25}$", "final_answer": "\\frac{87318}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2cc37839485281be", "problem_id": "a45dba5993de25e1", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $770. It is discounted by 40%. You buy 7 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 770 × (1 − 40/100) = 770 × 60/100 = 462.\nSubtotal = 7 × 462 = 3234.\nTotal with tax = 3234 × 108/100 = 87318/25.\nCheck: recomputing the formula yields 87318/25.\nFinal answer: $\\frac{87318}{25}$", "final_answer": "\\frac{87318}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d467dd64d51802a", "problem_id": "77059cff0e0e9a9a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} x^{2} + 2$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to x^{2} - x + 2.\n3) Then the limit is 2.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: 3.\nFinal answer: $3$", "final_answer": "3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d6591cf43dfb8853", "problem_id": "3bc7af426a09a19a", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the equation of the tangent line to $y=x^{3} + 3 x^{2} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)}$ at $x=-2$.", "output": "Differentiate: f'(x) = $3 x^{2} + 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\nSlope at x=-2: m = f'(-2) = $-2 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}$.\nPoint on curve: (-2, f(-2)) = (-2, $\\cos{\\left(4 \\right)} + e^{-1} + 8$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − \\cos{\\left(4 \\right)} + e^{-1} + 8 = -2 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}(x − -2).\nOptionally expand: y = $- 2 x + 2 x \\sin{\\left(4 \\right)} + \\frac{x}{2 e} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 4$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 2\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 4$", "final_answer": "y = - 2\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "adccf8a3cda95046", "problem_id": "93b58f8747b5969b", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x^{3} + 5 x^{2} + 11 x + 25$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{4} + 3 x^{3} + x^{2} + 3 x + 1$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $4 x^{3} + 9 x^{2} + 2 x + 3$.\n5) Evaluate at a: p'(2) = $75$.\nFinal answer: $75$", "final_answer": "75", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aad75a81256d8a40", "problem_id": "93b58f8747b5969b", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x^{3} + 5 x^{2} + 11 x + 25$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $x^{4} + 3 x^{3} + x^{2} + 3 x + 1$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $4 x^{3} + 9 x^{2} + 2 x + 3$.\nEvaluate at a: p'(2) = $75$.\nCheck: the identity gives $75$.\nFinal answer: $75$", "final_answer": "75", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d3609609acc08a88", "problem_id": "7027ce7f370fdc0a", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$6 x - 2 y$ = $-32$\n$x + 8 y$ = $-72$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $6 x - 2 y$ = $-32$.\n3) Multiply the second equation by $6$: $6 x + 48 y$ = $-432$.\n4) Subtract to eliminate x: $- 50 y$ = $400$.\n5) Solve for y: y = $-8$.\n6) Substitute back to find x: x = $-8$.\nFinal answer: $(x,y)=\\left(-8, -8\\right)$", "final_answer": "(x,y)=\\left(-8, -8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "349f1dc990cab7c4", "problem_id": "efeea33b1dba98bf", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{2} - \\frac{2 x}{3} + \\left(x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - \\frac{1}{3}}{x + 1}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 1}) = \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 1}$.\n3) So $f'(x) = \\frac{3 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{2} + 12 x + 6} + \\frac{6 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{2} + 12 x + 6} + \\frac{6 x^{2}}{6 x^{2} + 12 x + 6} + \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{2} + 12 x + 6} + \\frac{12 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{2} + 12 x + 6} + \\frac{12 x}{6 x^{2} + 12 x + 6} + \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 2 x + 1} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x + 2} - \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 1} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 1} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{2} + 12 x + 6} + \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{2} + 12 x + 6} - \\frac{2}{6 x^{2} + 12 x + 6} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 1}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 1}) = \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x + 1\\right)} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 1} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x + 1\\right)^{2}} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 1}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 6 x^{2} + 4 x + \\left(x + 1\\right) \\left(12 x + 3 \\left(x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4\\right) - 6 \\left(x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{6 \\left(x + 1\\right)^{2}}$.\nFinal answer: $\\frac{- 6 x^{2} + 4 x + \\left(x + 1\\right) \\left(12 x + 3 \\left(x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4\\right) - 6 \\left(x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{6 \\left(x + 1\\right)^{2}}$", "final_answer": "\\frac{- 6 x^{2} + 4 x + \\left(x + 1\\right) \\left(12 x + 3 \\left(x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4\\right) - 6 \\left(x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{6 \\left(x + 1\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "408c5358d21abbd6", "problem_id": "d96792a3e04f99f9", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 7 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=7, k=4, p=$\\frac{1}{6}$ and simplify.\n4) Here C(7,4) = 35.\nFinal answer: $\\frac{4375}{279936}$", "final_answer": "\\frac{4375}{279936}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "76b4524977f5256a", "problem_id": "30d6cf9a97495d00", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{x^{2} + 3}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 3}) = \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} - \\frac{2 x^{4}}{2 x^{4} + 12 x^{2} + 18} + \\frac{6 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} + \\frac{12 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} - \\frac{12 x^{2}}{2 x^{4} + 12 x^{2} + 18} + \\frac{36 x}{2 x^{4} + 12 x^{2} + 18} + \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 6 x^{2} + 9} + \\frac{9 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} + \\frac{18 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} - \\frac{18}{2 x^{4} + 12 x^{2} + 18} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 6} - \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3} + \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 3}) = - \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3\\right)^{2}} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x^{2} + 3\\right)} + \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 4 x \\left(- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} + 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x^{2} + 3\\right)^{2}}$.\nFinal answer: $\\frac{- 4 x \\left(- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} + 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x^{2} + 3\\right)^{2}}$", "final_answer": "\\frac{- 4 x \\left(- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} + 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x^{2} + 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7427416ff154c886", "problem_id": "d903d488291fd67c", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 2 x^{2} + 2 x + \\left(x^{2} + 3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{x^{2} + 3 x - 5}$.", "output": "1) Let f(x) = $\\frac{- 2 x^{2} + 2 x + \\left(x^{2} + 3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{x^{2} + 3 x - 5}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{2 \\left(- 2 x - 3\\right) \\left(- 2 x^{2} + 2 x + \\left(x^{2} + 3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(x^{2} + 3 x - 5\\right) \\left(- 8 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} + 3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 3 x - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x^{2} + 3 x - 5\\right)^{2}}$.\nFinal answer: $\\frac{\\left(- 4\\cdot x - 6\\right)\\cdot \\left(- 2\\cdot x^{2} + 2\\cdot x + \\left(x^{2} + 3\\cdot x - 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right) + \\left(x^{2} + 3\\cdot x - 5\\right)\\cdot \\left(- 8\\cdot x + \\left(4\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(x^{2} + 3\\cdot x - 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 6\\cdot x - 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\right)}{2\\cdot \\left(x^{2} + 3\\cdot x - 5\\right)^{2}}$", "final_answer": "\\frac{\\left(- 4\\cdot x - 6\\right)\\cdot \\left(- 2\\cdot x^{2} + 2\\cdot x + \\left(x^{2} + 3\\cdot x - 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right) + \\left(x^{2} + 3\\cdot x - 5\\right)\\cdot \\left(- 8\\cdot x + \\left(4\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(x^{2} + 3\\cdot x - 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 6\\cdot x - 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\right)}{2\\cdot \\left(x^{2} + 3\\cdot x - 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "24adee0e7d362604", "problem_id": "d903d488291fd67c", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 2 x^{2} + 2 x + \\left(x^{2} + 3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{x^{2} + 3 x - 5}$.", "output": "Let f(x) = $\\frac{- 2 x^{2} + 2 x + \\left(x^{2} + 3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{x^{2} + 3 x - 5}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{2 \\left(- 2 x - 3\\right) \\left(- 2 x^{2} + 2 x + \\left(x^{2} + 3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(x^{2} + 3 x - 5\\right) \\left(- 8 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} + 3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 3 x - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x^{2} + 3 x - 5\\right)^{2}}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\frac{\\left(- 4\\cdot x - 6\\right)\\cdot \\left(- 2\\cdot x^{2} + 2\\cdot x + \\left(x^{2} + 3\\cdot x - 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right) + \\left(x^{2} + 3\\cdot x - 5\\right)\\cdot \\left(- 8\\cdot x + \\left(4\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(x^{2} + 3\\cdot x - 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 6\\cdot x - 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\right)}{2\\cdot \\left(x^{2} + 3\\cdot x - 5\\right)^{2}}$", "final_answer": "\\frac{\\left(- 4\\cdot x - 6\\right)\\cdot \\left(- 2\\cdot x^{2} + 2\\cdot x + \\left(x^{2} + 3\\cdot x - 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right) + \\left(x^{2} + 3\\cdot x - 5\\right)\\cdot \\left(- 8\\cdot x + \\left(4\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(x^{2} + 3\\cdot x - 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 6\\cdot x - 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\right)}{2\\cdot \\left(x^{2} + 3\\cdot x - 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1b65a0b94891e805", "problem_id": "5e225562f6fb4d29", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 9 times, what is the probability of exactly 6 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=9, k=6, p=$\\frac{1}{5}$ and simplify.\nHere C(9,6) = 84.\nCheck: expression simplifies to $\\frac{5376}{1953125}$.\nFinal answer: $\\frac{5376}{1953125}$", "final_answer": "\\frac{5376}{1953125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3a72f760bf0b7e03", "problem_id": "5e225562f6fb4d29", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 9 times, what is the probability of exactly 6 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=9, k=6, p=$\\frac{1}{5}$ and simplify.\nFinal answer: $\\frac{5376}{1953125}$", "final_answer": "\\frac{5376}{1953125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "25f1a9fe81c81e0b", "problem_id": "f8bed3583b7ad7c9", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Two friends split $2170 in the ratio 9:5. How much does each receive?", "output": "1) Total ratio parts = 9+5 = 14.\n2) Each part = 2170 ÷ 14 = 155.\n3) First share = 9 × 155 = 1395.\n4) Second share = 5 × 155 = 775.\nFinal answer: $775$", "final_answer": "775", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b73036e15175b01a", "problem_id": "f8bed3583b7ad7c9", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $2170 in the ratio 9:5. How much does each receive?", "output": "Total ratio parts = 9+5 = 14.\nEach part = 2170 ÷ 14 = 155.\nFirst share = 9 × 155 = 1395.\nSecond share = 5 × 155 = 775.\nCheck: 1395+775=2170.\nFinal answer: $775$", "final_answer": "775", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4924aaed27d85c8b", "problem_id": "d1f5457009ed2ff5", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$4 x + y$ = $-24$\n$- 2 x + 3 y$ = $-2$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $- 8 x - 2 y$ = $48$.\n3) Multiply the second equation by $4$: $- 8 x + 12 y$ = $-8$.\n4) Subtract to eliminate x: $- 14 y$ = $56$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, -4\\right)$", "final_answer": "(x,y)=\\left(-5, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5c4f5e60c9d048a2", "problem_id": "d1f5457009ed2ff5", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$4 x + y$ = $-24$\n$- 2 x + 3 y$ = $-2$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $- 8 x - 2 y$ = $48$.\nMultiply the second equation by $4$: $- 8 x + 12 y$ = $-8$.\nSubtract to eliminate x: $- 14 y$ = $56$.\nSolve for y: y = $-4$.\nSubstitute back to find x: x = $-5$.\nCheck: plug (x,y)=(-5,-4) into both equations.\nFinal answer: $(x,y)=\\left(-5, -4\\right)$", "final_answer": "(x,y)=\\left(-5, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3775d034e0421097", "problem_id": "bab7b9519a5912d6", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A store sells a backpack for $134. It is discounted by 5%. You buy 4 backpacks. What is the total cost?", "output": "1) Discounted price = 134 × (1 − 5/100) = 134 × 95/100 = 1273/10.\n2) Subtotal = 4 × 1273/10 = 2546/5.\nFinal answer: $\\frac{2546}{5}$", "final_answer": "\\frac{2546}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4e3f55a824191980", "problem_id": "bab7b9519a5912d6", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A store sells a backpack for $134. It is discounted by 5%. You buy 4 backpacks. What is the total cost?", "output": "Discounted price = 134 × (1 − 5/100) = 134 × 95/100 = 1273/10.\nSubtotal = 4 × 1273/10 = 2546/5.\nFinal answer: $\\frac{2546}{5}$", "final_answer": "\\frac{2546}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a7d850180a4b61db", "problem_id": "d021ae6ed3bc7191", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A cyclist rides at 74 miles per hour for 2 hours, then at 51 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 74 × 2 = 148 miles.\n2) Second leg distance = 51 × 1 = 51 miles.\n3) Total distance = 148 + 51 = 199 miles.\nFinal answer: $199$", "final_answer": "199", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "24463afcf9435ab3", "problem_id": "d021ae6ed3bc7191", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 74 miles per hour for 2 hours, then at 51 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 74 × 2 = 148 miles.\nSecond leg distance = 51 × 1 = 51 miles.\nTotal distance = 148 + 51 = 199 miles.\nCheck: adding both legs gives 199.\nFinal answer: $199$", "final_answer": "199", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bfe86e784ea6bc69", "problem_id": "303add7924f2b2a3", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡1 (mod 8), x≡1 (mod 7). Give the solution modulo 56.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 56.\n3) Compute the CRT solution to get x ≡ 1 (mod 56).\nFinal answer: $x\\equiv 1\\pmod{56}$", "final_answer": "x\\equiv 1\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "516becb7b8b01e04", "problem_id": "303add7924f2b2a3", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡1 (mod 8), x≡1 (mod 7). Give the solution modulo 56.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 56.\nCompute the CRT solution to get x ≡ 1 (mod 56).\nCheck: 1 mod 8=1 and mod 7=1.\nFinal answer: $x\\equiv 1\\pmod{56}$", "final_answer": "x\\equiv 1\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "67f053f823f3ade4", "problem_id": "0f17fd6cc29f1ffb", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$- 7 x + 8 y$ = $17$\n$- 9 x + y$ = $59$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-9$: $63 x - 72 y$ = $-153$.\n3) Multiply the second equation by $-7$: $63 x - 7 y$ = $-413$.\n4) Subtract to eliminate x: $- 65 y$ = $260$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $-7$.\nFinal answer: $(x,y)=\\left(-7, -4\\right)$", "final_answer": "(x,y)=\\left(-7, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c1bc2163bd335191", "problem_id": "d3b067b3b14a6674", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{3} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\sin{\\left(x \\right)}) = \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\sin{\\left(x \\right)}) = \\cos{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.\nFinal answer: $- 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$", "final_answer": "- 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3f26dd5505520216", "problem_id": "58d8bc8923988970", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 3 x - 40$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} + 3 x - 40$ = $\\left(x - 5\\right) \\left(x + 8\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $5$.\nFrom the second factor: x = $-8$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{5, -8\\}$", "final_answer": "x \\in \\{5, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7cfabd59b2870668", "problem_id": "58d8bc8923988970", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 3 x - 40$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} + 3 x - 40$ = $\\left(x - 5\\right) \\left(x + 8\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $5$.\n5) From the second factor: x = $-8$.\nFinal answer: $x \\in \\{5, -8\\}$", "final_answer": "x \\in \\{5, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1d77ffdee940a1f5", "problem_id": "3e092c40a57764d2", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$9 x$ = $81$\n$- 5 x + 9 y$ = $-108$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-5$: $- 45 x$ = $-405$.\nMultiply the second equation by $9$: $- 45 x + 81 y$ = $-972$.\nSubtract to eliminate x: $- 81 y$ = $567$.\nSolve for y: y = $-7$.\nSubstitute back to find x: x = $9$.\nCheck: plug (x,y)=(9,-7) into both equations.\nFinal answer: $(x,y)=\\left(9, -7\\right)$", "final_answer": "(x,y)=\\left(9, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2cccc6b3354e58d7", "problem_id": "3e092c40a57764d2", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$9 x$ = $81$\n$- 5 x + 9 y$ = $-108$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-5$: $- 45 x$ = $-405$.\n3) Multiply the second equation by $9$: $- 45 x + 81 y$ = $-972$.\n4) Subtract to eliminate x: $- 81 y$ = $567$.\n5) Solve for y: y = $-7$.\n6) Substitute back to find x: x = $9$.\nFinal answer: $(x,y)=\\left(9, -7\\right)$", "final_answer": "(x,y)=\\left(9, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0fe400dcb73790a6", "problem_id": "f66b6cee2c8587c1", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Worker A finishes a job in 11 hours and Worker B finishes it in 15 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/11 job/hour. B's rate = 1/15 job/hour.\n2) Combined rate = 1/11 + 1/15 = 26/165 job/hour.\n3) Time = 1 ÷ (combined rate) = 165/26 hours.\nFinal answer: $\\frac{165}{26}$", "final_answer": "\\frac{165}{26}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "28a1e200ba0f2e3c", "problem_id": "f66b6cee2c8587c1", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Worker A finishes a job in 11 hours and Worker B finishes it in 15 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/11 job/hour. B's rate = 1/15 job/hour.\nCombined rate = 1/11 + 1/15 = 26/165 job/hour.\nTime = 1 ÷ (combined rate) = 165/26 hours.\nCheck: (combined rate)×(time) = 26/165×165/26 = 1.\nFinal answer: $\\frac{165}{26}$", "final_answer": "\\frac{165}{26}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c2949373400c9659", "problem_id": "280b77fbbeb75302", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Two friends split $310 in the ratio 7:3. How much does each receive?", "output": "Total ratio parts = 7+3 = 10.\nEach part = 310 ÷ 10 = 31.\nFirst share = 7 × 31 = 217.\nFinal answer: $93$", "final_answer": "93", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b2c2c28cfbdf486e", "problem_id": "280b77fbbeb75302", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Two friends split $310 in the ratio 7:3. How much does each receive?", "output": "Total ratio parts = 7+3 = 10.\nEach part = 310 ÷ 10 = 31.\nFirst share = 7 × 31 = 217.\nSecond share = 3 × 31 = 93.\nCheck: 217+93=310.\nFinal answer: $93$", "final_answer": "93", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b4c4a6c77072a71c", "problem_id": "7805031a1348fd52", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 2 \\sqrt{13} x + 2 x - 3 + 2 \\sqrt{13}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$2 - 2 \\sqrt{13}$, c=$-3 + 2 \\sqrt{13}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $68 - 16 \\sqrt{13}$, so √Δ = $\\sqrt{68 - 16 \\sqrt{13}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $-1 + \\sqrt{17 - 4 \\sqrt{13}} + \\sqrt{13}$.\n7) x2 = $- \\sqrt{17 - 4 \\sqrt{13}} - 1 + \\sqrt{13}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{-1 + \\sqrt{17 - 4\\cdot \\sqrt{13}} + \\sqrt{13}, - \\sqrt{17 - 4\\cdot \\sqrt{13}} - 1 + \\sqrt{13}\\}$", "final_answer": "x \\in \\{-1 + \\sqrt{17 - 4\\cdot \\sqrt{13}} + \\sqrt{13}, - \\sqrt{17 - 4\\cdot \\sqrt{13}} - 1 + \\sqrt{13}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a3539f1c812f938c", "problem_id": "6f3a478956ce613a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} \\sin{\\left(x \\right)} + x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 \\cos{\\left(2 x \\right)}) = - 2 \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 \\cos{\\left(2 x \\right)}) = - 4 \\sin{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$.\nFinal answer: $x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$", "final_answer": "x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7dd34dae9758615c", "problem_id": "dc8ce28e8ffe363f", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- 7 x - 2 y$ = $-15$\n$- x + 8 y$ = $31$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $7 x + 2 y$ = $15$.\nMultiply the second equation by $-7$: $7 x - 56 y$ = $-217$.\nSubtract to eliminate x: $58 y$ = $232$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $1$.\nCheck: plug (x,y)=(1,4) into both equations.\nFinal answer: $(x,y)=\\left(1, 4\\right)$", "final_answer": "(x,y)=\\left(1, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "965f7dd0bb9812c3", "problem_id": "dc8ce28e8ffe363f", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$- 7 x - 2 y$ = $-15$\n$- x + 8 y$ = $31$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $7 x + 2 y$ = $15$.\n3) Multiply the second equation by $-7$: $7 x - 56 y$ = $-217$.\n4) Subtract to eliminate x: $58 y$ = $232$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $1$.\nFinal answer: $(x,y)=\\left(1, 4\\right)$", "final_answer": "(x,y)=\\left(1, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72e271f40ca40bf7", "problem_id": "e9da35ad4cb225e4", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the congruences: x≡1 (mod 5), x≡2 (mod 11). Give the solution modulo 55.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 55.\n3) Compute the CRT solution to get x ≡ 46 (mod 55).\nFinal answer: $x\\equiv 46\\pmod{55}$", "final_answer": "x\\equiv 46\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0a47f6e32291858a", "problem_id": "77bf007d2ad0bb7e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} x - 3$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to -3.\n3) Then the limit is -3.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -6.\nFinal answer: $-6$", "final_answer": "-6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2b058b35303814b0", "problem_id": "dc9f8cb8d642c668", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the multiplicative inverse of 15 modulo 17.", "output": "1) We want x such that 15x ≡ 1 (mod 17).\n2) Use extended Euclid to find s,t with 15s + 17t = 1.\n3) One Bezout identity is: 15·(8) + 17·(-7) = 1.\n4) Reduce s modulo 17: x ≡ 8 (mod 17).\nFinal answer: $x\\equiv 8\\pmod{17}$", "final_answer": "x\\equiv 8\\pmod{17}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b0fc95541fb172d6", "problem_id": "dc9f8cb8d642c668", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Find the multiplicative inverse of 15 modulo 17.", "output": "We want x such that 15x ≡ 1 (mod 17).\nUse extended Euclid to find s,t with 15s + 17t = 1.\nOne Bezout identity is: 15·(8) + 17·(-7) = 1.\nFinal answer: $x\\equiv 8\\pmod{17}$", "final_answer": "x\\equiv 8\\pmod{17}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2f78c30a223c5620", "problem_id": "89d6d184019b9d06", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(3 x^{2} + 1\\right) \\cos{\\left(3 x + 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\cos{\\left(3 x + 2 \\right)}) = - \\sin{\\left(3 x + 2 \\right)}$.\n3) So $f'(x) = - 9 x^{2} \\sin{\\left(3 x + 2 \\right)} + x e^{x} + 6 x \\cos{\\left(3 x + 2 \\right)} - 2 e^{x} - \\sin{\\left(3 x + 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\cos{\\left(3 x + 2 \\right)}) = - 3 \\sin{\\left(3 x + 2 \\right)}$.\nTherefore the correct derivative is $f'(x) = 6 x \\cos{\\left(3 x + 2 \\right)} + \\left(x - 3\\right) e^{x} - 3 \\left(3 x^{2} + 1\\right) \\sin{\\left(3 x + 2 \\right)} + e^{x}$.\nFinal answer: $6 x \\cos{\\left(3 x + 2 \\right)} + \\left(x - 3\\right) e^{x} - 3 \\left(3 x^{2} + 1\\right) \\sin{\\left(3 x + 2 \\right)} + e^{x}$", "final_answer": "6 x \\cos{\\left(3 x + 2 \\right)} + \\left(x - 3\\right) e^{x} - 3 \\left(3 x^{2} + 1\\right) \\sin{\\left(3 x + 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "157ac746dd9ba2bd", "problem_id": "c5788cc5e1fe6176", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} + 2 x^{2} + x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 2$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $- 6 x^{2} + 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 1$.\n2) Slope at x=-2: m = f'(-2) = $-31 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}$.\n3) Point on curve: (-2, f(-2)) = (-2, $\\cos{\\left(4 \\right)} + e^{-1} + 24$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(4 \\right)} + e^{-1} + 24 = -31 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}(x − -2).\n6) Optionally expand: y = $- 31 x + 2 x \\sin{\\left(4 \\right)} + \\frac{x}{2 e} - 38 + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$.\nFinal answer: $y = - 31\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 38 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$", "final_answer": "y = - 31\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 38 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "77c81460af524bbf", "problem_id": "41c8235c257dfb53", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(- 3 x^{3} + 3 x^{2} - x\\right) \\cos{\\left(x^{2} + 3 x + 1 \\right)}$.", "output": "Let f(x) = $\\left(x - 3\\right) e^{x} + \\left(- 3 x^{3} + 3 x^{2} - x\\right) \\cos{\\left(x^{2} + 3 x + 1 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- x \\left(2 x + 3\\right) \\left(- 3 x^{2} + 3 x - 1\\right) \\sin{\\left(x^{2} + 3 x + 1 \\right)} + \\left(x - 3\\right) e^{x} + \\left(- 9 x^{2} + 6 x - 1\\right) \\cos{\\left(x^{2} + 3 x + 1 \\right)} + e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- x\\cdot \\left(2\\cdot x + 3\\right)\\cdot \\left(- 3\\cdot x^{2} + 3\\cdot x - 1\\right)\\cdot \\sin{\\left(x^{2} + 3\\cdot x + 1 \\right)} + \\left(x - 3\\right)\\cdot e^{x} + \\left(- 9\\cdot x^{2} + 6\\cdot x - 1\\right)\\cdot \\cos{\\left(x^{2} + 3\\cdot x + 1 \\right)} + e^{x}$", "final_answer": "- x\\cdot \\left(2\\cdot x + 3\\right)\\cdot \\left(- 3\\cdot x^{2} + 3\\cdot x - 1\\right)\\cdot \\sin{\\left(x^{2} + 3\\cdot x + 1 \\right)} + \\left(x - 3\\right)\\cdot e^{x} + \\left(- 9\\cdot x^{2} + 6\\cdot x - 1\\right)\\cdot \\cos{\\left(x^{2} + 3\\cdot x + 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a7cf342125217416", "problem_id": "41c8235c257dfb53", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(- 3 x^{3} + 3 x^{2} - x\\right) \\cos{\\left(x^{2} + 3 x + 1 \\right)}$.", "output": "1) Let f(x) = $\\left(x - 3\\right) e^{x} + \\left(- 3 x^{3} + 3 x^{2} - x\\right) \\cos{\\left(x^{2} + 3 x + 1 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- x \\left(2 x + 3\\right) \\left(- 3 x^{2} + 3 x - 1\\right) \\sin{\\left(x^{2} + 3 x + 1 \\right)} + \\left(x - 3\\right) e^{x} + \\left(- 9 x^{2} + 6 x - 1\\right) \\cos{\\left(x^{2} + 3 x + 1 \\right)} + e^{x}$.\nFinal answer: $- x\\cdot \\left(2\\cdot x + 3\\right)\\cdot \\left(- 3\\cdot x^{2} + 3\\cdot x - 1\\right)\\cdot \\sin{\\left(x^{2} + 3\\cdot x + 1 \\right)} + \\left(x - 3\\right)\\cdot e^{x} + \\left(- 9\\cdot x^{2} + 6\\cdot x - 1\\right)\\cdot \\cos{\\left(x^{2} + 3\\cdot x + 1 \\right)} + e^{x}$", "final_answer": "- x\\cdot \\left(2\\cdot x + 3\\right)\\cdot \\left(- 3\\cdot x^{2} + 3\\cdot x - 1\\right)\\cdot \\sin{\\left(x^{2} + 3\\cdot x + 1 \\right)} + \\left(x - 3\\right)\\cdot e^{x} + \\left(- 9\\cdot x^{2} + 6\\cdot x - 1\\right)\\cdot \\cos{\\left(x^{2} + 3\\cdot x + 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "24d5b585f395be49", "problem_id": "50f1693d576029df", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A store sells a backpack for $175. It is discounted by 5%. You buy 7 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 175 × (1 − 5/100) = 175 × 95/100 = 665/4.\n2) Subtotal = 7 × 665/4 = 4655/4.\n3) Total with tax = 4655/4 × 110/100 = 10241/8.\nFinal answer: $\\frac{10241}{8}$", "final_answer": "\\frac{10241}{8}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5daa5027d273b87c", "problem_id": "50f1693d576029df", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $175. It is discounted by 5%. You buy 7 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 175 × (1 − 5/100) = 175 × 95/100 = 665/4.\nSubtotal = 7 × 665/4 = 4655/4.\nTotal with tax = 4655/4 × 110/100 = 10241/8.\nCheck: recomputing the formula yields 10241/8.\nFinal answer: $\\frac{10241}{8}$", "final_answer": "\\frac{10241}{8}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e8dcf2de93d84353", "problem_id": "dda97b0d0e18c503", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 15 red balls and 14 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(29,9).\nFavorable samples: choose k reds and n−k blues: C(15,3)·C(14,6).\nCompute totals: C(29,9) = 10015005, and favorable = 1366365.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{91}{667}$.\nFinal answer: $\\frac{91}{667}$", "final_answer": "\\frac{91}{667}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "916879d7b05259af", "problem_id": "dda97b0d0e18c503", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 15 red balls and 14 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(29,9).\n3) Favorable samples: choose k reds and n−k blues: C(15,3)·C(14,6).\n4) Compute totals: C(29,9) = 10015005, and favorable = 1366365.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{91}{667}$", "final_answer": "\\frac{91}{667}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "187bc418e3490460", "problem_id": "4a446934ecc8c3b4", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $768 in the ratio 9:7. How much does each receive?", "output": "1) Total ratio parts = 9+7 = 16.\n2) Each part = 768 ÷ 16 = 48.\n3) First share = 9 × 48 = 432.\n4) Second share = 7 × 48 = 336.\nFinal answer: $336$", "final_answer": "336", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6e7fb64a68dd7396", "problem_id": "4a446934ecc8c3b4", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Two friends split $768 in the ratio 9:7. How much does each receive?", "output": "Total ratio parts = 9+7 = 16.\nEach part = 768 ÷ 16 = 48.\nFirst share = 9 × 48 = 432.\nFinal answer: $336$", "final_answer": "336", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2e26aab438a02195", "problem_id": "a9f23f6b54415941", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many binary strings of length 18 contain exactly 6 ones with no two ones adjacent?", "output": "1) Reserve 5 mandatory zeros between the 6 ones.\n2) This leaves n−k+1 = 13 valid slots for the ones.\n3) Choose the slots: C(13,6) = 1716.\nFinal answer: $1716$", "final_answer": "1716", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e9a6c921c6f02fa", "problem_id": "2aff119ccb94f44f", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 \\sin{\\left(x \\right)}) = 3 \\cos{\\left(x \\right)}$.\n3) So $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 \\sin{\\left(x \\right)}) = 3 \\cos{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$", "final_answer": "x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "16cdaef6855acb61", "problem_id": "6e2a853bbcc936f3", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{11 x}{2} - \\sqrt{13} x + \\frac{9}{2} + \\frac{9 \\sqrt{13}}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{11}{2} - \\sqrt{13}$, c=$\\frac{9}{2} + \\frac{9 \\sqrt{13}}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{101}{4} - 7 \\sqrt{13}$, so √Δ = $\\sqrt{\\frac{101}{4} - 7 \\sqrt{13}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{101 - 28 \\sqrt{13}}}{4} + \\frac{\\sqrt{13}}{2} + \\frac{11}{4}$.\nx2 = $- \\frac{\\sqrt{101 - 28 \\sqrt{13}}}{4} + \\frac{\\sqrt{13}}{2} + \\frac{11}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{101 - 28\\cdot \\sqrt{13}}}{4} + \\frac{\\sqrt{13}}{2} + \\frac{11}{4}, - \\frac{\\sqrt{101 - 28\\cdot \\sqrt{13}}}{4} + \\frac{\\sqrt{13}}{2} + \\frac{11}{4}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{101 - 28\\cdot \\sqrt{13}}}{4} + \\frac{\\sqrt{13}}{2} + \\frac{11}{4}, - \\frac{\\sqrt{101 - 28\\cdot \\sqrt{13}}}{4} + \\frac{\\sqrt{13}}{2} + \\frac{11}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2237509573280c7e", "problem_id": "6e2a853bbcc936f3", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{11 x}{2} - \\sqrt{13} x + \\frac{9}{2} + \\frac{9 \\sqrt{13}}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{11}{2} - \\sqrt{13}$, c=$\\frac{9}{2} + \\frac{9 \\sqrt{13}}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{101}{4} - 7 \\sqrt{13}$, so √Δ = $\\sqrt{\\frac{101}{4} - 7 \\sqrt{13}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{101 - 28 \\sqrt{13}}}{4} + \\frac{\\sqrt{13}}{2} + \\frac{11}{4}$.\n7) x2 = $- \\frac{\\sqrt{101 - 28 \\sqrt{13}}}{4} + \\frac{\\sqrt{13}}{2} + \\frac{11}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{101 - 28\\cdot \\sqrt{13}}}{4} + \\frac{\\sqrt{13}}{2} + \\frac{11}{4}, - \\frac{\\sqrt{101 - 28\\cdot \\sqrt{13}}}{4} + \\frac{\\sqrt{13}}{2} + \\frac{11}{4}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{101 - 28\\cdot \\sqrt{13}}}{4} + \\frac{\\sqrt{13}}{2} + \\frac{11}{4}, - \\frac{\\sqrt{101 - 28\\cdot \\sqrt{13}}}{4} + \\frac{\\sqrt{13}}{2} + \\frac{11}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ab5e7fc758ce2200", "problem_id": "82f49cd34d98cb42", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x^{3} - x^{2} - 3 x - 3$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{4} + x^{3} - x^{2} + 3 x - 2$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 4 x^{3} + 3 x^{2} - 2 x + 3$.\n5) Evaluate at a: p'(2) = $-21$.\nFinal answer: $-21$", "final_answer": "-21", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a9c8d0e994da35af", "problem_id": "82f49cd34d98cb42", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x^{3} - x^{2} - 3 x - 3$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{4} + x^{3} - x^{2} + 3 x - 2$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 4 x^{3} + 3 x^{2} - 2 x + 3$.\nEvaluate at a: p'(2) = $-21$.\nCheck: the identity gives $-21$.\nFinal answer: $-21$", "final_answer": "-21", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eed5c0b78517385c", "problem_id": "1098642163774e02", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $- 2 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $2 \\left(- x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}\\right)$.\nFinal answer: $- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c742bb33581f938a", "problem_id": "1098642163774e02", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $- 2 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $2 \\left(- x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}\\right)$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1fc2601bd9ffccc6", "problem_id": "31b35dfabb6e5a2f", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 11 red balls and 13 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(24,10).\n3) Favorable samples: choose k reds and n−k blues: C(11,5)·C(13,5).\n4) Compute totals: C(24,10) = 1961256, and favorable = 594594.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{9009}{29716}$", "final_answer": "\\frac{9009}{29716}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a388405879a18f75", "problem_id": "31b35dfabb6e5a2f", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 11 red balls and 13 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(24,10).\nFavorable samples: choose k reds and n−k blues: C(11,5)·C(13,5).\nCompute totals: C(24,10) = 1961256, and favorable = 594594.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{9009}{29716}$.\nFinal answer: $\\frac{9009}{29716}$", "final_answer": "\\frac{9009}{29716}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "938b64c672434455", "problem_id": "26b817389bf7d12f", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} 3 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $- \\pi + \\frac{\\pi^{3}}{4}$.\n4) Simplify the result.\nFinal answer: $- \\pi + \\frac{\\pi^{3}}{4}$", "final_answer": "- \\pi + \\frac{\\pi^{3}}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c2b4b192b467d419", "problem_id": "26b817389bf7d12f", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} 3 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $- \\pi + \\frac{\\pi^{3}}{4}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\pi + \\frac{\\pi^{3}}{4}$", "final_answer": "- \\pi + \\frac{\\pi^{3}}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a6dad0713d1c77e", "problem_id": "f086f8b09da45ee3", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the linear system:\n$x + 4 y$ = $-23$\n$- 6 x - 4 y$ = $38$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-6$: $- 6 x - 24 y$ = $138$.\nMultiply the second equation by $1$: $- 6 x - 4 y$ = $38$.\nFinal answer: $(x,y)=\\left(-3, -5\\right)$", "final_answer": "(x,y)=\\left(-3, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e4ff5d4d8b395b13", "problem_id": "f086f8b09da45ee3", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$x + 4 y$ = $-23$\n$- 6 x - 4 y$ = $38$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-6$: $- 6 x - 24 y$ = $138$.\n3) Multiply the second equation by $1$: $- 6 x - 4 y$ = $38$.\n4) Subtract to eliminate x: $- 20 y$ = $100$.\n5) Solve for y: y = $-5$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, -5\\right)$", "final_answer": "(x,y)=\\left(-3, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "da685e476e0ca53c", "problem_id": "b2ec461983e788b3", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 45 x + 162$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} - 45 x + 162$ = $3 \\left(x - 9\\right) \\left(x - 6\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $9$.\n5) From the second factor: x = $6$.\nFinal answer: $x \\in \\{9, 6\\}$", "final_answer": "x \\in \\{9, 6\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0a32c2c4a6990e1a", "problem_id": "b2ec461983e788b3", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 45 x + 162$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} - 45 x + 162$ = $3 \\left(x - 9\\right) \\left(x - 6\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $9$.\nFrom the second factor: x = $6$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{9, 6\\}$", "final_answer": "x \\in \\{9, 6\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "adc1d5a0f86b8dfd", "problem_id": "3ef06467c47a3153", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{7 x}{4} - \\frac{\\sqrt{7} x}{3} + \\frac{\\sqrt{7}}{4} + \\frac{3}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{7}{4} - \\frac{\\sqrt{7}}{3}$, c=$\\frac{\\sqrt{7}}{4} + \\frac{3}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{\\sqrt{7}}{6} + \\frac{121}{144}$, so √Δ = $\\sqrt{\\frac{\\sqrt{7}}{6} + \\frac{121}{144}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{7}}{6} + \\frac{\\sqrt{24 \\sqrt{7} + 121}}{24} + \\frac{7}{8}$.\n7) x2 = $- \\frac{\\sqrt{24 \\sqrt{7} + 121}}{24} + \\frac{\\sqrt{7}}{6} + \\frac{7}{8}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{7}}{6} + \\frac{\\sqrt{24\\cdot \\sqrt{7} + 121}}{24} + \\frac{7}{8}, - \\frac{\\sqrt{24\\cdot \\sqrt{7} + 121}}{24} + \\frac{\\sqrt{7}}{6} + \\frac{7}{8}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{7}}{6} + \\frac{\\sqrt{24\\cdot \\sqrt{7} + 121}}{24} + \\frac{7}{8}, - \\frac{\\sqrt{24\\cdot \\sqrt{7} + 121}}{24} + \\frac{\\sqrt{7}}{6} + \\frac{7}{8}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "645c893b7970a744", "problem_id": "3ef06467c47a3153", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{7 x}{4} - \\frac{\\sqrt{7} x}{3} + \\frac{\\sqrt{7}}{4} + \\frac{3}{4}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{7}{4} - \\frac{\\sqrt{7}}{3}$, c=$\\frac{\\sqrt{7}}{4} + \\frac{3}{4}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{\\sqrt{7}}{6} + \\frac{121}{144}$, so √Δ = $\\sqrt{\\frac{\\sqrt{7}}{6} + \\frac{121}{144}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{7}}{6} + \\frac{\\sqrt{24 \\sqrt{7} + 121}}{24} + \\frac{7}{8}$.\nx2 = $- \\frac{\\sqrt{24 \\sqrt{7} + 121}}{24} + \\frac{\\sqrt{7}}{6} + \\frac{7}{8}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{7}}{6} + \\frac{\\sqrt{24\\cdot \\sqrt{7} + 121}}{24} + \\frac{7}{8}, - \\frac{\\sqrt{24\\cdot \\sqrt{7} + 121}}{24} + \\frac{\\sqrt{7}}{6} + \\frac{7}{8}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{7}}{6} + \\frac{\\sqrt{24\\cdot \\sqrt{7} + 121}}{24} + \\frac{7}{8}, - \\frac{\\sqrt{24\\cdot \\sqrt{7} + 121}}{24} + \\frac{\\sqrt{7}}{6} + \\frac{7}{8}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7c49605549f2db40", "problem_id": "3e22eefde22cdcb3", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=35$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=35 and k=5.\n4) Compute C(39,4) = 82251.\nFinal answer: $82251$", "final_answer": "82251", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "baf027414b05159f", "problem_id": "3e22eefde22cdcb3", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=35$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=35 and k=5.\nCompute C(39,4) = 82251.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $82251$", "final_answer": "82251", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b229bcc2ca4f7039", "problem_id": "d90368256c43506c", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "An urn has 30 red balls and 16 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(46,11).\n3) Favorable samples: choose k reds and n−k blues: C(30,2)·C(16,9).\n4) Compute totals: C(46,11) = 13340783196, and favorable = 4976400.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{31900}{85517841}$", "final_answer": "\\frac{31900}{85517841}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c4c1e816b2303096", "problem_id": "d90368256c43506c", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 30 red balls and 16 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(46,11).\nFavorable samples: choose k reds and n−k blues: C(30,2)·C(16,9).\nCompute totals: C(46,11) = 13340783196, and favorable = 4976400.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{31900}{85517841}$.\nFinal answer: $\\frac{31900}{85517841}$", "final_answer": "\\frac{31900}{85517841}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "feb164a88c303395", "problem_id": "57179ae8e06efeb2", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{10} x - 3 x + 2 + 2 \\sqrt{10}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\sqrt{10} - 3$, c=$2 + 2 \\sqrt{10}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $11 - 2 \\sqrt{10}$, so √Δ = $\\sqrt{11 - 2 \\sqrt{10}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{11 - 2 \\sqrt{10}}}{2} + \\frac{3}{2} + \\frac{\\sqrt{10}}{2}$.\nx2 = $- \\frac{\\sqrt{11 - 2 \\sqrt{10}}}{2} + \\frac{3}{2} + \\frac{\\sqrt{10}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{11 - 2\\cdot \\sqrt{10}}}{2} + \\frac{3}{2} + \\frac{\\sqrt{10}}{2}, - \\frac{\\sqrt{11 - 2\\cdot \\sqrt{10}}}{2} + \\frac{3}{2} + \\frac{\\sqrt{10}}{2}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{11 - 2\\cdot \\sqrt{10}}}{2} + \\frac{3}{2} + \\frac{\\sqrt{10}}{2}, - \\frac{\\sqrt{11 - 2\\cdot \\sqrt{10}}}{2} + \\frac{3}{2} + \\frac{\\sqrt{10}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "04777083f6e085ef", "problem_id": "57179ae8e06efeb2", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{10} x - 3 x + 2 + 2 \\sqrt{10}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\sqrt{10} - 3$, c=$2 + 2 \\sqrt{10}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $11 - 2 \\sqrt{10}$, so √Δ = $\\sqrt{11 - 2 \\sqrt{10}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{11 - 2 \\sqrt{10}}}{2} + \\frac{3}{2} + \\frac{\\sqrt{10}}{2}$.\n7) x2 = $- \\frac{\\sqrt{11 - 2 \\sqrt{10}}}{2} + \\frac{3}{2} + \\frac{\\sqrt{10}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{11 - 2\\cdot \\sqrt{10}}}{2} + \\frac{3}{2} + \\frac{\\sqrt{10}}{2}, - \\frac{\\sqrt{11 - 2\\cdot \\sqrt{10}}}{2} + \\frac{3}{2} + \\frac{\\sqrt{10}}{2}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{11 - 2\\cdot \\sqrt{10}}}{2} + \\frac{3}{2} + \\frac{\\sqrt{10}}{2}, - \\frac{\\sqrt{11 - 2\\cdot \\sqrt{10}}}{2} + \\frac{3}{2} + \\frac{\\sqrt{10}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eb4f1125c67fb8c9", "problem_id": "3898f2a07ea5a533", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$7 x - 7 y$ = $-112$\n$- 10 x + 2 y$ = $104$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-10$: $- 70 x + 70 y$ = $1120$.\n3) Multiply the second equation by $7$: $- 70 x + 14 y$ = $728$.\n4) Subtract to eliminate x: $56 y$ = $392$.\n5) Solve for y: y = $7$.\n6) Substitute back to find x: x = $-9$.\nFinal answer: $(x,y)=\\left(-9, 7\\right)$", "final_answer": "(x,y)=\\left(-9, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9d80b47fcafd6c0d", "problem_id": "3898f2a07ea5a533", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$7 x - 7 y$ = $-112$\n$- 10 x + 2 y$ = $104$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-10$: $- 70 x + 70 y$ = $1120$.\nMultiply the second equation by $7$: $- 70 x + 14 y$ = $728$.\nSubtract to eliminate x: $56 y$ = $392$.\nSolve for y: y = $7$.\nSubstitute back to find x: x = $-9$.\nCheck: plug (x,y)=(-9,7) into both equations.\nFinal answer: $(x,y)=\\left(-9, 7\\right)$", "final_answer": "(x,y)=\\left(-9, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3061c1be0275650d", "problem_id": "84bace18374e5b1b", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$- 2 x - y$ = $-3$\n$3 x$ = $0$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $3$: $- 6 x - 3 y$ = $-9$.\n3) Multiply the second equation by $-2$: $- 6 x$ = $0$.\n4) Subtract to eliminate x: $- 3 y$ = $-9$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, 3\\right)$", "final_answer": "(x,y)=\\left(0, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aa85b427ab51ce6c", "problem_id": "84bace18374e5b1b", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- 2 x - y$ = $-3$\n$3 x$ = $0$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $- 6 x - 3 y$ = $-9$.\nMultiply the second equation by $-2$: $- 6 x$ = $0$.\nSubtract to eliminate x: $- 3 y$ = $-9$.\nSolve for y: y = $3$.\nSubstitute back to find x: x = $0$.\nCheck: plug (x,y)=(0,3) into both equations.\nFinal answer: $(x,y)=\\left(0, 3\\right)$", "final_answer": "(x,y)=\\left(0, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5330fd61b22526fa", "problem_id": "070b8d8e920c171e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- x^{2} + 3 x - 1\\right) \\cos{\\left(3 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\cos{\\left(3 x \\right)}) = \\sin{\\left(3 x \\right)}$.\n3) So $f'(x) = 3 x^{2} \\sin{\\left(3 x \\right)} + 2 x e^{x} - 9 x \\sin{\\left(3 x \\right)} - 2 x \\cos{\\left(3 x \\right)} + 4 e^{x} + \\sin{\\left(3 x \\right)} + 3 \\cos{\\left(3 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\cos{\\left(3 x \\right)}) = 3 \\sin{\\left(3 x \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(3 - 2 x\\right) \\cos{\\left(3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 3 \\left(x \\left(3 - x\\right) - 1\\right) \\sin{\\left(3 x \\right)} + 2 e^{x}$.\nFinal answer: $\\left(3 - 2 x\\right) \\cos{\\left(3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 3 \\left(x \\left(3 - x\\right) - 1\\right) \\sin{\\left(3 x \\right)} + 2 e^{x}$", "final_answer": "\\left(3 - 2 x\\right) \\cos{\\left(3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 3 \\left(x \\left(3 - x\\right) - 1\\right) \\sin{\\left(3 x \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "72284f014c53e1fb", "problem_id": "4d555e3f0bec517a", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 2 x^{2} - 2 x - 1$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $2 x^{3} - 3 x + 2$ and a = -1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $6 x^{2} - 3$.\n5) Evaluate at a: p'(-1) = $3$.\nFinal answer: $3$", "final_answer": "3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "666a817a249cec22", "problem_id": "6cfd91707ecbde8a", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the linear system:\n$- 7 x - 4 y$ = $-16$\n$5 x - 5 y$ = $-20$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $5$: $- 35 x - 20 y$ = $-80$.\n3) Multiply the second equation by $-7$: $- 35 x + 35 y$ = $140$.\n4) Subtract to eliminate x: $- 55 y$ = $-220$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, 4\\right)$", "final_answer": "(x,y)=\\left(0, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4f50acd2ba62fa87", "problem_id": "6cfd91707ecbde8a", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the linear system:\n$- 7 x - 4 y$ = $-16$\n$5 x - 5 y$ = $-20$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $5$: $- 35 x - 20 y$ = $-80$.\nMultiply the second equation by $-7$: $- 35 x + 35 y$ = $140$.\nFinal answer: $(x,y)=\\left(0, 4\\right)$", "final_answer": "(x,y)=\\left(0, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aaa4ad48b2b94c42", "problem_id": "6bc9ff43fcfeed7f", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Two friends split $1222 in the ratio 5:8. How much does each receive?", "output": "1) Total ratio parts = 5+8 = 13.\n2) Each part = 1222 ÷ 13 = 94.\n3) First share = 5 × 94 = 470.\n4) Second share = 8 × 94 = 752.\nFinal answer: $752$", "final_answer": "752", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a2a15da74c108afc", "problem_id": "6bc9ff43fcfeed7f", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Two friends split $1222 in the ratio 5:8. How much does each receive?", "output": "Total ratio parts = 5+8 = 13.\nEach part = 1222 ÷ 13 = 94.\nFirst share = 5 × 94 = 470.\nSecond share = 8 × 94 = 752.\nCheck: 470+752=1222.\nFinal answer: $752$", "final_answer": "752", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e203e6371d7038e1", "problem_id": "8b9f2aac71a529c6", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{2 x^{3} + 3 x^{2} - 2 x + \\left(x^{2} + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{x^{2} + 5}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 5}) = \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 5}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 20 x^{2} + 50} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 20 x^{2} + 50} + \\frac{4 x^{4}}{2 x^{4} + 20 x^{2} + 50} + \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 10 x^{2} + 25} + \\frac{10 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 20 x^{2} + 50} + \\frac{20 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 20 x^{2} + 50} + \\frac{64 x^{2}}{2 x^{4} + 20 x^{2} + 50} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 10} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 5} - \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 5} + \\frac{48 x}{2 x^{4} + 20 x^{2} + 50} - \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 5} + \\frac{25 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 20 x^{2} + 50} + \\frac{50 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 20 x^{2} + 50} - \\frac{20}{2 x^{4} + 20 x^{2} + 50}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 5}) = - \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 5\\right)^{2}} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x^{2} + 5\\right)} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 5} + \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 5}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 4 x \\left(2 x^{3} + 3 x^{2} - 2 x + \\left(x^{2} + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x^{2} + 5\\right) \\left(12 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 12 x + \\left(x^{2} + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 4\\right)}{2 \\left(x^{2} + 5\\right)^{2}}$.\nFinal answer: $\\frac{- 4 x \\left(2 x^{3} + 3 x^{2} - 2 x + \\left(x^{2} + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x^{2} + 5\\right) \\left(12 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 12 x + \\left(x^{2} + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 4\\right)}{2 \\left(x^{2} + 5\\right)^{2}}$", "final_answer": "\\frac{- 4 x \\left(2 x^{3} + 3 x^{2} - 2 x + \\left(x^{2} + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x^{2} + 5\\right) \\left(12 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 12 x + \\left(x^{2} + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 4\\right)}{2 \\left(x^{2} + 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "87be00011915c918", "problem_id": "2abe0497289107d8", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the congruences: x≡0 (mod 4), x≡3 (mod 11). Give the solution modulo 44.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 44.\n3) Compute the CRT solution to get x ≡ 36 (mod 44).\nFinal answer: $x\\equiv 36\\pmod{44}$", "final_answer": "x\\equiv 36\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "74b036da83ca7c31", "problem_id": "e93d19a3f2acd408", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 19 times, what is the probability of exactly 13 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=19, k=13, p=$\\frac{1}{2}$ and simplify.\n4) Here C(19,13) = 27132.\nFinal answer: $\\frac{6783}{131072}$", "final_answer": "\\frac{6783}{131072}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d525582d6b8a7587", "problem_id": "2024877873f6879b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - 3 x^{3} + 8 x^{2} - 19 x + 35$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{3} + 8 x^{2} - 20 x + 35.\n3) Then the limit is 131.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 129.\nFinal answer: $129$", "final_answer": "129", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "bcf04f560757f5dd", "problem_id": "4633164e7150ec41", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the linear system:\n$- 2 x + y$ = $3$\n$- 2 x + 4 y$ = $6$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $4 x - 2 y$ = $-6$.\n3) Multiply the second equation by $-2$: $4 x - 8 y$ = $-12$.\n4) Subtract to eliminate x: $6 y$ = $6$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, 1\\right)$", "final_answer": "(x,y)=\\left(-1, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "54ba45724528a4aa", "problem_id": "4633164e7150ec41", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the linear system:\n$- 2 x + y$ = $3$\n$- 2 x + 4 y$ = $6$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $4 x - 2 y$ = $-6$.\nMultiply the second equation by $-2$: $4 x - 8 y$ = $-12$.\nFinal answer: $(x,y)=\\left(-1, 1\\right)$", "final_answer": "(x,y)=\\left(-1, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1b7de4d4bd0ca9cc", "problem_id": "1e55f53a615bc923", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A cyclist rides at 44 miles per hour for 1/2 hours, then at 67 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 44 × 1/2 = 22 miles.\n2) Second leg distance = 67 × 1 = 67 miles.\n3) Total distance = 22 + 67 = 89 miles.\nFinal answer: $89$", "final_answer": "89", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4802583840d1a09c", "problem_id": "1e55f53a615bc923", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A cyclist rides at 44 miles per hour for 1/2 hours, then at 67 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 44 × 1/2 = 22 miles.\nSecond leg distance = 67 × 1 = 67 miles.\nTotal distance = 22 + 67 = 89 miles.\nCheck: adding both legs gives 89.\nFinal answer: $89$", "final_answer": "89", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1fbea6f1d12e0641", "problem_id": "207c0b128c686018", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- \\frac{2 x^{3}}{3} - \\frac{x^{2}}{3} + x + \\left(x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1}{x - 2}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 2}) = \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 2}$.\n3) So $f'(x) = - \\frac{8 x^{3}}{6 x^{2} - 24 x + 24} + \\frac{3 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{2} - 24 x + 24} + \\frac{6 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{2} - 24 x + 24} + \\frac{22 x^{2}}{6 x^{2} - 24 x + 24} - \\frac{12 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{2} - 24 x + 24} - \\frac{24 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{2} - 24 x + 24} + \\frac{8 x}{6 x^{2} - 24 x + 24} + \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 4 x + 4} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 4} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 2} - \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 2} + \\frac{12 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{2} - 24 x + 24} + \\frac{24 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{2} - 24 x + 24} - \\frac{6}{6 x^{2} - 24 x + 24} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 2}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 2}) = \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x - 2\\right)} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 2} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x - 2\\right)^{2}} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 2}$.\nTherefore the correct derivative is $f'(x) = \\frac{4 x^{3} + 2 x^{2} - 6 x + \\left(x - 2\\right) \\left(- 12 x^{2} - 4 x + 3 \\left(x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x - 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 6 \\left(x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6}{6 \\left(x - 2\\right)^{2}}$.\nFinal answer: $\\frac{4 x^{3} + 2 x^{2} - 6 x + \\left(x - 2\\right) \\left(- 12 x^{2} - 4 x + 3 \\left(x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x - 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 6 \\left(x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6}{6 \\left(x - 2\\right)^{2}}$", "final_answer": "\\frac{4 x^{3} + 2 x^{2} - 6 x + \\left(x - 2\\right) \\left(- 12 x^{2} - 4 x + 3 \\left(x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x - 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 6 \\left(x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6}{6 \\left(x - 2\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3291facb8f608575", "problem_id": "1656211171844e13", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "You want 68 liters of a 35% salt solution. You have a 30% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 68−x liters is 40% solution.\n2) Salt equation: 30x + 40(68−x) = 35·68.\n3) (30-40)x = (35-40)·68 ⇒ x = 34.\n4) Then 68−x = 34.\nFinal answer: $34 L of 30% and 34 L of 40%$", "final_answer": "34 L of 30% and 34 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ef803da18daae653", "problem_id": "ddd3020fbe34c2c5", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A cyclist rides at 54 miles per hour for 3/2 hours, then at 85 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 54 × 3/2 = 81 miles.\n2) Second leg distance = 85 × 5/2 = 425/2 miles.\n3) Total distance = 81 + 425/2 = 587/2 miles.\nFinal answer: $\\frac{587}{2}$", "final_answer": "\\frac{587}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1c3b4690da767885", "problem_id": "ddd3020fbe34c2c5", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 54 miles per hour for 3/2 hours, then at 85 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 54 × 3/2 = 81 miles.\nSecond leg distance = 85 × 5/2 = 425/2 miles.\nTotal distance = 81 + 425/2 = 587/2 miles.\nCheck: adding both legs gives 587/2.\nFinal answer: $\\frac{587}{2}$", "final_answer": "\\frac{587}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4bd76101e3677fe9", "problem_id": "38439effa759c11a", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A store sells a backpack for $52. It is discounted by 30%. You buy 2 backpacks. What is the total cost?", "output": "1) Discounted price = 52 × (1 − 30/100) = 52 × 70/100 = 182/5.\n2) Subtotal = 2 × 182/5 = 364/5.\nFinal answer: $\\frac{364}{5}$", "final_answer": "\\frac{364}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d467dd64d51802a", "problem_id": "77059cff0e0e9a9a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} x^{2} + 2$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to x^{2} - x + 2.\n3) Then the limit is 2.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: 3.\nFinal answer: $3$", "final_answer": "3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0c0ee824b5d238e2", "problem_id": "6a7ef6a41000840e", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 14 times, what is the probability of exactly 9 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=14, k=9, p=$\\frac{2}{5}$ and simplify.\n4) Here C(14,9) = 2002.\nFinal answer: $\\frac{249080832}{6103515625}$", "final_answer": "\\frac{249080832}{6103515625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3fc24b2edd4fd877", "problem_id": "6a7ef6a41000840e", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 14 times, what is the probability of exactly 9 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=14, k=9, p=$\\frac{2}{5}$ and simplify.\nHere C(14,9) = 2002.\nCheck: expression simplifies to $\\frac{249080832}{6103515625}$.\nFinal answer: $\\frac{249080832}{6103515625}$", "final_answer": "\\frac{249080832}{6103515625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "22239e8ba4ac53e8", "problem_id": "79a07a30947240cd", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "You want 25 liters of a 30% salt solution. You have a 10% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 25−x liters is 60% solution.\n2) Salt equation: 10x + 60(25−x) = 30·25.\n3) (10-60)x = (30-60)·25 ⇒ x = 15.\n4) Then 25−x = 10.\nFinal answer: $15 L of 10% and 10 L of 60%$", "final_answer": "15 L of 10% and 10 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c9fa084c79c97f83", "problem_id": "79a07a30947240cd", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "You want 25 liters of a 30% salt solution. You have a 10% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 25−x liters is 60% solution.\nSalt equation: 10x + 60(25−x) = 30·25.\n(10-60)x = (30-60)·25 ⇒ x = 15.\nFinal answer: $15 L of 10% and 10 L of 60%$", "final_answer": "15 L of 10% and 10 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9d080822be2cf4f1", "problem_id": "9fd93e05f75ea9dd", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(2 x^{2} + 2 x - 3\\right) \\cos{\\left(3 x^{2} + 2 x \\right)}$.", "output": "Let f(x) = $\\left(x + 1\\right) e^{x} + \\left(2 x^{2} + 2 x - 3\\right) \\cos{\\left(3 x^{2} + 2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\left(x + 1\\right) e^{x} + 2 \\left(2 x + 1\\right) \\cos{\\left(3 x^{2} + 2 x \\right)} - 2 \\left(3 x + 1\\right) \\left(2 x^{2} + 2 x - 3\\right) \\sin{\\left(3 x^{2} + 2 x \\right)} + e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\left(x + 1\\right)\\cdot e^{x} + \\left(4\\cdot x + 2\\right)\\cdot \\cos{\\left(3\\cdot x^{2} + 2\\cdot x \\right)} - \\left(6\\cdot x + 2\\right)\\cdot \\left(2\\cdot x^{2} + 2\\cdot x - 3\\right)\\cdot \\sin{\\left(3\\cdot x^{2} + 2\\cdot x \\right)} + e^{x}$", "final_answer": "\\left(x + 1\\right)\\cdot e^{x} + \\left(4\\cdot x + 2\\right)\\cdot \\cos{\\left(3\\cdot x^{2} + 2\\cdot x \\right)} - \\left(6\\cdot x + 2\\right)\\cdot \\left(2\\cdot x^{2} + 2\\cdot x - 3\\right)\\cdot \\sin{\\left(3\\cdot x^{2} + 2\\cdot x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "180caa4ba7e4294b", "problem_id": "9fd93e05f75ea9dd", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(2 x^{2} + 2 x - 3\\right) \\cos{\\left(3 x^{2} + 2 x \\right)}$.", "output": "1) Let f(x) = $\\left(x + 1\\right) e^{x} + \\left(2 x^{2} + 2 x - 3\\right) \\cos{\\left(3 x^{2} + 2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\left(x + 1\\right) e^{x} + 2 \\left(2 x + 1\\right) \\cos{\\left(3 x^{2} + 2 x \\right)} - 2 \\left(3 x + 1\\right) \\left(2 x^{2} + 2 x - 3\\right) \\sin{\\left(3 x^{2} + 2 x \\right)} + e^{x}$.\nFinal answer: $\\left(x + 1\\right)\\cdot e^{x} + \\left(4\\cdot x + 2\\right)\\cdot \\cos{\\left(3\\cdot x^{2} + 2\\cdot x \\right)} - \\left(6\\cdot x + 2\\right)\\cdot \\left(2\\cdot x^{2} + 2\\cdot x - 3\\right)\\cdot \\sin{\\left(3\\cdot x^{2} + 2\\cdot x \\right)} + e^{x}$", "final_answer": "\\left(x + 1\\right)\\cdot e^{x} + \\left(4\\cdot x + 2\\right)\\cdot \\cos{\\left(3\\cdot x^{2} + 2\\cdot x \\right)} - \\left(6\\cdot x + 2\\right)\\cdot \\left(2\\cdot x^{2} + 2\\cdot x - 3\\right)\\cdot \\sin{\\left(3\\cdot x^{2} + 2\\cdot x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2ed4daecab7e26a7", "problem_id": "c576674aaded5ee4", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$.\nFinal answer: $x^{2}\\cdot \\cos{\\left(x \\right)} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + 3\\cdot x\\cdot \\cos{\\left(x \\right)} + 3\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)}$", "final_answer": "x^{2}\\cdot \\cos{\\left(x \\right)} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + 3\\cdot x\\cdot \\cos{\\left(x \\right)} + 3\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8bdc46d49e3b394a", "problem_id": "232140c6d24e0324", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} x^{2} - 6 x + 22$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to x^{2} - 7 x + 22.\n3) Then the limit is 66.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: 62.\nFinal answer: $62$", "final_answer": "62", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "50dee5ed9fbc480e", "problem_id": "1a80d1e449aa40e9", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Two friends split $658 in the ratio 9:5. How much does each receive?", "output": "1) Total ratio parts = 9+5 = 14.\n2) Each part = 658 ÷ 14 = 47.\n3) First share = 9 × 47 = 423.\n4) Second share = 5 × 47 = 235.\nFinal answer: $235$", "final_answer": "235", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3b84be976dc37bab", "problem_id": "1a80d1e449aa40e9", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $658 in the ratio 9:5. How much does each receive?", "output": "Total ratio parts = 9+5 = 14.\nEach part = 658 ÷ 14 = 47.\nFirst share = 9 × 47 = 423.\nSecond share = 5 × 47 = 235.\nCheck: 423+235=658.\nFinal answer: $235$", "final_answer": "235", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0b565671c4c51039", "problem_id": "6bd9e2b60fe78f70", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$- 12 x - y$ = $73$\n$6 x - 5 y$ = $-31$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $6$: $- 72 x - 6 y$ = $438$.\n3) Multiply the second equation by $-12$: $- 72 x + 60 y$ = $372$.\n4) Subtract to eliminate x: $- 66 y$ = $66$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $-6$.\nFinal answer: $(x,y)=\\left(-6, -1\\right)$", "final_answer": "(x,y)=\\left(-6, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e89a2b826ca5c79a", "problem_id": "3861645a96784d24", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - 2 x^{3} - 7 x^{2} - 19 x - 57$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{3} - 7 x^{2} - 20 x - 57.\n3) Then the limit is -234.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: -231.\nFinal answer: $-231$", "final_answer": "-231", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d55536f28c18778d", "problem_id": "acaa0a04b8391da0", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} - 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 3$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $6 x^{2} + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 3$.\n2) Slope at x=-2: m = f'(-2) = $2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e} + 21$.\n3) Point on curve: (-2, f(-2)) = (-2, $-13 + \\cos{\\left(4 \\right)} + e^{-1}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -13 + \\cos{\\left(4 \\right)} + e^{-1} = 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e} + 21(x − -2).\n6) Optionally expand: y = $2 x \\sin{\\left(4 \\right)} + \\frac{x}{2 e} + 21 x + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 29$.\nFinal answer: $y = 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 21\\cdot x + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 29$", "final_answer": "y = 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 21\\cdot x + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 29", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4dab0cfb02ccbb6d", "problem_id": "86ad5e3e1f9c98d9", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - x^{2} + 5 x - 10$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} + 4 x - 10.\n3) Then the limit is -22.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -24.\nFinal answer: $-24$", "final_answer": "-24", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "bf046a91bb127a57", "problem_id": "949302d9b644b0ab", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Find the multiplicative inverse of 24 modulo 37.", "output": "We want x such that 24x ≡ 1 (mod 37).\nUse extended Euclid to find s,t with 24s + 37t = 1.\nOne Bezout identity is: 24·(17) + 37·(-11) = 1.\nFinal answer: $x\\equiv 17\\pmod{37}$", "final_answer": "x\\equiv 17\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "69b216438116da3c", "problem_id": "949302d9b644b0ab", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the multiplicative inverse of 24 modulo 37.", "output": "1) We want x such that 24x ≡ 1 (mod 37).\n2) Use extended Euclid to find s,t with 24s + 37t = 1.\n3) One Bezout identity is: 24·(17) + 37·(-11) = 1.\n4) Reduce s modulo 37: x ≡ 17 (mod 37).\nFinal answer: $x\\equiv 17\\pmod{37}$", "final_answer": "x\\equiv 17\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8ad5fb157e711ed7", "problem_id": "fd3ad7221698a125", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 21 red balls and 27 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(48,8).\n3) Favorable samples: choose k reds and n−k blues: C(21,3)·C(27,5).\n4) Compute totals: C(48,8) = 377348994, and favorable = 107370900.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{259350}{911471}$", "final_answer": "\\frac{259350}{911471}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3cebcfc8a192980a", "problem_id": "8becc4ad099fc928", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 3 x^{2} + 12 x - 35$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} + 11 x - 35.\n3) Then the limit is -95.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -98.\nFinal answer: $-98$", "final_answer": "-98", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "9a9bc62fb4989bf4", "problem_id": "d2d7af1102308ea3", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$- x + y$ = $0$\n$- y$ = $-2$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $-1$: $y$ = $2$.\n4) Subtract to eliminate x: $- y$ = $-2$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $2$.\nFinal answer: $(x,y)=\\left(2, 2\\right)$", "final_answer": "(x,y)=\\left(2, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "21811bd11b2d38ac", "problem_id": "d2d7af1102308ea3", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- x + y$ = $0$\n$- y$ = $-2$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $-1$: $y$ = $2$.\nSubtract to eliminate x: $- y$ = $-2$.\nSolve for y: y = $2$.\nSubstitute back to find x: x = $2$.\nCheck: plug (x,y)=(2,2) into both equations.\nFinal answer: $(x,y)=\\left(2, 2\\right)$", "final_answer": "(x,y)=\\left(2, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bb3bc7bc02e7323a", "problem_id": "ef6fd8487ea7d04b", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A cyclist rides at 25 miles per hour for 1 hours, then at 48 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 25 × 1 = 25 miles.\n2) Second leg distance = 48 × 1 = 48 miles.\n3) Total distance = 25 + 48 = 73 miles.\nFinal answer: $73$", "final_answer": "73", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "97b368e1ab361245", "problem_id": "ef6fd8487ea7d04b", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 25 miles per hour for 1 hours, then at 48 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 25 × 1 = 25 miles.\nSecond leg distance = 48 × 1 = 48 miles.\nTotal distance = 25 + 48 = 73 miles.\nCheck: adding both legs gives 73.\nFinal answer: $73$", "final_answer": "73", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8b7fc975a68dbf29", "problem_id": "ec3295382b14f1e2", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{3} + 2 x^{2} + 3 x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{3 x + 4}$.", "output": "Let f(x) = $\\frac{- 3 x^{3} + 2 x^{2} + 3 x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{3 x + 4}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{18 x^{3} - 12 x^{2} - 18 x + \\left(3 x + 4\\right) \\left(- 18 x^{2} + 8 x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 6 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6}{2 \\left(3 x + 4\\right)^{2}}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\frac{18\\cdot x^{3} - 12\\cdot x^{2} - 18\\cdot x + \\left(3\\cdot x + 4\\right)\\cdot \\left(- 18\\cdot x^{2} + 8\\cdot x + \\left(3\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x + 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6\\right) - \\left(18\\cdot x + 24\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 6}{2\\cdot \\left(3\\cdot x + 4\\right)^{2}}$", "final_answer": "\\frac{18\\cdot x^{3} - 12\\cdot x^{2} - 18\\cdot x + \\left(3\\cdot x + 4\\right)\\cdot \\left(- 18\\cdot x^{2} + 8\\cdot x + \\left(3\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x + 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6\\right) - \\left(18\\cdot x + 24\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 6}{2\\cdot \\left(3\\cdot x + 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "31c777d955fbd8d6", "problem_id": "ec3295382b14f1e2", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{3} + 2 x^{2} + 3 x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{3 x + 4}$.", "output": "1) Let f(x) = $\\frac{- 3 x^{3} + 2 x^{2} + 3 x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{3 x + 4}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{18 x^{3} - 12 x^{2} - 18 x + \\left(3 x + 4\\right) \\left(- 18 x^{2} + 8 x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 6 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6}{2 \\left(3 x + 4\\right)^{2}}$.\nFinal answer: $\\frac{18\\cdot x^{3} - 12\\cdot x^{2} - 18\\cdot x + \\left(3\\cdot x + 4\\right)\\cdot \\left(- 18\\cdot x^{2} + 8\\cdot x + \\left(3\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x + 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6\\right) - \\left(18\\cdot x + 24\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 6}{2\\cdot \\left(3\\cdot x + 4\\right)^{2}}$", "final_answer": "\\frac{18\\cdot x^{3} - 12\\cdot x^{2} - 18\\cdot x + \\left(3\\cdot x + 4\\right)\\cdot \\left(- 18\\cdot x^{2} + 8\\cdot x + \\left(3\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x + 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6\\right) - \\left(18\\cdot x + 24\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 6}{2\\cdot \\left(3\\cdot x + 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "434e4184b382a80a", "problem_id": "dba2c63b636fd922", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the equation of the tangent line to $y=- 4 x^{3} - 4 x^{2} + x$ at $x=0$.", "output": "1) Differentiate: f'(x) = $- 12 x^{2} - 8 x + 1$.\n2) Slope at x=0: m = f'(0) = $1$.\n3) Point on curve: (0, f(0)) = (0, $0$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 0 = 1(x − 0).\n6) Optionally expand: y = $x$.\nFinal answer: $y = x$", "final_answer": "y = x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "27a9abeb8c25c04a", "problem_id": "dba2c63b636fd922", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Find the equation of the tangent line to $y=- 4 x^{3} - 4 x^{2} + x$ at $x=0$.", "output": "Differentiate: f'(x) = $- 12 x^{2} - 8 x + 1$.\nSlope at x=0: m = f'(0) = $1$.\nPoint on curve: (0, f(0)) = (0, $0$).\nFinal answer: $y = x$", "final_answer": "y = x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "37ee20e15be73d89", "problem_id": "0954e66808e05141", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(2 x^{3} - x^{2}\\right) \\cos{\\left(3 x^{2} + 2 x + 3 \\right)}$.", "output": "1) Let f(x) = $2 \\left(x + 2\\right) e^{x} + \\left(2 x^{3} - x^{2}\\right) \\cos{\\left(3 x^{2} + 2 x + 3 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $2 \\left(- x^{2} \\left(2 x - 1\\right) \\left(3 x + 1\\right) \\sin{\\left(3 x^{2} + 2 x + 3 \\right)} + x \\left(3 x - 1\\right) \\cos{\\left(3 x^{2} + 2 x + 3 \\right)} + \\left(x + 2\\right) e^{x} + e^{x}\\right)$.\nFinal answer: $- 2\\cdot x^{2}\\cdot \\left(2\\cdot x - 1\\right)\\cdot \\left(3\\cdot x + 1\\right)\\cdot \\sin{\\left(3\\cdot x^{2} + 2\\cdot x + 3 \\right)} + 2\\cdot x\\cdot \\left(3\\cdot x - 1\\right)\\cdot \\cos{\\left(3\\cdot x^{2} + 2\\cdot x + 3 \\right)} + 2\\cdot \\left(x + 2\\right)\\cdot e^{x} + 2\\cdot e^{x}$", "final_answer": "- 2\\cdot x^{2}\\cdot \\left(2\\cdot x - 1\\right)\\cdot \\left(3\\cdot x + 1\\right)\\cdot \\sin{\\left(3\\cdot x^{2} + 2\\cdot x + 3 \\right)} + 2\\cdot x\\cdot \\left(3\\cdot x - 1\\right)\\cdot \\cos{\\left(3\\cdot x^{2} + 2\\cdot x + 3 \\right)} + 2\\cdot \\left(x + 2\\right)\\cdot e^{x} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "81b2ec9036d155a2", "problem_id": "0954e66808e05141", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(2 x^{3} - x^{2}\\right) \\cos{\\left(3 x^{2} + 2 x + 3 \\right)}$.", "output": "Let f(x) = $2 \\left(x + 2\\right) e^{x} + \\left(2 x^{3} - x^{2}\\right) \\cos{\\left(3 x^{2} + 2 x + 3 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $2 \\left(- x^{2} \\left(2 x - 1\\right) \\left(3 x + 1\\right) \\sin{\\left(3 x^{2} + 2 x + 3 \\right)} + x \\left(3 x - 1\\right) \\cos{\\left(3 x^{2} + 2 x + 3 \\right)} + \\left(x + 2\\right) e^{x} + e^{x}\\right)$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 2\\cdot x^{2}\\cdot \\left(2\\cdot x - 1\\right)\\cdot \\left(3\\cdot x + 1\\right)\\cdot \\sin{\\left(3\\cdot x^{2} + 2\\cdot x + 3 \\right)} + 2\\cdot x\\cdot \\left(3\\cdot x - 1\\right)\\cdot \\cos{\\left(3\\cdot x^{2} + 2\\cdot x + 3 \\right)} + 2\\cdot \\left(x + 2\\right)\\cdot e^{x} + 2\\cdot e^{x}$", "final_answer": "- 2\\cdot x^{2}\\cdot \\left(2\\cdot x - 1\\right)\\cdot \\left(3\\cdot x + 1\\right)\\cdot \\sin{\\left(3\\cdot x^{2} + 2\\cdot x + 3 \\right)} + 2\\cdot x\\cdot \\left(3\\cdot x - 1\\right)\\cdot \\cos{\\left(3\\cdot x^{2} + 2\\cdot x + 3 \\right)} + 2\\cdot \\left(x + 2\\right)\\cdot e^{x} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c34a9090e45a72f3", "problem_id": "d53b1d30f306da4f", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$3 x + y$ = $-27$\n$x$ = $-6$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $3 x + y$ = $-27$.\nMultiply the second equation by $3$: $3 x$ = $-18$.\nSubtract to eliminate x: $y$ = $-9$.\nSolve for y: y = $-9$.\nSubstitute back to find x: x = $-6$.\nCheck: plug (x,y)=(-6,-9) into both equations.\nFinal answer: $(x,y)=\\left(-6, -9\\right)$", "final_answer": "(x,y)=\\left(-6, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cd76d749784528f8", "problem_id": "d53b1d30f306da4f", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$3 x + y$ = $-27$\n$x$ = $-6$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $3 x + y$ = $-27$.\n3) Multiply the second equation by $3$: $3 x$ = $-18$.\n4) Subtract to eliminate x: $y$ = $-9$.\n5) Solve for y: y = $-9$.\n6) Substitute back to find x: x = $-6$.\nFinal answer: $(x,y)=\\left(-6, -9\\right)$", "final_answer": "(x,y)=\\left(-6, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7328c1b35c219e3d", "problem_id": "ff9a680dc10de5d7", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 20 x + 42$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} - 20 x + 42$ = $2 \\left(x - 7\\right) \\left(x - 3\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $7$.\nFrom the second factor: x = $3$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{7, 3\\}$", "final_answer": "x \\in \\{7, 3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "01b08540426994ba", "problem_id": "ff9a680dc10de5d7", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 20 x + 42$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 20 x + 42$ = $2 \\left(x - 7\\right) \\left(x - 3\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $7$.\n5) From the second factor: x = $3$.\nFinal answer: $x \\in \\{7, 3\\}$", "final_answer": "x \\in \\{7, 3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "95961c3cc538ee64", "problem_id": "0e94698edb22bdb2", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 2 x - 7$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $2 x^{2} - x - 1$ and a = -3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $4 x - 1$.\n5) Evaluate at a: p'(-3) = $-13$.\nFinal answer: $-13$", "final_answer": "-13", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d7b7b32205640caf", "problem_id": "0e94698edb22bdb2", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 2 x - 7$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{2} - x - 1$ and a = -3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-13$", "final_answer": "-13", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0956f36f1f13c8f7", "problem_id": "5e2f03f6be863841", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "You want 57 liters of a 35% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 57−x liters is 70% solution.\n2) Salt equation: 20x + 70(57−x) = 35·57.\n3) (20-70)x = (35-70)·57 ⇒ x = 399/10.\n4) Then 57−x = 171/10.\nFinal answer: $399/10 L of 20% and 171/10 L of 70%$", "final_answer": "399/10 L of 20% and 171/10 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ce8215541b038765", "problem_id": "5e2f03f6be863841", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "You want 57 liters of a 35% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 57−x liters is 70% solution.\nSalt equation: 20x + 70(57−x) = 35·57.\n(20-70)x = (35-70)·57 ⇒ x = 399/10.\nThen 57−x = 171/10.\nCheck: 20·399/10 + 70·171/10 = 35·57.\nFinal answer: $399/10 L of 20% and 171/10 L of 70%$", "final_answer": "399/10 L of 20% and 171/10 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "184547b59b5d4026", "problem_id": "be0560585710a62c", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{13} x + \\frac{16 x}{3} - 4 \\sqrt{13} + \\frac{16}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{16}{3} - 3 \\sqrt{13}$, c=$\\frac{16}{3} - 4 \\sqrt{13}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{1117}{9} - 16 \\sqrt{13}$, so √Δ = $\\sqrt{\\frac{1117}{9} - 16 \\sqrt{13}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{8}{3} + \\frac{\\sqrt{1117 - 144 \\sqrt{13}}}{6} + \\frac{3 \\sqrt{13}}{2}$.\n7) x2 = $- \\frac{\\sqrt{1117 - 144 \\sqrt{13}}}{6} - \\frac{8}{3} + \\frac{3 \\sqrt{13}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{8}{3} + \\frac{\\sqrt{1117 - 144\\cdot \\sqrt{13}}}{6} + \\frac{3\\cdot \\sqrt{13}}{2}, - \\frac{\\sqrt{1117 - 144\\cdot \\sqrt{13}}}{6} - \\frac{8}{3} + \\frac{3\\cdot \\sqrt{13}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{8}{3} + \\frac{\\sqrt{1117 - 144\\cdot \\sqrt{13}}}{6} + \\frac{3\\cdot \\sqrt{13}}{2}, - \\frac{\\sqrt{1117 - 144\\cdot \\sqrt{13}}}{6} - \\frac{8}{3} + \\frac{3\\cdot \\sqrt{13}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2fa44281ca4743ff", "problem_id": "da68ca0158e4bdcc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{4} - 2 x^{2} - 2 x + 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x) = - 2 x$.\n3) So $f'(x) = - 4 x^{3} - 6 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x) = -2$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- 2 x^{3} - 2 x - 1\\right)$.\nFinal answer: $2 \\left(- 2 x^{3} - 2 x - 1\\right)$", "final_answer": "2 \\left(- 2 x^{3} - 2 x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3273b6893d7dbe84", "problem_id": "6dcea0065e18491d", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A cyclist rides at 66 miles per hour for 2 hours, then at 86 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 66 × 2 = 132 miles.\n2) Second leg distance = 86 × 1 = 86 miles.\n3) Total distance = 132 + 86 = 218 miles.\nFinal answer: $218$", "final_answer": "218", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "20a3135e88c8d1c3", "problem_id": "6dcea0065e18491d", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 66 miles per hour for 2 hours, then at 86 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 66 × 2 = 132 miles.\nSecond leg distance = 86 × 1 = 86 miles.\nTotal distance = 132 + 86 = 218 miles.\nCheck: adding both legs gives 218.\nFinal answer: $218$", "final_answer": "218", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "22040285f017124e", "problem_id": "254bd645249d6e03", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "You want 32 liters of a 25% salt solution. You have a 15% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 32−x liters is 50% solution.\n2) Salt equation: 15x + 50(32−x) = 25·32.\n3) (15-50)x = (25-50)·32 ⇒ x = 160/7.\n4) Then 32−x = 64/7.\nFinal answer: $160/7 L of 15% and 64/7 L of 50%$", "final_answer": "160/7 L of 15% and 64/7 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "de33eb24aa4874b0", "problem_id": "254bd645249d6e03", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 32 liters of a 25% salt solution. You have a 15% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 32−x liters is 50% solution.\nSalt equation: 15x + 50(32−x) = 25·32.\n(15-50)x = (25-50)·32 ⇒ x = 160/7.\nThen 32−x = 64/7.\nCheck: 15·160/7 + 50·64/7 = 25·32.\nFinal answer: $160/7 L of 15% and 64/7 L of 50%$", "final_answer": "160/7 L of 15% and 64/7 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "648d83a52518430c", "problem_id": "adfe202ac964bfdc", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{4} + 3 x^{3} + 3 x^{2} - 3 x + 1$.", "output": "1) Differentiate term-by-term:\n2) d/dx(1) = 0.\n3) d/dx(- 3 x) = -3.\n4) d/dx(3 x^{2}) = 6 x.\n5) d/dx(3 x^{3}) = 9 x^{2}.\n6) d/dx(4 x^{4}) = 16 x^{3}.\n7) Combine: f'(x) = 16 x^{3} + 9 x^{2} + 6 x - 3.\nFinal answer: $16\\cdot x^{3} + 9\\cdot x^{2} + 6\\cdot x - 3$", "final_answer": "16\\cdot x^{3} + 9\\cdot x^{2} + 6\\cdot x - 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "549b394356aae775", "problem_id": "adfe202ac964bfdc", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{4} + 3 x^{3} + 3 x^{2} - 3 x + 1$.", "output": "Differentiate term-by-term:\nd/dx(1) = 0.\nd/dx(- 3 x) = -3.\nFinal answer: $16\\cdot x^{3} + 9\\cdot x^{2} + 6\\cdot x - 3$", "final_answer": "16\\cdot x^{3} + 9\\cdot x^{2} + 6\\cdot x - 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ad333b69a1c9a8cf", "problem_id": "30f0a13b5f4c0c0d", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- 2 x^{2} + 2 x + 3\\right) \\cos{\\left(x^{2} - 3 x \\right)}$.", "output": "1) Let f(x) = $2 \\left(x + 1\\right) e^{x} + \\left(- 2 x^{2} + 2 x + 3\\right) \\cos{\\left(x^{2} - 3 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $2 \\left(1 - 2 x\\right) \\cos{\\left(x^{2} - 3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - \\left(2 x - 3\\right) \\left(- 2 x^{2} + 2 x + 3\\right) \\sin{\\left(x^{2} - 3 x \\right)} + 2 e^{x}$.\nFinal answer: $\\left(2 - 4\\cdot x\\right)\\cdot \\cos{\\left(x^{2} - 3\\cdot x \\right)} - \\left(2\\cdot x - 3\\right)\\cdot \\left(- 2\\cdot x^{2} + 2\\cdot x + 3\\right)\\cdot \\sin{\\left(x^{2} - 3\\cdot x \\right)} + \\left(2\\cdot x + 2\\right)\\cdot e^{x} + 2\\cdot e^{x}$", "final_answer": "\\left(2 - 4\\cdot x\\right)\\cdot \\cos{\\left(x^{2} - 3\\cdot x \\right)} - \\left(2\\cdot x - 3\\right)\\cdot \\left(- 2\\cdot x^{2} + 2\\cdot x + 3\\right)\\cdot \\sin{\\left(x^{2} - 3\\cdot x \\right)} + \\left(2\\cdot x + 2\\right)\\cdot e^{x} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9a8636ced87abbdf", "problem_id": "74e4d58d58099a1a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(x^{3} + x^{2} + 2\\right) \\cos{\\left(3 x + 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x^{3} \\cos{\\left(3 x + 1 \\right)}) = - x^{3} \\sin{\\left(3 x + 1 \\right)}$.\n3) So $f'(x) = - x^{3} \\sin{\\left(3 x + 1 \\right)} - 3 x^{2} \\sin{\\left(3 x + 1 \\right)} + x e^{x} + 2 x \\cos{\\left(3 x + 1 \\right)} + 2 e^{x} - 6 \\sin{\\left(3 x + 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x^{3} \\cos{\\left(3 x + 1 \\right)}) = - 3 x^{3} \\sin{\\left(3 x + 1 \\right)} + 3 x^{2} \\cos{\\left(3 x + 1 \\right)}$.\nTherefore the correct derivative is $f'(x) = x \\left(3 x + 2\\right) \\cos{\\left(3 x + 1 \\right)} + \\left(x + 1\\right) e^{x} - 3 \\left(x^{2} \\left(x + 1\\right) + 2\\right) \\sin{\\left(3 x + 1 \\right)} + e^{x}$.\nFinal answer: $x \\left(3 x + 2\\right) \\cos{\\left(3 x + 1 \\right)} + \\left(x + 1\\right) e^{x} - 3 \\left(x^{2} \\left(x + 1\\right) + 2\\right) \\sin{\\left(3 x + 1 \\right)} + e^{x}$", "final_answer": "x \\left(3 x + 2\\right) \\cos{\\left(3 x + 1 \\right)} + \\left(x + 1\\right) e^{x} - 3 \\left(x^{2} \\left(x + 1\\right) + 2\\right) \\sin{\\left(3 x + 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "45f5e0ac83db0d46", "problem_id": "aaa146a38fe8ea5e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{2} + 2 x - 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(4 x^{2}) = 8 x^{2}$.\n3) So $f'(x) = 8 x^{2} + 2$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(4 x^{2}) = 8 x$.\nTherefore the correct derivative is $f'(x) = 2 \\left(4 x + 1\\right)$.\nFinal answer: $2 \\left(4 x + 1\\right)$", "final_answer": "2 \\left(4 x + 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "9af7e9950e36b5d9", "problem_id": "82ae4982f9415c89", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$- x + 4 y$ = $27$\n$- 5 x - 8 y$ = $-61$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-5$: $5 x - 20 y$ = $-135$.\nMultiply the second equation by $-1$: $5 x + 8 y$ = $61$.\nSubtract to eliminate x: $- 28 y$ = $-196$.\nSolve for y: y = $7$.\nSubstitute back to find x: x = $1$.\nCheck: plug (x,y)=(1,7) into both equations.\nFinal answer: $(x,y)=\\left(1, 7\\right)$", "final_answer": "(x,y)=\\left(1, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c0a7517db31bb7e0", "problem_id": "ac35ecde25d79c8b", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$- 8 x - 8 y$ = $-8$\n$7 x - 3 y$ = $-53$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $7$: $- 56 x - 56 y$ = $-56$.\n3) Multiply the second equation by $-8$: $- 56 x + 24 y$ = $424$.\n4) Subtract to eliminate x: $- 80 y$ = $-480$.\n5) Solve for y: y = $6$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, 6\\right)$", "final_answer": "(x,y)=\\left(-5, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "67bdf2a40481a3d1", "problem_id": "ac35ecde25d79c8b", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the linear system:\n$- 8 x - 8 y$ = $-8$\n$7 x - 3 y$ = $-53$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $7$: $- 56 x - 56 y$ = $-56$.\nMultiply the second equation by $-8$: $- 56 x + 24 y$ = $424$.\nFinal answer: $(x,y)=\\left(-5, 6\\right)$", "final_answer": "(x,y)=\\left(-5, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "450d5d41c9492660", "problem_id": "30f0a13b5f4c0c0d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- 2 x^{2} + 2 x + 3\\right) \\cos{\\left(x^{2} - 3 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\n3) So $f'(x) = 4 x^{3} \\sin{\\left(x^{2} - 3 x \\right)} - 10 x^{2} \\sin{\\left(x^{2} - 3 x \\right)} + 2 x e^{x} - 4 x \\cos{\\left(x^{2} - 3 x \\right)} + 4 e^{x} + 9 \\sin{\\left(x^{2} - 3 x \\right)} + 2 \\cos{\\left(x^{2} - 3 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(1 - 2 x\\right) \\cos{\\left(x^{2} - 3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - \\left(2 x - 3\\right) \\left(2 x \\left(1 - x\\right) + 3\\right) \\sin{\\left(x^{2} - 3 x \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(1 - 2 x\\right) \\cos{\\left(x^{2} - 3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - \\left(2 x - 3\\right) \\left(2 x \\left(1 - x\\right) + 3\\right) \\sin{\\left(x^{2} - 3 x \\right)} + 2 e^{x}$", "final_answer": "2 \\left(1 - 2 x\\right) \\cos{\\left(x^{2} - 3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - \\left(2 x - 3\\right) \\left(2 x \\left(1 - x\\right) + 3\\right) \\sin{\\left(x^{2} - 3 x \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "35249639bb14600f", "problem_id": "7e88f037989853b4", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the equation of the tangent line to $y=x^{3} - x^{2} + 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=1$.", "output": "1) Differentiate: f'(x) = $3 x^{2} - 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 2$.\n2) Slope at x=1: m = f'(1) = $- 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 3$.\n3) Point on curve: (1, f(1)) = (1, $\\cos{\\left(2 \\right)} + 1 + e^{\\frac{1}{2}}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(2 \\right)} + 1 + e^{\\frac{1}{2}} = - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 3(x − 1).\n6) Optionally expand: y = $- 2 x \\sin{\\left(2 \\right)} + \\frac{x e^{\\frac{1}{2}}}{2} + 3 x - 2 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 3\\cdot x - 2 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 3\\cdot x - 2 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4efb1ef3a0daa2d7", "problem_id": "e45059e2a9920e5f", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} + 2 x^{2} + 2 \\sin{\\left(x \\right)} - 3$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $9 x^{2} + 4 x + 2 \\cos{\\left(x \\right)}$.\n2) Slope at x=-1: m = f'(-1) = $2 \\cos{\\left(1 \\right)} + 5$.\n3) Point on curve: (-1, f(-1)) = (-1, $-4 - 2 \\sin{\\left(1 \\right)}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -4 - 2 \\sin{\\left(1 \\right)} = 2 \\cos{\\left(1 \\right)} + 5(x − -1).\n6) Optionally expand: y = $2 x \\cos{\\left(1 \\right)} + 5 x - 2 \\sin{\\left(1 \\right)} + 1 + 2 \\cos{\\left(1 \\right)}$.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 5\\cdot x - 2\\cdot \\sin{\\left(1 \\right)} + 1 + 2\\cdot \\cos{\\left(1 \\right)}$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 5\\cdot x - 2\\cdot \\sin{\\left(1 \\right)} + 1 + 2\\cdot \\cos{\\left(1 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d9b626c34c3e4ca4", "problem_id": "e45059e2a9920e5f", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} + 2 x^{2} + 2 \\sin{\\left(x \\right)} - 3$ at $x=-1$.", "output": "Differentiate: f'(x) = $9 x^{2} + 4 x + 2 \\cos{\\left(x \\right)}$.\nSlope at x=-1: m = f'(-1) = $2 \\cos{\\left(1 \\right)} + 5$.\nPoint on curve: (-1, f(-1)) = (-1, $-4 - 2 \\sin{\\left(1 \\right)}$).\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 5\\cdot x - 2\\cdot \\sin{\\left(1 \\right)} + 1 + 2\\cdot \\cos{\\left(1 \\right)}$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 5\\cdot x - 2\\cdot \\sin{\\left(1 \\right)} + 1 + 2\\cdot \\cos{\\left(1 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9b10467a87bd7b15", "problem_id": "e8c277daa985567a", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A store sells a backpack for $136. It is discounted by 25%. You buy 4 backpacks. What is the total cost?", "output": "1) Discounted price = 136 × (1 − 25/100) = 136 × 75/100 = 102.\n2) Subtotal = 4 × 102 = 408.\nFinal answer: $408$", "final_answer": "408", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fd52f38d7348716b", "problem_id": "e8c277daa985567a", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A store sells a backpack for $136. It is discounted by 25%. You buy 4 backpacks. What is the total cost?", "output": "Discounted price = 136 × (1 − 25/100) = 136 × 75/100 = 102.\nSubtotal = 4 × 102 = 408.\nCheck: recomputing the formula yields 408.\nFinal answer: $408$", "final_answer": "408", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ad79afc43faf3c26", "problem_id": "da644ade44cebd53", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$2 x + 3 y$ = $4$\n$x + 2 y$ = $3$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $2 x + 3 y$ = $4$.\n3) Multiply the second equation by $2$: $2 x + 4 y$ = $6$.\n4) Subtract to eliminate x: $- y$ = $-2$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, 2\\right)$", "final_answer": "(x,y)=\\left(-1, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "36d00614f06a1518", "problem_id": "da644ade44cebd53", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$2 x + 3 y$ = $4$\n$x + 2 y$ = $3$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $2 x + 3 y$ = $4$.\nMultiply the second equation by $2$: $2 x + 4 y$ = $6$.\nSubtract to eliminate x: $- y$ = $-2$.\nSolve for y: y = $2$.\nSubstitute back to find x: x = $-1$.\nCheck: plug (x,y)=(-1,2) into both equations.\nFinal answer: $(x,y)=\\left(-1, 2\\right)$", "final_answer": "(x,y)=\\left(-1, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3b201f9bc41d612d", "problem_id": "5d9d5aa0692fd009", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "An urn has 29 red balls and 15 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(44,12).\n3) Favorable samples: choose k reds and n−k blues: C(29,5)·C(15,7).\n4) Compute totals: C(44,12) = 21090682613, and favorable = 764188425.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{763425}{21069613}$", "final_answer": "\\frac{763425}{21069613}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "298bf4cd2402cc00", "problem_id": "fd2bcac6548591c0", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\left(x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)}\\right) + 3 \\sin{\\left(x \\right)}) = \\left(x^{2} + x + 3\\right) \\cos{\\left(x \\right)}$.\n3) So $f'(x) = x^{2} \\cos{\\left(x \\right)} + x \\cos{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\left(x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)}\\right) + 3 \\sin{\\left(x \\right)}) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} + 3 \\cos{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$", "final_answer": "x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0b80100f2c4203d7", "problem_id": "aa1d1f8459c18efb", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 11 times, what is the probability of exactly 9 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=9, p=$\\frac{3}{8}$ and simplify.\n4) Here C(11,9) = 55.\nFinal answer: $\\frac{27064125}{8589934592}$", "final_answer": "\\frac{27064125}{8589934592}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "32ca39bdbe1820b6", "problem_id": "aa1d1f8459c18efb", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 11 times, what is the probability of exactly 9 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=11, k=9, p=$\\frac{3}{8}$ and simplify.\nHere C(11,9) = 55.\nCheck: expression simplifies to $\\frac{27064125}{8589934592}$.\nFinal answer: $\\frac{27064125}{8589934592}$", "final_answer": "\\frac{27064125}{8589934592}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "50bc450dbcd6c26f", "problem_id": "9f245a525961241d", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $693 in the ratio 5:6. How much does each receive?", "output": "Total ratio parts = 5+6 = 11.\nEach part = 693 ÷ 11 = 63.\nFirst share = 5 × 63 = 315.\nSecond share = 6 × 63 = 378.\nCheck: 315+378=693.\nFinal answer: $378$", "final_answer": "378", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "00e5fb4a23958b62", "problem_id": "9f245a525961241d", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Two friends split $693 in the ratio 5:6. How much does each receive?", "output": "1) Total ratio parts = 5+6 = 11.\n2) Each part = 693 ÷ 11 = 63.\n3) First share = 5 × 63 = 315.\n4) Second share = 6 × 63 = 378.\nFinal answer: $378$", "final_answer": "378", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "60f9e4e9d617eec6", "problem_id": "e5506c8b4bf4b1db", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "You want 14 liters of a 35% salt solution. You have a 25% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 14−x liters is 40% solution.\n2) Salt equation: 25x + 40(14−x) = 35·14.\n3) (25-40)x = (35-40)·14 ⇒ x = 14/3.\n4) Then 14−x = 28/3.\nFinal answer: $14/3 L of 25% and 28/3 L of 40%$", "final_answer": "14/3 L of 25% and 28/3 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4ff51c92e5d7d9a9", "problem_id": "e5506c8b4bf4b1db", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "You want 14 liters of a 35% salt solution. You have a 25% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 14−x liters is 40% solution.\nSalt equation: 25x + 40(14−x) = 35·14.\n(25-40)x = (35-40)·14 ⇒ x = 14/3.\nFinal answer: $14/3 L of 25% and 28/3 L of 40%$", "final_answer": "14/3 L of 25% and 28/3 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f7373b6cce7667f", "problem_id": "fe52a866b9ebe549", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Worker A finishes a job in 15 hours and Worker B finishes it in 5 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/15 job/hour. B's rate = 1/5 job/hour.\nCombined rate = 1/15 + 1/5 = 4/15 job/hour.\nTime = 1 ÷ (combined rate) = 15/4 hours.\nCheck: (combined rate)×(time) = 4/15×15/4 = 1.\nFinal answer: $\\frac{15}{4}$", "final_answer": "\\frac{15}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "da302139c1eaaaa1", "problem_id": "fe52a866b9ebe549", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Worker A finishes a job in 15 hours and Worker B finishes it in 5 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/15 job/hour. B's rate = 1/5 job/hour.\n2) Combined rate = 1/15 + 1/5 = 4/15 job/hour.\n3) Time = 1 ÷ (combined rate) = 15/4 hours.\nFinal answer: $\\frac{15}{4}$", "final_answer": "\\frac{15}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "541af50c7892a548", "problem_id": "4ff89c98b3c00527", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$- 6 x - y$ = $50$\n$9 y$ = $-72$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $-6$: $- 54 y$ = $432$.\n4) Subtract to eliminate x: $54 y$ = $-432$.\n5) Solve for y: y = $-8$.\n6) Substitute back to find x: x = $-7$.\nFinal answer: $(x,y)=\\left(-7, -8\\right)$", "final_answer": "(x,y)=\\left(-7, -8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "12e8d070741c8d1d", "problem_id": "7dd28420924931e1", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{3} + \\frac{2 x^{2}}{3} + \\frac{x}{3} + \\left(x^{2} + x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - \\frac{1}{3}}{x^{2} + x + 1}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + x\\right) + 1}) = \\frac{e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + x + 1}$.\n3) So $f'(x) = \\frac{3 x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{6 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{6 x^{4}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{6 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{12 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{12 x^{3}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{9 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{18 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{20 x^{2}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{12 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{12 x}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 1} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{4}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 1} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 2 x + 2} + \\frac{e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + x + 1} - \\frac{e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} + x\\right) + 1}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + x\\right) + 1}) = \\frac{\\left(- 2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(x^{2} + x\\right) + 1\\right)^{2}} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(x^{2} + x\\right) + 1\\right)} + \\frac{e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} + x\\right) + 1}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(- 2 x - 1\\right) \\left(3 x^{3} + 2 x^{2} + x + 3 \\left(x \\left(x + 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) + \\left(x \\left(x + 1\\right) + 1\\right) \\left(18 x^{2} + 8 x + 6 \\left(2 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3 \\left(x \\left(x + 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x \\left(x + 1\\right) + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{6 \\left(x \\left(x + 1\\right) + 1\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(- 2 x - 1\\right) \\left(3 x^{3} + 2 x^{2} + x + 3 \\left(x \\left(x + 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) + \\left(x \\left(x + 1\\right) + 1\\right) \\left(18 x^{2} + 8 x + 6 \\left(2 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3 \\left(x \\left(x + 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x \\left(x + 1\\right) + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{6 \\left(x \\left(x + 1\\right) + 1\\right)^{2}}$", "final_answer": "\\frac{2 \\left(- 2 x - 1\\right) \\left(3 x^{3} + 2 x^{2} + x + 3 \\left(x \\left(x + 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) + \\left(x \\left(x + 1\\right) + 1\\right) \\left(18 x^{2} + 8 x + 6 \\left(2 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3 \\left(x \\left(x + 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x \\left(x + 1\\right) + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{6 \\left(x \\left(x + 1\\right) + 1\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "4bfb115ddb66a3ff", "problem_id": "4fac38ffce57c0f1", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 10 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=2, p=$\\frac{3}{8}$ and simplify.\n4) Here C(10,2) = 45.\nFinal answer: $\\frac{158203125}{1073741824}$", "final_answer": "\\frac{158203125}{1073741824}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fdae75d245ea2490", "problem_id": "4fac38ffce57c0f1", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 10 times, what is the probability of exactly 2 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=2, p=$\\frac{3}{8}$ and simplify.\nHere C(10,2) = 45.\nCheck: expression simplifies to $\\frac{158203125}{1073741824}$.\nFinal answer: $\\frac{158203125}{1073741824}$", "final_answer": "\\frac{158203125}{1073741824}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "defa47a7e8292d71", "problem_id": "ef0f19eeebf9f03b", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 14 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=14, k=3, p=$\\frac{4}{5}$ and simplify.\n4) Here C(14,3) = 364.\nFinal answer: $\\frac{23296}{6103515625}$", "final_answer": "\\frac{23296}{6103515625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c3b8ed2c804a6ec7", "problem_id": "ef0f19eeebf9f03b", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 14 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=14, k=3, p=$\\frac{4}{5}$ and simplify.\nFinal answer: $\\frac{23296}{6103515625}$", "final_answer": "\\frac{23296}{6103515625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2c388243b1ebfc5a", "problem_id": "9fd1ca712c4d4c01", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$- x + 3 y$ = $-7$\n$3 x - 2 y$ = $7$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $- 3 x + 9 y$ = $-21$.\nMultiply the second equation by $-1$: $- 3 x + 2 y$ = $-7$.\nSubtract to eliminate x: $7 y$ = $-14$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $1$.\nCheck: plug (x,y)=(1,-2) into both equations.\nFinal answer: $(x,y)=\\left(1, -2\\right)$", "final_answer": "(x,y)=\\left(1, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "342d51a31627224b", "problem_id": "2412cdcdf3201af4", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x^{3} - 5 x^{2} - 12 x - 22$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{3} - 5 x^{2} - 13 x - 22.\n3) Then the limit is -84.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -82.\nFinal answer: $-82$", "final_answer": "-82", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "88a0a0c2f20f4bb2", "problem_id": "06b8b1a755c39988", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "An urn has 22 red balls and 30 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(52,10).\n3) Favorable samples: choose k reds and n−k blues: C(22,1)·C(30,9).\n4) Compute totals: C(52,10) = 15820024220, and favorable = 314757300.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{4785}{240499}$", "final_answer": "\\frac{4785}{240499}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6dae3f76c41e5a33", "problem_id": "06b8b1a755c39988", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 22 red balls and 30 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(52,10).\nFavorable samples: choose k reds and n−k blues: C(22,1)·C(30,9).\nCompute totals: C(52,10) = 15820024220, and favorable = 314757300.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{4785}{240499}$.\nFinal answer: $\\frac{4785}{240499}$", "final_answer": "\\frac{4785}{240499}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ebc29bd00d21601b", "problem_id": "52485d481c3bdc6b", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the multiplicative inverse of 29 modulo 37.", "output": "We want x such that 29x ≡ 1 (mod 37).\nUse extended Euclid to find s,t with 29s + 37t = 1.\nOne Bezout identity is: 29·(-14) + 37·(11) = 1.\nReduce s modulo 37: x ≡ 23 (mod 37).\nCheck: 29·23 mod 37 = 1.\nFinal answer: $x\\equiv 23\\pmod{37}$", "final_answer": "x\\equiv 23\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4e4a379123c70625", "problem_id": "52485d481c3bdc6b", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the multiplicative inverse of 29 modulo 37.", "output": "1) We want x such that 29x ≡ 1 (mod 37).\n2) Use extended Euclid to find s,t with 29s + 37t = 1.\n3) One Bezout identity is: 29·(-14) + 37·(11) = 1.\n4) Reduce s modulo 37: x ≡ 23 (mod 37).\nFinal answer: $x\\equiv 23\\pmod{37}$", "final_answer": "x\\equiv 23\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "671c9c5681fa594d", "problem_id": "ade2a881c24ae976", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{4} + x^{3} + x - 2$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-2) = 0.\n3) d/dx(x) = 1.\n4) d/dx(x^{3}) = 3 x^{2}.\n5) d/dx(3 x^{4}) = 12 x^{3}.\n6) Combine: f'(x) = 12 x^{3} + 3 x^{2} + 1.\nFinal answer: $12\\cdot x^{3} + 3\\cdot x^{2} + 1$", "final_answer": "12\\cdot x^{3} + 3\\cdot x^{2} + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a842ef1ba1c20770", "problem_id": "ade2a881c24ae976", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{4} + x^{3} + x - 2$.", "output": "Differentiate term-by-term:\nd/dx(-2) = 0.\nd/dx(x) = 1.\nFinal answer: $12\\cdot x^{3} + 3\\cdot x^{2} + 1$", "final_answer": "12\\cdot x^{3} + 3\\cdot x^{2} + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a4cfe9f206e1ea9", "problem_id": "04cba17e7a1b9d99", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A store sells a backpack for $196. It is discounted by 30%. You buy 8 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 196 × (1 − 30/100) = 196 × 70/100 = 686/5.\n2) Subtotal = 8 × 686/5 = 5488/5.\n3) Total with tax = 5488/5 × 105/100 = 28812/25.\nFinal answer: $\\frac{28812}{25}$", "final_answer": "\\frac{28812}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0afc23fef071af9d", "problem_id": "04cba17e7a1b9d99", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "A store sells a backpack for $196. It is discounted by 30%. You buy 8 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 196 × (1 − 30/100) = 196 × 70/100 = 686/5.\nSubtotal = 8 × 686/5 = 5488/5.\nTotal with tax = 5488/5 × 105/100 = 28812/25.\nFinal answer: $\\frac{28812}{25}$", "final_answer": "\\frac{28812}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7dd455e3f906ec66", "problem_id": "666beb519a0a8e75", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $304. It is discounted by 10%. You buy 7 backpacks. What is the total cost?", "output": "Discounted price = 304 × (1 − 10/100) = 304 × 90/100 = 1368/5.\nSubtotal = 7 × 1368/5 = 9576/5.\nCheck: recomputing the formula yields 9576/5.\nFinal answer: $\\frac{9576}{5}$", "final_answer": "\\frac{9576}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "28c1570a107781ae", "problem_id": "666beb519a0a8e75", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A store sells a backpack for $304. It is discounted by 10%. You buy 7 backpacks. What is the total cost?", "output": "1) Discounted price = 304 × (1 − 10/100) = 304 × 90/100 = 1368/5.\n2) Subtotal = 7 × 1368/5 = 9576/5.\nFinal answer: $\\frac{9576}{5}$", "final_answer": "\\frac{9576}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9cccd89bd23116ae", "problem_id": "1caf36e71918de2f", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{2} - 4 x + 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 4 x) = - 4 x$.\n3) So $f'(x) = 4 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 4 x) = -4$.\nTherefore the correct derivative is $f'(x) = 4 \\left(2 x - 1\\right)$.\nFinal answer: $4 \\left(2 x - 1\\right)$", "final_answer": "4 \\left(2 x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "386446d462f83eca", "problem_id": "b5230199385e2098", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 6 x - 80$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} + 6 x - 80$ = $2 \\left(x - 5\\right) \\left(x + 8\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $5$.\n5) From the second factor: x = $-8$.\nFinal answer: $x \\in \\{5, -8\\}$", "final_answer": "x \\in \\{5, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c8cd5c9f54ebf779", "problem_id": "d175300b9b8daab6", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} 3 x^{4} + 4 x^{3} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{4} + 4 x^{3} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\pi).\n3) Subtract: F(\\pi) − F(- \\pi) = $\\frac{6 \\pi \\left(5 + \\pi^{4}\\right)}{5}$.\n4) Simplify the result.\nFinal answer: $\\frac{6\\cdot \\pi\\cdot \\left(5 + \\pi^{4}\\right)}{5}$", "final_answer": "\\frac{6\\cdot \\pi\\cdot \\left(5 + \\pi^{4}\\right)}{5}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4db93b573051724b", "problem_id": "d175300b9b8daab6", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} 3 x^{4} + 4 x^{3} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{4} + 4 x^{3} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3$.\nEvaluate at the bounds: F(\\pi) and F(- \\pi).\nSubtract: F(\\pi) − F(- \\pi) = $\\frac{6 \\pi \\left(5 + \\pi^{4}\\right)}{5}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{6\\cdot \\pi\\cdot \\left(5 + \\pi^{4}\\right)}{5}$", "final_answer": "\\frac{6\\cdot \\pi\\cdot \\left(5 + \\pi^{4}\\right)}{5}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e8cc1d25638e85d2", "problem_id": "6b3f01daa22a1bbb", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Two friends split $45 in the ratio 1:2. How much does each receive?", "output": "1) Total ratio parts = 1+2 = 3.\n2) Each part = 45 ÷ 3 = 15.\n3) First share = 1 × 15 = 15.\n4) Second share = 2 × 15 = 30.\nFinal answer: $30$", "final_answer": "30", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "098daf54b22c36b0", "problem_id": "e07c8f3c1d5f8a5b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - 3 x^{3} - 12 x^{2} - 38 x - 113$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{3} - 12 x^{2} - 39 x - 113.\n3) Then the limit is -419.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: -416.\nFinal answer: $-416$", "final_answer": "-416", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "4a12813da7e6ecf7", "problem_id": "29c969a7a907a40e", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "You want 38 liters of a 40% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 38−x liters is 70% solution.\n2) Salt equation: 25x + 70(38−x) = 40·38.\n3) (25-70)x = (40-70)·38 ⇒ x = 76/3.\n4) Then 38−x = 38/3.\nFinal answer: $76/3 L of 25% and 38/3 L of 70%$", "final_answer": "76/3 L of 25% and 38/3 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f476be2ea28d7372", "problem_id": "29c969a7a907a40e", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "You want 38 liters of a 40% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 38−x liters is 70% solution.\nSalt equation: 25x + 70(38−x) = 40·38.\n(25-70)x = (40-70)·38 ⇒ x = 76/3.\nThen 38−x = 38/3.\nCheck: 25·76/3 + 70·38/3 = 40·38.\nFinal answer: $76/3 L of 25% and 38/3 L of 70%$", "final_answer": "76/3 L of 25% and 38/3 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "69802948c71a38b6", "problem_id": "d48e18648ead720d", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A store sells a backpack for $20. It is discounted by 5%. You buy 6 backpacks. What is the total cost?", "output": "1) Discounted price = 20 × (1 − 5/100) = 20 × 95/100 = 19.\n2) Subtotal = 6 × 19 = 114.\nFinal answer: $114$", "final_answer": "114", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fec159333f952cfe", "problem_id": "f94bb98cea397d4b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{3} \\sin{\\left(x \\right)} + x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x^{2} \\sin{\\left(x \\right)}) = x^{2} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = 3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x^{2} \\sin{\\left(x \\right)}) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$", "final_answer": "3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "18acdee3ad5814fd", "problem_id": "cef86fd128c12c46", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 42 x + 120$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} + 42 x + 120$ = $3 \\left(x + 4\\right) \\left(x + 10\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-4$.\n5) From the second factor: x = $-10$.\nFinal answer: $x \\in \\{-4, -10\\}$", "final_answer": "x \\in \\{-4, -10\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b90721d86ebdff62", "problem_id": "27b3ab511b08e064", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} - 2 x - 5$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to - 3 x - 5.\n3) Then the limit is -8.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: -7.\nFinal answer: $-7$", "final_answer": "-7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ebb552e1e98b509b", "problem_id": "439069759cabcc6b", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_7=36$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_7=22.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=22 and k=7.\nCompute C(28,6) = 376740.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $376740$", "final_answer": "376740", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bf93502ecd12eb9c", "problem_id": "439069759cabcc6b", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_7=36$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_7=22.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=22 and k=7.\n4) Compute C(28,6) = 376740.\nFinal answer: $376740$", "final_answer": "376740", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "edc9a9737f63d651", "problem_id": "89c044b384df9592", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$x + 2 y$ = $-8$\n$2 x - y$ = $-1$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $2 x + 4 y$ = $-16$.\n3) Multiply the second equation by $1$: $2 x - y$ = $-1$.\n4) Subtract to eliminate x: $5 y$ = $-15$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, -3\\right)$", "final_answer": "(x,y)=\\left(-2, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "596f62a60923bf1e", "problem_id": "89c044b384df9592", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the linear system:\n$x + 2 y$ = $-8$\n$2 x - y$ = $-1$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $2$: $2 x + 4 y$ = $-16$.\nMultiply the second equation by $1$: $2 x - y$ = $-1$.\nFinal answer: $(x,y)=\\left(-2, -3\\right)$", "final_answer": "(x,y)=\\left(-2, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fab42d46bc73d8dd", "problem_id": "cbfc2742c4c3ebdf", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- \\frac{x^{2}}{2} + x + \\left(x^{2} - x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\frac{1}{2}}{x^{2} - x + 1}$.", "output": "1) Let f(x) = $\\frac{- \\frac{x^{2}}{2} + x + \\left(x^{2} - x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\frac{1}{2}}{x^{2} - x + 1}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{\\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + 2 \\left(x^{2} - x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(x^{2} - x + 1\\right) \\left(- 2 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} - x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - x + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{2 \\left(x^{2} - x + 1\\right)^{2}}$.\nFinal answer: $\\frac{\\left(1 - 2\\cdot x\\right)\\cdot \\left(- x^{2} + 2\\cdot x + \\left(2\\cdot x^{2} - 2\\cdot x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 1\\right) + \\left(x^{2} - x + 1\\right)\\cdot \\left(- 2\\cdot x + \\left(4\\cdot x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(x^{2} - x + 1\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} - 2\\cdot x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\right)}{2\\cdot \\left(x^{2} - x + 1\\right)^{2}}$", "final_answer": "\\frac{\\left(1 - 2\\cdot x\\right)\\cdot \\left(- x^{2} + 2\\cdot x + \\left(2\\cdot x^{2} - 2\\cdot x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 1\\right) + \\left(x^{2} - x + 1\\right)\\cdot \\left(- 2\\cdot x + \\left(4\\cdot x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(x^{2} - x + 1\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} - 2\\cdot x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\right)}{2\\cdot \\left(x^{2} - x + 1\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b8b2e231a06573f0", "problem_id": "cbfc2742c4c3ebdf", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- \\frac{x^{2}}{2} + x + \\left(x^{2} - x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\frac{1}{2}}{x^{2} - x + 1}$.", "output": "Let f(x) = $\\frac{- \\frac{x^{2}}{2} + x + \\left(x^{2} - x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\frac{1}{2}}{x^{2} - x + 1}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{\\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + 2 \\left(x^{2} - x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(x^{2} - x + 1\\right) \\left(- 2 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} - x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - x + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{2 \\left(x^{2} - x + 1\\right)^{2}}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\frac{\\left(1 - 2\\cdot x\\right)\\cdot \\left(- x^{2} + 2\\cdot x + \\left(2\\cdot x^{2} - 2\\cdot x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 1\\right) + \\left(x^{2} - x + 1\\right)\\cdot \\left(- 2\\cdot x + \\left(4\\cdot x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(x^{2} - x + 1\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} - 2\\cdot x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\right)}{2\\cdot \\left(x^{2} - x + 1\\right)^{2}}$", "final_answer": "\\frac{\\left(1 - 2\\cdot x\\right)\\cdot \\left(- x^{2} + 2\\cdot x + \\left(2\\cdot x^{2} - 2\\cdot x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 1\\right) + \\left(x^{2} - x + 1\\right)\\cdot \\left(- 2\\cdot x + \\left(4\\cdot x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(x^{2} - x + 1\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} - 2\\cdot x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\right)}{2\\cdot \\left(x^{2} - x + 1\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "53946ec7541f2945", "problem_id": "cb09f9741b7199b8", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(2 x^{2} - 3 x + 2\\right) \\cos{\\left(2 x + 1 \\right)}$.", "output": "1) Let f(x) = $\\left(x + 2\\right) e^{x} + \\left(2 x^{2} - 3 x + 2\\right) \\cos{\\left(2 x + 1 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\left(x + 2\\right) e^{x} + \\left(4 x - 3\\right) \\cos{\\left(2 x + 1 \\right)} - 2 \\left(2 x^{2} - 3 x + 2\\right) \\sin{\\left(2 x + 1 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 2\\right)\\cdot e^{x} + \\left(4\\cdot x - 3\\right)\\cdot \\cos{\\left(2\\cdot x + 1 \\right)} - \\left(4\\cdot x^{2} - 6\\cdot x + 4\\right)\\cdot \\sin{\\left(2\\cdot x + 1 \\right)} + e^{x}$", "final_answer": "\\left(x + 2\\right)\\cdot e^{x} + \\left(4\\cdot x - 3\\right)\\cdot \\cos{\\left(2\\cdot x + 1 \\right)} - \\left(4\\cdot x^{2} - 6\\cdot x + 4\\right)\\cdot \\sin{\\left(2\\cdot x + 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6cc5b83a990271c3", "problem_id": "cb09f9741b7199b8", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(2 x^{2} - 3 x + 2\\right) \\cos{\\left(2 x + 1 \\right)}$.", "output": "Let f(x) = $\\left(x + 2\\right) e^{x} + \\left(2 x^{2} - 3 x + 2\\right) \\cos{\\left(2 x + 1 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\left(x + 2\\right) e^{x} + \\left(4 x - 3\\right) \\cos{\\left(2 x + 1 \\right)} - 2 \\left(2 x^{2} - 3 x + 2\\right) \\sin{\\left(2 x + 1 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 2\\right)\\cdot e^{x} + \\left(4\\cdot x - 3\\right)\\cdot \\cos{\\left(2\\cdot x + 1 \\right)} - \\left(4\\cdot x^{2} - 6\\cdot x + 4\\right)\\cdot \\sin{\\left(2\\cdot x + 1 \\right)} + e^{x}$", "final_answer": "\\left(x + 2\\right)\\cdot e^{x} + \\left(4\\cdot x - 3\\right)\\cdot \\cos{\\left(2\\cdot x + 1 \\right)} - \\left(4\\cdot x^{2} - 6\\cdot x + 4\\right)\\cdot \\sin{\\left(2\\cdot x + 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "236115495c9f08a8", "problem_id": "6c51de97f8e91639", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{3} x + \\frac{5 x}{4} - \\frac{9 \\sqrt{3}}{4} - \\frac{9}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{5}{4} - \\sqrt{3}$, c=$- \\frac{9 \\sqrt{3}}{4} - \\frac{9}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{13 \\sqrt{3}}{2} + \\frac{217}{16}$, so √Δ = $\\sqrt{\\frac{13 \\sqrt{3}}{2} + \\frac{217}{16}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{5}{8} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{104 \\sqrt{3} + 217}}{8}$.\n7) x2 = $- \\frac{\\sqrt{104 \\sqrt{3} + 217}}{8} - \\frac{5}{8} + \\frac{\\sqrt{3}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{5}{8} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{104\\cdot \\sqrt{3} + 217}}{8}, - \\frac{\\sqrt{104\\cdot \\sqrt{3} + 217}}{8} - \\frac{5}{8} + \\frac{\\sqrt{3}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{5}{8} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{104\\cdot \\sqrt{3} + 217}}{8}, - \\frac{\\sqrt{104\\cdot \\sqrt{3} + 217}}{8} - \\frac{5}{8} + \\frac{\\sqrt{3}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "137533654b70efa3", "problem_id": "6c51de97f8e91639", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{3} x + \\frac{5 x}{4} - \\frac{9 \\sqrt{3}}{4} - \\frac{9}{4}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{5}{4} - \\sqrt{3}$, c=$- \\frac{9 \\sqrt{3}}{4} - \\frac{9}{4}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{- \\frac{5}{8} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{104\\cdot \\sqrt{3} + 217}}{8}, - \\frac{\\sqrt{104\\cdot \\sqrt{3} + 217}}{8} - \\frac{5}{8} + \\frac{\\sqrt{3}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{5}{8} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{104\\cdot \\sqrt{3} + 217}}{8}, - \\frac{\\sqrt{104\\cdot \\sqrt{3} + 217}}{8} - \\frac{5}{8} + \\frac{\\sqrt{3}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ffa156d85f587e69", "problem_id": "e5c3973e4d7fe225", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "A store sells a backpack for $41. It is discounted by 15%. You buy 4 backpacks. What is the total cost?", "output": "Discounted price = 41 × (1 − 15/100) = 41 × 85/100 = 697/20.\nSubtotal = 4 × 697/20 = 697/5.\nFinal answer: $\\frac{697}{5}$", "final_answer": "\\frac{697}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3986b239613aece2", "problem_id": "e5c3973e4d7fe225", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A store sells a backpack for $41. It is discounted by 15%. You buy 4 backpacks. What is the total cost?", "output": "1) Discounted price = 41 × (1 − 15/100) = 41 × 85/100 = 697/20.\n2) Subtotal = 4 × 697/20 = 697/5.\nFinal answer: $\\frac{697}{5}$", "final_answer": "\\frac{697}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "43e2883aa48f4a83", "problem_id": "d62e0e0ca7029ce5", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} - x - 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x^{2}) = 2 x^{2}$.\n3) So $f'(x) = 2 x^{2} - 1$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x^{2}) = 2 x$.\nTherefore the correct derivative is $f'(x) = 2 x - 1$.\nFinal answer: $2 x - 1$", "final_answer": "2 x - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "1aca23d8e7f4a41c", "problem_id": "76ce63b064bca52e", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{3 x^{2} + \\left(x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{x^{2} - x + 5}$.", "output": "1) Let f(x) = $\\frac{3 x^{2} + \\left(x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{x^{2} - x + 5}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{2 \\left(1 - 2 x\\right) \\left(3 x^{2} + \\left(x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} - x + 5\\right) \\left(12 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)}\\right)}{2 \\left(x^{2} - x + 5\\right)^{2}}$.\nFinal answer: $\\frac{\\left(2 - 4\\cdot x\\right)\\cdot \\left(3\\cdot x^{2} + \\left(x^{2} - x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} - x + 5\\right)\\cdot \\left(12\\cdot x + \\left(4\\cdot x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(x^{2} - x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} - 2\\cdot x + 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)}\\right)}{2\\cdot \\left(x^{2} - x + 5\\right)^{2}}$", "final_answer": "\\frac{\\left(2 - 4\\cdot x\\right)\\cdot \\left(3\\cdot x^{2} + \\left(x^{2} - x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} - x + 5\\right)\\cdot \\left(12\\cdot x + \\left(4\\cdot x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(x^{2} - x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} - 2\\cdot x + 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)}\\right)}{2\\cdot \\left(x^{2} - x + 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "37be484e7ee77f99", "problem_id": "c70b1fa0bce0719e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} 2 - x$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to 2 - 2 x.\n3) Then the limit is 10.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: 6.\nFinal answer: $6$", "final_answer": "6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "1da51048693a24e4", "problem_id": "a3debb79282a3d12", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{2 \\sqrt{6} x}{3} - \\frac{13 x}{12} - \\frac{\\sqrt{6}}{6} - \\frac{1}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{2 \\sqrt{6}}{3} - \\frac{13}{12}$, c=$- \\frac{\\sqrt{6}}{6} - \\frac{1}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{19 \\sqrt{6}}{9} + \\frac{745}{144}$, so √Δ = $\\sqrt{\\frac{19 \\sqrt{6}}{9} + \\frac{745}{144}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{13}{24} + \\frac{\\sqrt{6}}{3} + \\frac{\\sqrt{304 \\sqrt{6} + 745}}{24}$.\n7) x2 = $- \\frac{\\sqrt{304 \\sqrt{6} + 745}}{24} + \\frac{13}{24} + \\frac{\\sqrt{6}}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{13}{24} + \\frac{\\sqrt{6}}{3} + \\frac{\\sqrt{304\\cdot \\sqrt{6} + 745}}{24}, - \\frac{\\sqrt{304\\cdot \\sqrt{6} + 745}}{24} + \\frac{13}{24} + \\frac{\\sqrt{6}}{3}\\}$", "final_answer": "x \\in \\{\\frac{13}{24} + \\frac{\\sqrt{6}}{3} + \\frac{\\sqrt{304\\cdot \\sqrt{6} + 745}}{24}, - \\frac{\\sqrt{304\\cdot \\sqrt{6} + 745}}{24} + \\frac{13}{24} + \\frac{\\sqrt{6}}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1a8215a53d1df5da", "problem_id": "ca31ff128a0e0062", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "You want 69 liters of a 25% salt solution. You have a 10% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 69−x liters is 40% solution.\n2) Salt equation: 10x + 40(69−x) = 25·69.\n3) (10-40)x = (25-40)·69 ⇒ x = 69/2.\n4) Then 69−x = 69/2.\nFinal answer: $69/2 L of 10% and 69/2 L of 40%$", "final_answer": "69/2 L of 10% and 69/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b700113abd50a755", "problem_id": "ca31ff128a0e0062", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 69 liters of a 25% salt solution. You have a 10% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 69−x liters is 40% solution.\nSalt equation: 10x + 40(69−x) = 25·69.\n(10-40)x = (25-40)·69 ⇒ x = 69/2.\nThen 69−x = 69/2.\nCheck: 10·69/2 + 40·69/2 = 25·69.\nFinal answer: $69/2 L of 10% and 69/2 L of 40%$", "final_answer": "69/2 L of 10% and 69/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "59f364df4fb091ef", "problem_id": "0a9dff5398569e7b", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A store sells a backpack for $740. It is discounted by 40%. You buy 7 backpacks. What is the total cost?", "output": "1) Discounted price = 740 × (1 − 40/100) = 740 × 60/100 = 444.\n2) Subtotal = 7 × 444 = 3108.\nFinal answer: $3108$", "final_answer": "3108", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1c852b2d5320caa2", "problem_id": "e4305ec474fba739", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x^{3} \\sin{\\left(x \\right)}) = x^{3} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = x^{3} \\cos{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x^{3} \\sin{\\left(x \\right)}) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.\nFinal answer: $x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$", "final_answer": "x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "4e284426c12f555f", "problem_id": "210951f11d859829", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 10 red balls and 11 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(21,9).\n3) Favorable samples: choose k reds and n−k blues: C(10,2)·C(11,7).\n4) Compute totals: C(21,9) = 293930, and favorable = 14850.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1485}{29393}$", "final_answer": "\\frac{1485}{29393}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5abc15d8339e46be", "problem_id": "210951f11d859829", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 10 red balls and 11 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(21,9).\nFavorable samples: choose k reds and n−k blues: C(10,2)·C(11,7).\nCompute totals: C(21,9) = 293930, and favorable = 14850.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1485}{29393}$.\nFinal answer: $\\frac{1485}{29393}$", "final_answer": "\\frac{1485}{29393}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72e8fb189c19fca4", "problem_id": "7586aebb5f0c8596", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A store sells a backpack for $515. It is discounted by 25%. You buy 8 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 515 × (1 − 25/100) = 515 × 75/100 = 1545/4.\nSubtotal = 8 × 1545/4 = 3090.\nTotal with tax = 3090 × 105/100 = 6489/2.\nCheck: recomputing the formula yields 6489/2.\nFinal answer: $\\frac{6489}{2}$", "final_answer": "\\frac{6489}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d65377f29f401c11", "problem_id": "7586aebb5f0c8596", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A store sells a backpack for $515. It is discounted by 25%. You buy 8 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 515 × (1 − 25/100) = 515 × 75/100 = 1545/4.\n2) Subtotal = 8 × 1545/4 = 3090.\n3) Total with tax = 3090 × 105/100 = 6489/2.\nFinal answer: $\\frac{6489}{2}$", "final_answer": "\\frac{6489}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d78d319321ad7378", "problem_id": "e871423f383ea0e8", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 9 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=9, k=2, p=$\\frac{1}{5}$ and simplify.\n4) Here C(9,2) = 36.\nFinal answer: $\\frac{589824}{1953125}$", "final_answer": "\\frac{589824}{1953125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b179f1d5532b0187", "problem_id": "8ec01072a4ea0cf1", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - x^{2} - 4 x - 10$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} - 5 x - 10.\n3) Then the limit is -34.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: -31.\nFinal answer: $-31$", "final_answer": "-31", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "005af739e14ab0bb", "problem_id": "7b00be037fec103b", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$x + 7 y$ = $-24$\n$- 7 x + 4 y$ = $-44$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-7$: $- 7 x - 49 y$ = $168$.\n3) Multiply the second equation by $1$: $- 7 x + 4 y$ = $-44$.\n4) Subtract to eliminate x: $- 53 y$ = $212$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, -4\\right)$", "final_answer": "(x,y)=\\left(4, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "56c06874255edb6c", "problem_id": "7b00be037fec103b", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$x + 7 y$ = $-24$\n$- 7 x + 4 y$ = $-44$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-7$: $- 7 x - 49 y$ = $168$.\nMultiply the second equation by $1$: $- 7 x + 4 y$ = $-44$.\nSubtract to eliminate x: $- 53 y$ = $212$.\nSolve for y: y = $-4$.\nSubstitute back to find x: x = $4$.\nCheck: plug (x,y)=(4,-4) into both equations.\nFinal answer: $(x,y)=\\left(4, -4\\right)$", "final_answer": "(x,y)=\\left(4, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d14452940b4c188d", "problem_id": "24bbc59ba0f74ee4", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "You want 49 liters of a 45% salt solution. You have a 15% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 49−x liters is 50% solution.\n2) Salt equation: 15x + 50(49−x) = 45·49.\n3) (15-50)x = (45-50)·49 ⇒ x = 7.\n4) Then 49−x = 42.\nFinal answer: $7 L of 15% and 42 L of 50%$", "final_answer": "7 L of 15% and 42 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9bfa9025c636e345", "problem_id": "24bbc59ba0f74ee4", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "You want 49 liters of a 45% salt solution. You have a 15% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 49−x liters is 50% solution.\nSalt equation: 15x + 50(49−x) = 45·49.\n(15-50)x = (45-50)·49 ⇒ x = 7.\nThen 49−x = 42.\nCheck: 15·7 + 50·42 = 45·49.\nFinal answer: $7 L of 15% and 42 L of 50%$", "final_answer": "7 L of 15% and 42 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dbd7046eb6a99fa9", "problem_id": "c3cd457315d63785", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the multiplicative inverse of 46 modulo 53.", "output": "1) We want x such that 46x ≡ 1 (mod 53).\n2) Use extended Euclid to find s,t with 46s + 53t = 1.\n3) One Bezout identity is: 46·(15) + 53·(-13) = 1.\n4) Reduce s modulo 53: x ≡ 15 (mod 53).\nFinal answer: $x\\equiv 15\\pmod{53}$", "final_answer": "x\\equiv 15\\pmod{53}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0b5ce5ce80937896", "problem_id": "23eafac488bc3cdd", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} x - 2$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to -2.\n3) Then the limit is -2.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -4.\nFinal answer: $-4$", "final_answer": "-4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c93890819a5d1093", "problem_id": "268995b8a49b5fbb", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Worker A finishes a job in 8 hours and Worker B finishes it in 11 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/8 job/hour. B's rate = 1/11 job/hour.\nCombined rate = 1/8 + 1/11 = 19/88 job/hour.\nTime = 1 ÷ (combined rate) = 88/19 hours.\nCheck: (combined rate)×(time) = 19/88×88/19 = 1.\nFinal answer: $\\frac{88}{19}$", "final_answer": "\\frac{88}{19}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "64e07c51e02c4820", "problem_id": "268995b8a49b5fbb", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Worker A finishes a job in 8 hours and Worker B finishes it in 11 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/8 job/hour. B's rate = 1/11 job/hour.\n2) Combined rate = 1/8 + 1/11 = 19/88 job/hour.\n3) Time = 1 ÷ (combined rate) = 88/19 hours.\nFinal answer: $\\frac{88}{19}$", "final_answer": "\\frac{88}{19}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3b53a4217fc8851c", "problem_id": "813044c6fff92796", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$- 2 x + 4 y$ = $-24$\n$4 x + 3 y$ = $4$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $- 8 x + 16 y$ = $-96$.\n3) Multiply the second equation by $-2$: $- 8 x - 6 y$ = $-8$.\n4) Subtract to eliminate x: $22 y$ = $-88$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, -4\\right)$", "final_answer": "(x,y)=\\left(4, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "026c0879c3c5fd67", "problem_id": "813044c6fff92796", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$- 2 x + 4 y$ = $-24$\n$4 x + 3 y$ = $4$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $- 8 x + 16 y$ = $-96$.\nMultiply the second equation by $-2$: $- 8 x - 6 y$ = $-8$.\nSubtract to eliminate x: $22 y$ = $-88$.\nSolve for y: y = $-4$.\nSubstitute back to find x: x = $4$.\nCheck: plug (x,y)=(4,-4) into both equations.\nFinal answer: $(x,y)=\\left(4, -4\\right)$", "final_answer": "(x,y)=\\left(4, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fbaf34337140c3c2", "problem_id": "e299be075e05a933", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "You want 28 liters of a 30% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 28−x liters is 50% solution.\n2) Salt equation: 25x + 50(28−x) = 30·28.\n3) (25-50)x = (30-50)·28 ⇒ x = 112/5.\n4) Then 28−x = 28/5.\nFinal answer: $112/5 L of 25% and 28/5 L of 50%$", "final_answer": "112/5 L of 25% and 28/5 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "09fe020a284257e7", "problem_id": "e299be075e05a933", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "You want 28 liters of a 30% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 28−x liters is 50% solution.\nSalt equation: 25x + 50(28−x) = 30·28.\n(25-50)x = (30-50)·28 ⇒ x = 112/5.\nFinal answer: $112/5 L of 25% and 28/5 L of 50%$", "final_answer": "112/5 L of 25% and 28/5 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f3cf25a4d3492d0c", "problem_id": "8654b4f6f741c4c3", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$2 x + 4 y$ = $-16$\n$- 3 x + 2 y$ = $8$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $- 6 x - 12 y$ = $48$.\nMultiply the second equation by $2$: $- 6 x + 4 y$ = $16$.\nSubtract to eliminate x: $- 16 y$ = $32$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $-4$.\nCheck: plug (x,y)=(-4,-2) into both equations.\nFinal answer: $(x,y)=\\left(-4, -2\\right)$", "final_answer": "(x,y)=\\left(-4, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fef0ecab9d9aeb15", "problem_id": "8654b4f6f741c4c3", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the linear system:\n$2 x + 4 y$ = $-16$\n$- 3 x + 2 y$ = $8$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $- 6 x - 12 y$ = $48$.\nMultiply the second equation by $2$: $- 6 x + 4 y$ = $16$.\nFinal answer: $(x,y)=\\left(-4, -2\\right)$", "final_answer": "(x,y)=\\left(-4, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0c06f0c7e44f2ac4", "problem_id": "22a3089edc84880c", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x - 3$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{2} - x + 1$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 2 x - 1$.\n5) Evaluate at a: p'(2) = $-5$.\nFinal answer: $-5$", "final_answer": "-5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "894212ea276310e3", "problem_id": "22a3089edc84880c", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x - 3$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{2} - x + 1$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-5$", "final_answer": "-5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "365fdbdc0faa74bc", "problem_id": "507eb9a4a288b565", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{3} + x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $2 x^{3} + x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 2$.\nCompute an antiderivative: F(x) = $\\frac{x^{4}}{2} + \\frac{x^{2}}{2} - 2 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $C + \\frac{x^{4}}{2} + \\frac{x^{2}}{2} - 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{4}}{2} + \\frac{x^{2}}{2} - 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a42300b4ab774253", "problem_id": "507eb9a4a288b565", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{3} + x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $2 x^{3} + x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 2$.\n3) Compute an antiderivative: F(x) = $\\frac{x^{4}}{2} + \\frac{x^{2}}{2} - 2 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{x^{4}}{2} + \\frac{x^{2}}{2} - 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{4}}{2} + \\frac{x^{2}}{2} - 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f6588f39ae746c65", "problem_id": "3fcba1c3b40c80b2", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡0 (mod 8), x≡3 (mod 11). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 80 (mod 88).\nFinal answer: $x\\equiv 80\\pmod{88}$", "final_answer": "x\\equiv 80\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fdc8cd2ecd03d912", "problem_id": "3fcba1c3b40c80b2", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡0 (mod 8), x≡3 (mod 11). Give the solution modulo 88.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 88.\nCompute the CRT solution to get x ≡ 80 (mod 88).\nCheck: 80 mod 8=0 and mod 11=3.\nFinal answer: $x\\equiv 80\\pmod{88}$", "final_answer": "x\\equiv 80\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1e6c8382933c96eb", "problem_id": "1e3f5d34be19a547", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 13 red balls and 11 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(24,5).\n3) Favorable samples: choose k reds and n−k blues: C(13,2)·C(11,3).\n4) Compute totals: C(24,5) = 42504, and favorable = 12870.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{195}{644}$", "final_answer": "\\frac{195}{644}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8b4f130aae9f1942", "problem_id": "1e3f5d34be19a547", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 13 red balls and 11 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(24,5).\nFavorable samples: choose k reds and n−k blues: C(13,2)·C(11,3).\nCompute totals: C(24,5) = 42504, and favorable = 12870.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{195}{644}$.\nFinal answer: $\\frac{195}{644}$", "final_answer": "\\frac{195}{644}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5f3209c8739b895c", "problem_id": "3451b6d8ff262c62", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $- x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- x^{3} \\cos{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- x^{3}\\cdot \\cos{\\left(x \\right)} - 3\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + \\cos{\\left(x \\right)}$", "final_answer": "- x^{3}\\cdot \\cos{\\left(x \\right)} - 3\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "481b51e4325115e4", "problem_id": "3451b6d8ff262c62", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $- x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- x^{3} \\cos{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.\nFinal answer: $- x^{3}\\cdot \\cos{\\left(x \\right)} - 3\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + \\cos{\\left(x \\right)}$", "final_answer": "- x^{3}\\cdot \\cos{\\left(x \\right)} - 3\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fb83c9f787435399", "problem_id": "2860b9294822295a", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_4=26$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=26 and k=4.\n4) Compute C(29,3) = 3654.\nFinal answer: $3654$", "final_answer": "3654", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "20f32edef04d4911", "problem_id": "edc00fcb23b7330a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} x^{3} + 7 x^{2} + 27 x + 109$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to x^{3} + 7 x^{2} + 26 x + 109.\n3) Then the limit is 389.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: 393.\nFinal answer: $393$", "final_answer": "393", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d75e10c420cd6358", "problem_id": "4ed30e1497631f7c", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $684 in the ratio 5:7. How much does each receive?", "output": "1) Total ratio parts = 5+7 = 12.\n2) Each part = 684 ÷ 12 = 57.\n3) First share = 5 × 57 = 285.\n4) Second share = 7 × 57 = 399.\nFinal answer: $399$", "final_answer": "399", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d2e9f8d0a262e965", "problem_id": "4ed30e1497631f7c", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Two friends split $684 in the ratio 5:7. How much does each receive?", "output": "Total ratio parts = 5+7 = 12.\nEach part = 684 ÷ 12 = 57.\nFirst share = 5 × 57 = 285.\nFinal answer: $399$", "final_answer": "399", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e40eda7c3d9f3017", "problem_id": "7bea3cb924117883", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡2 (mod 3), x≡1 (mod 8). Give the solution modulo 24.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 24.\n3) Compute the CRT solution to get x ≡ 17 (mod 24).\nFinal answer: $x\\equiv 17\\pmod{24}$", "final_answer": "x\\equiv 17\\pmod{24}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "95996384c857b606", "problem_id": "7bea3cb924117883", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 3), x≡1 (mod 8). Give the solution modulo 24.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 24.\nCompute the CRT solution to get x ≡ 17 (mod 24).\nCheck: 17 mod 3=2 and mod 8=1.\nFinal answer: $x\\equiv 17\\pmod{24}$", "final_answer": "x\\equiv 17\\pmod{24}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e912ec89f834558c", "problem_id": "f0d97fdc17f3b402", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$3 x + 7 y$ = $-65$\n$10 x - 10 y$ = $50$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $10$: $30 x + 70 y$ = $-650$.\nMultiply the second equation by $3$: $30 x - 30 y$ = $150$.\nSubtract to eliminate x: $100 y$ = $-800$.\nSolve for y: y = $-8$.\nSubstitute back to find x: x = $-3$.\nCheck: plug (x,y)=(-3,-8) into both equations.\nFinal answer: $(x,y)=\\left(-3, -8\\right)$", "final_answer": "(x,y)=\\left(-3, -8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "889c410ff9c69e4a", "problem_id": "f0d97fdc17f3b402", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$3 x + 7 y$ = $-65$\n$10 x - 10 y$ = $50$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $10$: $30 x + 70 y$ = $-650$.\n3) Multiply the second equation by $3$: $30 x - 30 y$ = $150$.\n4) Subtract to eliminate x: $100 y$ = $-800$.\n5) Solve for y: y = $-8$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, -8\\right)$", "final_answer": "(x,y)=\\left(-3, -8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "97c92459d2e712b5", "problem_id": "9b7be8182a320b01", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many binary strings of length 14 contain exactly 4 ones with no two ones adjacent?", "output": "Reserve 3 mandatory zeros between the 4 ones.\nThis leaves n−k+1 = 11 valid slots for the ones.\nChoose the slots: C(11,4) = 330.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $330$", "final_answer": "330", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c5e73ad23bf3eba0", "problem_id": "9b7be8182a320b01", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many binary strings of length 14 contain exactly 4 ones with no two ones adjacent?", "output": "1) Reserve 3 mandatory zeros between the 4 ones.\n2) This leaves n−k+1 = 11 valid slots for the ones.\n3) Choose the slots: C(11,4) = 330.\nFinal answer: $330$", "final_answer": "330", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "45f08c95121583ca", "problem_id": "10352392b4981094", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} - 2 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\nSubtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{\\pi^{3}}{12} - \\frac{\\pi^{2}}{8} + \\frac{\\pi}{2} + 2$.\nFinal answer: $- \\frac{\\pi^{3}}{12} - \\frac{\\pi^{2}}{8} + \\frac{\\pi}{2} + 2$", "final_answer": "- \\frac{\\pi^{3}}{12} - \\frac{\\pi^{2}}{8} + \\frac{\\pi}{2} + 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1b3081feebca8d6b", "problem_id": "10352392b4981094", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} - 2 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\n3) Subtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{\\pi^{3}}{12} - \\frac{\\pi^{2}}{8} + \\frac{\\pi}{2} + 2$.\n4) Simplify the result.\nFinal answer: $- \\frac{\\pi^{3}}{12} - \\frac{\\pi^{2}}{8} + \\frac{\\pi}{2} + 2$", "final_answer": "- \\frac{\\pi^{3}}{12} - \\frac{\\pi^{2}}{8} + \\frac{\\pi}{2} + 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ca6f9828192b9863", "problem_id": "3edc1adad42da0c8", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "You want 30 liters of a 45% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 30−x liters is 70% solution.\nSalt equation: 20x + 70(30−x) = 45·30.\n(20-70)x = (45-70)·30 ⇒ x = 15.\nThen 30−x = 15.\nCheck: 20·15 + 70·15 = 45·30.\nFinal answer: $15 L of 20% and 15 L of 70%$", "final_answer": "15 L of 20% and 15 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6c04d19f5ba0adb7", "problem_id": "3edc1adad42da0c8", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "You want 30 liters of a 45% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 30−x liters is 70% solution.\n2) Salt equation: 20x + 70(30−x) = 45·30.\n3) (20-70)x = (45-70)·30 ⇒ x = 15.\n4) Then 30−x = 15.\nFinal answer: $15 L of 20% and 15 L of 70%$", "final_answer": "15 L of 20% and 15 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "81c7bd9024a2c397", "problem_id": "22cbb7d6762ab061", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(- x^{2} - x - 2\\right) \\cos{\\left(x - 3 \\right)}$.", "output": "1) Let f(x) = $2 \\left(x + 2\\right) e^{x} + \\left(- x^{2} - x - 2\\right) \\cos{\\left(x - 3 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\left(- 2 x - 1\\right) \\cos{\\left(x - 3 \\right)} + 2 \\left(x + 2\\right) e^{x} - \\left(- x^{2} - x - 2\\right) \\sin{\\left(x - 3 \\right)} + 2 e^{x}$.\nFinal answer: $\\left(- 2\\cdot x - 1\\right)\\cdot \\cos{\\left(x - 3 \\right)} + \\left(2\\cdot x + 4\\right)\\cdot e^{x} - \\left(- x^{2} - x - 2\\right)\\cdot \\sin{\\left(x - 3 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(- 2\\cdot x - 1\\right)\\cdot \\cos{\\left(x - 3 \\right)} + \\left(2\\cdot x + 4\\right)\\cdot e^{x} - \\left(- x^{2} - x - 2\\right)\\cdot \\sin{\\left(x - 3 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "99d9ce6de6a37730", "problem_id": "22cbb7d6762ab061", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(- x^{2} - x - 2\\right) \\cos{\\left(x - 3 \\right)}$.", "output": "Let f(x) = $2 \\left(x + 2\\right) e^{x} + \\left(- x^{2} - x - 2\\right) \\cos{\\left(x - 3 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\left(- 2 x - 1\\right) \\cos{\\left(x - 3 \\right)} + 2 \\left(x + 2\\right) e^{x} - \\left(- x^{2} - x - 2\\right) \\sin{\\left(x - 3 \\right)} + 2 e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\left(- 2\\cdot x - 1\\right)\\cdot \\cos{\\left(x - 3 \\right)} + \\left(2\\cdot x + 4\\right)\\cdot e^{x} - \\left(- x^{2} - x - 2\\right)\\cdot \\sin{\\left(x - 3 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(- 2\\cdot x - 1\\right)\\cdot \\cos{\\left(x - 3 \\right)} + \\left(2\\cdot x + 4\\right)\\cdot e^{x} - \\left(- x^{2} - x - 2\\right)\\cdot \\sin{\\left(x - 3 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff94b0da7a1b04f4", "problem_id": "0208c76e2c55f4be", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{3}$. If it is flipped 12 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=4, p=$\\frac{1}{3}$ and simplify.\n4) Here C(12,4) = 495.\nFinal answer: $\\frac{14080}{59049}$", "final_answer": "\\frac{14080}{59049}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "466ccb9d345aecef", "problem_id": "0b02fa7f2099f57b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - 2 x - 8$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to - 3 x - 8.\n3) Then the limit is -20.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: -16.\nFinal answer: $-16$", "final_answer": "-16", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "33f1c4c811d2139a", "problem_id": "1bb9af3d82255237", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$x + 8 y$ = $-50$\n$- 6 x + 5 y$ = $-71$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-6$: $- 6 x - 48 y$ = $300$.\n3) Multiply the second equation by $1$: $- 6 x + 5 y$ = $-71$.\n4) Subtract to eliminate x: $- 53 y$ = $371$.\n5) Solve for y: y = $-7$.\n6) Substitute back to find x: x = $6$.\nFinal answer: $(x,y)=\\left(6, -7\\right)$", "final_answer": "(x,y)=\\left(6, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dbccbd2482089a52", "problem_id": "1bb9af3d82255237", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$x + 8 y$ = $-50$\n$- 6 x + 5 y$ = $-71$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-6$: $- 6 x - 48 y$ = $300$.\nMultiply the second equation by $1$: $- 6 x + 5 y$ = $-71$.\nSubtract to eliminate x: $- 53 y$ = $371$.\nSolve for y: y = $-7$.\nSubstitute back to find x: x = $6$.\nCheck: plug (x,y)=(6,-7) into both equations.\nFinal answer: $(x,y)=\\left(6, -7\\right)$", "final_answer": "(x,y)=\\left(6, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7dee763a17d0e02e", "problem_id": "06e4bc2eebfcb779", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} 2 x^{3} + x^{2} + 4 x + 1$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $2 x^{4} - x^{3} + 3 x^{2} - 3 x - 2$ and a = 1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $8 x^{3} - 3 x^{2} + 6 x - 3$.\n5) Evaluate at a: p'(1) = $8$.\nFinal answer: $8$", "final_answer": "8", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2b26a85421504b60", "problem_id": "06e4bc2eebfcb779", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} 2 x^{3} + x^{2} + 4 x + 1$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{4} - x^{3} + 3 x^{2} - 3 x - 2$ and a = 1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $8 x^{3} - 3 x^{2} + 6 x - 3$.\nEvaluate at a: p'(1) = $8$.\nCheck: the identity gives $8$.\nFinal answer: $8$", "final_answer": "8", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1e1cae6426bba7de", "problem_id": "4f5b09861a905c03", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "You want 65 liters of a 45% salt solution. You have a 30% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 65−x liters is 50% solution.\nSalt equation: 30x + 50(65−x) = 45·65.\n(30-50)x = (45-50)·65 ⇒ x = 65/4.\nFinal answer: $65/4 L of 30% and 195/4 L of 50%$", "final_answer": "65/4 L of 30% and 195/4 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8849313bf5f375ee", "problem_id": "4f5b09861a905c03", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "You want 65 liters of a 45% salt solution. You have a 30% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 65−x liters is 50% solution.\n2) Salt equation: 30x + 50(65−x) = 45·65.\n3) (30-50)x = (45-50)·65 ⇒ x = 65/4.\n4) Then 65−x = 195/4.\nFinal answer: $65/4 L of 30% and 195/4 L of 50%$", "final_answer": "65/4 L of 30% and 195/4 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b1bf195548c8c9d9", "problem_id": "9ef96f653160a7af", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "How many binary strings of length 25 contain exactly 10 ones with no two ones adjacent?", "output": "1) Reserve 9 mandatory zeros between the 10 ones.\n2) This leaves n−k+1 = 16 valid slots for the ones.\n3) Choose the slots: C(16,10) = 8008.\nFinal answer: $8008$", "final_answer": "8008", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "14c74c9efefaf1df", "problem_id": "ded2825e9d1f2dcf", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 2 x + 1$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to x + 1.\n3) Then the limit is 0.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: -1.\nFinal answer: $-1$", "final_answer": "-1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c1f90993805b5212", "problem_id": "11f1bf55f71d6729", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} - 2 x^{3} - x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{3} - x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\nEvaluate at the bounds: F(\\pi) and F(- \\pi).\nSubtract: F(\\pi) − F(- \\pi) = $- \\frac{2 \\pi^{3}}{3}$.\nFinal answer: $- \\frac{2\\cdot \\pi^{3}}{3}$", "final_answer": "- \\frac{2\\cdot \\pi^{3}}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "85797bfc1787ca7f", "problem_id": "11f1bf55f71d6729", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} - 2 x^{3} - x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{3} - x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\pi).\n3) Subtract: F(\\pi) − F(- \\pi) = $- \\frac{2 \\pi^{3}}{3}$.\n4) Simplify the result.\nFinal answer: $- \\frac{2\\cdot \\pi^{3}}{3}$", "final_answer": "- \\frac{2\\cdot \\pi^{3}}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "827a910a0f7a2a66", "problem_id": "93eef7a87d357969", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡4 (mod 8), x≡4 (mod 5). Give the solution modulo 40.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 40.\nCompute the CRT solution to get x ≡ 4 (mod 40).\nCheck: 4 mod 8=4 and mod 5=4.\nFinal answer: $x\\equiv 4\\pmod{40}$", "final_answer": "x\\equiv 4\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "976b148bff28ea1e", "problem_id": "93eef7a87d357969", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the congruences: x≡4 (mod 8), x≡4 (mod 5). Give the solution modulo 40.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 40.\n3) Compute the CRT solution to get x ≡ 4 (mod 40).\nFinal answer: $x\\equiv 4\\pmod{40}$", "final_answer": "x\\equiv 4\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ad6d72bb3bf874d7", "problem_id": "ceeb0e86087a79f9", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡0 (mod 4), x≡10 (mod 11). Give the solution modulo 44.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 44.\nCompute the CRT solution to get x ≡ 32 (mod 44).\nCheck: 32 mod 4=0 and mod 11=10.\nFinal answer: $x\\equiv 32\\pmod{44}$", "final_answer": "x\\equiv 32\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "621aa543cb992d54", "problem_id": "ceeb0e86087a79f9", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡0 (mod 4), x≡10 (mod 11). Give the solution modulo 44.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 44.\n3) Compute the CRT solution to get x ≡ 32 (mod 44).\nFinal answer: $x\\equiv 32\\pmod{44}$", "final_answer": "x\\equiv 32\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d102e14b8c9af4a5", "problem_id": "64f7bdef29dc90da", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} 3 x^{3} - 3 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{3} - 3 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2$.\nEvaluate at the bounds: F(2 \\pi) and F(0).\nSubtract: F(2 \\pi) − F(0) = $2 \\pi \\left(- 4 \\pi^{2} - 2 + \\pi + 6 \\pi^{3}\\right)$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $2\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} - 2 + \\pi + 6\\cdot \\pi^{3}\\right)$", "final_answer": "2\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} - 2 + \\pi + 6\\cdot \\pi^{3}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0d3ce905a9f97d45", "problem_id": "64f7bdef29dc90da", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} 3 x^{3} - 3 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{3} - 3 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(0).\n3) Subtract: F(2 \\pi) − F(0) = $2 \\pi \\left(- 4 \\pi^{2} - 2 + \\pi + 6 \\pi^{3}\\right)$.\n4) Simplify the result.\nFinal answer: $2\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} - 2 + \\pi + 6\\cdot \\pi^{3}\\right)$", "final_answer": "2\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} - 2 + \\pi + 6\\cdot \\pi^{3}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4262608ce1ee343d", "problem_id": "88a3f8b1c4d2dc72", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Worker A finishes a job in 9 hours and Worker B finishes it in 8 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/9 job/hour. B's rate = 1/8 job/hour.\n2) Combined rate = 1/9 + 1/8 = 17/72 job/hour.\n3) Time = 1 ÷ (combined rate) = 72/17 hours.\nFinal answer: $\\frac{72}{17}$", "final_answer": "\\frac{72}{17}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d44b70977f5720f4", "problem_id": "88a3f8b1c4d2dc72", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Worker A finishes a job in 9 hours and Worker B finishes it in 8 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/9 job/hour. B's rate = 1/8 job/hour.\nCombined rate = 1/9 + 1/8 = 17/72 job/hour.\nTime = 1 ÷ (combined rate) = 72/17 hours.\nFinal answer: $\\frac{72}{17}$", "final_answer": "\\frac{72}{17}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a0e9845f4c064d6", "problem_id": "5b0eac0d9a358a1a", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 62 miles per hour for 3/2 hours, then at 23 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 62 × 3/2 = 93 miles.\nSecond leg distance = 23 × 3 = 69 miles.\nTotal distance = 93 + 69 = 162 miles.\nCheck: adding both legs gives 162.\nFinal answer: $162$", "final_answer": "162", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7e805b21a2e57653", "problem_id": "5b0eac0d9a358a1a", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A cyclist rides at 62 miles per hour for 3/2 hours, then at 23 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 62 × 3/2 = 93 miles.\n2) Second leg distance = 23 × 3 = 69 miles.\n3) Total distance = 93 + 69 = 162 miles.\nFinal answer: $162$", "final_answer": "162", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1a24d8472417230d", "problem_id": "071117aaf9c64e80", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Find the multiplicative inverse of 9 modulo 13.", "output": "1) We want x such that 9x ≡ 1 (mod 13).\n2) Use extended Euclid to find s,t with 9s + 13t = 1.\n3) One Bezout identity is: 9·(3) + 13·(-2) = 1.\n4) Reduce s modulo 13: x ≡ 3 (mod 13).\nFinal answer: $x\\equiv 3\\pmod{13}$", "final_answer": "x\\equiv 3\\pmod{13}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a8ca08385148d39a", "problem_id": "071117aaf9c64e80", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 9 modulo 13.", "output": "We want x such that 9x ≡ 1 (mod 13).\nUse extended Euclid to find s,t with 9s + 13t = 1.\nOne Bezout identity is: 9·(3) + 13·(-2) = 1.\nReduce s modulo 13: x ≡ 3 (mod 13).\nCheck: 9·3 mod 13 = 1.\nFinal answer: $x\\equiv 3\\pmod{13}$", "final_answer": "x\\equiv 3\\pmod{13}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "71672d0c4aa6846c", "problem_id": "1cfad1636f9bd3b1", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{11} x + 6 x - \\sqrt{11} + 5$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$6 - \\sqrt{11}$, c=$5 - \\sqrt{11}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $27 - 8 \\sqrt{11}$, so √Δ = $\\sqrt{27 - 8 \\sqrt{11}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $-3 + \\frac{\\sqrt{27 - 8 \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}$.\n7) x2 = $-3 - \\frac{\\sqrt{27 - 8 \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{-3 + \\frac{\\sqrt{27 - 8\\cdot \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}, -3 - \\frac{\\sqrt{27 - 8\\cdot \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}\\}$", "final_answer": "x \\in \\{-3 + \\frac{\\sqrt{27 - 8\\cdot \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}, -3 - \\frac{\\sqrt{27 - 8\\cdot \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f342284d111bf3b9", "problem_id": "1cfad1636f9bd3b1", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{11} x + 6 x - \\sqrt{11} + 5$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$6 - \\sqrt{11}$, c=$5 - \\sqrt{11}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $27 - 8 \\sqrt{11}$, so √Δ = $\\sqrt{27 - 8 \\sqrt{11}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $-3 + \\frac{\\sqrt{27 - 8 \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}$.\nx2 = $-3 - \\frac{\\sqrt{27 - 8 \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-3 + \\frac{\\sqrt{27 - 8\\cdot \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}, -3 - \\frac{\\sqrt{27 - 8\\cdot \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}\\}$", "final_answer": "x \\in \\{-3 + \\frac{\\sqrt{27 - 8\\cdot \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}, -3 - \\frac{\\sqrt{27 - 8\\cdot \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "511b3f4bee62d884", "problem_id": "35f63850d2c45553", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "You want 15 liters of a 50% salt solution. You have a 30% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 15−x liters is 70% solution.\n2) Salt equation: 30x + 70(15−x) = 50·15.\n3) (30-70)x = (50-70)·15 ⇒ x = 15/2.\n4) Then 15−x = 15/2.\nFinal answer: $15/2 L of 30% and 15/2 L of 70%$", "final_answer": "15/2 L of 30% and 15/2 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "69cfeba60396a9ee", "problem_id": "35f63850d2c45553", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "You want 15 liters of a 50% salt solution. You have a 30% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 15−x liters is 70% solution.\nSalt equation: 30x + 70(15−x) = 50·15.\n(30-70)x = (50-70)·15 ⇒ x = 15/2.\nFinal answer: $15/2 L of 30% and 15/2 L of 70%$", "final_answer": "15/2 L of 30% and 15/2 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "02cfc65e5a1b03b8", "problem_id": "b37b7a01e38512fb", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A cyclist rides at 34 miles per hour for 1/2 hours, then at 70 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 34 × 1/2 = 17 miles.\n2) Second leg distance = 70 × 5/2 = 175 miles.\n3) Total distance = 17 + 175 = 192 miles.\nFinal answer: $192$", "final_answer": "192", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1d8d4da999ab52e3", "problem_id": "b37b7a01e38512fb", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A cyclist rides at 34 miles per hour for 1/2 hours, then at 70 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 34 × 1/2 = 17 miles.\nSecond leg distance = 70 × 5/2 = 175 miles.\nTotal distance = 17 + 175 = 192 miles.\nCheck: adding both legs gives 192.\nFinal answer: $192$", "final_answer": "192", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "de88d6cf8b493fe6", "problem_id": "7c79c8c25f15235f", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $455 in the ratio 7:6. How much does each receive?", "output": "Total ratio parts = 7+6 = 13.\nEach part = 455 ÷ 13 = 35.\nFirst share = 7 × 35 = 245.\nSecond share = 6 × 35 = 210.\nCheck: 245+210=455.\nFinal answer: $210$", "final_answer": "210", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fc9729277abbedc6", "problem_id": "7c79c8c25f15235f", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Two friends split $455 in the ratio 7:6. How much does each receive?", "output": "1) Total ratio parts = 7+6 = 13.\n2) Each part = 455 ÷ 13 = 35.\n3) First share = 7 × 35 = 245.\n4) Second share = 6 × 35 = 210.\nFinal answer: $210$", "final_answer": "210", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "347068b444d8df0a", "problem_id": "e48621923d646dc5", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} - 4 x^{3} + x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 4 x^{3} + x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\n2) Evaluate at the bounds: F(\\pi) and F(0).\n3) Subtract: F(\\pi) − F(0) = $\\pi^{2} \\left(- \\pi^{2} + \\frac{\\pi}{3} + \\frac{3}{2}\\right)$.\n4) Simplify the result.\nFinal answer: $\\pi^{2}\\cdot \\left(- \\pi^{2} + \\frac{\\pi}{3} + \\frac{3}{2}\\right)$", "final_answer": "\\pi^{2}\\cdot \\left(- \\pi^{2} + \\frac{\\pi}{3} + \\frac{3}{2}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0fc07bb6e4144e02", "problem_id": "e48621923d646dc5", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} - 4 x^{3} + x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 4 x^{3} + x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nEvaluate at the bounds: F(\\pi) and F(0).\nSubtract: F(\\pi) − F(0) = $\\pi^{2} \\left(- \\pi^{2} + \\frac{\\pi}{3} + \\frac{3}{2}\\right)$.\nFinal answer: $\\pi^{2}\\cdot \\left(- \\pi^{2} + \\frac{\\pi}{3} + \\frac{3}{2}\\right)$", "final_answer": "\\pi^{2}\\cdot \\left(- \\pi^{2} + \\frac{\\pi}{3} + \\frac{3}{2}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e223852e7c87ed76", "problem_id": "77fbed54651aa902", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} - x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\pi).\n3) Subtract: F(\\pi) − F(- \\pi) = $\\frac{2 \\pi \\left(6 - \\pi^{2}\\right)}{3}$.\n4) Simplify the result.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(6 - \\pi^{2}\\right)}{3}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(6 - \\pi^{2}\\right)}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6d85e7c015993855", "problem_id": "77fbed54651aa902", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} - x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\nEvaluate at the bounds: F(\\pi) and F(- \\pi).\nSubtract: F(\\pi) − F(- \\pi) = $\\frac{2 \\pi \\left(6 - \\pi^{2}\\right)}{3}$.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(6 - \\pi^{2}\\right)}{3}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(6 - \\pi^{2}\\right)}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f88e6921b62010b7", "problem_id": "2ac87d426699ce1d", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A cyclist rides at 30 miles per hour for 1/2 hours, then at 65 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 30 × 1/2 = 15 miles.\n2) Second leg distance = 65 × 5/2 = 325/2 miles.\n3) Total distance = 15 + 325/2 = 355/2 miles.\nFinal answer: $\\frac{355}{2}$", "final_answer": "\\frac{355}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "952ff1e6a285bb0e", "problem_id": "2ac87d426699ce1d", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A cyclist rides at 30 miles per hour for 1/2 hours, then at 65 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 30 × 1/2 = 15 miles.\nSecond leg distance = 65 × 5/2 = 325/2 miles.\nTotal distance = 15 + 325/2 = 355/2 miles.\nFinal answer: $\\frac{355}{2}$", "final_answer": "\\frac{355}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "228cd2a6958b9f7c", "problem_id": "30b1fab2e37dc270", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{3} + 3 x^{2} - 3 x - 2$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(3 x^{3}) = 9 x^{3}$.\n3) So $f'(x) = 9 x^{3} + 6 x - 3$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(3 x^{3}) = 9 x^{2}$.\nTherefore the correct derivative is $f'(x) = 3 \\left(3 x^{2} + 2 x - 1\\right)$.\nFinal answer: $3 \\left(3 x^{2} + 2 x - 1\\right)$", "final_answer": "3 \\left(3 x^{2} + 2 x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "35f9ad1c0989a6fd", "problem_id": "64c914777d65c03c", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + \\frac{6 x}{5} - \\frac{48}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$\\frac{6}{5}$, c=$- \\frac{48}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{2916}{25}$, so √Δ = $\\frac{54}{5}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{8}{5}$.\n7) x2 = $-2$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{8}{5}, -2\\}$", "final_answer": "x \\in \\{\\frac{8}{5}, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5879dbc9537ee7c5", "problem_id": "218011b181ba8159", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A cyclist rides at 57 miles per hour for 2 hours, then at 66 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 57 × 2 = 114 miles.\n2) Second leg distance = 66 × 1 = 66 miles.\n3) Total distance = 114 + 66 = 180 miles.\nFinal answer: $180$", "final_answer": "180", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aff3715e93eac44c", "problem_id": "218011b181ba8159", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A cyclist rides at 57 miles per hour for 2 hours, then at 66 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 57 × 2 = 114 miles.\nSecond leg distance = 66 × 1 = 66 miles.\nTotal distance = 114 + 66 = 180 miles.\nFinal answer: $180$", "final_answer": "180", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "930403cfb7641da2", "problem_id": "5da22607f8cfb2b5", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - x^{2} + 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 3$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 9 x^{2} - 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 2$.\n2) Slope at x=2: m = f'(2) = $-38 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\n3) Point on curve: (2, f(2)) = (2, $-21 + \\cos{\\left(4 \\right)} + e$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -21 + \\cos{\\left(4 \\right)} + e = -38 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\n6) Optionally expand: y = $- 38 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 55$.\nFinal answer: $y = - 38\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 55$", "final_answer": "y = - 38\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 55", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "037911f14e7a4b7b", "problem_id": "5da22607f8cfb2b5", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - x^{2} + 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 3$ at $x=2$.", "output": "Differentiate: f'(x) = $- 9 x^{2} - 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 2$.\nSlope at x=2: m = f'(2) = $-38 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\nPoint on curve: (2, f(2)) = (2, $-21 + \\cos{\\left(4 \\right)} + e$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -21 + \\cos{\\left(4 \\right)} + e = -38 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\nOptionally expand: y = $- 38 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 55$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 38\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 55$", "final_answer": "y = - 38\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 55", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0939fb2d1b2e1cac", "problem_id": "78172a1024ad9bd8", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Two friends split $1177 in the ratio 8:3. How much does each receive?", "output": "1) Total ratio parts = 8+3 = 11.\n2) Each part = 1177 ÷ 11 = 107.\n3) First share = 8 × 107 = 856.\n4) Second share = 3 × 107 = 321.\nFinal answer: $321$", "final_answer": "321", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e574dae443c671c6", "problem_id": "78172a1024ad9bd8", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Two friends split $1177 in the ratio 8:3. How much does each receive?", "output": "Total ratio parts = 8+3 = 11.\nEach part = 1177 ÷ 11 = 107.\nFirst share = 8 × 107 = 856.\nSecond share = 3 × 107 = 321.\nCheck: 856+321=1177.\nFinal answer: $321$", "final_answer": "321", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "88d73fc0a8d8ce3b", "problem_id": "63bed2671c6cdb57", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x \\left(- x^{2} - x + 1\\right)$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{4} + x^{3} + 3 x^{2} - 2 x - 3$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-10$", "final_answer": "-10", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "73d8ea93d9ba2db9", "problem_id": "63bed2671c6cdb57", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x \\left(- x^{2} - x + 1\\right)$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{4} + x^{3} + 3 x^{2} - 2 x - 3$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 4 x^{3} + 3 x^{2} + 6 x - 2$.\nEvaluate at a: p'(2) = $-10$.\nCheck: the identity gives $-10$.\nFinal answer: $-10$", "final_answer": "-10", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f5be41c349334ca8", "problem_id": "303abfbf507cb79d", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{2} \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $- 3 x^{2} \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $- 3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3966a9cb41cd4a73", "problem_id": "303abfbf507cb79d", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{2} \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $- 3 x^{2} \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4ab34b5b8e5f44eb", "problem_id": "5437bf005bdd9a90", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{4} - x^{2} + 3 x + 4$.", "output": "1) Differentiate term-by-term:\n2) d/dx(4) = 0.\n3) d/dx(- x^{2}) = - 2 x.\n4) d/dx(- 2 x^{4}) = - 8 x^{3}.\n5) d/dx(3 x) = 3.\n6) Combine: f'(x) = - 8 x^{3} - 2 x + 3.\nFinal answer: $- 8\\cdot x^{3} - 2\\cdot x + 3$", "final_answer": "- 8\\cdot x^{3} - 2\\cdot x + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b65c71a0e0a9cc79", "problem_id": "5437bf005bdd9a90", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{4} - x^{2} + 3 x + 4$.", "output": "Differentiate term-by-term:\nd/dx(4) = 0.\nd/dx(- x^{2}) = - 2 x.\nd/dx(- 2 x^{4}) = - 8 x^{3}.\nd/dx(3 x) = 3.\nCombine: f'(x) = - 8 x^{3} - 2 x + 3.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 8\\cdot x^{3} - 2\\cdot x + 3$", "final_answer": "- 8\\cdot x^{3} - 2\\cdot x + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7bc222d649aea5fc", "problem_id": "f22661851ca28a79", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$- x + y$ = $4$\n$- x + 4 y$ = $13$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $x - y$ = $-4$.\n3) Multiply the second equation by $-1$: $x - 4 y$ = $-13$.\n4) Subtract to eliminate x: $3 y$ = $9$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, 3\\right)$", "final_answer": "(x,y)=\\left(-1, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3d48c7028561c8e6", "problem_id": "f22661851ca28a79", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- x + y$ = $4$\n$- x + 4 y$ = $13$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $x - y$ = $-4$.\nMultiply the second equation by $-1$: $x - 4 y$ = $-13$.\nSubtract to eliminate x: $3 y$ = $9$.\nSolve for y: y = $3$.\nSubstitute back to find x: x = $-1$.\nCheck: plug (x,y)=(-1,3) into both equations.\nFinal answer: $(x,y)=\\left(-1, 3\\right)$", "final_answer": "(x,y)=\\left(-1, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "908bc19af4f3e33a", "problem_id": "69779afad3f6b4e0", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the multiplicative inverse of 5 modulo 53.", "output": "1) We want x such that 5x ≡ 1 (mod 53).\n2) Use extended Euclid to find s,t with 5s + 53t = 1.\n3) One Bezout identity is: 5·(-21) + 53·(2) = 1.\n4) Reduce s modulo 53: x ≡ 32 (mod 53).\nFinal answer: $x\\equiv 32\\pmod{53}$", "final_answer": "x\\equiv 32\\pmod{53}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a3a5b0e45ec74cc2", "problem_id": "69779afad3f6b4e0", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the multiplicative inverse of 5 modulo 53.", "output": "We want x such that 5x ≡ 1 (mod 53).\nUse extended Euclid to find s,t with 5s + 53t = 1.\nOne Bezout identity is: 5·(-21) + 53·(2) = 1.\nReduce s modulo 53: x ≡ 32 (mod 53).\nCheck: 5·32 mod 53 = 1.\nFinal answer: $x\\equiv 32\\pmod{53}$", "final_answer": "x\\equiv 32\\pmod{53}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e462be170412a08f", "problem_id": "90ce192073de0a89", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{4} - 2 x^{3} + x^{2} - 2 x$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x^{2}) = 2 x^{2}$.\n3) So $f'(x) = 8 x^{3} - 4 x^{2} - 2$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x^{2}) = 2 x$.\nTherefore the correct derivative is $f'(x) = 2 \\left(4 x^{3} - 3 x^{2} + x - 1\\right)$.\nFinal answer: $2 \\left(4 x^{3} - 3 x^{2} + x - 1\\right)$", "final_answer": "2 \\left(4 x^{3} - 3 x^{2} + x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7afea8db3f758f88", "problem_id": "2662bf8213d18dc7", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 28 x + 98$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} + 28 x + 98$ = $2 \\left(x + 7\\right)^{2}$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $-7$.\nFrom the second factor: x = $-7$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-7, -7\\}$", "final_answer": "x \\in \\{-7, -7\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a999f4b167374bd", "problem_id": "09bb96136799cbc5", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 11 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=3, p=$\\frac{1}{6}$ and simplify.\n4) Here C(11,3) = 165.\nFinal answer: $\\frac{21484375}{120932352}$", "final_answer": "\\frac{21484375}{120932352}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "06cf0e3af2eb42f2", "problem_id": "09bb96136799cbc5", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 11 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=11, k=3, p=$\\frac{1}{6}$ and simplify.\nHere C(11,3) = 165.\nCheck: expression simplifies to $\\frac{21484375}{120932352}$.\nFinal answer: $\\frac{21484375}{120932352}$", "final_answer": "\\frac{21484375}{120932352}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "97b0b64a1f366dea", "problem_id": "c828ad35e0b90740", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A store sells a backpack for $139. It is discounted by 35%. You buy 7 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 139 × (1 − 35/100) = 139 × 65/100 = 1807/20.\n2) Subtotal = 7 × 1807/20 = 12649/20.\n3) Total with tax = 12649/20 × 107/100 = 1353443/2000.\nFinal answer: $\\frac{1353443}{2000}$", "final_answer": "\\frac{1353443}{2000}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b90721d86ebdff62", "problem_id": "27b3ab511b08e064", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} - 2 x - 5$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to - 3 x - 5.\n3) Then the limit is -8.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: -7.\nFinal answer: $-7$", "final_answer": "-7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a6030b71966d7944", "problem_id": "61b229ca8d0e7541", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$4 x + y$ = $24$\n$- 8 x + 2 y$ = $-32$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-8$: $- 32 x - 8 y$ = $-192$.\n3) Multiply the second equation by $4$: $- 32 x + 8 y$ = $-128$.\n4) Subtract to eliminate x: $- 16 y$ = $-64$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, 4\\right)$", "final_answer": "(x,y)=\\left(5, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db8da358982dd1b1", "problem_id": "f159eca792120ea4", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} 2 x^{3} + 11 x^{2} + 41 x + 162$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} + 11 x^{2} + 40 x + 162.\n3) Then the limit is 626.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: 630.\nFinal answer: $630$", "final_answer": "630", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "33c3a6dd19ad429e", "problem_id": "d3e7f61b09125fef", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 37 liters of a 40% salt solution. You have a 10% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 37−x liters is 70% solution.\nSalt equation: 10x + 70(37−x) = 40·37.\n(10-70)x = (40-70)·37 ⇒ x = 37/2.\nThen 37−x = 37/2.\nCheck: 10·37/2 + 70·37/2 = 40·37.\nFinal answer: $37/2 L of 10% and 37/2 L of 70%$", "final_answer": "37/2 L of 10% and 37/2 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eb710ff278f0d898", "problem_id": "d3e7f61b09125fef", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "You want 37 liters of a 40% salt solution. You have a 10% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 37−x liters is 70% solution.\n2) Salt equation: 10x + 70(37−x) = 40·37.\n3) (10-70)x = (40-70)·37 ⇒ x = 37/2.\n4) Then 37−x = 37/2.\nFinal answer: $37/2 L of 10% and 37/2 L of 70%$", "final_answer": "37/2 L of 10% and 37/2 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f0d00d390e96d98f", "problem_id": "7d7b25e413a29328", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Two friends split $2890 in the ratio 9:8. How much does each receive?", "output": "$1360$", "final_answer": "1360", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ea613f46ee0c94ba", "problem_id": "7d7b25e413a29328", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Two friends split $2890 in the ratio 9:8. How much does each receive?", "output": "1) Total ratio parts = 9+8 = 17.\n2) Each part = 2890 ÷ 17 = 170.\n3) First share = 9 × 170 = 1530.\n4) Second share = 8 × 170 = 1360.\nFinal answer: $1360$", "final_answer": "1360", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b11749be0fa3ee55", "problem_id": "85c407aa9b2b469e", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A store sells a backpack for $92. It is discounted by 20%. You buy 8 backpacks. What is the total cost?", "output": "1) Discounted price = 92 × (1 − 20/100) = 92 × 80/100 = 368/5.\n2) Subtotal = 8 × 368/5 = 2944/5.\nFinal answer: $\\frac{2944}{5}$", "final_answer": "\\frac{2944}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f74c7b5ca1a5d75c", "problem_id": "85c407aa9b2b469e", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "A store sells a backpack for $92. It is discounted by 20%. You buy 8 backpacks. What is the total cost?", "output": "Discounted price = 92 × (1 − 20/100) = 92 × 80/100 = 368/5.\nSubtotal = 8 × 368/5 = 2944/5.\nFinal answer: $\\frac{2944}{5}$", "final_answer": "\\frac{2944}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3e731e5581ff0ff3", "problem_id": "004f576f93d56dd4", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} - 3 x^{2} + 4 x + 2$ at $x=-1$.", "output": "Differentiate: f'(x) = $- 6 x^{2} - 6 x + 4$.\nSlope at x=-1: m = f'(-1) = $4$.\nPoint on curve: (-1, f(-1)) = (-1, $-3$).\nFinal answer: $y = 4\\cdot x + 1$", "final_answer": "y = 4\\cdot x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "95caafb4e5f141c0", "problem_id": "004f576f93d56dd4", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} - 3 x^{2} + 4 x + 2$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $- 6 x^{2} - 6 x + 4$.\n2) Slope at x=-1: m = f'(-1) = $4$.\n3) Point on curve: (-1, f(-1)) = (-1, $-3$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -3 = 4(x − -1).\n6) Optionally expand: y = $4 x + 1$.\nFinal answer: $y = 4\\cdot x + 1$", "final_answer": "y = 4\\cdot x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7a0af439a9dd50c5", "problem_id": "98e765fee2d42e08", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} - 4 x + 1$.", "output": "Differentiate term-by-term:\nd/dx(1) = 0.\nd/dx(- 4 x) = -4.\nFinal answer: $- 4\\cdot x - 4$", "final_answer": "- 4\\cdot x - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b90c65dfb8fc88ad", "problem_id": "98e765fee2d42e08", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} - 4 x + 1$.", "output": "Differentiate term-by-term:\nd/dx(1) = 0.\nd/dx(- 4 x) = -4.\nd/dx(- 2 x^{2}) = - 4 x.\nCombine: f'(x) = 4 \\left(- x - 1\\right).\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 4\\cdot x - 4$", "final_answer": "- 4\\cdot x - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a8db60db86b83d6", "problem_id": "7a8c0fbd658f65c3", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$5 x + 2 y$ = $-27$\n$- 5 x$ = $45$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-5$: $- 25 x - 10 y$ = $135$.\n3) Multiply the second equation by $5$: $- 25 x$ = $225$.\n4) Subtract to eliminate x: $- 10 y$ = $-90$.\n5) Solve for y: y = $9$.\n6) Substitute back to find x: x = $-9$.\nFinal answer: $(x,y)=\\left(-9, 9\\right)$", "final_answer": "(x,y)=\\left(-9, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a7ccd4bf131ef0e0", "problem_id": "7a8c0fbd658f65c3", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$5 x + 2 y$ = $-27$\n$- 5 x$ = $45$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-5$: $- 25 x - 10 y$ = $135$.\nMultiply the second equation by $5$: $- 25 x$ = $225$.\nSubtract to eliminate x: $- 10 y$ = $-90$.\nSolve for y: y = $9$.\nSubstitute back to find x: x = $-9$.\nCheck: plug (x,y)=(-9,9) into both equations.\nFinal answer: $(x,y)=\\left(-9, 9\\right)$", "final_answer": "(x,y)=\\left(-9, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5dea40f63970a607", "problem_id": "a9cd6b3176b6ba56", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A cyclist rides at 66 miles per hour for 2 hours, then at 44 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 66 × 2 = 132 miles.\n2) Second leg distance = 44 × 2 = 88 miles.\n3) Total distance = 132 + 88 = 220 miles.\nFinal answer: $220$", "final_answer": "220", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fc9afbd1875dc500", "problem_id": "a9cd6b3176b6ba56", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A cyclist rides at 66 miles per hour for 2 hours, then at 44 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 66 × 2 = 132 miles.\nSecond leg distance = 44 × 2 = 88 miles.\nTotal distance = 132 + 88 = 220 miles.\nCheck: adding both legs gives 220.\nFinal answer: $220$", "final_answer": "220", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4378e96f2ca2c76b", "problem_id": "bf45112650714123", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A store sells a backpack for $470. It is discounted by 35%. You buy 4 backpacks. What is the total cost?", "output": "1) Discounted price = 470 × (1 − 35/100) = 470 × 65/100 = 611/2.\n2) Subtotal = 4 × 611/2 = 1222.\nFinal answer: $1222$", "final_answer": "1222", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2075f34f888e9964", "problem_id": "86b258ce2d1631ea", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(- x^{3} - 2 x^{2} + 3\\right) \\cos{\\left(x + 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 2 x^{2} \\cos{\\left(x + 1 \\right)}) = 2 x^{2} \\sin{\\left(x + 1 \\right)}$.\n3) So $f'(x) = x^{3} \\sin{\\left(x + 1 \\right)} + 2 x^{2} \\sin{\\left(x + 1 \\right)} - 3 x^{2} \\cos{\\left(x + 1 \\right)} + x e^{x} - 2 e^{x} - 3 \\sin{\\left(x + 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 2 x^{2} \\cos{\\left(x + 1 \\right)}) = 2 x^{2} \\sin{\\left(x + 1 \\right)} - 4 x \\cos{\\left(x + 1 \\right)}$.\nTherefore the correct derivative is $f'(x) = x \\left(- 3 x - 4\\right) \\cos{\\left(x + 1 \\right)} + \\left(x - 3\\right) e^{x} - \\left(x^{2} \\left(- x - 2\\right) + 3\\right) \\sin{\\left(x + 1 \\right)} + e^{x}$.\nFinal answer: $x \\left(- 3 x - 4\\right) \\cos{\\left(x + 1 \\right)} + \\left(x - 3\\right) e^{x} - \\left(x^{2} \\left(- x - 2\\right) + 3\\right) \\sin{\\left(x + 1 \\right)} + e^{x}$", "final_answer": "x \\left(- 3 x - 4\\right) \\cos{\\left(x + 1 \\right)} + \\left(x - 3\\right) e^{x} - \\left(x^{2} \\left(- x - 2\\right) + 3\\right) \\sin{\\left(x + 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "45a11ef7d3c217ef", "problem_id": "4e2a2e48874dbb11", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} x^{3} + x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $x^{3} + x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{15 \\pi^{4}}{64} - \\frac{3 \\pi^{2}}{8} + 5 + \\frac{3 \\pi^{3}}{8} + 6 \\pi$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{15\\cdot \\pi^{4}}{64} - \\frac{3\\cdot \\pi^{2}}{8} + 5 + \\frac{3\\cdot \\pi^{3}}{8} + 6\\cdot \\pi$", "final_answer": "- \\frac{15\\cdot \\pi^{4}}{64} - \\frac{3\\cdot \\pi^{2}}{8} + 5 + \\frac{3\\cdot \\pi^{3}}{8} + 6\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d457bab4348cfe9a", "problem_id": "4e2a2e48874dbb11", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} x^{3} + x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $x^{3} + x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{15 \\pi^{4}}{64} - \\frac{3 \\pi^{2}}{8} + 5 + \\frac{3 \\pi^{3}}{8} + 6 \\pi$.\n4) Simplify the result.\nFinal answer: $- \\frac{15\\cdot \\pi^{4}}{64} - \\frac{3\\cdot \\pi^{2}}{8} + 5 + \\frac{3\\cdot \\pi^{3}}{8} + 6\\cdot \\pi$", "final_answer": "- \\frac{15\\cdot \\pi^{4}}{64} - \\frac{3\\cdot \\pi^{2}}{8} + 5 + \\frac{3\\cdot \\pi^{3}}{8} + 6\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3014d33bca4182af", "problem_id": "1865096cee8d2bff", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A store sells a backpack for $74. It is discounted by 5%. You buy 12 backpacks. What is the total cost?", "output": "Discounted price = 74 × (1 − 5/100) = 74 × 95/100 = 703/10.\nSubtotal = 12 × 703/10 = 4218/5.\nFinal answer: $\\frac{4218}{5}$", "final_answer": "\\frac{4218}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "041814deb38679e4", "problem_id": "1865096cee8d2bff", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A store sells a backpack for $74. It is discounted by 5%. You buy 12 backpacks. What is the total cost?", "output": "1) Discounted price = 74 × (1 − 5/100) = 74 × 95/100 = 703/10.\n2) Subtotal = 12 × 703/10 = 4218/5.\nFinal answer: $\\frac{4218}{5}$", "final_answer": "\\frac{4218}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c098bb5a2ce42096", "problem_id": "bc07347abcdf219c", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 5 - x$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{2} + 3 x + 2$ and a = -2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $3 - 2 x$.\n5) Evaluate at a: p'(-2) = $7$.\nFinal answer: $7$", "final_answer": "7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a6ccf828ace270da", "problem_id": "bc07347abcdf219c", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 5 - x$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{2} + 3 x + 2$ and a = -2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $3 - 2 x$.\nEvaluate at a: p'(-2) = $7$.\nCheck: the identity gives $7$.\nFinal answer: $7$", "final_answer": "7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1562ef54f05fa0c5", "problem_id": "e6da9c5be5f716d9", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 8 red balls and 10 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(18,4).\n3) Favorable samples: choose k reds and n−k blues: C(8,1)·C(10,3).\n4) Compute totals: C(18,4) = 3060, and favorable = 960.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{16}{51}$", "final_answer": "\\frac{16}{51}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a12d5ecfd7d304d5", "problem_id": "e6da9c5be5f716d9", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 8 red balls and 10 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(18,4).\nFavorable samples: choose k reds and n−k blues: C(8,1)·C(10,3).\nCompute totals: C(18,4) = 3060, and favorable = 960.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{16}{51}$.\nFinal answer: $\\frac{16}{51}$", "final_answer": "\\frac{16}{51}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "61702d51281d86d2", "problem_id": "b2aacd895985cd28", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} + 3 x^{2} + x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 2$ at $x=1$.", "output": "1) Differentiate: f'(x) = $6 x^{2} + 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 1$.\n2) Slope at x=1: m = f'(1) = $- 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 13$.\n3) Point on curve: (1, f(1)) = (1, $\\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} + 8$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} + 8 = - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 13(x − 1).\n6) Optionally expand: y = $- 2 x \\sin{\\left(2 \\right)} + \\frac{x e^{\\frac{1}{2}}}{2} + 13 x - 5 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 13\\cdot x - 5 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 13\\cdot x - 5 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "046e1bcf49cbce69", "problem_id": "b2aacd895985cd28", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} + 3 x^{2} + x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 2$ at $x=1$.", "output": "Differentiate: f'(x) = $6 x^{2} + 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 1$.\nSlope at x=1: m = f'(1) = $- 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 13$.\nPoint on curve: (1, f(1)) = (1, $\\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} + 8$).\nFinal answer: $y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 13\\cdot x - 5 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 13\\cdot x - 5 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dc38a8ab53679730", "problem_id": "ef784d37dff8d546", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the multiplicative inverse of 4 modulo 31.", "output": "1) We want x such that 4x ≡ 1 (mod 31).\n2) Use extended Euclid to find s,t with 4s + 31t = 1.\n3) One Bezout identity is: 4·(8) + 31·(-1) = 1.\n4) Reduce s modulo 31: x ≡ 8 (mod 31).\nFinal answer: $x\\equiv 8\\pmod{31}$", "final_answer": "x\\equiv 8\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c8539b1103410c32", "problem_id": "ef784d37dff8d546", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Find the multiplicative inverse of 4 modulo 31.", "output": "We want x such that 4x ≡ 1 (mod 31).\nUse extended Euclid to find s,t with 4s + 31t = 1.\nOne Bezout identity is: 4·(8) + 31·(-1) = 1.\nFinal answer: $x\\equiv 8\\pmod{31}$", "final_answer": "x\\equiv 8\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "56300c5acd5f9520", "problem_id": "ca5fb40e8694572a", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} - 2 x^{3} + x^{2} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{3} + x^{2} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{15 \\pi^{4}}{32} - 3 \\pi - 1 + \\frac{3 \\pi^{3}}{8} + \\frac{3 \\pi^{2}}{2}$.\n4) Simplify the result.\nFinal answer: $- \\frac{15\\cdot \\pi^{4}}{32} - 3\\cdot \\pi - 1 + \\frac{3\\cdot \\pi^{3}}{8} + \\frac{3\\cdot \\pi^{2}}{2}$", "final_answer": "- \\frac{15\\cdot \\pi^{4}}{32} - 3\\cdot \\pi - 1 + \\frac{3\\cdot \\pi^{3}}{8} + \\frac{3\\cdot \\pi^{2}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "519d8662c77ad2d3", "problem_id": "ca5fb40e8694572a", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} - 2 x^{3} + x^{2} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{3} + x^{2} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2$.\nEvaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{15 \\pi^{4}}{32} - 3 \\pi - 1 + \\frac{3 \\pi^{3}}{8} + \\frac{3 \\pi^{2}}{2}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{15\\cdot \\pi^{4}}{32} - 3\\cdot \\pi - 1 + \\frac{3\\cdot \\pi^{3}}{8} + \\frac{3\\cdot \\pi^{2}}{2}$", "final_answer": "- \\frac{15\\cdot \\pi^{4}}{32} - 3\\cdot \\pi - 1 + \\frac{3\\cdot \\pi^{3}}{8} + \\frac{3\\cdot \\pi^{2}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "79470f8ea643e61e", "problem_id": "e725b9b3a9f46c53", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(2 x^{2} + 2 x + 1\\right) \\cos{\\left(2 x^{2} + 3 \\right)}$.", "output": "1) Let f(x) = $\\left(x + 2\\right) e^{x} + \\left(2 x^{2} + 2 x + 1\\right) \\cos{\\left(2 x^{2} + 3 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- 4 x \\left(2 x^{2} + 2 x + 1\\right) \\sin{\\left(2 x^{2} + 3 \\right)} + \\left(x + 2\\right) e^{x} + 2 \\left(2 x + 1\\right) \\cos{\\left(2 x^{2} + 3 \\right)} + e^{x}$.\nFinal answer: $- 4\\cdot x\\cdot \\left(2\\cdot x^{2} + 2\\cdot x + 1\\right)\\cdot \\sin{\\left(2\\cdot x^{2} + 3 \\right)} + \\left(x + 2\\right)\\cdot e^{x} + \\left(4\\cdot x + 2\\right)\\cdot \\cos{\\left(2\\cdot x^{2} + 3 \\right)} + e^{x}$", "final_answer": "- 4\\cdot x\\cdot \\left(2\\cdot x^{2} + 2\\cdot x + 1\\right)\\cdot \\sin{\\left(2\\cdot x^{2} + 3 \\right)} + \\left(x + 2\\right)\\cdot e^{x} + \\left(4\\cdot x + 2\\right)\\cdot \\cos{\\left(2\\cdot x^{2} + 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b1edda3e48597c80", "problem_id": "e725b9b3a9f46c53", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(2 x^{2} + 2 x + 1\\right) \\cos{\\left(2 x^{2} + 3 \\right)}$.", "output": "Let f(x) = $\\left(x + 2\\right) e^{x} + \\left(2 x^{2} + 2 x + 1\\right) \\cos{\\left(2 x^{2} + 3 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- 4 x \\left(2 x^{2} + 2 x + 1\\right) \\sin{\\left(2 x^{2} + 3 \\right)} + \\left(x + 2\\right) e^{x} + 2 \\left(2 x + 1\\right) \\cos{\\left(2 x^{2} + 3 \\right)} + e^{x}$.\nFinal answer: $- 4\\cdot x\\cdot \\left(2\\cdot x^{2} + 2\\cdot x + 1\\right)\\cdot \\sin{\\left(2\\cdot x^{2} + 3 \\right)} + \\left(x + 2\\right)\\cdot e^{x} + \\left(4\\cdot x + 2\\right)\\cdot \\cos{\\left(2\\cdot x^{2} + 3 \\right)} + e^{x}$", "final_answer": "- 4\\cdot x\\cdot \\left(2\\cdot x^{2} + 2\\cdot x + 1\\right)\\cdot \\sin{\\left(2\\cdot x^{2} + 3 \\right)} + \\left(x + 2\\right)\\cdot e^{x} + \\left(4\\cdot x + 2\\right)\\cdot \\cos{\\left(2\\cdot x^{2} + 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0fe99f9fac52262b", "problem_id": "937915b32299dd62", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{3} - 3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $2 x^{3} - 3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}$.\nCompute an antiderivative: F(x) = $\\frac{x^{4}}{2} - x^{3} - \\frac{3 x^{2}}{2} - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $C + \\frac{x^{4}}{2} - x^{3} - \\frac{3\\cdot x^{2}}{2} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{4}}{2} - x^{3} - \\frac{3\\cdot x^{2}}{2} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bd697e3ef09e90fc", "problem_id": "937915b32299dd62", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{3} - 3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $2 x^{3} - 3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}$.\n3) Compute an antiderivative: F(x) = $\\frac{x^{4}}{2} - x^{3} - \\frac{3 x^{2}}{2} - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{x^{4}}{2} - x^{3} - \\frac{3\\cdot x^{2}}{2} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{4}}{2} - x^{3} - \\frac{3\\cdot x^{2}}{2} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a0c7562c7afd8b3", "problem_id": "f3ec14017ae852a6", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡0 (mod 7), x≡9 (mod 11). Give the solution modulo 77.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 77.\n3) Compute the CRT solution to get x ≡ 42 (mod 77).\nFinal answer: $x\\equiv 42\\pmod{77}$", "final_answer": "x\\equiv 42\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "994d67b9a5833e89", "problem_id": "f3ec14017ae852a6", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡0 (mod 7), x≡9 (mod 11). Give the solution modulo 77.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 77.\nCompute the CRT solution to get x ≡ 42 (mod 77).\nCheck: 42 mod 7=0 and mod 11=9.\nFinal answer: $x\\equiv 42\\pmod{77}$", "final_answer": "x\\equiv 42\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e86fc5994f8ff9fe", "problem_id": "0f6d095eb8f8c8f3", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{9 x}{2} - \\frac{3 \\sqrt{5} x}{2} + \\frac{9}{2} + \\frac{9 \\sqrt{5}}{4}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{9}{2} - \\frac{3 \\sqrt{5}}{2}$, c=$\\frac{9}{2} + \\frac{9 \\sqrt{5}}{4}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{9 \\sqrt{5}}{2} + \\frac{27}{2}$, so √Δ = $\\sqrt{\\frac{9 \\sqrt{5}}{2} + \\frac{27}{2}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{3 \\sqrt{5}}{4} + \\frac{9}{4} + \\frac{3 \\sqrt{2 \\sqrt{5} + 6}}{4}$.\nx2 = $- \\frac{3 \\sqrt{2 \\sqrt{5} + 6}}{4} + \\frac{3 \\sqrt{5}}{4} + \\frac{9}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{3\\cdot \\sqrt{5}}{4} + \\frac{9}{4} + \\frac{3\\cdot \\sqrt{2\\cdot \\sqrt{5} + 6}}{4}, - \\frac{3\\cdot \\sqrt{2\\cdot \\sqrt{5} + 6}}{4} + \\frac{3\\cdot \\sqrt{5}}{4} + \\frac{9}{4}\\}$", "final_answer": "x \\in \\{\\frac{3\\cdot \\sqrt{5}}{4} + \\frac{9}{4} + \\frac{3\\cdot \\sqrt{2\\cdot \\sqrt{5} + 6}}{4}, - \\frac{3\\cdot \\sqrt{2\\cdot \\sqrt{5} + 6}}{4} + \\frac{3\\cdot \\sqrt{5}}{4} + \\frac{9}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7d78f2e80259af29", "problem_id": "0f6d095eb8f8c8f3", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{9 x}{2} - \\frac{3 \\sqrt{5} x}{2} + \\frac{9}{2} + \\frac{9 \\sqrt{5}}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{9}{2} - \\frac{3 \\sqrt{5}}{2}$, c=$\\frac{9}{2} + \\frac{9 \\sqrt{5}}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{9 \\sqrt{5}}{2} + \\frac{27}{2}$, so √Δ = $\\sqrt{\\frac{9 \\sqrt{5}}{2} + \\frac{27}{2}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{3 \\sqrt{5}}{4} + \\frac{9}{4} + \\frac{3 \\sqrt{2 \\sqrt{5} + 6}}{4}$.\n7) x2 = $- \\frac{3 \\sqrt{2 \\sqrt{5} + 6}}{4} + \\frac{3 \\sqrt{5}}{4} + \\frac{9}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{3\\cdot \\sqrt{5}}{4} + \\frac{9}{4} + \\frac{3\\cdot \\sqrt{2\\cdot \\sqrt{5} + 6}}{4}, - \\frac{3\\cdot \\sqrt{2\\cdot \\sqrt{5} + 6}}{4} + \\frac{3\\cdot \\sqrt{5}}{4} + \\frac{9}{4}\\}$", "final_answer": "x \\in \\{\\frac{3\\cdot \\sqrt{5}}{4} + \\frac{9}{4} + \\frac{3\\cdot \\sqrt{2\\cdot \\sqrt{5} + 6}}{4}, - \\frac{3\\cdot \\sqrt{2\\cdot \\sqrt{5} + 6}}{4} + \\frac{3\\cdot \\sqrt{5}}{4} + \\frac{9}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "124cccd47c2d531d", "problem_id": "3e82dc15f90c8a72", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$12 x + 8 y$ = $8$\n$4 x + 11 y$ = $-39$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $48 x + 32 y$ = $32$.\n3) Multiply the second equation by $12$: $48 x + 132 y$ = $-468$.\n4) Subtract to eliminate x: $- 100 y$ = $500$.\n5) Solve for y: y = $-5$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, -5\\right)$", "final_answer": "(x,y)=\\left(4, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "af73134c2db36ffd", "problem_id": "d37a8dd6d1d2c67d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{3} - 3 x^{2} - 2 x - 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- x^{3}) = - 3 x^{3}$.\n3) So $f'(x) = - 3 x^{3} - 6 x - 2$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- x^{3}) = - 3 x^{2}$.\nTherefore the correct derivative is $f'(x) = - 3 x^{2} - 6 x - 2$.\nFinal answer: $- 3 x^{2} - 6 x - 2$", "final_answer": "- 3 x^{2} - 6 x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a30f1b006994af3b", "problem_id": "bb7fc4227c8302ce", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Two friends split $2520 in the ratio 7:8. How much does each receive?", "output": "1) Total ratio parts = 7+8 = 15.\n2) Each part = 2520 ÷ 15 = 168.\n3) First share = 7 × 168 = 1176.\n4) Second share = 8 × 168 = 1344.\nFinal answer: $1344$", "final_answer": "1344", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9a3c21b0a0e2bb49", "problem_id": "bb7fc4227c8302ce", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $2520 in the ratio 7:8. How much does each receive?", "output": "Total ratio parts = 7+8 = 15.\nEach part = 2520 ÷ 15 = 168.\nFirst share = 7 × 168 = 1176.\nSecond share = 8 × 168 = 1344.\nCheck: 1176+1344=2520.\nFinal answer: $1344$", "final_answer": "1344", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "af514ab331c2a87e", "problem_id": "a6eca9856ab0e0cd", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{15 x}{2} - \\sqrt{3} x + \\frac{3 \\sqrt{3}}{2} + 9$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{15}{2} - \\sqrt{3}$, c=$\\frac{3 \\sqrt{3}}{2} + 9$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $9 \\sqrt{3} + \\frac{93}{4}$, so √Δ = $\\sqrt{9 \\sqrt{3} + \\frac{93}{4}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{36 \\sqrt{3} + 93}}{4} + \\frac{15}{4}$.\n7) x2 = $- \\frac{\\sqrt{36 \\sqrt{3} + 93}}{4} + \\frac{\\sqrt{3}}{2} + \\frac{15}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{36\\cdot \\sqrt{3} + 93}}{4} + \\frac{15}{4}, - \\frac{\\sqrt{36\\cdot \\sqrt{3} + 93}}{4} + \\frac{\\sqrt{3}}{2} + \\frac{15}{4}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{36\\cdot \\sqrt{3} + 93}}{4} + \\frac{15}{4}, - \\frac{\\sqrt{36\\cdot \\sqrt{3} + 93}}{4} + \\frac{\\sqrt{3}}{2} + \\frac{15}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b179f1d5532b0187", "problem_id": "8ec01072a4ea0cf1", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - x^{2} - 4 x - 10$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} - 5 x - 10.\n3) Then the limit is -34.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: -31.\nFinal answer: $-31$", "final_answer": "-31", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0e366212f638079e", "problem_id": "6fde8532c961c4a0", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_6=27$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_6=21.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=21 and k=6.\nCompute C(26,5) = 65780.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $65780$", "final_answer": "65780", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "93a4892536b7eee8", "problem_id": "6fde8532c961c4a0", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_6=27$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_6=21.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=21 and k=6.\nFinal answer: $65780$", "final_answer": "65780", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "59084780d6589cc1", "problem_id": "127e9bd1a7b813a3", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A cyclist rides at 48 miles per hour for 1 hours, then at 90 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 48 × 1 = 48 miles.\n2) Second leg distance = 90 × 1 = 90 miles.\n3) Total distance = 48 + 90 = 138 miles.\nFinal answer: $138$", "final_answer": "138", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ee778e1238423ad9", "problem_id": "127e9bd1a7b813a3", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A cyclist rides at 48 miles per hour for 1 hours, then at 90 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 48 × 1 = 48 miles.\nSecond leg distance = 90 × 1 = 90 miles.\nTotal distance = 48 + 90 = 138 miles.\nCheck: adding both legs gives 138.\nFinal answer: $138$", "final_answer": "138", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b7a2c4997dd7fc8e", "problem_id": "76ce63b064bca52e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{3 x^{2} + \\left(x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{x^{2} - x + 5}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 5}) = \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 5}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} - \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} - \\frac{4 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} + \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 11 x^{2} - 10 x + 25} + \\frac{11 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} + \\frac{22 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} - \\frac{6 x^{2}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 11 x^{2} - 10 x + 25} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 2 x + 10} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 5} - \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 5} - \\frac{10 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} - \\frac{20 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} + \\frac{72 x}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} - \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 5} + \\frac{25 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} + \\frac{50 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} - \\frac{6}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 5}) = \\frac{x^{2} \\left(1 - 2 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(x^{2} - x\\right) + 5\\right)^{2}} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(x^{2} - x\\right) + 5\\right)} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 5} + \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 5}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(1 - 2 x\\right) \\left(3 x^{2} + \\left(x \\left(x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x \\left(x - 1\\right) + 5\\right) \\left(12 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)}\\right)}{2 \\left(x \\left(x - 1\\right) + 5\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(1 - 2 x\\right) \\left(3 x^{2} + \\left(x \\left(x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x \\left(x - 1\\right) + 5\\right) \\left(12 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)}\\right)}{2 \\left(x \\left(x - 1\\right) + 5\\right)^{2}}$", "final_answer": "\\frac{2 \\left(1 - 2 x\\right) \\left(3 x^{2} + \\left(x \\left(x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x \\left(x - 1\\right) + 5\\right) \\left(12 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)}\\right)}{2 \\left(x \\left(x - 1\\right) + 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d1e00a7b39291a15", "problem_id": "717dc5816b679cda", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 14 red balls and 15 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(29,6).\nFavorable samples: choose k reds and n−k blues: C(14,1)·C(15,5).\nCompute totals: C(29,6) = 475020, and favorable = 42042.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{77}{870}$.\nFinal answer: $\\frac{77}{870}$", "final_answer": "\\frac{77}{870}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a236e5427c808215", "problem_id": "80f53ffa2a56b3ea", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} - 3 x - 4$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to - 4 x - 4.\n3) Then the limit is -8.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: -7.\nFinal answer: $-7$", "final_answer": "-7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b38b06f60bf842a1", "problem_id": "cdd93f02711e0d47", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$- 7 x + 6 y$ = $30$\n$7 x - y$ = $-5$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $7$: $- 49 x + 42 y$ = $210$.\n3) Multiply the second equation by $-7$: $- 49 x + 7 y$ = $35$.\n4) Subtract to eliminate x: $35 y$ = $175$.\n5) Solve for y: y = $5$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, 5\\right)$", "final_answer": "(x,y)=\\left(0, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d136b14fd3a7078", "problem_id": "25e6266d2c9e8484", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(- 3 x^{3} + 3 x^{2} - 3 x + 3\\right) \\cos{\\left(2 x + 3 \\right)}$.", "output": "1) Let f(x) = $2 \\left(x + 2\\right) e^{x} + \\left(- 3 x^{3} + 3 x^{2} - 3 x + 3\\right) \\cos{\\left(2 x + 3 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $2 \\left(x + 2\\right) e^{x} + 3 \\left(- 3 x^{2} + 2 x - 1\\right) \\cos{\\left(2 x + 3 \\right)} - 6 \\left(- x^{3} + x^{2} - x + 1\\right) \\sin{\\left(2 x + 3 \\right)} + 2 e^{x}$.\nFinal answer: $\\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(- 9\\cdot x^{2} + 6\\cdot x - 3\\right)\\cdot \\cos{\\left(2\\cdot x + 3 \\right)} - \\left(- 6\\cdot x^{3} + 6\\cdot x^{2} - 6\\cdot x + 6\\right)\\cdot \\sin{\\left(2\\cdot x + 3 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(- 9\\cdot x^{2} + 6\\cdot x - 3\\right)\\cdot \\cos{\\left(2\\cdot x + 3 \\right)} - \\left(- 6\\cdot x^{3} + 6\\cdot x^{2} - 6\\cdot x + 6\\right)\\cdot \\sin{\\left(2\\cdot x + 3 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "793a7b892da965c1", "problem_id": "25e6266d2c9e8484", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(- 3 x^{3} + 3 x^{2} - 3 x + 3\\right) \\cos{\\left(2 x + 3 \\right)}$.", "output": "Let f(x) = $2 \\left(x + 2\\right) e^{x} + \\left(- 3 x^{3} + 3 x^{2} - 3 x + 3\\right) \\cos{\\left(2 x + 3 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $2 \\left(x + 2\\right) e^{x} + 3 \\left(- 3 x^{2} + 2 x - 1\\right) \\cos{\\left(2 x + 3 \\right)} - 6 \\left(- x^{3} + x^{2} - x + 1\\right) \\sin{\\left(2 x + 3 \\right)} + 2 e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(- 9\\cdot x^{2} + 6\\cdot x - 3\\right)\\cdot \\cos{\\left(2\\cdot x + 3 \\right)} - \\left(- 6\\cdot x^{3} + 6\\cdot x^{2} - 6\\cdot x + 6\\right)\\cdot \\sin{\\left(2\\cdot x + 3 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(- 9\\cdot x^{2} + 6\\cdot x - 3\\right)\\cdot \\cos{\\left(2\\cdot x + 3 \\right)} - \\left(- 6\\cdot x^{3} + 6\\cdot x^{2} - 6\\cdot x + 6\\right)\\cdot \\sin{\\left(2\\cdot x + 3 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "63c5a0bacdc9efd6", "problem_id": "e997ddb897072c3c", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=23$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=23 and k=7.\n4) Compute C(29,6) = 475020.\nFinal answer: $475020$", "final_answer": "475020", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f7ed84a2ea5e4a57", "problem_id": "e997ddb897072c3c", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=23$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=23 and k=7.\nCompute C(29,6) = 475020.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $475020$", "final_answer": "475020", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0a47ee9fd8deafcf", "problem_id": "92a6f970e28b10cf", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} - 4 x^{3} - 3 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 4 x^{3} - 3 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\pi).\n3) Subtract: F(\\pi) − F(- \\pi) = $2 \\pi \\left(1 - \\pi^{2}\\right)$.\n4) Simplify the result.\nFinal answer: $2\\cdot \\pi\\cdot \\left(1 - \\pi^{2}\\right)$", "final_answer": "2\\cdot \\pi\\cdot \\left(1 - \\pi^{2}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6b6a7d423b979f10", "problem_id": "92a6f970e28b10cf", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} - 4 x^{3} - 3 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 4 x^{3} - 3 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\nEvaluate at the bounds: F(\\pi) and F(- \\pi).\nSubtract: F(\\pi) − F(- \\pi) = $2 \\pi \\left(1 - \\pi^{2}\\right)$.\nFinal answer: $2\\cdot \\pi\\cdot \\left(1 - \\pi^{2}\\right)$", "final_answer": "2\\cdot \\pi\\cdot \\left(1 - \\pi^{2}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec866f5fc1718102", "problem_id": "4b3bcb34a77fd78a", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "You want 64 liters of a 42% salt solution. You have a 25% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 64−x liters is 60% solution.\n2) Salt equation: 25x + 60(64−x) = 42·64.\n3) (25-60)x = (42-60)·64 ⇒ x = 1152/35.\n4) Then 64−x = 1088/35.\nFinal answer: $1152/35 L of 25% and 1088/35 L of 60%$", "final_answer": "1152/35 L of 25% and 1088/35 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dddab979f0d002cc", "problem_id": "4b3bcb34a77fd78a", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "You want 64 liters of a 42% salt solution. You have a 25% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 64−x liters is 60% solution.\nSalt equation: 25x + 60(64−x) = 42·64.\n(25-60)x = (42-60)·64 ⇒ x = 1152/35.\nThen 64−x = 1088/35.\nCheck: 25·1152/35 + 60·1088/35 = 42·64.\nFinal answer: $1152/35 L of 25% and 1088/35 L of 60%$", "final_answer": "1152/35 L of 25% and 1088/35 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "627ca6604de2e750", "problem_id": "5abae00375360cc1", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the congruences: x≡3 (mod 5), x≡11 (mod 13). Give the solution modulo 65.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 65.\n3) Compute the CRT solution to get x ≡ 63 (mod 65).\nFinal answer: $x\\equiv 63\\pmod{65}$", "final_answer": "x\\equiv 63\\pmod{65}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "66bdf4ee420d0a8e", "problem_id": "5abae00375360cc1", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡3 (mod 5), x≡11 (mod 13). Give the solution modulo 65.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 65.\nCompute the CRT solution to get x ≡ 63 (mod 65).\nCheck: 63 mod 5=3 and mod 13=11.\nFinal answer: $x\\equiv 63\\pmod{65}$", "final_answer": "x\\equiv 63\\pmod{65}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "efcbaa2f82ef7f25", "problem_id": "7df43ead91d49429", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=42$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=42 and k=5.\n4) Compute C(46,4) = 163185.\nFinal answer: $163185$", "final_answer": "163185", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b9e59f1f922c8a6", "problem_id": "7df43ead91d49429", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=42$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=42 and k=5.\nFinal answer: $163185$", "final_answer": "163185", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8ab560c892f1718d", "problem_id": "45d7f4740e0d7b41", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 15 liters of a 30% salt solution. You have a 10% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 15−x liters is 40% solution.\nSalt equation: 10x + 40(15−x) = 30·15.\n(10-40)x = (30-40)·15 ⇒ x = 5.\nThen 15−x = 10.\nCheck: 10·5 + 40·10 = 30·15.\nFinal answer: $5 L of 10% and 10 L of 40%$", "final_answer": "5 L of 10% and 10 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f9fb20fb3145cd62", "problem_id": "b708677091c5e5b4", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(2 x^{3} - x^{2} + x - 2\\right) \\cos{\\left(2 x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x e^{x}) = 2 x e^{x}$.\n3) So $f'(x) = - 4 x^{3} \\sin{\\left(2 x + 3 \\right)} + 2 x^{2} \\sin{\\left(2 x + 3 \\right)} + 6 x^{2} \\cos{\\left(2 x + 3 \\right)} + 2 x e^{x} - 2 x \\sin{\\left(2 x + 3 \\right)} - 2 x \\cos{\\left(2 x + 3 \\right)} + 4 e^{x} + 4 \\sin{\\left(2 x + 3 \\right)} + \\cos{\\left(2 x + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x e^{x}) = 2 x e^{x} + 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x + 2\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) + 1\\right) - 2\\right) \\sin{\\left(2 x + 3 \\right)} + \\left(6 x^{2} - 2 x + 1\\right) \\cos{\\left(2 x + 3 \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(x + 2\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) + 1\\right) - 2\\right) \\sin{\\left(2 x + 3 \\right)} + \\left(6 x^{2} - 2 x + 1\\right) \\cos{\\left(2 x + 3 \\right)} + 2 e^{x}$", "final_answer": "2 \\left(x + 2\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) + 1\\right) - 2\\right) \\sin{\\left(2 x + 3 \\right)} + \\left(6 x^{2} - 2 x + 1\\right) \\cos{\\left(2 x + 3 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "798e29f57b415c87", "problem_id": "5a0ecfa98aaeef83", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Two friends split $403 in the ratio 8:5. How much does each receive?", "output": "1) Total ratio parts = 8+5 = 13.\n2) Each part = 403 ÷ 13 = 31.\n3) First share = 8 × 31 = 248.\n4) Second share = 5 × 31 = 155.\nFinal answer: $155$", "final_answer": "155", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "59e5520e7f0b9418", "problem_id": "5a0ecfa98aaeef83", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Two friends split $403 in the ratio 8:5. How much does each receive?", "output": "Total ratio parts = 8+5 = 13.\nEach part = 403 ÷ 13 = 31.\nFirst share = 8 × 31 = 248.\nFinal answer: $155$", "final_answer": "155", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8f0fb9b107af9999", "problem_id": "9ee979406022b475", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - x^{2} - 3 x - 4\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- x^{2} - 3 x - 4$.\n3) Compute an antiderivative: F(x) = $- \\frac{x^{3}}{3} - \\frac{3 x^{2}}{2} - 4 x$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{x^{3}}{3} - \\frac{3\\cdot x^{2}}{2} - 4\\cdot x$", "final_answer": "C - \\frac{x^{3}}{3} - \\frac{3\\cdot x^{2}}{2} - 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aef90bb428ad5dab", "problem_id": "9ee979406022b475", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - x^{2} - 3 x - 4\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- x^{2} - 3 x - 4$.\nCompute an antiderivative: F(x) = $- \\frac{x^{3}}{3} - \\frac{3 x^{2}}{2} - 4 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - \\frac{x^{3}}{3} - \\frac{3\\cdot x^{2}}{2} - 4\\cdot x$", "final_answer": "C - \\frac{x^{3}}{3} - \\frac{3\\cdot x^{2}}{2} - 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e74cf533f116857", "problem_id": "c6487d96884aaeb6", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 36 x + 81$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} + 36 x + 81$ = $3 \\left(x + 3\\right) \\left(x + 9\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $-3$.\nFrom the second factor: x = $-9$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-3, -9\\}$", "final_answer": "x \\in \\{-3, -9\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a776cea87a638d01", "problem_id": "c6487d96884aaeb6", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 36 x + 81$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} + 36 x + 81$ = $3 \\left(x + 3\\right) \\left(x + 9\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-3$.\n5) From the second factor: x = $-9$.\nFinal answer: $x \\in \\{-3, -9\\}$", "final_answer": "x \\in \\{-3, -9\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c067467656288aa0", "problem_id": "7cf0726f1b6da4fd", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡4 (mod 5), x≡1 (mod 13). Give the solution modulo 65.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 65.\n3) Compute the CRT solution to get x ≡ 14 (mod 65).\nFinal answer: $x\\equiv 14\\pmod{65}$", "final_answer": "x\\equiv 14\\pmod{65}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c915b79c4a9e37f4", "problem_id": "7cf0726f1b6da4fd", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the congruences: x≡4 (mod 5), x≡1 (mod 13). Give the solution modulo 65.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 65.\nCompute the CRT solution to get x ≡ 14 (mod 65).\nFinal answer: $x\\equiv 14\\pmod{65}$", "final_answer": "x\\equiv 14\\pmod{65}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "31cb8fb89e69bf75", "problem_id": "a2834fb9e51a07b3", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - x^{3} + x + 5$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to 5 - x^{3}.\n3) Then the limit is -22.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: -19.\nFinal answer: $-19$", "final_answer": "-19", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "edc4f87864dc78cf", "problem_id": "3e6a3d0e23d52778", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A store sells a backpack for $69. It is discounted by 15%. You buy 7 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 69 × (1 − 15/100) = 69 × 85/100 = 1173/20.\n2) Subtotal = 7 × 1173/20 = 8211/20.\n3) Total with tax = 8211/20 × 108/100 = 221697/500.\nFinal answer: $\\frac{221697}{500}$", "final_answer": "\\frac{221697}{500}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b319d2f2d3877e33", "problem_id": "f10ba129c6073897", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_10=32$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_10=12.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=12 and k=10.\n4) Compute C(21,9) = 293930.\nFinal answer: $293930$", "final_answer": "293930", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7de94c9a4e056ef8", "problem_id": "f10ba129c6073897", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_10=32$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_10=12.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=12 and k=10.\nCompute C(21,9) = 293930.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $293930$", "final_answer": "293930", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0b60975b192cc736", "problem_id": "47e433c993c39b43", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - \\frac{x}{5} - \\frac{2}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$- \\frac{1}{5}$, c=$- \\frac{2}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{121}{25}$, so √Δ = $\\frac{11}{5}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{2}{5}$.\n7) x2 = $- \\frac{1}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{2}{5}, - \\frac{1}{3}\\}$", "final_answer": "x \\in \\{\\frac{2}{5}, - \\frac{1}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6b6968c0dfa328bc", "problem_id": "47e433c993c39b43", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - \\frac{x}{5} - \\frac{2}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$- \\frac{1}{5}$, c=$- \\frac{2}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{2}{5}, - \\frac{1}{3}\\}$", "final_answer": "x \\in \\{\\frac{2}{5}, - \\frac{1}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "90dbd9d7872bdbc0", "problem_id": "be6aeea4d4f6cbc0", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "A cyclist rides at 34 miles per hour for 3/2 hours, then at 20 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "$71$", "final_answer": "71", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f57f5952d9fce4fa", "problem_id": "c0fbe97de884fdee", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} + 4 x + 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x^{3}) = 3 x^{3}$.\n3) So $f'(x) = 3 x^{3} + 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x^{3}) = 3 x^{2}$.\nTherefore the correct derivative is $f'(x) = 3 x^{2} + 4$.\nFinal answer: $3 x^{2} + 4$", "final_answer": "3 x^{2} + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6b6b1d03b19ccd0a", "problem_id": "b731384d31e8ccd3", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the linear system:\n$- 11 x + 12 y$ = $57$\n$11 x + 6 y$ = $-21$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $11$: $- 121 x + 132 y$ = $627$.\nMultiply the second equation by $-11$: $- 121 x - 66 y$ = $231$.\nFinal answer: $(x,y)=\\left(-3, 2\\right)$", "final_answer": "(x,y)=\\left(-3, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a403640815dca851", "problem_id": "b731384d31e8ccd3", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$- 11 x + 12 y$ = $57$\n$11 x + 6 y$ = $-21$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $11$: $- 121 x + 132 y$ = $627$.\n3) Multiply the second equation by $-11$: $- 121 x - 66 y$ = $231$.\n4) Subtract to eliminate x: $198 y$ = $396$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, 2\\right)$", "final_answer": "(x,y)=\\left(-3, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "faccf687ad4737ba", "problem_id": "4515d91ae660cecc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - 2 x^{2} + 3 x - 8$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{2} + 2 x - 8.\n3) Then the limit is -20.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -22.\nFinal answer: $-22$", "final_answer": "-22", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e070c9d7d01ba831", "problem_id": "42c4d83b80d9fd42", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A store sells a backpack for $120. It is discounted by 10%. You buy 7 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 120 × (1 − 10/100) = 120 × 90/100 = 108.\n2) Subtotal = 7 × 108 = 756.\n3) Total with tax = 756 × 105/100 = 3969/5.\nFinal answer: $\\frac{3969}{5}$", "final_answer": "\\frac{3969}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "69735f047320f23b", "problem_id": "3f28178ada29b9c7", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $- 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot x\\cdot \\cos{\\left(x \\right)} - 2\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot x\\cdot \\cos{\\left(x \\right)} - 2\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2fa8de539c9af64b", "problem_id": "3f28178ada29b9c7", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $- 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot x\\cdot \\cos{\\left(x \\right)} - 2\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot x\\cdot \\cos{\\left(x \\right)} - 2\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a8815508677c6751", "problem_id": "0658028f6d39597a", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A cyclist rides at 39 miles per hour for 1/2 hours, then at 21 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 39 × 1/2 = 39/2 miles.\n2) Second leg distance = 21 × 5/2 = 105/2 miles.\n3) Total distance = 39/2 + 105/2 = 72 miles.\nFinal answer: $72$", "final_answer": "72", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3b9f9e8888283d5b", "problem_id": "0658028f6d39597a", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A cyclist rides at 39 miles per hour for 1/2 hours, then at 21 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 39 × 1/2 = 39/2 miles.\nSecond leg distance = 21 × 5/2 = 105/2 miles.\nTotal distance = 39/2 + 105/2 = 72 miles.\nCheck: adding both legs gives 72.\nFinal answer: $72$", "final_answer": "72", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "baa6c0078c00f638", "problem_id": "3b9c7e537a4ab7f5", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} x^{3} - 3 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $x^{3} - 3 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4$.\n2) Evaluate at the bounds: F(2 \\pi) and F(0).\n3) Subtract: F(2 \\pi) − F(0) = $2 \\pi \\left(- 4 \\pi^{2} + 4 + 3 \\pi + 2 \\pi^{3}\\right)$.\n4) Simplify the result.\nFinal answer: $2\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} + 4 + 3\\cdot \\pi + 2\\cdot \\pi^{3}\\right)$", "final_answer": "2\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} + 4 + 3\\cdot \\pi + 2\\cdot \\pi^{3}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8c0e1cfe19a4f8f7", "problem_id": "3b9c7e537a4ab7f5", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} x^{3} - 3 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $x^{3} - 3 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4$.\nEvaluate at the bounds: F(2 \\pi) and F(0).\nSubtract: F(2 \\pi) − F(0) = $2 \\pi \\left(- 4 \\pi^{2} + 4 + 3 \\pi + 2 \\pi^{3}\\right)$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $2\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} + 4 + 3\\cdot \\pi + 2\\cdot \\pi^{3}\\right)$", "final_answer": "2\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} + 4 + 3\\cdot \\pi + 2\\cdot \\pi^{3}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "df202bb723a04fcc", "problem_id": "3787fad3e50f603c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} + 4$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(3 x^{2}) = 6 x^{2}$.\n3) So $f'(x) = 6 x^{2}$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(3 x^{2}) = 6 x$.\nTherefore the correct derivative is $f'(x) = 6 x$.\nFinal answer: $6 x$", "final_answer": "6 x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e0ddcbb5097d3eb2", "problem_id": "5f345a491bfed909", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the equation of the tangent line to $y=x^{3} - x^{2} - x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=1$.", "output": "1) Differentiate: f'(x) = $3 x^{2} - 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 1$.\n2) Slope at x=1: m = f'(1) = $- 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}$.\n3) Point on curve: (1, f(1)) = (1, $\\cos{\\left(2 \\right)} + e^{\\frac{1}{2}}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} = - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}(x − 1).\n6) Optionally expand: y = $- 2 x \\sin{\\left(2 \\right)} + \\frac{x e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3c6c8ffa6642322b", "problem_id": "989f1c79fa0e677c", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Two friends split $663 in the ratio 5:8. How much does each receive?", "output": "1) Total ratio parts = 5+8 = 13.\n2) Each part = 663 ÷ 13 = 51.\n3) First share = 5 × 51 = 255.\n4) Second share = 8 × 51 = 408.\nFinal answer: $408$", "final_answer": "408", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6d4ead8b8163746f", "problem_id": "989f1c79fa0e677c", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Two friends split $663 in the ratio 5:8. How much does each receive?", "output": "Total ratio parts = 5+8 = 13.\nEach part = 663 ÷ 13 = 51.\nFirst share = 5 × 51 = 255.\nSecond share = 8 × 51 = 408.\nCheck: 255+408=663.\nFinal answer: $408$", "final_answer": "408", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d024a04cbe7092a4", "problem_id": "6fc1a67842e2075b", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A store sells a backpack for $306. It is discounted by 35%. You buy 3 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 306 × (1 − 35/100) = 306 × 65/100 = 1989/10.\n2) Subtotal = 3 × 1989/10 = 5967/10.\n3) Total with tax = 5967/10 × 105/100 = 125307/200.\nFinal answer: $\\frac{125307}{200}$", "final_answer": "\\frac{125307}{200}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "418ded65e0df8f39", "problem_id": "6fc1a67842e2075b", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A store sells a backpack for $306. It is discounted by 35%. You buy 3 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 306 × (1 − 35/100) = 306 × 65/100 = 1989/10.\nSubtotal = 3 × 1989/10 = 5967/10.\nTotal with tax = 5967/10 × 105/100 = 125307/200.\nCheck: recomputing the formula yields 125307/200.\nFinal answer: $\\frac{125307}{200}$", "final_answer": "\\frac{125307}{200}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7385a930b418386c", "problem_id": "f97b5b0a45595681", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 2 \\sqrt{11} x - \\frac{31 x}{5} + \\frac{6}{5} + \\frac{2 \\sqrt{11}}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- 2 \\sqrt{11} - \\frac{31}{5}$, c=$\\frac{6}{5} + \\frac{2 \\sqrt{11}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{116 \\sqrt{11}}{5} + \\frac{1941}{25}$, so √Δ = $\\sqrt{\\frac{116 \\sqrt{11}}{5} + \\frac{1941}{25}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{31}{10} + \\sqrt{11} + \\frac{\\sqrt{580 \\sqrt{11} + 1941}}{10}$.\n7) x2 = $- \\frac{\\sqrt{580 \\sqrt{11} + 1941}}{10} + \\frac{31}{10} + \\sqrt{11}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{31}{10} + \\sqrt{11} + \\frac{\\sqrt{580\\cdot \\sqrt{11} + 1941}}{10}, - \\frac{\\sqrt{580\\cdot \\sqrt{11} + 1941}}{10} + \\frac{31}{10} + \\sqrt{11}\\}$", "final_answer": "x \\in \\{\\frac{31}{10} + \\sqrt{11} + \\frac{\\sqrt{580\\cdot \\sqrt{11} + 1941}}{10}, - \\frac{\\sqrt{580\\cdot \\sqrt{11} + 1941}}{10} + \\frac{31}{10} + \\sqrt{11}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "87ebd2f5291f7517", "problem_id": "4130f56657689103", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(2 x^{2} - 3 x - 3\\right) \\cos{\\left(3 x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 3 e^{x}) = - 3 e^{x}$.\n3) So $f'(x) = - 6 x^{2} \\sin{\\left(3 x + 3 \\right)} + x e^{x} + 9 x \\sin{\\left(3 x + 3 \\right)} + 4 x \\cos{\\left(3 x + 3 \\right)} - 2 e^{x} + 9 \\sin{\\left(3 x + 3 \\right)} - 3 \\cos{\\left(3 x + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 3 e^{x}) = - 3 e^{x}$.\nTherefore the correct derivative is $f'(x) = \\left(x - 3\\right) e^{x} + \\left(4 x - 3\\right) \\cos{\\left(3 x + 3 \\right)} - 3 \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x + 3 \\right)} + e^{x}$.\nFinal answer: $\\left(x - 3\\right) e^{x} + \\left(4 x - 3\\right) \\cos{\\left(3 x + 3 \\right)} - 3 \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x + 3 \\right)} + e^{x}$", "final_answer": "\\left(x - 3\\right) e^{x} + \\left(4 x - 3\\right) \\cos{\\left(3 x + 3 \\right)} - 3 \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x + 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0009eee6c3645d0e", "problem_id": "15053477f9f44ed9", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$- 4 x - 7 y$ = $13$\n$- 3 y$ = $21$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $-4$: $12 y$ = $-84$.\n4) Subtract to eliminate x: $- 12 y$ = $84$.\n5) Solve for y: y = $-7$.\n6) Substitute back to find x: x = $9$.\nFinal answer: $(x,y)=\\left(9, -7\\right)$", "final_answer": "(x,y)=\\left(9, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a82c3d2e5d41ebe", "problem_id": "15053477f9f44ed9", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$- 4 x - 7 y$ = $13$\n$- 3 y$ = $21$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $-4$: $12 y$ = $-84$.\nSubtract to eliminate x: $- 12 y$ = $84$.\nSolve for y: y = $-7$.\nSubstitute back to find x: x = $9$.\nCheck: plug (x,y)=(9,-7) into both equations.\nFinal answer: $(x,y)=\\left(9, -7\\right)$", "final_answer": "(x,y)=\\left(9, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "13fe79b916f5002c", "problem_id": "5657bd67d9d2a4a2", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} \\sin{\\left(x \\right)} - 3 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $3 x^{2} \\sin{\\left(x \\right)} - 3 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x\\cdot \\sin{\\left(x \\right)} - 3\\cdot x\\cdot \\cos{\\left(x \\right)} - 3\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x\\cdot \\sin{\\left(x \\right)} - 3\\cdot x\\cdot \\cos{\\left(x \\right)} - 3\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "957e3dd2bc39bf5d", "problem_id": "5657bd67d9d2a4a2", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} \\sin{\\left(x \\right)} - 3 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $3 x^{2} \\sin{\\left(x \\right)} - 3 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x\\cdot \\sin{\\left(x \\right)} - 3\\cdot x\\cdot \\cos{\\left(x \\right)} - 3\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x\\cdot \\sin{\\left(x \\right)} - 3\\cdot x\\cdot \\cos{\\left(x \\right)} - 3\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4eaedb0ccdd36de3", "problem_id": "d3d1e01cc5a03496", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{10} x + \\frac{x}{2} - \\frac{3}{2} + \\sqrt{10}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{1}{2} - \\sqrt{10}$, c=$- \\frac{3}{2} + \\sqrt{10}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{65}{4} - 5 \\sqrt{10}$, so √Δ = $\\sqrt{\\frac{65}{4} - 5 \\sqrt{10}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{1}{4} + \\frac{\\sqrt{65 - 20 \\sqrt{10}}}{4} + \\frac{\\sqrt{10}}{2}$.\nx2 = $- \\frac{\\sqrt{65 - 20 \\sqrt{10}}}{4} - \\frac{1}{4} + \\frac{\\sqrt{10}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{1}{4} + \\frac{\\sqrt{65 - 20\\cdot \\sqrt{10}}}{4} + \\frac{\\sqrt{10}}{2}, - \\frac{\\sqrt{65 - 20\\cdot \\sqrt{10}}}{4} - \\frac{1}{4} + \\frac{\\sqrt{10}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{1}{4} + \\frac{\\sqrt{65 - 20\\cdot \\sqrt{10}}}{4} + \\frac{\\sqrt{10}}{2}, - \\frac{\\sqrt{65 - 20\\cdot \\sqrt{10}}}{4} - \\frac{1}{4} + \\frac{\\sqrt{10}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "754e0e34b0c5303b", "problem_id": "9078a693041d9ffc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} x^{2} - 4 x + 7$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to x^{2} - 5 x + 7.\n3) Then the limit is 21.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 19.\nFinal answer: $19$", "final_answer": "19", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6e02063b19fcb652", "problem_id": "c4ce4623662acf09", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A store sells a backpack for $73. It is discounted by 30%. You buy 6 backpacks. What is the total cost?", "output": "1) Discounted price = 73 × (1 − 30/100) = 73 × 70/100 = 511/10.\n2) Subtotal = 6 × 511/10 = 1533/5.\nFinal answer: $\\frac{1533}{5}$", "final_answer": "\\frac{1533}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c6f5c734d2190cd8", "problem_id": "c4ce4623662acf09", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A store sells a backpack for $73. It is discounted by 30%. You buy 6 backpacks. What is the total cost?", "output": "Discounted price = 73 × (1 − 30/100) = 73 × 70/100 = 511/10.\nSubtotal = 6 × 511/10 = 1533/5.\nFinal answer: $\\frac{1533}{5}$", "final_answer": "\\frac{1533}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "71811a0705726340", "problem_id": "a79e6db3c10c3f59", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_5=35$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_5=20.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=20 and k=5.\n4) Compute C(24,4) = 10626.\nFinal answer: $10626$", "final_answer": "10626", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4c1c09f46235d4f3", "problem_id": "a79e6db3c10c3f59", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_5=35$?", "output": "Let y_i=x_i-3, then y_i≥0 and y_1+…+y_5=20.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=20 and k=5.\nFinal answer: $10626$", "final_answer": "10626", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bf516aa969893009", "problem_id": "39e2c5e6341de2e8", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$4 y$ = $12$\n$- 4 x - y$ = $-15$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $- 16 y$ = $-48$.\n3) Multiply the second equation by $0$: $0$ = $0$.\n4) Subtract to eliminate x: $- 16 y$ = $-48$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, 3\\right)$", "final_answer": "(x,y)=\\left(3, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5abe86162b785a6e", "problem_id": "39e2c5e6341de2e8", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the linear system:\n$4 y$ = $12$\n$- 4 x - y$ = $-15$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $- 16 y$ = $-48$.\nMultiply the second equation by $0$: $0$ = $0$.\nFinal answer: $(x,y)=\\left(3, 3\\right)$", "final_answer": "(x,y)=\\left(3, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cdf8081c508bfbf7", "problem_id": "9d4cd9c05c99030c", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x^{3} + 4 x^{2} + 7 x + 13$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $x^{4} + 2 x^{3} - x^{2} - x + 3$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $4 x^{3} + 6 x^{2} - 2 x - 1$.\nEvaluate at a: p'(2) = $51$.\nCheck: the identity gives $51$.\nFinal answer: $51$", "final_answer": "51", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5663757701e35988", "problem_id": "9d4cd9c05c99030c", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x^{3} + 4 x^{2} + 7 x + 13$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{4} + 2 x^{3} - x^{2} - x + 3$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $4 x^{3} + 6 x^{2} - 2 x - 1$.\n5) Evaluate at a: p'(2) = $51$.\nFinal answer: $51$", "final_answer": "51", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "62346817d80d88e2", "problem_id": "0db212cd5ba3e4ce", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 13 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=13, k=8, p=$\\frac{3}{7}$ and simplify.\nFinal answer: $\\frac{8646663168}{96889010407}$", "final_answer": "\\frac{8646663168}{96889010407}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bfa748e8ea430622", "problem_id": "0db212cd5ba3e4ce", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 13 times, what is the probability of exactly 8 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=13, k=8, p=$\\frac{3}{7}$ and simplify.\n4) Here C(13,8) = 1287.\nFinal answer: $\\frac{8646663168}{96889010407}$", "final_answer": "\\frac{8646663168}{96889010407}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "edc850f3c33e1f79", "problem_id": "79d23148dd294ced", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A store sells a backpack for $103. It is discounted by 35%. You buy 7 backpacks. What is the total cost?", "output": "1) Discounted price = 103 × (1 − 35/100) = 103 × 65/100 = 1339/20.\n2) Subtotal = 7 × 1339/20 = 9373/20.\nFinal answer: $\\frac{9373}{20}$", "final_answer": "\\frac{9373}{20}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "49f5507992d51aac", "problem_id": "79d23148dd294ced", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $103. It is discounted by 35%. You buy 7 backpacks. What is the total cost?", "output": "Discounted price = 103 × (1 − 35/100) = 103 × 65/100 = 1339/20.\nSubtotal = 7 × 1339/20 = 9373/20.\nCheck: recomputing the formula yields 9373/20.\nFinal answer: $\\frac{9373}{20}$", "final_answer": "\\frac{9373}{20}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "20a34a80c6bcc4db", "problem_id": "c9b917bb8a13198b", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Two friends split $2512 in the ratio 8:8. How much does each receive?", "output": "1) Total ratio parts = 8+8 = 16.\n2) Each part = 2512 ÷ 16 = 157.\n3) First share = 8 × 157 = 1256.\n4) Second share = 8 × 157 = 1256.\nFinal answer: $1256$", "final_answer": "1256", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "546602681a6ee0be", "problem_id": "c9b917bb8a13198b", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $2512 in the ratio 8:8. How much does each receive?", "output": "Total ratio parts = 8+8 = 16.\nEach part = 2512 ÷ 16 = 157.\nFirst share = 8 × 157 = 1256.\nSecond share = 8 × 157 = 1256.\nCheck: 1256+1256=2512.\nFinal answer: $1256$", "final_answer": "1256", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "97a3f11808ab72e1", "problem_id": "18ba61bf42d77710", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 10 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=5, p=$\\frac{1}{10}$ and simplify.\nHere C(10,5) = 252.\nCheck: expression simplifies to $\\frac{3720087}{2500000000}$.\nFinal answer: $\\frac{3720087}{2500000000}$", "final_answer": "\\frac{3720087}{2500000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b8427d921e4b0e5d", "problem_id": "18ba61bf42d77710", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 10 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=5, p=$\\frac{1}{10}$ and simplify.\n4) Here C(10,5) = 252.\nFinal answer: $\\frac{3720087}{2500000000}$", "final_answer": "\\frac{3720087}{2500000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9d87d97e1c5ff34c", "problem_id": "2421ca0f487cbab0", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the equation of the tangent line to $y=2 x^{2} + 2 \\sin{\\left(x \\right)} + 1$ at $x=0$.", "output": "1) Differentiate: f'(x) = $4 x + 2 \\cos{\\left(x \\right)}$.\n2) Slope at x=0: m = f'(0) = $2$.\n3) Point on curve: (0, f(0)) = (0, $1$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 1 = 2(x − 0).\n6) Optionally expand: y = $2 x + 1$.\nFinal answer: $y = 2\\cdot x + 1$", "final_answer": "y = 2\\cdot x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e00e99a5f9d2ce96", "problem_id": "2421ca0f487cbab0", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{2} + 2 \\sin{\\left(x \\right)} + 1$ at $x=0$.", "output": "Differentiate: f'(x) = $4 x + 2 \\cos{\\left(x \\right)}$.\nSlope at x=0: m = f'(0) = $2$.\nPoint on curve: (0, f(0)) = (0, $1$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 1 = 2(x − 0).\nOptionally expand: y = $2 x + 1$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 2\\cdot x + 1$", "final_answer": "y = 2\\cdot x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "538d5c8395456a55", "problem_id": "1c6ad58a6a03a235", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Worker A finishes a job in 18 hours and Worker B finishes it in 16 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/18 job/hour. B's rate = 1/16 job/hour.\nCombined rate = 1/18 + 1/16 = 17/144 job/hour.\nTime = 1 ÷ (combined rate) = 144/17 hours.\nCheck: (combined rate)×(time) = 17/144×144/17 = 1.\nFinal answer: $\\frac{144}{17}$", "final_answer": "\\frac{144}{17}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "516a60c82bdf07bc", "problem_id": "1c6ad58a6a03a235", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Worker A finishes a job in 18 hours and Worker B finishes it in 16 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/18 job/hour. B's rate = 1/16 job/hour.\n2) Combined rate = 1/18 + 1/16 = 17/144 job/hour.\n3) Time = 1 ÷ (combined rate) = 144/17 hours.\nFinal answer: $\\frac{144}{17}$", "final_answer": "\\frac{144}{17}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cc3e230b30885100", "problem_id": "21eb279526c9da1c", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_9=35$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_9=8.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=8 and k=9.\n4) Compute C(16,8) = 12870.\nFinal answer: $12870$", "final_answer": "12870", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "147cf9c84ba3ceaf", "problem_id": "21eb279526c9da1c", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_9=35$?", "output": "Let y_i=x_i-3, then y_i≥0 and y_1+…+y_9=8.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=8 and k=9.\nCompute C(16,8) = 12870.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $12870$", "final_answer": "12870", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "12721dd0f9ff07ef", "problem_id": "0238029570559c90", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 16 times, what is the probability of exactly 10 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=16, k=10, p=$\\frac{1}{2}$ and simplify.\n4) Here C(16,10) = 8008.\nFinal answer: $\\frac{1001}{8192}$", "final_answer": "\\frac{1001}{8192}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b3f5c7c15fb0a6d8", "problem_id": "0238029570559c90", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 16 times, what is the probability of exactly 10 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=16, k=10, p=$\\frac{1}{2}$ and simplify.\nFinal answer: $\\frac{1001}{8192}$", "final_answer": "\\frac{1001}{8192}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "104a716b79501100", "problem_id": "eb9c97c9146cc5a8", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 15 times, what is the probability of exactly 7 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=15, k=7, p=$\\frac{1}{6}$ and simplify.\n4) Here C(15,7) = 6435.\nFinal answer: $\\frac{279296875}{52242776064}$", "final_answer": "\\frac{279296875}{52242776064}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f1c8af3a8a7f43be", "problem_id": "eb9c97c9146cc5a8", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 15 times, what is the probability of exactly 7 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=15, k=7, p=$\\frac{1}{6}$ and simplify.\nFinal answer: $\\frac{279296875}{52242776064}$", "final_answer": "\\frac{279296875}{52242776064}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3ec4c24c603f92fb", "problem_id": "552ed28d63f74062", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Worker A finishes a job in 19 hours and Worker B finishes it in 21 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/19 job/hour. B's rate = 1/21 job/hour.\nCombined rate = 1/19 + 1/21 = 40/399 job/hour.\nTime = 1 ÷ (combined rate) = 399/40 hours.\nCheck: (combined rate)×(time) = 40/399×399/40 = 1.\nFinal answer: $\\frac{399}{40}$", "final_answer": "\\frac{399}{40}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "71055cf489f87e49", "problem_id": "552ed28d63f74062", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Worker A finishes a job in 19 hours and Worker B finishes it in 21 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/19 job/hour. B's rate = 1/21 job/hour.\n2) Combined rate = 1/19 + 1/21 = 40/399 job/hour.\n3) Time = 1 ÷ (combined rate) = 399/40 hours.\nFinal answer: $\\frac{399}{40}$", "final_answer": "\\frac{399}{40}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b82b207f712cb150", "problem_id": "6e9095838528a6d5", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 3$.\n3) Compute an antiderivative: F(x) = $- x^{3} + x^{2} + 3 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C - x^{3} + x^{2} + 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - x^{3} + x^{2} + 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "39730e07450e8ff4", "problem_id": "6e9095838528a6d5", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 3$.\nCompute an antiderivative: F(x) = $- x^{3} + x^{2} + 3 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $C - x^{3} + x^{2} + 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - x^{3} + x^{2} + 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5461862a395ffba2", "problem_id": "b6e082056d2780db", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} - x^{3} - 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{3} - 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\nEvaluate at the bounds: F(2 \\pi) and F(0).\nSubtract: F(2 \\pi) − F(0) = $\\frac{2 \\pi \\left(- 6 \\pi^{3} - 8 \\pi^{2} + 6 + 9 \\pi\\right)}{3}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(- 6\\cdot \\pi^{3} - 8\\cdot \\pi^{2} + 6 + 9\\cdot \\pi\\right)}{3}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(- 6\\cdot \\pi^{3} - 8\\cdot \\pi^{2} + 6 + 9\\cdot \\pi\\right)}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dd485e202d8c7c62", "problem_id": "b6e082056d2780db", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} - x^{3} - 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{3} - 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(0).\n3) Subtract: F(2 \\pi) − F(0) = $\\frac{2 \\pi \\left(- 6 \\pi^{3} - 8 \\pi^{2} + 6 + 9 \\pi\\right)}{3}$.\n4) Simplify the result.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(- 6\\cdot \\pi^{3} - 8\\cdot \\pi^{2} + 6 + 9\\cdot \\pi\\right)}{3}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(- 6\\cdot \\pi^{3} - 8\\cdot \\pi^{2} + 6 + 9\\cdot \\pi\\right)}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "44c7b8f779c91561", "problem_id": "f0d2e46114adce79", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A store sells a backpack for $90. It is discounted by 5%. You buy 6 backpacks. What is the total cost?", "output": "Discounted price = 90 × (1 − 5/100) = 90 × 95/100 = 171/2.\nSubtotal = 6 × 171/2 = 513.\nCheck: recomputing the formula yields 513.\nFinal answer: $513$", "final_answer": "513", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1b2b07b45a346566", "problem_id": "f0d2e46114adce79", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A store sells a backpack for $90. It is discounted by 5%. You buy 6 backpacks. What is the total cost?", "output": "1) Discounted price = 90 × (1 − 5/100) = 90 × 95/100 = 171/2.\n2) Subtotal = 6 × 171/2 = 513.\nFinal answer: $513$", "final_answer": "513", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ce00c6a0dd64e196", "problem_id": "3a8d3312e95e714b", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many binary strings of length 10 contain exactly 2 ones with no two ones adjacent?", "output": "Reserve 1 mandatory zeros between the 2 ones.\nThis leaves n−k+1 = 9 valid slots for the ones.\nChoose the slots: C(9,2) = 36.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $36$", "final_answer": "36", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2763ae4de0e9c2a5", "problem_id": "3a8d3312e95e714b", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "How many binary strings of length 10 contain exactly 2 ones with no two ones adjacent?", "output": "1) Reserve 1 mandatory zeros between the 2 ones.\n2) This leaves n−k+1 = 9 valid slots for the ones.\n3) Choose the slots: C(9,2) = 36.\nFinal answer: $36$", "final_answer": "36", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "649b7f0d0694cd88", "problem_id": "66a21acda650baa6", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} 2 x^{4} - 2 x^{3} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $2 x^{4} - 2 x^{3} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\nEvaluate at the bounds: F(2 \\pi) and F(0).\nSubtract: F(2 \\pi) − F(0) = $\\frac{2 \\pi \\left(- 20 \\pi^{3} - 20 \\pi + 5 + 32 \\pi^{4}\\right)}{5}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(- 20\\cdot \\pi^{3} - 20\\cdot \\pi + 5 + 32\\cdot \\pi^{4}\\right)}{5}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(- 20\\cdot \\pi^{3} - 20\\cdot \\pi + 5 + 32\\cdot \\pi^{4}\\right)}{5}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e2b0577e7d59ef20", "problem_id": "66a21acda650baa6", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} 2 x^{4} - 2 x^{3} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{4} - 2 x^{3} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\n2) Evaluate at the bounds: F(2 \\pi) and F(0).\n3) Subtract: F(2 \\pi) − F(0) = $\\frac{2 \\pi \\left(- 20 \\pi^{3} - 20 \\pi + 5 + 32 \\pi^{4}\\right)}{5}$.\n4) Simplify the result.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(- 20\\cdot \\pi^{3} - 20\\cdot \\pi + 5 + 32\\cdot \\pi^{4}\\right)}{5}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(- 20\\cdot \\pi^{3} - 20\\cdot \\pi + 5 + 32\\cdot \\pi^{4}\\right)}{5}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db9aa32c70cac141", "problem_id": "bf43eee1b28b6b14", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_5=18$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_5=13.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=13 and k=5.\nFinal answer: $2380$", "final_answer": "2380", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7a31b78a0a1cb8d0", "problem_id": "bf43eee1b28b6b14", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_5=18$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_5=13.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=13 and k=5.\n4) Compute C(17,4) = 2380.\nFinal answer: $2380$", "final_answer": "2380", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "50f006e2da298b24", "problem_id": "ff8ce59cf7682148", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the multiplicative inverse of 29 modulo 47.", "output": "1) We want x such that 29x ≡ 1 (mod 47).\n2) Use extended Euclid to find s,t with 29s + 47t = 1.\n3) One Bezout identity is: 29·(13) + 47·(-8) = 1.\n4) Reduce s modulo 47: x ≡ 13 (mod 47).\nFinal answer: $x\\equiv 13\\pmod{47}$", "final_answer": "x\\equiv 13\\pmod{47}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a00156136a39cf53", "problem_id": "ff8ce59cf7682148", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the multiplicative inverse of 29 modulo 47.", "output": "We want x such that 29x ≡ 1 (mod 47).\nUse extended Euclid to find s,t with 29s + 47t = 1.\nOne Bezout identity is: 29·(13) + 47·(-8) = 1.\nReduce s modulo 47: x ≡ 13 (mod 47).\nCheck: 29·13 mod 47 = 1.\nFinal answer: $x\\equiv 13\\pmod{47}$", "final_answer": "x\\equiv 13\\pmod{47}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "61fbbf29bd4e4d07", "problem_id": "08197c8f5507fd90", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $70 in the ratio 2:8. How much does each receive?", "output": "1) Total ratio parts = 2+8 = 10.\n2) Each part = 70 ÷ 10 = 7.\n3) First share = 2 × 7 = 14.\n4) Second share = 8 × 7 = 56.\nFinal answer: $56$", "final_answer": "56", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "468efe002a437e5e", "problem_id": "08197c8f5507fd90", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Two friends split $70 in the ratio 2:8. How much does each receive?", "output": "Total ratio parts = 2+8 = 10.\nEach part = 70 ÷ 10 = 7.\nFirst share = 2 × 7 = 14.\nFinal answer: $56$", "final_answer": "56", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d030c897256c383b", "problem_id": "bd7da8a461e60e58", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "You want 59 liters of a 30% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 59−x liters is 50% solution.\n2) Salt equation: 25x + 50(59−x) = 30·59.\n3) (25-50)x = (30-50)·59 ⇒ x = 236/5.\n4) Then 59−x = 59/5.\nFinal answer: $236/5 L of 25% and 59/5 L of 50%$", "final_answer": "236/5 L of 25% and 59/5 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6c249bf4548d4098", "problem_id": "0639cd931b08127e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x - 1$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 3 x - 1.\n3) Then the limit is -7.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -5.\nFinal answer: $-5$", "final_answer": "-5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "30ba04b2a4eea2a5", "problem_id": "5ab91e0620fac600", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- x + 4 y$ = $-7$\n$- 2 y$ = $4$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $-1$: $2 y$ = $-4$.\nSubtract to eliminate x: $- 2 y$ = $4$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $-1$.\nCheck: plug (x,y)=(-1,-2) into both equations.\nFinal answer: $(x,y)=\\left(-1, -2\\right)$", "final_answer": "(x,y)=\\left(-1, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "23820a2edf91b37d", "problem_id": "5ab91e0620fac600", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$- x + 4 y$ = $-7$\n$- 2 y$ = $4$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $-1$: $2 y$ = $-4$.\n4) Subtract to eliminate x: $- 2 y$ = $4$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, -2\\right)$", "final_answer": "(x,y)=\\left(-1, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bd9670fca4bb5026", "problem_id": "434fb9ff87e5e1ff", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{3} - 3 x^{2} + 2 x + \\left(x^{2} + 3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{x^{2} + 3 x + 7}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{7 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) + 7}) = \\frac{7 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3 x + 7}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} - \\frac{2 x^{4}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{6 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{12 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} - \\frac{12 x^{3}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{23 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{46 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} - \\frac{64 x^{2}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{42 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{84 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} - \\frac{92 x}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{14 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 6 x^{3} + 23 x^{2} + 42 x + 49} + \\frac{49 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{98 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{16}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{21 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 6 x^{3} + 23 x^{2} + 42 x + 49} - \\frac{7 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 6 x + 14} + \\frac{7 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3 x + 7} - \\frac{7 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) + 7}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{7 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) + 7}) = \\frac{7 \\left(- 2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(x^{2} + 3 x\\right) + 7\\right)^{2}} + \\frac{7 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(x^{2} + 3 x\\right) + 7\\right)} + \\frac{7 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) + 7}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(- 2 x - 3\\right) \\left(- x^{3} - 3 x^{2} + 2 x + \\left(x \\left(x + 3\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2\\right) + \\left(x \\left(x + 3\\right) + 7\\right) \\left(- 6 x^{2} - 12 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x + 3\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x + 3\\right) + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x + 3\\right) + 7\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(- 2 x - 3\\right) \\left(- x^{3} - 3 x^{2} + 2 x + \\left(x \\left(x + 3\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2\\right) + \\left(x \\left(x + 3\\right) + 7\\right) \\left(- 6 x^{2} - 12 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x + 3\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x + 3\\right) + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x + 3\\right) + 7\\right)^{2}}$", "final_answer": "\\frac{2 \\left(- 2 x - 3\\right) \\left(- x^{3} - 3 x^{2} + 2 x + \\left(x \\left(x + 3\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2\\right) + \\left(x \\left(x + 3\\right) + 7\\right) \\left(- 6 x^{2} - 12 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x + 3\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x + 3\\right) + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x + 3\\right) + 7\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "811ab2d87d41c060", "problem_id": "f6f0abb105185a2b", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} - 3 x^{4} - 2 x^{3} - x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{4} - 2 x^{3} - x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $- \\frac{3 \\pi^{5}}{80} - 2 \\pi - \\frac{\\pi^{3}}{12} + 4$.\n4) Simplify the result.\nFinal answer: $- \\frac{3\\cdot \\pi^{5}}{80} - 2\\cdot \\pi - \\frac{\\pi^{3}}{12} + 4$", "final_answer": "- \\frac{3\\cdot \\pi^{5}}{80} - 2\\cdot \\pi - \\frac{\\pi^{3}}{12} + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "71b438aa9142372d", "problem_id": "de799e4a9a4ed599", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡10 (mod 11), x≡6 (mod 13). Give the solution modulo 143.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 143.\n3) Compute the CRT solution to get x ≡ 32 (mod 143).\nFinal answer: $x\\equiv 32\\pmod{143}$", "final_answer": "x\\equiv 32\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "441f3beacd4f8fb1", "problem_id": "de799e4a9a4ed599", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡10 (mod 11), x≡6 (mod 13). Give the solution modulo 143.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 143.\nCompute the CRT solution to get x ≡ 32 (mod 143).\nCheck: 32 mod 11=10 and mod 13=6.\nFinal answer: $x\\equiv 32\\pmod{143}$", "final_answer": "x\\equiv 32\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8a57b6fc6bcb9391", "problem_id": "2eae6a3d1f410b8c", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} \\sin{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $- 2 x^{3} \\sin{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} + 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} + 3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x\\cdot \\sin{\\left(x \\right)} - x\\cdot \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} + 3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x\\cdot \\sin{\\left(x \\right)} - x\\cdot \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e1bcf5157e18e0a9", "problem_id": "2eae6a3d1f410b8c", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} \\sin{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $- 2 x^{3} \\sin{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} + 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} + 3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x\\cdot \\sin{\\left(x \\right)} - x\\cdot \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} + 3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x\\cdot \\sin{\\left(x \\right)} - x\\cdot \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "edf064a9f6af0ff6", "problem_id": "2c2e2aa854cff8ad", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_8=18$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_8=2.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=2 and k=8.\n4) Compute C(9,7) = 36.\nFinal answer: $36$", "final_answer": "36", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e791a9079bf4eeb9", "problem_id": "2c2e2aa854cff8ad", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_8=18$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_8=2.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=2 and k=8.\nCompute C(9,7) = 36.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $36$", "final_answer": "36", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "74698a650e99a14f", "problem_id": "d937a38ef77d1952", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} 4 x^{3} + 4 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $4 x^{3} + 4 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{3 \\pi^{2}}{4} - \\frac{3 \\pi}{2} - 1 + \\frac{3 \\pi^{3}}{2} + \\frac{15 \\pi^{4}}{16}$.\n4) Simplify the result.\nFinal answer: $- \\frac{3\\cdot \\pi^{2}}{4} - \\frac{3\\cdot \\pi}{2} - 1 + \\frac{3\\cdot \\pi^{3}}{2} + \\frac{15\\cdot \\pi^{4}}{16}$", "final_answer": "- \\frac{3\\cdot \\pi^{2}}{4} - \\frac{3\\cdot \\pi}{2} - 1 + \\frac{3\\cdot \\pi^{3}}{2} + \\frac{15\\cdot \\pi^{4}}{16}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "76760376b94c71f7", "problem_id": "d937a38ef77d1952", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} 4 x^{3} + 4 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $4 x^{3} + 4 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1$.\nEvaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{3 \\pi^{2}}{4} - \\frac{3 \\pi}{2} - 1 + \\frac{3 \\pi^{3}}{2} + \\frac{15 \\pi^{4}}{16}$.\nFinal answer: $- \\frac{3\\cdot \\pi^{2}}{4} - \\frac{3\\cdot \\pi}{2} - 1 + \\frac{3\\cdot \\pi^{3}}{2} + \\frac{15\\cdot \\pi^{4}}{16}$", "final_answer": "- \\frac{3\\cdot \\pi^{2}}{4} - \\frac{3\\cdot \\pi}{2} - 1 + \\frac{3\\cdot \\pi^{3}}{2} + \\frac{15\\cdot \\pi^{4}}{16}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cbfa212a9a8a7039", "problem_id": "dc8e48a4999df69b", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A store sells a backpack for $178. It is discounted by 15%. You buy 5 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 178 × (1 − 15/100) = 178 × 85/100 = 1513/10.\n2) Subtotal = 5 × 1513/10 = 1513/2.\n3) Total with tax = 1513/2 × 109/100 = 164917/200.\nFinal answer: $\\frac{164917}{200}$", "final_answer": "\\frac{164917}{200}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b7e7736c966dcfc", "problem_id": "dc8e48a4999df69b", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A store sells a backpack for $178. It is discounted by 15%. You buy 5 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 178 × (1 − 15/100) = 178 × 85/100 = 1513/10.\nSubtotal = 5 × 1513/10 = 1513/2.\nTotal with tax = 1513/2 × 109/100 = 164917/200.\nCheck: recomputing the formula yields 164917/200.\nFinal answer: $\\frac{164917}{200}$", "final_answer": "\\frac{164917}{200}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d7df7dec77d0fe1e", "problem_id": "e5b35a0d889e8303", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "You want 20 liters of a 35% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 20−x liters is 70% solution.\n2) Salt equation: 25x + 70(20−x) = 35·20.\n3) (25-70)x = (35-70)·20 ⇒ x = 140/9.\n4) Then 20−x = 40/9.\nFinal answer: $140/9 L of 25% and 40/9 L of 70%$", "final_answer": "140/9 L of 25% and 40/9 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "25606b9fca74f950", "problem_id": "e5b35a0d889e8303", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "You want 20 liters of a 35% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 20−x liters is 70% solution.\nSalt equation: 25x + 70(20−x) = 35·20.\n(25-70)x = (35-70)·20 ⇒ x = 140/9.\nThen 20−x = 40/9.\nCheck: 25·140/9 + 70·40/9 = 35·20.\nFinal answer: $140/9 L of 25% and 40/9 L of 70%$", "final_answer": "140/9 L of 25% and 40/9 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "609627b36608a982", "problem_id": "7253d4677acf777a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} + 3 x + \\left(3 x - 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{3 x - 7}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 7}) = \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 7}$.\n3) So $f'(x) = \\frac{9 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} - 84 x + 98} + \\frac{18 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} - 84 x + 98} - \\frac{6 x^{2}}{18 x^{2} - 84 x + 98} - \\frac{42 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} - 84 x + 98} - \\frac{84 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} - 84 x + 98} + \\frac{28 x}{18 x^{2} - 84 x + 98} + \\frac{9 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{2} - 42 x + 49} - \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x - 14} + \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 7} - \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 7} + \\frac{49 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} - 84 x + 98} + \\frac{98 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} - 84 x + 98} - \\frac{48}{18 x^{2} - 84 x + 98} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 7}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 7}) = \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(3 x - 7\\right)} + \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 7} - \\frac{9 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(3 x - 7\\right)^{2}} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 7}$.\nTherefore the correct derivative is $f'(x) = \\frac{6 x^{2} - 18 x + \\left(3 x - 7\\right) \\left(- 4 x + \\left(3 x - 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x - 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 6 \\left(3 x - 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6}{2 \\left(3 x - 7\\right)^{2}}$.\nFinal answer: $\\frac{6 x^{2} - 18 x + \\left(3 x - 7\\right) \\left(- 4 x + \\left(3 x - 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x - 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 6 \\left(3 x - 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6}{2 \\left(3 x - 7\\right)^{2}}$", "final_answer": "\\frac{6 x^{2} - 18 x + \\left(3 x - 7\\right) \\left(- 4 x + \\left(3 x - 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x - 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 6 \\left(3 x - 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6}{2 \\left(3 x - 7\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f3b309365b3f5427", "problem_id": "6a2e697e425d32bf", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Worker A finishes a job in 8 hours and Worker B finishes it in 22 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/8 job/hour. B's rate = 1/22 job/hour.\n2) Combined rate = 1/8 + 1/22 = 15/88 job/hour.\n3) Time = 1 ÷ (combined rate) = 88/15 hours.\nFinal answer: $\\frac{88}{15}$", "final_answer": "\\frac{88}{15}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4a5be54e74c94809", "problem_id": "c670496eff6dd649", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Two friends split $1512 in the ratio 8:4. How much does each receive?", "output": "1) Total ratio parts = 8+4 = 12.\n2) Each part = 1512 ÷ 12 = 126.\n3) First share = 8 × 126 = 1008.\n4) Second share = 4 × 126 = 504.\nFinal answer: $504$", "final_answer": "504", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f414b44ac33c5f2a", "problem_id": "c576674aaded5ee4", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 \\cos{\\left(2 x \\right)}) = - 2 \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 \\cos{\\left(2 x \\right)}) = - 4 \\sin{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$.\nFinal answer: $x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$", "final_answer": "x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "84bce4248ba36810", "problem_id": "6dd32e1a23ed656f", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "How many binary strings of length 23 contain exactly 5 ones with no two ones adjacent?", "output": "1) Reserve 4 mandatory zeros between the 5 ones.\n2) This leaves n−k+1 = 19 valid slots for the ones.\n3) Choose the slots: C(19,5) = 11628.\nFinal answer: $11628$", "final_answer": "11628", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "92248d6755569ade", "problem_id": "855a8eedf256ec62", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x^{2} + 5 x + 7$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to x^{2} + 4 x + 7.\n3) Then the limit is 19.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 21.\nFinal answer: $21$", "final_answer": "21", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2aafa07c34778b94", "problem_id": "ebde76c068c25356", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A cyclist rides at 62 miles per hour for 1/2 hours, then at 20 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 62 × 1/2 = 31 miles.\n2) Second leg distance = 20 × 3/2 = 30 miles.\n3) Total distance = 31 + 30 = 61 miles.\nFinal answer: $61$", "final_answer": "61", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "638ba1da2993fa78", "problem_id": "ebde76c068c25356", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A cyclist rides at 62 miles per hour for 1/2 hours, then at 20 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 62 × 1/2 = 31 miles.\nSecond leg distance = 20 × 3/2 = 30 miles.\nTotal distance = 31 + 30 = 61 miles.\nCheck: adding both legs gives 61.\nFinal answer: $61$", "final_answer": "61", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "877fcc6ffc441656", "problem_id": "72df392868d8caac", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A cyclist rides at 28 miles per hour for 1/2 hours, then at 60 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 28 × 1/2 = 14 miles.\nSecond leg distance = 60 × 2 = 120 miles.\nTotal distance = 14 + 120 = 134 miles.\nCheck: adding both legs gives 134.\nFinal answer: $134$", "final_answer": "134", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4275d0601d14da45", "problem_id": "72df392868d8caac", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "A cyclist rides at 28 miles per hour for 1/2 hours, then at 60 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 28 × 1/2 = 14 miles.\nSecond leg distance = 60 × 2 = 120 miles.\nTotal distance = 14 + 120 = 134 miles.\nFinal answer: $134$", "final_answer": "134", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "74659fe0bcff5547", "problem_id": "bb4bf42d15bc9f3f", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 17 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=17, k=5, p=$\\frac{1}{6}$ and simplify.\nFinal answer: $\\frac{377685546875}{4231664861184}$", "final_answer": "\\frac{377685546875}{4231664861184}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7088486cf399126c", "problem_id": "bb4bf42d15bc9f3f", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 17 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=17, k=5, p=$\\frac{1}{6}$ and simplify.\nHere C(17,5) = 6188.\nCheck: expression simplifies to $\\frac{377685546875}{4231664861184}$.\nFinal answer: $\\frac{377685546875}{4231664861184}$", "final_answer": "\\frac{377685546875}{4231664861184}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0518d7d50523a5c1", "problem_id": "3ac59d67e6e720c8", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡6 (mod 9), x≡0 (mod 13). Give the solution modulo 117.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 117.\nCompute the CRT solution to get x ≡ 78 (mod 117).\nCheck: 78 mod 9=6 and mod 13=0.\nFinal answer: $x\\equiv 78\\pmod{117}$", "final_answer": "x\\equiv 78\\pmod{117}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "39ea384122505063", "problem_id": "3ac59d67e6e720c8", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the congruences: x≡6 (mod 9), x≡0 (mod 13). Give the solution modulo 117.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 117.\n3) Compute the CRT solution to get x ≡ 78 (mod 117).\nFinal answer: $x\\equiv 78\\pmod{117}$", "final_answer": "x\\equiv 78\\pmod{117}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dbf4c4560407e6dd", "problem_id": "fb66368f2415621d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(- 3 x^{3} - x + 1\\right) \\cos{\\left(- x^{2} + x + 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x \\cos{\\left(\\left(- x^{2} + x\\right) + 2 \\right)}) = x \\sin{\\left(- x^{2} + x + 2 \\right)}$.\n3) So $f'(x) = - 6 x^{4} \\sin{\\left(- x^{2} + x + 2 \\right)} + 3 x^{3} \\sin{\\left(- x^{2} + x + 2 \\right)} - 9 x^{2} \\cos{\\left(\\left(- x^{2} + x\\right) + 2 \\right)} + 2 x e^{x} + 3 x \\sin{\\left(- x^{2} + x + 2 \\right)} + 6 e^{x} - \\sin{\\left(- x^{2} + x + 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x \\cos{\\left(\\left(- x^{2} + x\\right) + 2 \\right)}) = x \\left(1 - 2 x\\right) \\sin{\\left(- x^{2} + x + 2 \\right)} - \\cos{\\left(\\left(- x^{2} + x\\right) + 2 \\right)}$.\nTherefore the correct derivative is $f'(x) = - \\left(1 - 2 x\\right) \\left(x \\left(- 3 x^{2} - 1\\right) + 1\\right) \\sin{\\left(- x^{2} + x + 2 \\right)} + 2 \\left(x + 2\\right) e^{x} + \\left(- 9 x^{2} - 1\\right) \\cos{\\left(\\left(- x^{2} + x\\right) + 2 \\right)} + 2 e^{x}$.\nFinal answer: $- \\left(1 - 2 x\\right) \\left(x \\left(- 3 x^{2} - 1\\right) + 1\\right) \\sin{\\left(- x^{2} + x + 2 \\right)} + 2 \\left(x + 2\\right) e^{x} + \\left(- 9 x^{2} - 1\\right) \\cos{\\left(\\left(- x^{2} + x\\right) + 2 \\right)} + 2 e^{x}$", "final_answer": "- \\left(1 - 2 x\\right) \\left(x \\left(- 3 x^{2} - 1\\right) + 1\\right) \\sin{\\left(- x^{2} + x + 2 \\right)} + 2 \\left(x + 2\\right) e^{x} + \\left(- 9 x^{2} - 1\\right) \\cos{\\left(\\left(- x^{2} + x\\right) + 2 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d8bf5b37c1c95f1b", "problem_id": "df99127fc26c6b53", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{11 x}{6} + \\frac{1}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{11}{6}$, c=$\\frac{1}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{49}{36}$, so √Δ = $\\frac{7}{6}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{3}{2}$.\n7) x2 = $\\frac{1}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{3}{2}, \\frac{1}{3}\\}$", "final_answer": "x \\in \\{\\frac{3}{2}, \\frac{1}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7e627c1bc610dafd", "problem_id": "80f53ffa2a56b3ea", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} - 3 x - 4$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 3 x^{2} - x - 2$ and a = 1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 6 x - 1$.\n5) Evaluate at a: p'(1) = $-7$.\nFinal answer: $-7$", "final_answer": "-7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "90d8eda3db51587b", "problem_id": "80f53ffa2a56b3ea", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} - 3 x - 4$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 3 x^{2} - x - 2$ and a = 1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 6 x - 1$.\nEvaluate at a: p'(1) = $-7$.\nCheck: the identity gives $-7$.\nFinal answer: $-7$", "final_answer": "-7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b134b88c42acfe8b", "problem_id": "224cf25a1bf400f1", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$5 x + 2 y$ = $43$\n$7 x + 2 y$ = $61$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $7$: $35 x + 14 y$ = $301$.\n3) Multiply the second equation by $5$: $35 x + 10 y$ = $305$.\n4) Subtract to eliminate x: $4 y$ = $-4$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $9$.\nFinal answer: $(x,y)=\\left(9, -1\\right)$", "final_answer": "(x,y)=\\left(9, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d1c29e87287f00c8", "problem_id": "224cf25a1bf400f1", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$5 x + 2 y$ = $43$\n$7 x + 2 y$ = $61$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $7$: $35 x + 14 y$ = $301$.\nMultiply the second equation by $5$: $35 x + 10 y$ = $305$.\nSubtract to eliminate x: $4 y$ = $-4$.\nSolve for y: y = $-1$.\nSubstitute back to find x: x = $9$.\nCheck: plug (x,y)=(9,-1) into both equations.\nFinal answer: $(x,y)=\\left(9, -1\\right)$", "final_answer": "(x,y)=\\left(9, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3d2a23cce47892be", "problem_id": "8ec01072a4ea0cf1", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - x^{2} - 4 x - 10$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{3} - x^{2} + 2 x$ and a = 3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 3 x^{2} - 2 x + 2$.\n5) Evaluate at a: p'(3) = $-31$.\nFinal answer: $-31$", "final_answer": "-31", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "64a6dd9a4e2732ca", "problem_id": "8ec01072a4ea0cf1", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - x^{2} - 4 x - 10$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{3} - x^{2} + 2 x$ and a = 3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 3 x^{2} - 2 x + 2$.\nEvaluate at a: p'(3) = $-31$.\nCheck: the identity gives $-31$.\nFinal answer: $-31$", "final_answer": "-31", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2ee9ef01369eba19", "problem_id": "69c756309a1a3a77", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A store sells a backpack for $396. It is discounted by 30%. You buy 3 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 396 × (1 − 30/100) = 396 × 70/100 = 1386/5.\n2) Subtotal = 3 × 1386/5 = 4158/5.\n3) Total with tax = 4158/5 × 108/100 = 112266/125.\nFinal answer: $\\frac{112266}{125}$", "final_answer": "\\frac{112266}{125}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "058f3be12f6dbdf4", "problem_id": "69c756309a1a3a77", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $396. It is discounted by 30%. You buy 3 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 396 × (1 − 30/100) = 396 × 70/100 = 1386/5.\nSubtotal = 3 × 1386/5 = 4158/5.\nTotal with tax = 4158/5 × 108/100 = 112266/125.\nCheck: recomputing the formula yields 112266/125.\nFinal answer: $\\frac{112266}{125}$", "final_answer": "\\frac{112266}{125}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cb2501581cd5d687", "problem_id": "24993fdec186daaa", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 48 x + 180$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} + 48 x + 180$ = $3 \\left(x + 6\\right) \\left(x + 10\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $-6$.\nFrom the second factor: x = $-10$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-6, -10\\}$", "final_answer": "x \\in \\{-6, -10\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e606d46251106d6f", "problem_id": "24993fdec186daaa", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 48 x + 180$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} + 48 x + 180$ = $3 \\left(x + 6\\right) \\left(x + 10\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-6$.\n5) From the second factor: x = $-10$.\nFinal answer: $x \\in \\{-6, -10\\}$", "final_answer": "x \\in \\{-6, -10\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4e12232d19000564", "problem_id": "2eb2585ba545fa50", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡5 (mod 9), x≡6 (mod 10). Give the solution modulo 90.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 90.\n3) Compute the CRT solution to get x ≡ 86 (mod 90).\nFinal answer: $x\\equiv 86\\pmod{90}$", "final_answer": "x\\equiv 86\\pmod{90}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a0e81074308e1536", "problem_id": "2eb2585ba545fa50", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡5 (mod 9), x≡6 (mod 10). Give the solution modulo 90.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 90.\nCompute the CRT solution to get x ≡ 86 (mod 90).\nCheck: 86 mod 9=5 and mod 10=6.\nFinal answer: $x\\equiv 86\\pmod{90}$", "final_answer": "x\\equiv 86\\pmod{90}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a946f93269e5697a", "problem_id": "ff7d3b83be19fdbb", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Two friends split $2070 in the ratio 9:6. How much does each receive?", "output": "1) Total ratio parts = 9+6 = 15.\n2) Each part = 2070 ÷ 15 = 138.\n3) First share = 9 × 138 = 1242.\n4) Second share = 6 × 138 = 828.\nFinal answer: $828$", "final_answer": "828", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0697058d3265446d", "problem_id": "ff7d3b83be19fdbb", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Two friends split $2070 in the ratio 9:6. How much does each receive?", "output": "Total ratio parts = 9+6 = 15.\nEach part = 2070 ÷ 15 = 138.\nFirst share = 9 × 138 = 1242.\nSecond share = 6 × 138 = 828.\nCheck: 1242+828=2070.\nFinal answer: $828$", "final_answer": "828", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dbafa21d097aae89", "problem_id": "004c19f4ae989621", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A disease has prevalence $\\frac{1}{200}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{217}{4000}$.\n3) Compute P(D|+) = $\\frac{18}{217}$.\nFinal answer: $\\frac{18}{217}$", "final_answer": "\\frac{18}{217}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a8923b96ac63fe0", "problem_id": "004c19f4ae989621", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A disease has prevalence $\\frac{1}{200}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{217}{4000}$.\nCompute P(D|+) = $\\frac{18}{217}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{18}{217}$", "final_answer": "\\frac{18}{217}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "82cb07e2f287f9a8", "problem_id": "f13f6d4c28c4623f", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{5}$. If it is flipped 20 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=20, k=2, p=$\\frac{3}{5}$ and simplify.\n4) Here C(20,2) = 190.\nFinal answer: $\\frac{89653248}{19073486328125}$", "final_answer": "\\frac{89653248}{19073486328125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec9ff43afc123a0b", "problem_id": "3ef96dfabadc66d8", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 2 x^{3} + 5 x^{2} - 6 x + 8$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{3} + 5 x^{2} - 7 x + 8.\n3) Then the limit is 22.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: 21.\nFinal answer: $21$", "final_answer": "21", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "87cbc52984841941", "problem_id": "9b5ba7c7ad3deae7", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_9=41$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_9=32.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=32 and k=9.\nCompute C(40,8) = 76904685.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $76904685$", "final_answer": "76904685", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "035fc65b36d5dce0", "problem_id": "9b5ba7c7ad3deae7", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_9=41$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_9=32.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=32 and k=9.\n4) Compute C(40,8) = 76904685.\nFinal answer: $76904685$", "final_answer": "76904685", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "87909ef9c9abf067", "problem_id": "eb3dc200753b87c5", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 2 x - 2$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $2 x^{2}$ and a = -1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $4 x$.\n5) Evaluate at a: p'(-1) = $-4$.\nFinal answer: $-4$", "final_answer": "-4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "be5d6f14b23f4eb1", "problem_id": "eb3dc200753b87c5", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 2 x - 2$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{2}$ and a = -1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $4 x$.\nEvaluate at a: p'(-1) = $-4$.\nCheck: the identity gives $-4$.\nFinal answer: $-4$", "final_answer": "-4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4579ecd5ac579c9b", "problem_id": "7fa15a90c4f0d122", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$2 y$ = $14$\n$8 x - 11 y$ = $-77$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $8$: $16 y$ = $112$.\n3) Multiply the second equation by $0$: $0$ = $0$.\n4) Subtract to eliminate x: $16 y$ = $112$.\n5) Solve for y: y = $7$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, 7\\right)$", "final_answer": "(x,y)=\\left(0, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6575d09421344d12", "problem_id": "7fa15a90c4f0d122", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$2 y$ = $14$\n$8 x - 11 y$ = $-77$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $8$: $16 y$ = $112$.\nMultiply the second equation by $0$: $0$ = $0$.\nSubtract to eliminate x: $16 y$ = $112$.\nSolve for y: y = $7$.\nSubstitute back to find x: x = $0$.\nCheck: plug (x,y)=(0,7) into both equations.\nFinal answer: $(x,y)=\\left(0, 7\\right)$", "final_answer": "(x,y)=\\left(0, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "866d810b877b41a4", "problem_id": "48f630bab2e01587", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{32 x}{5} - 2 \\sqrt{10} x + 7 + \\frac{14 \\sqrt{10}}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{32}{5} - 2 \\sqrt{10}$, c=$7 + \\frac{14 \\sqrt{10}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{72 \\sqrt{10}}{5} + \\frac{1324}{25}$, so √Δ = $\\sqrt{\\frac{72 \\sqrt{10}}{5} + \\frac{1324}{25}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\sqrt{10} + \\frac{16}{5} + \\frac{\\sqrt{90 \\sqrt{10} + 331}}{5}$.\n7) x2 = $- \\frac{\\sqrt{90 \\sqrt{10} + 331}}{5} + \\sqrt{10} + \\frac{16}{5}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\sqrt{10} + \\frac{16}{5} + \\frac{\\sqrt{90\\cdot \\sqrt{10} + 331}}{5}, - \\frac{\\sqrt{90\\cdot \\sqrt{10} + 331}}{5} + \\sqrt{10} + \\frac{16}{5}\\}$", "final_answer": "x \\in \\{\\sqrt{10} + \\frac{16}{5} + \\frac{\\sqrt{90\\cdot \\sqrt{10} + 331}}{5}, - \\frac{\\sqrt{90\\cdot \\sqrt{10} + 331}}{5} + \\sqrt{10} + \\frac{16}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3fd744d84fd7e0c2", "problem_id": "48f630bab2e01587", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{32 x}{5} - 2 \\sqrt{10} x + 7 + \\frac{14 \\sqrt{10}}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{32}{5} - 2 \\sqrt{10}$, c=$7 + \\frac{14 \\sqrt{10}}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{72 \\sqrt{10}}{5} + \\frac{1324}{25}$, so √Δ = $\\sqrt{\\frac{72 \\sqrt{10}}{5} + \\frac{1324}{25}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\sqrt{10} + \\frac{16}{5} + \\frac{\\sqrt{90 \\sqrt{10} + 331}}{5}$.\nx2 = $- \\frac{\\sqrt{90 \\sqrt{10} + 331}}{5} + \\sqrt{10} + \\frac{16}{5}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\sqrt{10} + \\frac{16}{5} + \\frac{\\sqrt{90\\cdot \\sqrt{10} + 331}}{5}, - \\frac{\\sqrt{90\\cdot \\sqrt{10} + 331}}{5} + \\sqrt{10} + \\frac{16}{5}\\}$", "final_answer": "x \\in \\{\\sqrt{10} + \\frac{16}{5} + \\frac{\\sqrt{90\\cdot \\sqrt{10} + 331}}{5}, - \\frac{\\sqrt{90\\cdot \\sqrt{10} + 331}}{5} + \\sqrt{10} + \\frac{16}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "80d8f8b250d743a1", "problem_id": "460777c35de319f3", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "An urn has 13 red balls and 16 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(29,6).\n3) Favorable samples: choose k reds and n−k blues: C(13,3)·C(16,3).\n4) Compute totals: C(29,6) = 475020, and favorable = 160160.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{88}{261}$", "final_answer": "\\frac{88}{261}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7e1f7caa2a2d16f7", "problem_id": "460777c35de319f3", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 13 red balls and 16 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(29,6).\nFavorable samples: choose k reds and n−k blues: C(13,3)·C(16,3).\nCompute totals: C(29,6) = 475020, and favorable = 160160.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{88}{261}$.\nFinal answer: $\\frac{88}{261}$", "final_answer": "\\frac{88}{261}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8141b97060f97202", "problem_id": "b2edf834eb6ba921", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{3}$. If it is flipped 6 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=6, k=2, p=$\\frac{1}{3}$ and simplify.\n4) Here C(6,2) = 15.\nFinal answer: $\\frac{80}{243}$", "final_answer": "\\frac{80}{243}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2ae54833097afa4a", "problem_id": "b2edf834eb6ba921", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{3}$. If it is flipped 6 times, what is the probability of exactly 2 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=6, k=2, p=$\\frac{1}{3}$ and simplify.\nHere C(6,2) = 15.\nCheck: expression simplifies to $\\frac{80}{243}$.\nFinal answer: $\\frac{80}{243}$", "final_answer": "\\frac{80}{243}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3fc211a0499b5bea", "problem_id": "ebfa26b12415bca7", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "You want 13 liters of a 45% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 13−x liters is 70% solution.\nSalt equation: 20x + 70(13−x) = 45·13.\n(20-70)x = (45-70)·13 ⇒ x = 13/2.\nFinal answer: $13/2 L of 20% and 13/2 L of 70%$", "final_answer": "13/2 L of 20% and 13/2 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "62db9188065511fc", "problem_id": "ebfa26b12415bca7", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "You want 13 liters of a 45% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 13−x liters is 70% solution.\n2) Salt equation: 20x + 70(13−x) = 45·13.\n3) (20-70)x = (45-70)·13 ⇒ x = 13/2.\n4) Then 13−x = 13/2.\nFinal answer: $13/2 L of 20% and 13/2 L of 70%$", "final_answer": "13/2 L of 20% and 13/2 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c8e75d2c956f9ce9", "problem_id": "375df97973ecfade", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Worker A finishes a job in 17 hours and Worker B finishes it in 10 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/17 job/hour. B's rate = 1/10 job/hour.\n2) Combined rate = 1/17 + 1/10 = 27/170 job/hour.\n3) Time = 1 ÷ (combined rate) = 170/27 hours.\nFinal answer: $\\frac{170}{27}$", "final_answer": "\\frac{170}{27}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "beeb9c317e3a5de6", "problem_id": "375df97973ecfade", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Worker A finishes a job in 17 hours and Worker B finishes it in 10 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/17 job/hour. B's rate = 1/10 job/hour.\nCombined rate = 1/17 + 1/10 = 27/170 job/hour.\nTime = 1 ÷ (combined rate) = 170/27 hours.\nCheck: (combined rate)×(time) = 27/170×170/27 = 1.\nFinal answer: $\\frac{170}{27}$", "final_answer": "\\frac{170}{27}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "96afda2ff86e1866", "problem_id": "6125942be49f73c0", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$5 y$ = $-15$\n$- 6 x - 4 y$ = $48$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-6$: $- 30 y$ = $90$.\nMultiply the second equation by $0$: $0$ = $0$.\nSubtract to eliminate x: $- 30 y$ = $90$.\nSolve for y: y = $-3$.\nSubstitute back to find x: x = $-6$.\nCheck: plug (x,y)=(-6,-3) into both equations.\nFinal answer: $(x,y)=\\left(-6, -3\\right)$", "final_answer": "(x,y)=\\left(-6, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1b0d7eb7aab7c9dd", "problem_id": "6125942be49f73c0", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$5 y$ = $-15$\n$- 6 x - 4 y$ = $48$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-6$: $- 30 y$ = $90$.\n3) Multiply the second equation by $0$: $0$ = $0$.\n4) Subtract to eliminate x: $- 30 y$ = $90$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $-6$.\nFinal answer: $(x,y)=\\left(-6, -3\\right)$", "final_answer": "(x,y)=\\left(-6, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c0bdfee9fb5313cc", "problem_id": "76ac84f7d5160554", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 20 red balls and 11 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(31,8).\n3) Favorable samples: choose k reds and n−k blues: C(20,3)·C(11,5).\n4) Compute totals: C(31,8) = 7888725, and favorable = 526680.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{11704}{175305}$", "final_answer": "\\frac{11704}{175305}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "73f435289317c338", "problem_id": "76ac84f7d5160554", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "An urn has 20 red balls and 11 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(31,8).\nFavorable samples: choose k reds and n−k blues: C(20,3)·C(11,5).\nFinal answer: $\\frac{11704}{175305}$", "final_answer": "\\frac{11704}{175305}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "20ff3011b3918eb9", "problem_id": "80c98d8186a287b2", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$12 x$ = $24$\n$- 5 x + 5 y$ = $10$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-5$: $- 60 x$ = $-120$.\n3) Multiply the second equation by $12$: $- 60 x + 60 y$ = $120$.\n4) Subtract to eliminate x: $- 60 y$ = $-240$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $2$.\nFinal answer: $(x,y)=\\left(2, 4\\right)$", "final_answer": "(x,y)=\\left(2, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d848b6d97c96d367", "problem_id": "80c98d8186a287b2", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$12 x$ = $24$\n$- 5 x + 5 y$ = $10$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-5$: $- 60 x$ = $-120$.\nMultiply the second equation by $12$: $- 60 x + 60 y$ = $120$.\nSubtract to eliminate x: $- 60 y$ = $-240$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $2$.\nCheck: plug (x,y)=(2,4) into both equations.\nFinal answer: $(x,y)=\\left(2, 4\\right)$", "final_answer": "(x,y)=\\left(2, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ab78496102b8dd5e", "problem_id": "0bbfa95e7c773601", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Two friends split $742 in the ratio 5:9. How much does each receive?", "output": "Total ratio parts = 5+9 = 14.\nEach part = 742 ÷ 14 = 53.\nFirst share = 5 × 53 = 265.\nSecond share = 9 × 53 = 477.\nCheck: 265+477=742.\nFinal answer: $477$", "final_answer": "477", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f4d866c279c906e5", "problem_id": "0bbfa95e7c773601", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $742 in the ratio 5:9. How much does each receive?", "output": "1) Total ratio parts = 5+9 = 14.\n2) Each part = 742 ÷ 14 = 53.\n3) First share = 5 × 53 = 265.\n4) Second share = 9 × 53 = 477.\nFinal answer: $477$", "final_answer": "477", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c1d0c353f511ab74", "problem_id": "d1bb25cbbacd6e51", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{3 x}{2} - \\frac{9}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$- \\frac{3}{2}$, c=$- \\frac{9}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{81}{4}$, so √Δ = $\\frac{9}{2}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{3}{2}$.\n7) x2 = $- \\frac{3}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{3}{2}, - \\frac{3}{4}\\}$", "final_answer": "x \\in \\{\\frac{3}{2}, - \\frac{3}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b538152179830bb8", "problem_id": "d1bb25cbbacd6e51", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{3 x}{2} - \\frac{9}{4}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$- \\frac{3}{2}$, c=$- \\frac{9}{4}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{81}{4}$, so √Δ = $\\frac{9}{2}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{3}{2}$.\nx2 = $- \\frac{3}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{3}{2}, - \\frac{3}{4}\\}$", "final_answer": "x \\in \\{\\frac{3}{2}, - \\frac{3}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "775b7b564c6a67d0", "problem_id": "56ad6eb37304924e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 3 \\sin{\\left(x \\right)}) = - 3 \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 3 \\sin{\\left(x \\right)}) = - 3 \\cos{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$", "final_answer": "- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "bc1534b1ec92b3aa", "problem_id": "143396637e61f045", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A store sells a backpack for $306. It is discounted by 30%. You buy 11 backpacks. What is the total cost?", "output": "Discounted price = 306 × (1 − 30/100) = 306 × 70/100 = 1071/5.\nSubtotal = 11 × 1071/5 = 11781/5.\nCheck: recomputing the formula yields 11781/5.\nFinal answer: $\\frac{11781}{5}$", "final_answer": "\\frac{11781}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e2b4ade50eb3f32a", "problem_id": "4130f56657689103", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(2 x^{2} - 3 x - 3\\right) \\cos{\\left(3 x + 3 \\right)}$.", "output": "1) Let f(x) = $\\left(x - 3\\right) e^{x} + \\left(2 x^{2} - 3 x - 3\\right) \\cos{\\left(3 x + 3 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\left(x - 3\\right) e^{x} + \\left(4 x - 3\\right) \\cos{\\left(3 x + 3 \\right)} - 3 \\left(2 x^{2} - 3 x - 3\\right) \\sin{\\left(3 x + 3 \\right)} + e^{x}$.\nFinal answer: $\\left(x - 3\\right)\\cdot e^{x} + \\left(4\\cdot x - 3\\right)\\cdot \\cos{\\left(3\\cdot x + 3 \\right)} - \\left(6\\cdot x^{2} - 9\\cdot x - 9\\right)\\cdot \\sin{\\left(3\\cdot x + 3 \\right)} + e^{x}$", "final_answer": "\\left(x - 3\\right)\\cdot e^{x} + \\left(4\\cdot x - 3\\right)\\cdot \\cos{\\left(3\\cdot x + 3 \\right)} - \\left(6\\cdot x^{2} - 9\\cdot x - 9\\right)\\cdot \\sin{\\left(3\\cdot x + 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff835da143c9280d", "problem_id": "4130f56657689103", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(2 x^{2} - 3 x - 3\\right) \\cos{\\left(3 x + 3 \\right)}$.", "output": "Let f(x) = $\\left(x - 3\\right) e^{x} + \\left(2 x^{2} - 3 x - 3\\right) \\cos{\\left(3 x + 3 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\left(x - 3\\right) e^{x} + \\left(4 x - 3\\right) \\cos{\\left(3 x + 3 \\right)} - 3 \\left(2 x^{2} - 3 x - 3\\right) \\sin{\\left(3 x + 3 \\right)} + e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\left(x - 3\\right)\\cdot e^{x} + \\left(4\\cdot x - 3\\right)\\cdot \\cos{\\left(3\\cdot x + 3 \\right)} - \\left(6\\cdot x^{2} - 9\\cdot x - 9\\right)\\cdot \\sin{\\left(3\\cdot x + 3 \\right)} + e^{x}$", "final_answer": "\\left(x - 3\\right)\\cdot e^{x} + \\left(4\\cdot x - 3\\right)\\cdot \\cos{\\left(3\\cdot x + 3 \\right)} - \\left(6\\cdot x^{2} - 9\\cdot x - 9\\right)\\cdot \\sin{\\left(3\\cdot x + 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f844b8ad20fb69df", "problem_id": "6b4aaabc2cf40ec2", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 3 x + 2$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} + 3 x + 2$ = $\\left(x + 1\\right) \\left(x + 2\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-1$.\n5) From the second factor: x = $-2$.\nFinal answer: $x \\in \\{-1, -2\\}$", "final_answer": "x \\in \\{-1, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "610e591a5a6f3c50", "problem_id": "6b4aaabc2cf40ec2", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 3 x + 2$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} + 3 x + 2$ = $\\left(x + 1\\right) \\left(x + 2\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $-1$.\nFrom the second factor: x = $-2$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-1, -2\\}$", "final_answer": "x \\in \\{-1, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6b1c77fdf1bd32b7", "problem_id": "feef0ecab6329ded", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 2 \\sqrt{10} x - 5 x + 6 + 4 \\sqrt{10}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- 2 \\sqrt{10} - 5$, c=$6 + 4 \\sqrt{10}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $4 \\sqrt{10} + 41$, so √Δ = $\\sqrt{4 \\sqrt{10} + 41}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{5}{2} + \\sqrt{10} + \\frac{\\sqrt{4 \\sqrt{10} + 41}}{2}$.\n7) x2 = $- \\frac{\\sqrt{4 \\sqrt{10} + 41}}{2} + \\frac{5}{2} + \\sqrt{10}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{5}{2} + \\sqrt{10} + \\frac{\\sqrt{4\\cdot \\sqrt{10} + 41}}{2}, - \\frac{\\sqrt{4\\cdot \\sqrt{10} + 41}}{2} + \\frac{5}{2} + \\sqrt{10}\\}$", "final_answer": "x \\in \\{\\frac{5}{2} + \\sqrt{10} + \\frac{\\sqrt{4\\cdot \\sqrt{10} + 41}}{2}, - \\frac{\\sqrt{4\\cdot \\sqrt{10} + 41}}{2} + \\frac{5}{2} + \\sqrt{10}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0b8cc34b3368a8ca", "problem_id": "feef0ecab6329ded", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 2 \\sqrt{10} x - 5 x + 6 + 4 \\sqrt{10}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- 2 \\sqrt{10} - 5$, c=$6 + 4 \\sqrt{10}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $4 \\sqrt{10} + 41$, so √Δ = $\\sqrt{4 \\sqrt{10} + 41}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{5}{2} + \\sqrt{10} + \\frac{\\sqrt{4 \\sqrt{10} + 41}}{2}$.\nx2 = $- \\frac{\\sqrt{4 \\sqrt{10} + 41}}{2} + \\frac{5}{2} + \\sqrt{10}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{5}{2} + \\sqrt{10} + \\frac{\\sqrt{4\\cdot \\sqrt{10} + 41}}{2}, - \\frac{\\sqrt{4\\cdot \\sqrt{10} + 41}}{2} + \\frac{5}{2} + \\sqrt{10}\\}$", "final_answer": "x \\in \\{\\frac{5}{2} + \\sqrt{10} + \\frac{\\sqrt{4\\cdot \\sqrt{10} + 41}}{2}, - \\frac{\\sqrt{4\\cdot \\sqrt{10} + 41}}{2} + \\frac{5}{2} + \\sqrt{10}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "df202bb723a04fcc", "problem_id": "3787fad3e50f603c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} + 4$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(3 x^{2}) = 6 x^{2}$.\n3) So $f'(x) = 6 x^{2}$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(3 x^{2}) = 6 x$.\nTherefore the correct derivative is $f'(x) = 6 x$.\nFinal answer: $6 x$", "final_answer": "6 x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8d93cb5d5cfb8952", "problem_id": "8f915b3d6de19c87", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the linear system:\n$2 x + 8 y$ = $24$\n$- 3 x - 3 y$ = $0$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $- 6 x - 24 y$ = $-72$.\nMultiply the second equation by $2$: $- 6 x - 6 y$ = $0$.\nFinal answer: $(x,y)=\\left(-4, 4\\right)$", "final_answer": "(x,y)=\\left(-4, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9775d819bcf1e81c", "problem_id": "866bb530141ccf10", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{2 \\sqrt{3} x}{3} - \\frac{14 x}{15} + \\frac{1}{5} + \\frac{2 \\sqrt{3}}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{2 \\sqrt{3}}{3} - \\frac{14}{15}$, c=$\\frac{1}{5} + \\frac{2 \\sqrt{3}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{316}{225} - \\frac{16 \\sqrt{3}}{45}$, so √Δ = $\\sqrt{\\frac{316}{225} - \\frac{16 \\sqrt{3}}{45}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{79 - 20 \\sqrt{3}}}{15} + \\frac{7}{15} + \\frac{\\sqrt{3}}{3}$.\n7) x2 = $- \\frac{\\sqrt{79 - 20 \\sqrt{3}}}{15} + \\frac{7}{15} + \\frac{\\sqrt{3}}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{79 - 20\\cdot \\sqrt{3}}}{15} + \\frac{7}{15} + \\frac{\\sqrt{3}}{3}, - \\frac{\\sqrt{79 - 20\\cdot \\sqrt{3}}}{15} + \\frac{7}{15} + \\frac{\\sqrt{3}}{3}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{79 - 20\\cdot \\sqrt{3}}}{15} + \\frac{7}{15} + \\frac{\\sqrt{3}}{3}, - \\frac{\\sqrt{79 - 20\\cdot \\sqrt{3}}}{15} + \\frac{7}{15} + \\frac{\\sqrt{3}}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c03c4b39701ccfb6", "problem_id": "866bb530141ccf10", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{2 \\sqrt{3} x}{3} - \\frac{14 x}{15} + \\frac{1}{5} + \\frac{2 \\sqrt{3}}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{2 \\sqrt{3}}{3} - \\frac{14}{15}$, c=$\\frac{1}{5} + \\frac{2 \\sqrt{3}}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{316}{225} - \\frac{16 \\sqrt{3}}{45}$, so √Δ = $\\sqrt{\\frac{316}{225} - \\frac{16 \\sqrt{3}}{45}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{79 - 20 \\sqrt{3}}}{15} + \\frac{7}{15} + \\frac{\\sqrt{3}}{3}$.\nx2 = $- \\frac{\\sqrt{79 - 20 \\sqrt{3}}}{15} + \\frac{7}{15} + \\frac{\\sqrt{3}}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{79 - 20\\cdot \\sqrt{3}}}{15} + \\frac{7}{15} + \\frac{\\sqrt{3}}{3}, - \\frac{\\sqrt{79 - 20\\cdot \\sqrt{3}}}{15} + \\frac{7}{15} + \\frac{\\sqrt{3}}{3}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{79 - 20\\cdot \\sqrt{3}}}{15} + \\frac{7}{15} + \\frac{\\sqrt{3}}{3}, - \\frac{\\sqrt{79 - 20\\cdot \\sqrt{3}}}{15} + \\frac{7}{15} + \\frac{\\sqrt{3}}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4c79db24a4647ed3", "problem_id": "c2beab8a129c5d2e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} 3 x + 9$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to 2 x + 9.\n3) Then the limit is 15.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: 18.\nFinal answer: $18$", "final_answer": "18", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a945592a16396ade", "problem_id": "a2be8e0723a41315", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A store sells a backpack for $875. It is discounted by 5%. You buy 8 backpacks. What is the total cost?", "output": "1) Discounted price = 875 × (1 − 5/100) = 875 × 95/100 = 3325/4.\n2) Subtotal = 8 × 3325/4 = 6650.\nFinal answer: $6650$", "final_answer": "6650", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a70568a4c086eba", "problem_id": "aae1fb5eaa1aecc6", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} - 3 x^{2} + 11 x - 41$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} + 10 x - 41.\n3) Then the limit is -129.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: -133.\nFinal answer: $-133$", "final_answer": "-133", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "969d80cbe9144377", "problem_id": "bbf7a88755e47d62", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Worker A finishes a job in 6 hours and Worker B finishes it in 19 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/6 job/hour. B's rate = 1/19 job/hour.\n2) Combined rate = 1/6 + 1/19 = 25/114 job/hour.\n3) Time = 1 ÷ (combined rate) = 114/25 hours.\nFinal answer: $\\frac{114}{25}$", "final_answer": "\\frac{114}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "487b576ac067c2ff", "problem_id": "a00373c18df48976", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 x^{2} \\sin{\\left(x \\right)}) = 3 x^{2} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = 3 x^{2} \\cos{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 x^{2} \\sin{\\left(x \\right)}) = 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 3 \\left(x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)}\\right)$.\nFinal answer: $3 \\left(x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)}\\right)$", "final_answer": "3 \\left(x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "93102c7cda6ebbf9", "problem_id": "395a9ddecd8bc9bc", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 18 x - 48$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} + 18 x - 48$ = $3 \\left(x - 2\\right) \\left(x + 8\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $2$.\n5) From the second factor: x = $-8$.\nFinal answer: $x \\in \\{2, -8\\}$", "final_answer": "x \\in \\{2, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "47d3974499e5a484", "problem_id": "b604093e35b40bb6", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 8 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=8, k=5, p=$\\frac{1}{5}$ and simplify.\nFinal answer: $\\frac{3584}{390625}$", "final_answer": "\\frac{3584}{390625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "24e15d94957cb454", "problem_id": "b604093e35b40bb6", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 8 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=8, k=5, p=$\\frac{1}{5}$ and simplify.\n4) Here C(8,5) = 56.\nFinal answer: $\\frac{3584}{390625}$", "final_answer": "\\frac{3584}{390625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "576eccf0a5fe947d", "problem_id": "e86ce9884e9558b5", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the equation of the tangent line to $y=- x^{2} - 3 x + 2 \\sin{\\left(x \\right)} + 1$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 2 x + 2 \\cos{\\left(x \\right)} - 3$.\n2) Slope at x=2: m = f'(2) = $-7 + 2 \\cos{\\left(2 \\right)}$.\n3) Point on curve: (2, f(2)) = (2, $-9 + 2 \\sin{\\left(2 \\right)}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -9 + 2 \\sin{\\left(2 \\right)} = -7 + 2 \\cos{\\left(2 \\right)}(x − 2).\n6) Optionally expand: y = $- 7 x + 2 x \\cos{\\left(2 \\right)} - 4 \\cos{\\left(2 \\right)} + 2 \\sin{\\left(2 \\right)} + 5$.\nFinal answer: $y = - 7\\cdot x + 2\\cdot x\\cdot \\cos{\\left(2 \\right)} - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)} + 5$", "final_answer": "y = - 7\\cdot x + 2\\cdot x\\cdot \\cos{\\left(2 \\right)} - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)} + 5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9a54fb081e4c4369", "problem_id": "4dd4cc37b96485d5", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} + x - 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(2 x^{2}) = 4 x^{2}$.\n3) So $f'(x) = 4 x^{2} + 1$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(2 x^{2}) = 4 x$.\nTherefore the correct derivative is $f'(x) = 4 x + 1$.\nFinal answer: $4 x + 1$", "final_answer": "4 x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "355ff55b77dd55a8", "problem_id": "f87a945a9dd16e82", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$3 x - 12 y$ = $-27$\n$- 12 x - 4 y$ = $-100$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-12$: $- 36 x + 144 y$ = $324$.\n3) Multiply the second equation by $3$: $- 36 x - 12 y$ = $-300$.\n4) Subtract to eliminate x: $156 y$ = $624$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $7$.\nFinal answer: $(x,y)=\\left(7, 4\\right)$", "final_answer": "(x,y)=\\left(7, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a9a6788978b01f34", "problem_id": "f87a945a9dd16e82", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$3 x - 12 y$ = $-27$\n$- 12 x - 4 y$ = $-100$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-12$: $- 36 x + 144 y$ = $324$.\nMultiply the second equation by $3$: $- 36 x - 12 y$ = $-300$.\nSubtract to eliminate x: $156 y$ = $624$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $7$.\nCheck: plug (x,y)=(7,4) into both equations.\nFinal answer: $(x,y)=\\left(7, 4\\right)$", "final_answer": "(x,y)=\\left(7, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff1ee417e58fa584", "problem_id": "09900a8fe01de90d", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{2} - 4 x + 2\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 3 x^{2} - 4 x + 2$.\n3) Compute an antiderivative: F(x) = $- x^{3} - 2 x^{2} + 2 x$.\n4) Include the constant of integration C.\nFinal answer: $C - x^{3} - 2\\cdot x^{2} + 2\\cdot x$", "final_answer": "C - x^{3} - 2\\cdot x^{2} + 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d60dfec2e2b12d4", "problem_id": "4e5e51f53bf24457", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} 3 x^{3} + 10 x^{2} + 39 x + 153$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} + 10 x^{2} + 38 x + 153.\n3) Then the limit is 657.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: 661.\nFinal answer: $661$", "final_answer": "661", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "da2e30d3280cf795", "problem_id": "c5de6fe90af158c0", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 20 red balls and 23 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(43,9).\nFavorable samples: choose k reds and n−k blues: C(20,1)·C(23,8).\nCompute totals: C(43,9) = 563921995, and favorable = 9806280.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{103224}{5936021}$.\nFinal answer: $\\frac{103224}{5936021}$", "final_answer": "\\frac{103224}{5936021}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b898122276f724bf", "problem_id": "c5de6fe90af158c0", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 20 red balls and 23 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(43,9).\n3) Favorable samples: choose k reds and n−k blues: C(20,1)·C(23,8).\n4) Compute totals: C(43,9) = 563921995, and favorable = 9806280.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{103224}{5936021}$", "final_answer": "\\frac{103224}{5936021}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ad270c1621b2d434", "problem_id": "98d7a16d2251441d", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the congruences: x≡8 (mod 9), x≡10 (mod 13). Give the solution modulo 117.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 117.\nCompute the CRT solution to get x ≡ 62 (mod 117).\nFinal answer: $x\\equiv 62\\pmod{117}$", "final_answer": "x\\equiv 62\\pmod{117}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "005e4bc069ca28bc", "problem_id": "98d7a16d2251441d", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡8 (mod 9), x≡10 (mod 13). Give the solution modulo 117.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 117.\n3) Compute the CRT solution to get x ≡ 62 (mod 117).\nFinal answer: $x\\equiv 62\\pmod{117}$", "final_answer": "x\\equiv 62\\pmod{117}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a8cca2f84d30c55a", "problem_id": "35c878fe729aa815", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $167. It is discounted by 25%. You buy 14 backpacks. What is the total cost?", "output": "Discounted price = 167 × (1 − 25/100) = 167 × 75/100 = 501/4.\nSubtotal = 14 × 501/4 = 3507/2.\nCheck: recomputing the formula yields 3507/2.\nFinal answer: $\\frac{3507}{2}$", "final_answer": "\\frac{3507}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e89e2d311f043f3f", "problem_id": "35c878fe729aa815", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A store sells a backpack for $167. It is discounted by 25%. You buy 14 backpacks. What is the total cost?", "output": "1) Discounted price = 167 × (1 − 25/100) = 167 × 75/100 = 501/4.\n2) Subtotal = 14 × 501/4 = 3507/2.\nFinal answer: $\\frac{3507}{2}$", "final_answer": "\\frac{3507}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eafbbb10b927af5b", "problem_id": "b375721b77ebe12f", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A disease has prevalence $\\frac{2}{25}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{61}{500}$.\n3) Compute P(D|+) = $\\frac{38}{61}$.\nFinal answer: $\\frac{38}{61}$", "final_answer": "\\frac{38}{61}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e5824afb2c8cd8ed", "problem_id": "b375721b77ebe12f", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A disease has prevalence $\\frac{2}{25}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{61}{500}$.\nCompute P(D|+) = $\\frac{38}{61}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{38}{61}$", "final_answer": "\\frac{38}{61}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a7b4f82404e82ae1", "problem_id": "7827922cb5c9ded0", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A store sells a backpack for $48. It is discounted by 10%. You buy 6 backpacks. What is the total cost?", "output": "1) Discounted price = 48 × (1 − 10/100) = 48 × 90/100 = 216/5.\n2) Subtotal = 6 × 216/5 = 1296/5.\nFinal answer: $\\frac{1296}{5}$", "final_answer": "\\frac{1296}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b3eba6023a3db04a", "problem_id": "7827922cb5c9ded0", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A store sells a backpack for $48. It is discounted by 10%. You buy 6 backpacks. What is the total cost?", "output": "Discounted price = 48 × (1 − 10/100) = 48 × 90/100 = 216/5.\nSubtotal = 6 × 216/5 = 1296/5.\nFinal answer: $\\frac{1296}{5}$", "final_answer": "\\frac{1296}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f3fbbe0d67f55169", "problem_id": "95fc71173b41f3ee", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 2 \\sqrt{2} x + 5 x - 5 \\sqrt{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$5 - 2 \\sqrt{2}$, c=$- 5 \\sqrt{2}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $20 \\sqrt{2} + 33$, so √Δ = $\\sqrt{20 \\sqrt{2} + 33}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{5}{4} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{20 \\sqrt{2} + 33}}{4}$.\nx2 = $- \\frac{\\sqrt{20 \\sqrt{2} + 33}}{4} - \\frac{5}{4} + \\frac{\\sqrt{2}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{5}{4} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{20\\cdot \\sqrt{2} + 33}}{4}, - \\frac{\\sqrt{20\\cdot \\sqrt{2} + 33}}{4} - \\frac{5}{4} + \\frac{\\sqrt{2}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{5}{4} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{20\\cdot \\sqrt{2} + 33}}{4}, - \\frac{\\sqrt{20\\cdot \\sqrt{2} + 33}}{4} - \\frac{5}{4} + \\frac{\\sqrt{2}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a78fe802a0a96871", "problem_id": "95fc71173b41f3ee", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 2 \\sqrt{2} x + 5 x - 5 \\sqrt{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$5 - 2 \\sqrt{2}$, c=$- 5 \\sqrt{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $20 \\sqrt{2} + 33$, so √Δ = $\\sqrt{20 \\sqrt{2} + 33}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{5}{4} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{20 \\sqrt{2} + 33}}{4}$.\n7) x2 = $- \\frac{\\sqrt{20 \\sqrt{2} + 33}}{4} - \\frac{5}{4} + \\frac{\\sqrt{2}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{5}{4} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{20\\cdot \\sqrt{2} + 33}}{4}, - \\frac{\\sqrt{20\\cdot \\sqrt{2} + 33}}{4} - \\frac{5}{4} + \\frac{\\sqrt{2}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{5}{4} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{20\\cdot \\sqrt{2} + 33}}{4}, - \\frac{\\sqrt{20\\cdot \\sqrt{2} + 33}}{4} - \\frac{5}{4} + \\frac{\\sqrt{2}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0ef373ca342c069a", "problem_id": "0d972dd91842d810", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "You want 29 liters of a 30% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 29−x liters is 70% solution.\nSalt equation: 25x + 70(29−x) = 30·29.\n(25-70)x = (30-70)·29 ⇒ x = 232/9.\nThen 29−x = 29/9.\nCheck: 25·232/9 + 70·29/9 = 30·29.\nFinal answer: $232/9 L of 25% and 29/9 L of 70%$", "final_answer": "232/9 L of 25% and 29/9 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "29d5ba833a015cc2", "problem_id": "0d972dd91842d810", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "You want 29 liters of a 30% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 29−x liters is 70% solution.\n2) Salt equation: 25x + 70(29−x) = 30·29.\n3) (25-70)x = (30-70)·29 ⇒ x = 232/9.\n4) Then 29−x = 29/9.\nFinal answer: $232/9 L of 25% and 29/9 L of 70%$", "final_answer": "232/9 L of 25% and 29/9 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "45cdc8752da3b5dd", "problem_id": "003bf6b95b72a761", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 19 times, what is the probability of exactly 14 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=19, k=14, p=$\\frac{1}{10}$ and simplify.\nHere C(19,14) = 11628.\nCheck: expression simplifies to $\\frac{171655443}{2500000000000000000}$.\nFinal answer: $\\frac{171655443}{2500000000000000000}$", "final_answer": "\\frac{171655443}{2500000000000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "58aed7d5514a9ed6", "problem_id": "003bf6b95b72a761", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 19 times, what is the probability of exactly 14 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=19, k=14, p=$\\frac{1}{10}$ and simplify.\n4) Here C(19,14) = 11628.\nFinal answer: $\\frac{171655443}{2500000000000000000}$", "final_answer": "\\frac{171655443}{2500000000000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9957ef1c1940709f", "problem_id": "a099b5f27048c043", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$- 2 x + 3 y$ = $-22$\n$- 7 x - 6 y$ = $22$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-7$: $14 x - 21 y$ = $154$.\n3) Multiply the second equation by $-2$: $14 x + 12 y$ = $-44$.\n4) Subtract to eliminate x: $- 33 y$ = $198$.\n5) Solve for y: y = $-6$.\n6) Substitute back to find x: x = $2$.\nFinal answer: $(x,y)=\\left(2, -6\\right)$", "final_answer": "(x,y)=\\left(2, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7e19c892373cdba7", "problem_id": "af7b0f60f2a2007c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{3} - 3 x^{2} + x - 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x) = x$.\n3) So $f'(x) = 12 x^{2} - 5 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x) = 1$.\nTherefore the correct derivative is $f'(x) = 12 x^{2} - 6 x + 1$.\nFinal answer: $12 x^{2} - 6 x + 1$", "final_answer": "12 x^{2} - 6 x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5c1909896ba4c7d8", "problem_id": "28c0b7b14091c3f7", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} 2 x^{4} + 4 x^{3} - x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{4} + 4 x^{3} - x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{9 \\pi}{2} - \\frac{3 \\pi^{3}}{8} - \\frac{3 \\pi^{2}}{4} - 1 + \\frac{15 \\pi^{4}}{16} + \\frac{33 \\pi^{5}}{80}$.\n4) Simplify the result.\nFinal answer: $- \\frac{9\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{3}}{8} - \\frac{3\\cdot \\pi^{2}}{4} - 1 + \\frac{15\\cdot \\pi^{4}}{16} + \\frac{33\\cdot \\pi^{5}}{80}$", "final_answer": "- \\frac{9\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{3}}{8} - \\frac{3\\cdot \\pi^{2}}{4} - 1 + \\frac{15\\cdot \\pi^{4}}{16} + \\frac{33\\cdot \\pi^{5}}{80}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8f6c4cb8ed4285c0", "problem_id": "28c0b7b14091c3f7", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} 2 x^{4} + 4 x^{3} - x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $2 x^{4} + 4 x^{3} - x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\nEvaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{9 \\pi}{2} - \\frac{3 \\pi^{3}}{8} - \\frac{3 \\pi^{2}}{4} - 1 + \\frac{15 \\pi^{4}}{16} + \\frac{33 \\pi^{5}}{80}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{9\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{3}}{8} - \\frac{3\\cdot \\pi^{2}}{4} - 1 + \\frac{15\\cdot \\pi^{4}}{16} + \\frac{33\\cdot \\pi^{5}}{80}$", "final_answer": "- \\frac{9\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{3}}{8} - \\frac{3\\cdot \\pi^{2}}{4} - 1 + \\frac{15\\cdot \\pi^{4}}{16} + \\frac{33\\cdot \\pi^{5}}{80}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1f5eff5306185c7c", "problem_id": "8151f168017c9a34", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} x^{2} + 3 x + 11$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $x^{3} + 2 x - 2$ and a = 3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $29$", "final_answer": "29", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ce7c52cb56fc47f", "problem_id": "8151f168017c9a34", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} x^{2} + 3 x + 11$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{3} + 2 x - 2$ and a = 3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $3 x^{2} + 2$.\n5) Evaluate at a: p'(3) = $29$.\nFinal answer: $29$", "final_answer": "29", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "236566cb96b5bd8b", "problem_id": "047135e7fea93997", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Two friends split $715 in the ratio 7:4. How much does each receive?", "output": "Total ratio parts = 7+4 = 11.\nEach part = 715 ÷ 11 = 65.\nFirst share = 7 × 65 = 455.\nFinal answer: $260$", "final_answer": "260", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8050dbf5a5774774", "problem_id": "047135e7fea93997", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Two friends split $715 in the ratio 7:4. How much does each receive?", "output": "1) Total ratio parts = 7+4 = 11.\n2) Each part = 715 ÷ 11 = 65.\n3) First share = 7 × 65 = 455.\n4) Second share = 4 × 65 = 260.\nFinal answer: $260$", "final_answer": "260", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4ca245c4274ffa2a", "problem_id": "25a88429ed08aac1", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{4} - 2 x^{3} - x^{2} - 2 x - 4\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $2 x^{4} - 2 x^{3} - x^{2} - 2 x - 4$.\nCompute an antiderivative: F(x) = $\\frac{2 x^{5}}{5} - \\frac{x^{4}}{2} - \\frac{x^{3}}{3} - x^{2} - 4 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{2\\cdot x^{5}}{5} - \\frac{x^{4}}{2} - \\frac{x^{3}}{3} - x^{2} - 4\\cdot x$", "final_answer": "C + \\frac{2\\cdot x^{5}}{5} - \\frac{x^{4}}{2} - \\frac{x^{3}}{3} - x^{2} - 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bcb6ffab1e23d7ae", "problem_id": "25a88429ed08aac1", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{4} - 2 x^{3} - x^{2} - 2 x - 4\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $2 x^{4} - 2 x^{3} - x^{2} - 2 x - 4$.\n3) Compute an antiderivative: F(x) = $\\frac{2 x^{5}}{5} - \\frac{x^{4}}{2} - \\frac{x^{3}}{3} - x^{2} - 4 x$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{2\\cdot x^{5}}{5} - \\frac{x^{4}}{2} - \\frac{x^{3}}{3} - x^{2} - 4\\cdot x$", "final_answer": "C + \\frac{2\\cdot x^{5}}{5} - \\frac{x^{4}}{2} - \\frac{x^{3}}{3} - x^{2} - 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7941a858c353e3cb", "problem_id": "cef0f9f3cc7ba762", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡2 (mod 5), x≡0 (mod 8). Give the solution modulo 40.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 40.\nCompute the CRT solution to get x ≡ 32 (mod 40).\nCheck: 32 mod 5=2 and mod 8=0.\nFinal answer: $x\\equiv 32\\pmod{40}$", "final_answer": "x\\equiv 32\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "677c6a9412d7bf49", "problem_id": "cef0f9f3cc7ba762", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡2 (mod 5), x≡0 (mod 8). Give the solution modulo 40.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 40.\n3) Compute the CRT solution to get x ≡ 32 (mod 40).\nFinal answer: $x\\equiv 32\\pmod{40}$", "final_answer": "x\\equiv 32\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dcddaee31c755855", "problem_id": "4b392c0b0630177c", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{7}$. If it is flipped 10 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=4, p=$\\frac{4}{7}$ and simplify.\n4) Here C(10,4) = 210.\nFinal answer: $\\frac{5598720}{40353607}$", "final_answer": "\\frac{5598720}{40353607}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3a80101e731fcbac", "problem_id": "d425f2e063466925", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 2 x^{3} - 3 x^{2} + 8 x - 23$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} - 3 x^{2} + 7 x - 23.\n3) Then the limit is -125.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -128.\nFinal answer: $-128$", "final_answer": "-128", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7b3ec965bafcbb16", "problem_id": "efb4ecd06810cb5f", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A cyclist rides at 54 miles per hour for 1/2 hours, then at 73 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 54 × 1/2 = 27 miles.\n2) Second leg distance = 73 × 2 = 146 miles.\n3) Total distance = 27 + 146 = 173 miles.\nFinal answer: $173$", "final_answer": "173", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "08b2b0e6256fbdd0", "problem_id": "efb4ecd06810cb5f", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "A cyclist rides at 54 miles per hour for 1/2 hours, then at 73 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 54 × 1/2 = 27 miles.\nSecond leg distance = 73 × 2 = 146 miles.\nTotal distance = 27 + 146 = 173 miles.\nFinal answer: $173$", "final_answer": "173", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3edd244d6775897c", "problem_id": "5d281f9f99f3796d", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the multiplicative inverse of 12 modulo 43.", "output": "1) We want x such that 12x ≡ 1 (mod 43).\n2) Use extended Euclid to find s,t with 12s + 43t = 1.\n3) One Bezout identity is: 12·(18) + 43·(-5) = 1.\n4) Reduce s modulo 43: x ≡ 18 (mod 43).\nFinal answer: $x\\equiv 18\\pmod{43}$", "final_answer": "x\\equiv 18\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aaf8774944956639", "problem_id": "5d281f9f99f3796d", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the multiplicative inverse of 12 modulo 43.", "output": "We want x such that 12x ≡ 1 (mod 43).\nUse extended Euclid to find s,t with 12s + 43t = 1.\nOne Bezout identity is: 12·(18) + 43·(-5) = 1.\nReduce s modulo 43: x ≡ 18 (mod 43).\nCheck: 12·18 mod 43 = 1.\nFinal answer: $x\\equiv 18\\pmod{43}$", "final_answer": "x\\equiv 18\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0690f1a139a81aa7", "problem_id": "a6d73985fdff9722", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(2 x^{3} - x^{2} - 3 x - 1\\right) \\cos{\\left(2 x - 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x e^{x}) = x e^{x}$.\n3) So $f'(x) = - 4 x^{3} \\sin{\\left(2 x - 2 \\right)} + 2 x^{2} \\sin{\\left(2 x - 2 \\right)} + 6 x^{2} \\cos{\\left(2 x - 2 \\right)} + x e^{x} + 6 x \\sin{\\left(2 x - 2 \\right)} - 2 x \\cos{\\left(2 x - 2 \\right)} + e^{x} + 2 \\sin{\\left(2 x - 2 \\right)} - 3 \\cos{\\left(2 x - 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x e^{x}) = x e^{x} + e^{x}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 1\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) - 3\\right) - 1\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(6 x^{2} - 2 x - 3\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 1\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) - 3\\right) - 1\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(6 x^{2} - 2 x - 3\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}$", "final_answer": "\\left(x + 1\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) - 3\\right) - 1\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(6 x^{2} - 2 x - 3\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a506ce9be0e0a87f", "problem_id": "3df22199aa30d693", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "An urn has 9 red balls and 13 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(22,6).\n3) Favorable samples: choose k reds and n−k blues: C(9,2)·C(13,4).\n4) Compute totals: C(22,6) = 74613, and favorable = 25740.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{780}{2261}$", "final_answer": "\\frac{780}{2261}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "68fe0c605c168d20", "problem_id": "4154b2535c601c01", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 6 x - 8$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} + 6 x - 8$ = $2 \\left(x - 1\\right) \\left(x + 4\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $1$.\n5) From the second factor: x = $-4$.\nFinal answer: $x \\in \\{1, -4\\}$", "final_answer": "x \\in \\{1, -4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "571d5e1fb5ddc19b", "problem_id": "4154b2535c601c01", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 6 x - 8$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} + 6 x - 8$ = $2 \\left(x - 1\\right) \\left(x + 4\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $1$.\nFrom the second factor: x = $-4$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{1, -4\\}$", "final_answer": "x \\in \\{1, -4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f7c1a9705a57222b", "problem_id": "206b3e3d9dc7b849", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Two friends split $444 in the ratio 6:6. How much does each receive?", "output": "1) Total ratio parts = 6+6 = 12.\n2) Each part = 444 ÷ 12 = 37.\n3) First share = 6 × 37 = 222.\n4) Second share = 6 × 37 = 222.\nFinal answer: $222$", "final_answer": "222", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "453f622a9105df8e", "problem_id": "206b3e3d9dc7b849", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Two friends split $444 in the ratio 6:6. How much does each receive?", "output": "Total ratio parts = 6+6 = 12.\nEach part = 444 ÷ 12 = 37.\nFirst share = 6 × 37 = 222.\nFinal answer: $222$", "final_answer": "222", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cb4941fd2c602980", "problem_id": "33c5160dd60b3dbc", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡3 (mod 9), x≡3 (mod 7). Give the solution modulo 63.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 63.\n3) Compute the CRT solution to get x ≡ 3 (mod 63).\nFinal answer: $x\\equiv 3\\pmod{63}$", "final_answer": "x\\equiv 3\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9a0708b13215da01", "problem_id": "33c5160dd60b3dbc", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡3 (mod 9), x≡3 (mod 7). Give the solution modulo 63.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 63.\nCompute the CRT solution to get x ≡ 3 (mod 63).\nCheck: 3 mod 9=3 and mod 7=3.\nFinal answer: $x\\equiv 3\\pmod{63}$", "final_answer": "x\\equiv 3\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8b5863e05cfce646", "problem_id": "9b30ab9314867c74", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "A store sells a backpack for $75. It is discounted by 5%. You buy 7 backpacks. What is the total cost?", "output": "Discounted price = 75 × (1 − 5/100) = 75 × 95/100 = 285/4.\nSubtotal = 7 × 285/4 = 1995/4.\nFinal answer: $\\frac{1995}{4}$", "final_answer": "\\frac{1995}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6f93d1c9fd6e8ee6", "problem_id": "9b30ab9314867c74", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A store sells a backpack for $75. It is discounted by 5%. You buy 7 backpacks. What is the total cost?", "output": "1) Discounted price = 75 × (1 − 5/100) = 75 × 95/100 = 285/4.\n2) Subtotal = 7 × 285/4 = 1995/4.\nFinal answer: $\\frac{1995}{4}$", "final_answer": "\\frac{1995}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ba3c52af624a20cd", "problem_id": "3705c406fdd716e7", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A store sells a backpack for $116. It is discounted by 5%. You buy 13 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 116 × (1 − 5/100) = 116 × 95/100 = 551/5.\nSubtotal = 13 × 551/5 = 7163/5.\nTotal with tax = 7163/5 × 105/100 = 150423/100.\nCheck: recomputing the formula yields 150423/100.\nFinal answer: $\\frac{150423}{100}$", "final_answer": "\\frac{150423}{100}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5ba4b55c1814398a", "problem_id": "3705c406fdd716e7", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A store sells a backpack for $116. It is discounted by 5%. You buy 13 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 116 × (1 − 5/100) = 116 × 95/100 = 551/5.\n2) Subtotal = 13 × 551/5 = 7163/5.\n3) Total with tax = 7163/5 × 105/100 = 150423/100.\nFinal answer: $\\frac{150423}{100}$", "final_answer": "\\frac{150423}{100}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0943bcd1a71aa46f", "problem_id": "cad7394de25e46e7", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - x^{2} - x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- x^{2} - x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1$.\nCompute an antiderivative: F(x) = $- \\frac{x^{3}}{3} - \\frac{x^{2}}{2} - x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $C - \\frac{x^{3}}{3} - \\frac{x^{2}}{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{x^{3}}{3} - \\frac{x^{2}}{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d729f34efce1d16", "problem_id": "cad7394de25e46e7", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - x^{2} - x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- x^{2} - x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1$.\n3) Compute an antiderivative: F(x) = $- \\frac{x^{3}}{3} - \\frac{x^{2}}{2} - x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{x^{3}}{3} - \\frac{x^{2}}{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{x^{3}}{3} - \\frac{x^{2}}{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1fdec3e72596553a", "problem_id": "e56db95ffb001f01", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$- 4 x - y$ = $4$\n$- 2 x + 2 y$ = $2$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $8 x + 2 y$ = $-8$.\n3) Multiply the second equation by $-4$: $8 x - 8 y$ = $-8$.\n4) Subtract to eliminate x: $10 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, 0\\right)$", "final_answer": "(x,y)=\\left(-1, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "22e31ab7ce7337de", "problem_id": "e56db95ffb001f01", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the linear system:\n$- 4 x - y$ = $4$\n$- 2 x + 2 y$ = $2$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $8 x + 2 y$ = $-8$.\nMultiply the second equation by $-4$: $8 x - 8 y$ = $-8$.\nFinal answer: $(x,y)=\\left(-1, 0\\right)$", "final_answer": "(x,y)=\\left(-1, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fba339bd7806720a", "problem_id": "582e8e84c652ab52", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Worker A finishes a job in 6 hours and Worker B finishes it in 12 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/6 job/hour. B's rate = 1/12 job/hour.\n2) Combined rate = 1/6 + 1/12 = 1/4 job/hour.\n3) Time = 1 ÷ (combined rate) = 4 hours.\nFinal answer: $4$", "final_answer": "4", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d4752f4ce5293c84", "problem_id": "582e8e84c652ab52", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Worker A finishes a job in 6 hours and Worker B finishes it in 12 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/6 job/hour. B's rate = 1/12 job/hour.\nCombined rate = 1/6 + 1/12 = 1/4 job/hour.\nTime = 1 ÷ (combined rate) = 4 hours.\nFinal answer: $4$", "final_answer": "4", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "26332a68029dbb4b", "problem_id": "f4ef6dc7aaaccbad", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=44$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=44 and k=7.\n4) Compute C(50,6) = 15890700.\nFinal answer: $15890700$", "final_answer": "15890700", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c81d1bdbae26a6aa", "problem_id": "4de46a39da1882c5", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(- 2 x^{3} - 3\\right) \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\n3) So $f'(x) = 4 x^{3} \\sin{\\left(2 x \\right)} - 6 x^{2} \\cos{\\left(2 x \\right)} + x e^{x} + 3 e^{x} + 6 \\sin{\\left(2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = - 6 x^{2} \\cos{\\left(2 x \\right)} + \\left(x + 2\\right) e^{x} - 2 \\left(- 2 x^{3} - 3\\right) \\sin{\\left(2 x \\right)} + e^{x}$.\nFinal answer: $- 6 x^{2} \\cos{\\left(2 x \\right)} + \\left(x + 2\\right) e^{x} - 2 \\left(- 2 x^{3} - 3\\right) \\sin{\\left(2 x \\right)} + e^{x}$", "final_answer": "- 6 x^{2} \\cos{\\left(2 x \\right)} + \\left(x + 2\\right) e^{x} - 2 \\left(- 2 x^{3} - 3\\right) \\sin{\\left(2 x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3b5b47df4149df17", "problem_id": "12a9d64bf76d660c", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $660 in the ratio 2:4. How much does each receive?", "output": "Total ratio parts = 2+4 = 6.\nEach part = 660 ÷ 6 = 110.\nFirst share = 2 × 110 = 220.\nSecond share = 4 × 110 = 440.\nCheck: 220+440=660.\nFinal answer: $440$", "final_answer": "440", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "79285d80b317573e", "problem_id": "e9ea8694d5d5651d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - x^{3} + 2 x^{2} - 3 x + 4$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to - x^{3} + 2 x^{2} - 4 x + 4.\n3) Then the limit is 28.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 26.\nFinal answer: $26$", "final_answer": "26", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "393e8bc51e9e5cf1", "problem_id": "e3905de450773855", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} - x - 4$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{2} - 3 x + 1$ and a = 1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 2 x - 3$.\n5) Evaluate at a: p'(1) = $-5$.\nFinal answer: $-5$", "final_answer": "-5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2001c6b6aa7f1ad5", "problem_id": "e3905de450773855", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} - x - 4$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{2} - 3 x + 1$ and a = 1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 2 x - 3$.\nEvaluate at a: p'(1) = $-5$.\nCheck: the identity gives $-5$.\nFinal answer: $-5$", "final_answer": "-5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d932e4677c64b38", "problem_id": "c2df2badce1e5170", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=17$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=17 and k=6.\n4) Compute C(22,5) = 26334.\nFinal answer: $26334$", "final_answer": "26334", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0756bd68209ed269", "problem_id": "c2df2badce1e5170", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=17$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=17 and k=6.\nCompute C(22,5) = 26334.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $26334$", "final_answer": "26334", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b88b9cb5bf1c8714", "problem_id": "ef152cede9686fd3", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 2 x^{4} + 2 x^{3} + 4 x^{2} + 3 x - 2\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 2 x^{4} + 2 x^{3} + 4 x^{2} + 3 x - 2$.\n3) Compute an antiderivative: F(x) = $- \\frac{2 x^{5}}{5} + \\frac{x^{4}}{2} + \\frac{4 x^{3}}{3} + \\frac{3 x^{2}}{2} - 2 x$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{2\\cdot x^{5}}{5} + \\frac{x^{4}}{2} + \\frac{4\\cdot x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} - 2\\cdot x$", "final_answer": "C - \\frac{2\\cdot x^{5}}{5} + \\frac{x^{4}}{2} + \\frac{4\\cdot x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} - 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ce8758065b8e57ab", "problem_id": "ef152cede9686fd3", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 2 x^{4} + 2 x^{3} + 4 x^{2} + 3 x - 2\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 2 x^{4} + 2 x^{3} + 4 x^{2} + 3 x - 2$.\nCompute an antiderivative: F(x) = $- \\frac{2 x^{5}}{5} + \\frac{x^{4}}{2} + \\frac{4 x^{3}}{3} + \\frac{3 x^{2}}{2} - 2 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - \\frac{2\\cdot x^{5}}{5} + \\frac{x^{4}}{2} + \\frac{4\\cdot x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} - 2\\cdot x$", "final_answer": "C - \\frac{2\\cdot x^{5}}{5} + \\frac{x^{4}}{2} + \\frac{4\\cdot x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} - 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d7691cee62f185b1", "problem_id": "f73f1c0320ba34ae", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$7 x + y$ = $-62$\n$- 8 x$ = $72$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-8$: $- 56 x - 8 y$ = $496$.\n3) Multiply the second equation by $7$: $- 56 x$ = $504$.\n4) Subtract to eliminate x: $- 8 y$ = $-8$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $-9$.\nFinal answer: $(x,y)=\\left(-9, 1\\right)$", "final_answer": "(x,y)=\\left(-9, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "238adf181736dc1a", "problem_id": "f73f1c0320ba34ae", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$7 x + y$ = $-62$\n$- 8 x$ = $72$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-8$: $- 56 x - 8 y$ = $496$.\nMultiply the second equation by $7$: $- 56 x$ = $504$.\nSubtract to eliminate x: $- 8 y$ = $-8$.\nSolve for y: y = $1$.\nSubstitute back to find x: x = $-9$.\nCheck: plug (x,y)=(-9,1) into both equations.\nFinal answer: $(x,y)=\\left(-9, 1\\right)$", "final_answer": "(x,y)=\\left(-9, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "af055a4694f7acae", "problem_id": "0513b7ca4956f5c5", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the linear system:\n$- 3 x - 2 y$ = $-21$\n$- 4 x + 7 y$ = $1$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $12 x + 8 y$ = $84$.\nMultiply the second equation by $-3$: $12 x - 21 y$ = $-3$.\nFinal answer: $(x,y)=\\left(5, 3\\right)$", "final_answer": "(x,y)=\\left(5, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9c886ca1d5280908", "problem_id": "0513b7ca4956f5c5", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$- 3 x - 2 y$ = $-21$\n$- 4 x + 7 y$ = $1$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $12 x + 8 y$ = $84$.\n3) Multiply the second equation by $-3$: $12 x - 21 y$ = $-3$.\n4) Subtract to eliminate x: $29 y$ = $87$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, 3\\right)$", "final_answer": "(x,y)=\\left(5, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "46f049597400e7b1", "problem_id": "6c5ea18f59a92a36", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} 3 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $\\frac{\\pi \\left(-12 + \\pi^{2}\\right)}{4}$.\n4) Simplify the result.\nFinal answer: $\\frac{\\pi\\cdot \\left(-12 + \\pi^{2}\\right)}{4}$", "final_answer": "\\frac{\\pi\\cdot \\left(-12 + \\pi^{2}\\right)}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aeace12fc73bb02b", "problem_id": "6c5ea18f59a92a36", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} 3 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $\\frac{\\pi \\left(-12 + \\pi^{2}\\right)}{4}$.\nFinal answer: $\\frac{\\pi\\cdot \\left(-12 + \\pi^{2}\\right)}{4}$", "final_answer": "\\frac{\\pi\\cdot \\left(-12 + \\pi^{2}\\right)}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a308c764c6827fd5", "problem_id": "7a1f0e318e19a3c0", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{4} + x^{3} + 2 x^{2} - 2 x + 3$.", "output": "Differentiate term-by-term:\nd/dx(3) = 0.\nd/dx(x^{3}) = 3 x^{2}.\nd/dx(- 2 x) = -2.\nd/dx(- 2 x^{4}) = - 8 x^{3}.\nd/dx(2 x^{2}) = 4 x.\nCombine: f'(x) = - 8 x^{3} + 3 x^{2} + 4 x - 2.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 8\\cdot x^{3} + 3\\cdot x^{2} + 4\\cdot x - 2$", "final_answer": "- 8\\cdot x^{3} + 3\\cdot x^{2} + 4\\cdot x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aa06726ed52d8c53", "problem_id": "7a1f0e318e19a3c0", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{4} + x^{3} + 2 x^{2} - 2 x + 3$.", "output": "1) Differentiate term-by-term:\n2) d/dx(3) = 0.\n3) d/dx(x^{3}) = 3 x^{2}.\n4) d/dx(- 2 x) = -2.\n5) d/dx(- 2 x^{4}) = - 8 x^{3}.\n6) d/dx(2 x^{2}) = 4 x.\n7) Combine: f'(x) = - 8 x^{3} + 3 x^{2} + 4 x - 2.\nFinal answer: $- 8\\cdot x^{3} + 3\\cdot x^{2} + 4\\cdot x - 2$", "final_answer": "- 8\\cdot x^{3} + 3\\cdot x^{2} + 4\\cdot x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "37644af0d8520a8d", "problem_id": "1659839626b92f95", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 20 x + 50$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 20 x + 50$ = $2 \\left(x - 5\\right)^{2}$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $5$.\n5) From the second factor: x = $5$.\nFinal answer: $x \\in \\{5, 5\\}$", "final_answer": "x \\in \\{5, 5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f2a342bdb3e79f83", "problem_id": "1659839626b92f95", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 20 x + 50$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} - 20 x + 50$ = $2 \\left(x - 5\\right)^{2}$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $5$.\nFrom the second factor: x = $5$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{5, 5\\}$", "final_answer": "x \\in \\{5, 5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0b97c5a70b1421b9", "problem_id": "27ce7730fa31c2c6", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A cyclist rides at 22 miles per hour for 2 hours, then at 71 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 22 × 2 = 44 miles.\nSecond leg distance = 71 × 2 = 142 miles.\nTotal distance = 44 + 142 = 186 miles.\nCheck: adding both legs gives 186.\nFinal answer: $186$", "final_answer": "186", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db7f556d7e735fa9", "problem_id": "27ce7730fa31c2c6", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A cyclist rides at 22 miles per hour for 2 hours, then at 71 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 22 × 2 = 44 miles.\n2) Second leg distance = 71 × 2 = 142 miles.\n3) Total distance = 44 + 142 = 186 miles.\nFinal answer: $186$", "final_answer": "186", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ae1fdd17c194dfbd", "problem_id": "3ab2484b0e2bbffd", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "How many binary strings of length 18 contain exactly 7 ones with no two ones adjacent?", "output": "1) Reserve 6 mandatory zeros between the 7 ones.\n2) This leaves n−k+1 = 12 valid slots for the ones.\n3) Choose the slots: C(12,7) = 792.\nFinal answer: $792$", "final_answer": "792", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ed12678366c21512", "problem_id": "3ab2484b0e2bbffd", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many binary strings of length 18 contain exactly 7 ones with no two ones adjacent?", "output": "Reserve 6 mandatory zeros between the 7 ones.\nThis leaves n−k+1 = 12 valid slots for the ones.\nChoose the slots: C(12,7) = 792.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $792$", "final_answer": "792", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f9205aab24d01430", "problem_id": "da0bfbe0b1319385", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Worker A finishes a job in 12 hours and Worker B finishes it in 8 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/12 job/hour. B's rate = 1/8 job/hour.\n2) Combined rate = 1/12 + 1/8 = 5/24 job/hour.\n3) Time = 1 ÷ (combined rate) = 24/5 hours.\nFinal answer: $\\frac{24}{5}$", "final_answer": "\\frac{24}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1074f14bc64801c7", "problem_id": "da0bfbe0b1319385", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Worker A finishes a job in 12 hours and Worker B finishes it in 8 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/12 job/hour. B's rate = 1/8 job/hour.\nCombined rate = 1/12 + 1/8 = 5/24 job/hour.\nTime = 1 ÷ (combined rate) = 24/5 hours.\nCheck: (combined rate)×(time) = 5/24×24/5 = 1.\nFinal answer: $\\frac{24}{5}$", "final_answer": "\\frac{24}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f9f62607cdde662a", "problem_id": "2c452872b2952eb7", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡3 (mod 9), x≡1 (mod 5). Give the solution modulo 45.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 45.\nCompute the CRT solution to get x ≡ 21 (mod 45).\nCheck: 21 mod 9=3 and mod 5=1.\nFinal answer: $x\\equiv 21\\pmod{45}$", "final_answer": "x\\equiv 21\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1fe991058b967707", "problem_id": "2c452872b2952eb7", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡3 (mod 9), x≡1 (mod 5). Give the solution modulo 45.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 45.\n3) Compute the CRT solution to get x ≡ 21 (mod 45).\nFinal answer: $x\\equiv 21\\pmod{45}$", "final_answer": "x\\equiv 21\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fa1bcad140588183", "problem_id": "8b729437b1e0adfb", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 55 liters of a 45% salt solution. You have a 30% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 55−x liters is 60% solution.\nSalt equation: 30x + 60(55−x) = 45·55.\n(30-60)x = (45-60)·55 ⇒ x = 55/2.\nThen 55−x = 55/2.\nCheck: 30·55/2 + 60·55/2 = 45·55.\nFinal answer: $55/2 L of 30% and 55/2 L of 60%$", "final_answer": "55/2 L of 30% and 55/2 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "292b22f1b24161cb", "problem_id": "8b729437b1e0adfb", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "You want 55 liters of a 45% salt solution. You have a 30% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 55−x liters is 60% solution.\n2) Salt equation: 30x + 60(55−x) = 45·55.\n3) (30-60)x = (45-60)·55 ⇒ x = 55/2.\n4) Then 55−x = 55/2.\nFinal answer: $55/2 L of 30% and 55/2 L of 60%$", "final_answer": "55/2 L of 30% and 55/2 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "97a559ed1ffb69d2", "problem_id": "49120258eaad4316", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{3} x}{2} - 2 x + 1 + \\frac{3 \\sqrt{3}}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{3 \\sqrt{3}}{2} - 2$, c=$1 + \\frac{3 \\sqrt{3}}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{1 + \\frac{3\\cdot \\sqrt{3}}{2}, 1\\}$", "final_answer": "x \\in \\{1 + \\frac{3\\cdot \\sqrt{3}}{2}, 1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ebb4cae82b13c52b", "problem_id": "49120258eaad4316", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{3} x}{2} - 2 x + 1 + \\frac{3 \\sqrt{3}}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{3 \\sqrt{3}}{2} - 2$, c=$1 + \\frac{3 \\sqrt{3}}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{27}{4}$, so √Δ = $\\frac{3 \\sqrt{3}}{2}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $1 + \\frac{3 \\sqrt{3}}{2}$.\n7) x2 = $1$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{1 + \\frac{3\\cdot \\sqrt{3}}{2}, 1\\}$", "final_answer": "x \\in \\{1 + \\frac{3\\cdot \\sqrt{3}}{2}, 1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9298cd8f68c14454", "problem_id": "d478583577f020ce", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 10 x - 12$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 10 x - 12$ = $2 \\left(x - 6\\right) \\left(x + 1\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $6$.\n5) From the second factor: x = $-1$.\nFinal answer: $x \\in \\{6, -1\\}$", "final_answer": "x \\in \\{6, -1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a236e5427c808215", "problem_id": "80f53ffa2a56b3ea", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} - 3 x - 4$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to - 4 x - 4.\n3) Then the limit is -8.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: -7.\nFinal answer: $-7$", "final_answer": "-7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b65bf33217d75b7b", "problem_id": "068acf4ec6bf6899", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "You want 45 liters of a 25% salt solution. You have a 10% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 45−x liters is 40% solution.\nSalt equation: 10x + 40(45−x) = 25·45.\n(10-40)x = (25-40)·45 ⇒ x = 45/2.\nThen 45−x = 45/2.\nCheck: 10·45/2 + 40·45/2 = 25·45.\nFinal answer: $45/2 L of 10% and 45/2 L of 40%$", "final_answer": "45/2 L of 10% and 45/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "24cbde7299456994", "problem_id": "068acf4ec6bf6899", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "You want 45 liters of a 25% salt solution. You have a 10% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 45−x liters is 40% solution.\n2) Salt equation: 10x + 40(45−x) = 25·45.\n3) (10-40)x = (25-40)·45 ⇒ x = 45/2.\n4) Then 45−x = 45/2.\nFinal answer: $45/2 L of 10% and 45/2 L of 40%$", "final_answer": "45/2 L of 10% and 45/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "883dda80a2c73fdb", "problem_id": "0ab94f512a79c484", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 19 times, what is the probability of exactly 16 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=19, k=16, p=$\\frac{1}{10}$ and simplify.\n4) Here C(19,16) = 969.\nFinal answer: $\\frac{706401}{10000000000000000000}$", "final_answer": "\\frac{706401}{10000000000000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eb56f16b9230e1a6", "problem_id": "0ab94f512a79c484", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 19 times, what is the probability of exactly 16 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=19, k=16, p=$\\frac{1}{10}$ and simplify.\nHere C(19,16) = 969.\nCheck: expression simplifies to $\\frac{706401}{10000000000000000000}$.\nFinal answer: $\\frac{706401}{10000000000000000000}$", "final_answer": "\\frac{706401}{10000000000000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5944ba641824257a", "problem_id": "113f5b76c3899d9c", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A cyclist rides at 85 miles per hour for 1 hours, then at 80 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 85 × 1 = 85 miles.\n2) Second leg distance = 80 × 2 = 160 miles.\n3) Total distance = 85 + 160 = 245 miles.\nFinal answer: $245$", "final_answer": "245", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "faea09d068d73d36", "problem_id": "113f5b76c3899d9c", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 85 miles per hour for 1 hours, then at 80 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 85 × 1 = 85 miles.\nSecond leg distance = 80 × 2 = 160 miles.\nTotal distance = 85 + 160 = 245 miles.\nCheck: adding both legs gives 245.\nFinal answer: $245$", "final_answer": "245", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0c217c8a19159712", "problem_id": "0a1be025f503c38a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{2} + x + 3$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{2} + 3.\n3) Then the limit is 11.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 13.\nFinal answer: $13$", "final_answer": "13", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6f91917d162ce021", "problem_id": "d03d27ac9f88215e", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_10=41$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_10=11.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=11 and k=10.\n4) Compute C(20,9) = 167960.\nFinal answer: $167960$", "final_answer": "167960", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f57f07c562bd3ffb", "problem_id": "3ad34152c21e519a", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$3 x - 2 y$ = $-3$\n$- 2 x - 4 y$ = $2$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $- 6 x + 4 y$ = $6$.\n3) Multiply the second equation by $3$: $- 6 x - 12 y$ = $6$.\n4) Subtract to eliminate x: $16 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, 0\\right)$", "final_answer": "(x,y)=\\left(-1, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c350d3747237c3af", "problem_id": "3ad34152c21e519a", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$3 x - 2 y$ = $-3$\n$- 2 x - 4 y$ = $2$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $- 6 x + 4 y$ = $6$.\nMultiply the second equation by $3$: $- 6 x - 12 y$ = $6$.\nSubtract to eliminate x: $16 y$ = $0$.\nSolve for y: y = $0$.\nSubstitute back to find x: x = $-1$.\nCheck: plug (x,y)=(-1,0) into both equations.\nFinal answer: $(x,y)=\\left(-1, 0\\right)$", "final_answer": "(x,y)=\\left(-1, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "79839f57237d098d", "problem_id": "4582debe631a0032", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$x + 2 y$ = $9$\n$- 3 x + y$ = $-13$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $- 3 x - 6 y$ = $-27$.\n3) Multiply the second equation by $1$: $- 3 x + y$ = $-13$.\n4) Subtract to eliminate x: $- 7 y$ = $-14$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, 2\\right)$", "final_answer": "(x,y)=\\left(5, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ccf4a3f16e1567ea", "problem_id": "83a3b0f783dfd312", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 4 x^{3} + 3 x^{2} - 4 x - 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(3 x^{2}) = 6 x^{2}$.\n3) So $f'(x) = - 6 x^{2} - 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(3 x^{2}) = 6 x$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- 6 x^{2} + 3 x - 2\\right)$.\nFinal answer: $2 \\left(- 6 x^{2} + 3 x - 2\\right)$", "final_answer": "2 \\left(- 6 x^{2} + 3 x - 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "4759b40057e1a774", "problem_id": "19d224a19f416658", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 x^{3} - 7 x^{2} + 11 x - 23$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $3 x^{4} - x^{3} - 3 x^{2} - x + 3$ and a = -2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-97$", "final_answer": "-97", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bc21ec079c161bda", "problem_id": "19d224a19f416658", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 x^{3} - 7 x^{2} + 11 x - 23$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{4} - x^{3} - 3 x^{2} - x + 3$ and a = -2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $12 x^{3} - 3 x^{2} - 6 x - 1$.\n5) Evaluate at a: p'(-2) = $-97$.\nFinal answer: $-97$", "final_answer": "-97", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "54cdc77e229d3e32", "problem_id": "a6d73985fdff9722", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(2 x^{3} - x^{2} - 3 x - 1\\right) \\cos{\\left(2 x - 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x^{2} \\cos{\\left(2 x - 2 \\right)}) = x^{2} \\sin{\\left(2 x - 2 \\right)}$.\n3) So $f'(x) = - 4 x^{3} \\sin{\\left(2 x - 2 \\right)} + x^{2} \\sin{\\left(2 x - 2 \\right)} + 6 x^{2} \\cos{\\left(2 x - 2 \\right)} + x e^{x} + 6 x \\sin{\\left(2 x - 2 \\right)} + 2 e^{x} + 2 \\sin{\\left(2 x - 2 \\right)} - 3 \\cos{\\left(2 x - 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x^{2} \\cos{\\left(2 x - 2 \\right)}) = 2 x^{2} \\sin{\\left(2 x - 2 \\right)} - 2 x \\cos{\\left(2 x - 2 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 1\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) - 3\\right) - 1\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(6 x^{2} - 2 x - 3\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 1\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) - 3\\right) - 1\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(6 x^{2} - 2 x - 3\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}$", "final_answer": "\\left(x + 1\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) - 3\\right) - 1\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(6 x^{2} - 2 x - 3\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "abd7d797c342ac05", "problem_id": "cb88888163ab60ca", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - \\frac{9 x}{5} - \\frac{6}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$- \\frac{9}{5}$, c=$- \\frac{6}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{441}{25}$, so √Δ = $\\frac{21}{5}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $1$.\nx2 = $- \\frac{2}{5}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{1, - \\frac{2}{5}\\}$", "final_answer": "x \\in \\{1, - \\frac{2}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4b99e369cddf20f4", "problem_id": "3ef96dfabadc66d8", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 2 x^{3} + 5 x^{2} - 6 x + 8$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 2 x^{4} + 3 x^{3} - x^{2} + 2 x - 1$ and a = -1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 8 x^{3} + 9 x^{2} - 2 x + 2$.\n5) Evaluate at a: p'(-1) = $21$.\nFinal answer: $21$", "final_answer": "21", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "14146e314c15ffb2", "problem_id": "b6919d05747282d0", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 3 x^{2} + 5 x + 11$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{2} + 4 x + 11.\n3) Then the limit is 31.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 33.\nFinal answer: $33$", "final_answer": "33", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f994e39e5dcd5b51", "problem_id": "078645d4a36b6481", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 49$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} - 49$ = $\\left(x - 7\\right) \\left(x + 7\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $7$.\n5) From the second factor: x = $-7$.\nFinal answer: $x \\in \\{7, -7\\}$", "final_answer": "x \\in \\{7, -7\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "036c5bf730707591", "problem_id": "078645d4a36b6481", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 49$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} - 49$ = $\\left(x - 7\\right) \\left(x + 7\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $7$.\nFrom the second factor: x = $-7$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{7, -7\\}$", "final_answer": "x \\in \\{7, -7\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "12107ac2742ac5e9", "problem_id": "0d368b05b7e6b258", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Worker A finishes a job in 16 hours and Worker B finishes it in 11 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/16 job/hour. B's rate = 1/11 job/hour.\nCombined rate = 1/16 + 1/11 = 27/176 job/hour.\nTime = 1 ÷ (combined rate) = 176/27 hours.\nCheck: (combined rate)×(time) = 27/176×176/27 = 1.\nFinal answer: $\\frac{176}{27}$", "final_answer": "\\frac{176}{27}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "05ed858fccce412b", "problem_id": "0d368b05b7e6b258", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Worker A finishes a job in 16 hours and Worker B finishes it in 11 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/16 job/hour. B's rate = 1/11 job/hour.\n2) Combined rate = 1/16 + 1/11 = 27/176 job/hour.\n3) Time = 1 ÷ (combined rate) = 176/27 hours.\nFinal answer: $\\frac{176}{27}$", "final_answer": "\\frac{176}{27}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1296c9c48b2347d1", "problem_id": "0dafb2aff5ef62aa", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Worker A finishes a job in 16 hours and Worker B finishes it in 18 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/16 job/hour. B's rate = 1/18 job/hour.\n2) Combined rate = 1/16 + 1/18 = 17/144 job/hour.\n3) Time = 1 ÷ (combined rate) = 144/17 hours.\nFinal answer: $\\frac{144}{17}$", "final_answer": "\\frac{144}{17}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5e0b0a7dfd2bb57c", "problem_id": "0dafb2aff5ef62aa", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Worker A finishes a job in 16 hours and Worker B finishes it in 18 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/16 job/hour. B's rate = 1/18 job/hour.\nCombined rate = 1/16 + 1/18 = 17/144 job/hour.\nTime = 1 ÷ (combined rate) = 144/17 hours.\nFinal answer: $\\frac{144}{17}$", "final_answer": "\\frac{144}{17}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "05c47558a674bb5a", "problem_id": "005d417f258c82bc", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the linear system:\n$- 3 x - y$ = $1$\n$6 x + 11 y$ = $-11$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $6$: $- 18 x - 6 y$ = $6$.\n3) Multiply the second equation by $-3$: $- 18 x - 33 y$ = $33$.\n4) Subtract to eliminate x: $27 y$ = $-27$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, -1\\right)$", "final_answer": "(x,y)=\\left(0, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bd30bdb0ce9f9a97", "problem_id": "005d417f258c82bc", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- 3 x - y$ = $1$\n$6 x + 11 y$ = $-11$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $6$: $- 18 x - 6 y$ = $6$.\nMultiply the second equation by $-3$: $- 18 x - 33 y$ = $33$.\nSubtract to eliminate x: $27 y$ = $-27$.\nSolve for y: y = $-1$.\nSubstitute back to find x: x = $0$.\nCheck: plug (x,y)=(0,-1) into both equations.\nFinal answer: $(x,y)=\\left(0, -1\\right)$", "final_answer": "(x,y)=\\left(0, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b63efe3804f93be3", "problem_id": "db5c75d96fa02d1d", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{2 \\sqrt{7} x}{3} - \\frac{7 x}{15} - \\frac{4 \\sqrt{7}}{5} - 2$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{2 \\sqrt{7}}{3} - \\frac{7}{15}$, c=$- \\frac{4 \\sqrt{7}}{5} - 2$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{172 \\sqrt{7}}{45} + \\frac{2549}{225}$, so √Δ = $\\sqrt{\\frac{172 \\sqrt{7}}{45} + \\frac{2549}{225}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{7}{30} + \\frac{\\sqrt{7}}{3} + \\frac{\\sqrt{860 \\sqrt{7} + 2549}}{30}$.\n7) x2 = $- \\frac{\\sqrt{860 \\sqrt{7} + 2549}}{30} + \\frac{7}{30} + \\frac{\\sqrt{7}}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{7}{30} + \\frac{\\sqrt{7}}{3} + \\frac{\\sqrt{860\\cdot \\sqrt{7} + 2549}}{30}, - \\frac{\\sqrt{860\\cdot \\sqrt{7} + 2549}}{30} + \\frac{7}{30} + \\frac{\\sqrt{7}}{3}\\}$", "final_answer": "x \\in \\{\\frac{7}{30} + \\frac{\\sqrt{7}}{3} + \\frac{\\sqrt{860\\cdot \\sqrt{7} + 2549}}{30}, - \\frac{\\sqrt{860\\cdot \\sqrt{7} + 2549}}{30} + \\frac{7}{30} + \\frac{\\sqrt{7}}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7044b2acac857b12", "problem_id": "db5c75d96fa02d1d", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{2 \\sqrt{7} x}{3} - \\frac{7 x}{15} - \\frac{4 \\sqrt{7}}{5} - 2$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{2 \\sqrt{7}}{3} - \\frac{7}{15}$, c=$- \\frac{4 \\sqrt{7}}{5} - 2$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{172 \\sqrt{7}}{45} + \\frac{2549}{225}$, so √Δ = $\\sqrt{\\frac{172 \\sqrt{7}}{45} + \\frac{2549}{225}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{7}{30} + \\frac{\\sqrt{7}}{3} + \\frac{\\sqrt{860 \\sqrt{7} + 2549}}{30}$.\nx2 = $- \\frac{\\sqrt{860 \\sqrt{7} + 2549}}{30} + \\frac{7}{30} + \\frac{\\sqrt{7}}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{7}{30} + \\frac{\\sqrt{7}}{3} + \\frac{\\sqrt{860\\cdot \\sqrt{7} + 2549}}{30}, - \\frac{\\sqrt{860\\cdot \\sqrt{7} + 2549}}{30} + \\frac{7}{30} + \\frac{\\sqrt{7}}{3}\\}$", "final_answer": "x \\in \\{\\frac{7}{30} + \\frac{\\sqrt{7}}{3} + \\frac{\\sqrt{860\\cdot \\sqrt{7} + 2549}}{30}, - \\frac{\\sqrt{860\\cdot \\sqrt{7} + 2549}}{30} + \\frac{7}{30} + \\frac{\\sqrt{7}}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c7cee599c59823cc", "problem_id": "3d63dfef7d7afbb4", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{3} x + \\frac{17 x}{3} - 5 \\sqrt{3} + \\frac{20}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{17}{3} - 3 \\sqrt{3}$, c=$\\frac{20}{3} - 5 \\sqrt{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{292}{9} - 14 \\sqrt{3}$, so √Δ = $\\sqrt{\\frac{292}{9} - 14 \\sqrt{3}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{17}{6} + \\frac{\\sqrt{292 - 126 \\sqrt{3}}}{6} + \\frac{3 \\sqrt{3}}{2}$.\nx2 = $- \\frac{17}{6} - \\frac{\\sqrt{292 - 126 \\sqrt{3}}}{6} + \\frac{3 \\sqrt{3}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{17}{6} + \\frac{\\sqrt{292 - 126\\cdot \\sqrt{3}}}{6} + \\frac{3\\cdot \\sqrt{3}}{2}, - \\frac{17}{6} - \\frac{\\sqrt{292 - 126\\cdot \\sqrt{3}}}{6} + \\frac{3\\cdot \\sqrt{3}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{17}{6} + \\frac{\\sqrt{292 - 126\\cdot \\sqrt{3}}}{6} + \\frac{3\\cdot \\sqrt{3}}{2}, - \\frac{17}{6} - \\frac{\\sqrt{292 - 126\\cdot \\sqrt{3}}}{6} + \\frac{3\\cdot \\sqrt{3}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4c64f5e6e67fd84a", "problem_id": "3d63dfef7d7afbb4", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{3} x + \\frac{17 x}{3} - 5 \\sqrt{3} + \\frac{20}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{17}{3} - 3 \\sqrt{3}$, c=$\\frac{20}{3} - 5 \\sqrt{3}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{- \\frac{17}{6} + \\frac{\\sqrt{292 - 126\\cdot \\sqrt{3}}}{6} + \\frac{3\\cdot \\sqrt{3}}{2}, - \\frac{17}{6} - \\frac{\\sqrt{292 - 126\\cdot \\sqrt{3}}}{6} + \\frac{3\\cdot \\sqrt{3}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{17}{6} + \\frac{\\sqrt{292 - 126\\cdot \\sqrt{3}}}{6} + \\frac{3\\cdot \\sqrt{3}}{2}, - \\frac{17}{6} - \\frac{\\sqrt{292 - 126\\cdot \\sqrt{3}}}{6} + \\frac{3\\cdot \\sqrt{3}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "52b0a3d97657f8c8", "problem_id": "68e376ef4a67103c", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_9=26$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=26 and k=9.\nCompute C(34,8) = 18156204.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $18156204$", "final_answer": "18156204", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "600dfa7e30da9af1", "problem_id": "68e376ef4a67103c", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_9=26$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=26 and k=9.\n4) Compute C(34,8) = 18156204.\nFinal answer: $18156204$", "final_answer": "18156204", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fc8f58a87b819f50", "problem_id": "1bc921ff405d9c69", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Worker A finishes a job in 20 hours and Worker B finishes it in 22 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/20 job/hour. B's rate = 1/22 job/hour.\n2) Combined rate = 1/20 + 1/22 = 21/220 job/hour.\n3) Time = 1 ÷ (combined rate) = 220/21 hours.\nFinal answer: $\\frac{220}{21}$", "final_answer": "\\frac{220}{21}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "53fcd100f8e7ac6a", "problem_id": "1bc921ff405d9c69", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Worker A finishes a job in 20 hours and Worker B finishes it in 22 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/20 job/hour. B's rate = 1/22 job/hour.\nCombined rate = 1/20 + 1/22 = 21/220 job/hour.\nTime = 1 ÷ (combined rate) = 220/21 hours.\nCheck: (combined rate)×(time) = 21/220×220/21 = 1.\nFinal answer: $\\frac{220}{21}$", "final_answer": "\\frac{220}{21}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3955bd39b1f45ea9", "problem_id": "de8b62e6f47e6969", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 10 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=4, p=$\\frac{1}{6}$ and simplify.\n4) Here C(10,4) = 210.\nFinal answer: $\\frac{546875}{10077696}$", "final_answer": "\\frac{546875}{10077696}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c205ca843e65c3d7", "problem_id": "de8b62e6f47e6969", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 10 times, what is the probability of exactly 4 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=4, p=$\\frac{1}{6}$ and simplify.\nHere C(10,4) = 210.\nCheck: expression simplifies to $\\frac{546875}{10077696}$.\nFinal answer: $\\frac{546875}{10077696}$", "final_answer": "\\frac{546875}{10077696}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "01864582dc10d008", "problem_id": "4818e8e18b395890", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 12 red balls and 10 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(22,10).\n3) Favorable samples: choose k reds and n−k blues: C(12,3)·C(10,7).\n4) Compute totals: C(22,10) = 646646, and favorable = 26400.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1200}{29393}$", "final_answer": "\\frac{1200}{29393}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4a91dacdc5a5aaf5", "problem_id": "4818e8e18b395890", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "An urn has 12 red balls and 10 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(22,10).\nFavorable samples: choose k reds and n−k blues: C(12,3)·C(10,7).\nFinal answer: $\\frac{1200}{29393}$", "final_answer": "\\frac{1200}{29393}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "96e7a9ac15431d00", "problem_id": "a9087f6b652e2a86", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Worker A finishes a job in 10 hours and Worker B finishes it in 5 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/10 job/hour. B's rate = 1/5 job/hour.\n2) Combined rate = 1/10 + 1/5 = 3/10 job/hour.\n3) Time = 1 ÷ (combined rate) = 10/3 hours.\nFinal answer: $\\frac{10}{3}$", "final_answer": "\\frac{10}{3}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8064b69100447edd", "problem_id": "070b8d8e920c171e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- x^{2} + 3 x - 1\\right) \\cos{\\left(3 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\n3) So $f'(x) = 3 x^{2} \\sin{\\left(3 x \\right)} + 2 x e^{x} - 9 x \\sin{\\left(3 x \\right)} - 2 x \\cos{\\left(3 x \\right)} + 4 e^{x} + 3 \\sin{\\left(3 x \\right)} + 3 \\cos{\\left(3 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = \\left(3 - 2 x\\right) \\cos{\\left(3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 3 \\left(x \\left(3 - x\\right) - 1\\right) \\sin{\\left(3 x \\right)} + 2 e^{x}$.\nFinal answer: $\\left(3 - 2 x\\right) \\cos{\\left(3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 3 \\left(x \\left(3 - x\\right) - 1\\right) \\sin{\\left(3 x \\right)} + 2 e^{x}$", "final_answer": "\\left(3 - 2 x\\right) \\cos{\\left(3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 3 \\left(x \\left(3 - x\\right) - 1\\right) \\sin{\\left(3 x \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "33bb14a89889e2fc", "problem_id": "1cc03d8413e10524", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$- x - 3 y$ = $-1$\n$- x - 4 y$ = $0$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $x + 3 y$ = $1$.\n3) Multiply the second equation by $-1$: $x + 4 y$ = $0$.\n4) Subtract to eliminate x: $- y$ = $1$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, -1\\right)$", "final_answer": "(x,y)=\\left(4, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4b01379f1faca2f7", "problem_id": "1cc03d8413e10524", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$- x - 3 y$ = $-1$\n$- x - 4 y$ = $0$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $x + 3 y$ = $1$.\nMultiply the second equation by $-1$: $x + 4 y$ = $0$.\nSubtract to eliminate x: $- y$ = $1$.\nSolve for y: y = $-1$.\nSubstitute back to find x: x = $4$.\nCheck: plug (x,y)=(4,-1) into both equations.\nFinal answer: $(x,y)=\\left(4, -1\\right)$", "final_answer": "(x,y)=\\left(4, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8816895d7cc7c4f5", "problem_id": "3e6b998fbdd90eba", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the multiplicative inverse of 30 modulo 41.", "output": "1) We want x such that 30x ≡ 1 (mod 41).\n2) Use extended Euclid to find s,t with 30s + 41t = 1.\n3) One Bezout identity is: 30·(-15) + 41·(11) = 1.\n4) Reduce s modulo 41: x ≡ 26 (mod 41).\nFinal answer: $x\\equiv 26\\pmod{41}$", "final_answer": "x\\equiv 26\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "23967794f6357012", "problem_id": "3e6b998fbdd90eba", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 30 modulo 41.", "output": "We want x such that 30x ≡ 1 (mod 41).\nUse extended Euclid to find s,t with 30s + 41t = 1.\nOne Bezout identity is: 30·(-15) + 41·(11) = 1.\nReduce s modulo 41: x ≡ 26 (mod 41).\nCheck: 30·26 mod 41 = 1.\nFinal answer: $x\\equiv 26\\pmod{41}$", "final_answer": "x\\equiv 26\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72f4c35fb3dbfa4e", "problem_id": "8c4bb6b45b46a27f", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the multiplicative inverse of 6 modulo 31.", "output": "1) We want x such that 6x ≡ 1 (mod 31).\n2) Use extended Euclid to find s,t with 6s + 31t = 1.\n3) One Bezout identity is: 6·(-5) + 31·(1) = 1.\n4) Reduce s modulo 31: x ≡ 26 (mod 31).\nFinal answer: $x\\equiv 26\\pmod{31}$", "final_answer": "x\\equiv 26\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "578747ccbad37dc1", "problem_id": "8c4bb6b45b46a27f", "style": "answer_only", "instruction": "Final answer only.", "input": "Find the multiplicative inverse of 6 modulo 31.", "output": "$x\\equiv 26\\pmod{31}$", "final_answer": "x\\equiv 26\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f39875ad400e5b42", "problem_id": "819da4c9955a9db5", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Two friends split $912 in the ratio 9:7. How much does each receive?", "output": "1) Total ratio parts = 9+7 = 16.\n2) Each part = 912 ÷ 16 = 57.\n3) First share = 9 × 57 = 513.\n4) Second share = 7 × 57 = 399.\nFinal answer: $399$", "final_answer": "399", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6bcd647e0b35271b", "problem_id": "c94653b55cacc193", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} \\sin{\\left(x \\right)} - x^{2} \\sin{\\left(x \\right)} - 3 x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\cos{\\left(2 x \\right)}) = - \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = - 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\cos{\\left(2 x \\right)}) = - 2 \\sin{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$", "final_answer": "- 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3b25999556d0d247", "problem_id": "0b571a6346d5770f", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡0 (mod 4), x≡7 (mod 13). Give the solution modulo 52.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 52.\n3) Compute the CRT solution to get x ≡ 20 (mod 52).\nFinal answer: $x\\equiv 20\\pmod{52}$", "final_answer": "x\\equiv 20\\pmod{52}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d01272b354e4c7b2", "problem_id": "bfa05cd85d8d5b07", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 3 x^{3} - 8 x^{2} - 16 x - 30$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{3} - 8 x^{2} - 17 x - 30.\n3) Then the limit is -120.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -118.\nFinal answer: $-118$", "final_answer": "-118", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7e12256b5a45baa2", "problem_id": "e1b7522c5e581182", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$- 4 x + 2 y$ = $-2$\n$4 x + 4 y$ = $32$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $- 16 x + 8 y$ = $-8$.\n3) Multiply the second equation by $-4$: $- 16 x - 16 y$ = $-128$.\n4) Subtract to eliminate x: $24 y$ = $120$.\n5) Solve for y: y = $5$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, 5\\right)$", "final_answer": "(x,y)=\\left(3, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1df3117a986d667d", "problem_id": "e1b7522c5e581182", "style": "answer_only", "instruction": "Final answer only.", "input": "Solve the linear system:\n$- 4 x + 2 y$ = $-2$\n$4 x + 4 y$ = $32$", "output": "$(x,y)=\\left(3, 5\\right)$", "final_answer": "(x,y)=\\left(3, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9cfc802d52c292be", "problem_id": "f2e15f9be1389c93", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$4 x + 3 y$ = $6$\n$- 3 y$ = $-6$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $4$: $- 12 y$ = $-24$.\n4) Subtract to eliminate x: $12 y$ = $24$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, 2\\right)$", "final_answer": "(x,y)=\\left(0, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8901c8a3bf985016", "problem_id": "f2e15f9be1389c93", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the linear system:\n$4 x + 3 y$ = $6$\n$- 3 y$ = $-6$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $4$: $- 12 y$ = $-24$.\nFinal answer: $(x,y)=\\left(0, 2\\right)$", "final_answer": "(x,y)=\\left(0, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "09b4efe7cdebdfd4", "problem_id": "4fe79d7eb5823dc4", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Two friends split $1720 in the ratio 5:5. How much does each receive?", "output": "1) Total ratio parts = 5+5 = 10.\n2) Each part = 1720 ÷ 10 = 172.\n3) First share = 5 × 172 = 860.\n4) Second share = 5 × 172 = 860.\nFinal answer: $860$", "final_answer": "860", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c4534fb9e991908f", "problem_id": "4fe79d7eb5823dc4", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Two friends split $1720 in the ratio 5:5. How much does each receive?", "output": "Total ratio parts = 5+5 = 10.\nEach part = 1720 ÷ 10 = 172.\nFirst share = 5 × 172 = 860.\nFinal answer: $860$", "final_answer": "860", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "95a3cbc50f26ac5b", "problem_id": "4c587703018552c9", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many binary strings of length 20 contain exactly 8 ones with no two ones adjacent?", "output": "Reserve 7 mandatory zeros between the 8 ones.\nThis leaves n−k+1 = 13 valid slots for the ones.\nChoose the slots: C(13,8) = 1287.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $1287$", "final_answer": "1287", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "365d86a49a9d3f51", "problem_id": "4c587703018552c9", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "How many binary strings of length 20 contain exactly 8 ones with no two ones adjacent?", "output": "1) Reserve 7 mandatory zeros between the 8 ones.\n2) This leaves n−k+1 = 13 valid slots for the ones.\n3) Choose the slots: C(13,8) = 1287.\nFinal answer: $1287$", "final_answer": "1287", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f069134612ead58f", "problem_id": "c9f2d39d69a0765f", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Worker A finishes a job in 19 hours and Worker B finishes it in 14 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/19 job/hour. B's rate = 1/14 job/hour.\n2) Combined rate = 1/19 + 1/14 = 33/266 job/hour.\n3) Time = 1 ÷ (combined rate) = 266/33 hours.\nFinal answer: $\\frac{266}{33}$", "final_answer": "\\frac{266}{33}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5dbe6c157ecd8e9f", "problem_id": "c9f2d39d69a0765f", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Worker A finishes a job in 19 hours and Worker B finishes it in 14 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/19 job/hour. B's rate = 1/14 job/hour.\nCombined rate = 1/19 + 1/14 = 33/266 job/hour.\nTime = 1 ÷ (combined rate) = 266/33 hours.\nCheck: (combined rate)×(time) = 33/266×266/33 = 1.\nFinal answer: $\\frac{266}{33}$", "final_answer": "\\frac{266}{33}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a4d96dccfc9fb434", "problem_id": "020f03b83c4580d5", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "An urn has 7 red balls and 7 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(14,8).\n3) Favorable samples: choose k reds and n−k blues: C(7,2)·C(7,6).\n4) Compute totals: C(14,8) = 3003, and favorable = 147.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{7}{143}$", "final_answer": "\\frac{7}{143}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "987273d7633b0aa0", "problem_id": "020f03b83c4580d5", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 7 red balls and 7 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(14,8).\nFavorable samples: choose k reds and n−k blues: C(7,2)·C(7,6).\nCompute totals: C(14,8) = 3003, and favorable = 147.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{7}{143}$.\nFinal answer: $\\frac{7}{143}$", "final_answer": "\\frac{7}{143}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a438e42b921175fd", "problem_id": "12835a4598cf5287", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 17 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=17, k=8, p=$\\frac{3}{7}$ and simplify.\nFinal answer: $\\frac{41811420119040}{232630513987207}$", "final_answer": "\\frac{41811420119040}{232630513987207}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c9099c68e00c9141", "problem_id": "12835a4598cf5287", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 17 times, what is the probability of exactly 8 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=17, k=8, p=$\\frac{3}{7}$ and simplify.\n4) Here C(17,8) = 24310.\nFinal answer: $\\frac{41811420119040}{232630513987207}$", "final_answer": "\\frac{41811420119040}{232630513987207}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "529cc5bbc5f01edb", "problem_id": "ff592a07c5b4be07", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A store sells a backpack for $580. It is discounted by 20%. You buy 10 backpacks. What is the total cost?", "output": "Discounted price = 580 × (1 − 20/100) = 580 × 80/100 = 464.\nSubtotal = 10 × 464 = 4640.\nCheck: recomputing the formula yields 4640.\nFinal answer: $4640$", "final_answer": "4640", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "617d06ec9901c8e3", "problem_id": "ff592a07c5b4be07", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A store sells a backpack for $580. It is discounted by 20%. You buy 10 backpacks. What is the total cost?", "output": "1) Discounted price = 580 × (1 − 20/100) = 580 × 80/100 = 464.\n2) Subtotal = 10 × 464 = 4640.\nFinal answer: $4640$", "final_answer": "4640", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6e12a0d94abe3384", "problem_id": "d11ac21a01cfb3a5", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the multiplicative inverse of 9 modulo 17.", "output": "1) We want x such that 9x ≡ 1 (mod 17).\n2) Use extended Euclid to find s,t with 9s + 17t = 1.\n3) One Bezout identity is: 9·(2) + 17·(-1) = 1.\n4) Reduce s modulo 17: x ≡ 2 (mod 17).\nFinal answer: $x\\equiv 2\\pmod{17}$", "final_answer": "x\\equiv 2\\pmod{17}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "307c4612c42b7126", "problem_id": "d11ac21a01cfb3a5", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the multiplicative inverse of 9 modulo 17.", "output": "We want x such that 9x ≡ 1 (mod 17).\nUse extended Euclid to find s,t with 9s + 17t = 1.\nOne Bezout identity is: 9·(2) + 17·(-1) = 1.\nReduce s modulo 17: x ≡ 2 (mod 17).\nCheck: 9·2 mod 17 = 1.\nFinal answer: $x\\equiv 2\\pmod{17}$", "final_answer": "x\\equiv 2\\pmod{17}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a62b7e4302340bb4", "problem_id": "76b00a58211bd1aa", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int x^{3} + 2 x^{2} - x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $x^{3} + 2 x^{2} - x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1$.\n3) Compute an antiderivative: F(x) = $\\frac{x^{4}}{4} + \\frac{2 x^{3}}{3} - \\frac{x^{2}}{2} - x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{x^{4}}{4} + \\frac{2\\cdot x^{3}}{3} - \\frac{x^{2}}{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{4}}{4} + \\frac{2\\cdot x^{3}}{3} - \\frac{x^{2}}{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6bd2d3268eda9ecc", "problem_id": "76b00a58211bd1aa", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int x^{3} + 2 x^{2} - x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $x^{3} + 2 x^{2} - x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1$.\nCompute an antiderivative: F(x) = $\\frac{x^{4}}{4} + \\frac{2 x^{3}}{3} - \\frac{x^{2}}{2} - x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $C + \\frac{x^{4}}{4} + \\frac{2\\cdot x^{3}}{3} - \\frac{x^{2}}{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{4}}{4} + \\frac{2\\cdot x^{3}}{3} - \\frac{x^{2}}{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2b2c5e342319f14f", "problem_id": "923f98e9d013a2dd", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_3=18$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=18 and k=3.\n4) Compute C(20,2) = 190.\nFinal answer: $190$", "final_answer": "190", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "228cd2a6958b9f7c", "problem_id": "30b1fab2e37dc270", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{3} + 3 x^{2} - 3 x - 2$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(3 x^{3}) = 9 x^{3}$.\n3) So $f'(x) = 9 x^{3} + 6 x - 3$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(3 x^{3}) = 9 x^{2}$.\nTherefore the correct derivative is $f'(x) = 3 \\left(3 x^{2} + 2 x - 1\\right)$.\nFinal answer: $3 \\left(3 x^{2} + 2 x - 1\\right)$", "final_answer": "3 \\left(3 x^{2} + 2 x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "9a42e880af0e2c0d", "problem_id": "e79ed46de00e8176", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$2 x + 2 y$ = $-14$\n$4 x + 3 y$ = $-23$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $8 x + 8 y$ = $-56$.\nMultiply the second equation by $2$: $8 x + 6 y$ = $-46$.\nSubtract to eliminate x: $2 y$ = $-10$.\nSolve for y: y = $-5$.\nSubstitute back to find x: x = $-2$.\nCheck: plug (x,y)=(-2,-5) into both equations.\nFinal answer: $(x,y)=\\left(-2, -5\\right)$", "final_answer": "(x,y)=\\left(-2, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d2e53257fa48ae7a", "problem_id": "e79ed46de00e8176", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Solve the linear system:\n$2 x + 2 y$ = $-14$\n$4 x + 3 y$ = $-23$", "output": "$(x,y)=\\left(-2, -5\\right)$", "final_answer": "(x,y)=\\left(-2, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b97687c79debaa8e", "problem_id": "f0edbf272a0e2609", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} + x^{2} - 2 x + 4$.", "output": "1) Differentiate term-by-term:\n2) d/dx(4) = 0.\n3) d/dx(x^{2}) = 2 x.\n4) d/dx(- 2 x) = -2.\n5) d/dx(- 2 x^{3}) = - 6 x^{2}.\n6) Combine: f'(x) = 2 \\left(- 3 x^{2} + x - 1\\right).\nFinal answer: $- 6\\cdot x^{2} + 2\\cdot x - 2$", "final_answer": "- 6\\cdot x^{2} + 2\\cdot x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2076d6a8e8cbfecc", "problem_id": "f0edbf272a0e2609", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} + x^{2} - 2 x + 4$.", "output": "Differentiate term-by-term:\nd/dx(4) = 0.\nd/dx(x^{2}) = 2 x.\nd/dx(- 2 x) = -2.\nd/dx(- 2 x^{3}) = - 6 x^{2}.\nCombine: f'(x) = 2 \\left(- 3 x^{2} + x - 1\\right).\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 6\\cdot x^{2} + 2\\cdot x - 2$", "final_answer": "- 6\\cdot x^{2} + 2\\cdot x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dab0ecc975eb32d6", "problem_id": "140e2b0e481900a7", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$- x$ = $2$\n$- 2 x + y$ = $5$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $2 x$ = $-4$.\n3) Multiply the second equation by $-1$: $2 x - y$ = $-5$.\n4) Subtract to eliminate x: $y$ = $1$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, 1\\right)$", "final_answer": "(x,y)=\\left(-2, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ff3e2d14958746b", "problem_id": "8eb129c8ffc74457", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 x^{2} - 5 x + 8$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{2} - 6 x + 8.\n3) Then the limit is 32.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 30.\nFinal answer: $30$", "final_answer": "30", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b6b6ae8ebbfe743e", "problem_id": "0b578320ce4bd2d0", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many binary strings of length 19 contain exactly 9 ones with no two ones adjacent?", "output": "1) Reserve 8 mandatory zeros between the 9 ones.\n2) This leaves n−k+1 = 11 valid slots for the ones.\n3) Choose the slots: C(11,9) = 55.\nFinal answer: $55$", "final_answer": "55", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a2ac5a9cb2aaf20", "problem_id": "d54f72c07c949c2c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x^{2} - 3 x - 9$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{2} - 4 x - 9.\n3) Then the limit is -25.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -23.\nFinal answer: $-23$", "final_answer": "-23", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "33e64acd5781c07e", "problem_id": "a0468febc42385af", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Two friends split $1456 in the ratio 9:7. How much does each receive?", "output": "1) Total ratio parts = 9+7 = 16.\n2) Each part = 1456 ÷ 16 = 91.\n3) First share = 9 × 91 = 819.\n4) Second share = 7 × 91 = 637.\nFinal answer: $637$", "final_answer": "637", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a06bff08cf264be1", "problem_id": "a0468febc42385af", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Two friends split $1456 in the ratio 9:7. How much does each receive?", "output": "Total ratio parts = 9+7 = 16.\nEach part = 1456 ÷ 16 = 91.\nFirst share = 9 × 91 = 819.\nSecond share = 7 × 91 = 637.\nCheck: 819+637=1456.\nFinal answer: $637$", "final_answer": "637", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f63bb0478e934ff9", "problem_id": "477d6852b5ce2b95", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the multiplicative inverse of 2 modulo 31.", "output": "1) We want x such that 2x ≡ 1 (mod 31).\n2) Use extended Euclid to find s,t with 2s + 31t = 1.\n3) One Bezout identity is: 2·(-15) + 31·(1) = 1.\n4) Reduce s modulo 31: x ≡ 16 (mod 31).\nFinal answer: $x\\equiv 16\\pmod{31}$", "final_answer": "x\\equiv 16\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "297999477fdb5085", "problem_id": "84ef03e2676efdbb", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - x$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to - 2 x.\n3) Then the limit is -6.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: -3.\nFinal answer: $-3$", "final_answer": "-3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ffd6fa11866437cc", "problem_id": "21eb070656446cd4", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 11 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=3, p=$\\frac{4}{5}$ and simplify.\n4) Here C(11,3) = 165.\nFinal answer: $\\frac{2112}{9765625}$", "final_answer": "\\frac{2112}{9765625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dba5146c5d261e58", "problem_id": "21eb070656446cd4", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 11 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=11, k=3, p=$\\frac{4}{5}$ and simplify.\nHere C(11,3) = 165.\nCheck: expression simplifies to $\\frac{2112}{9765625}$.\nFinal answer: $\\frac{2112}{9765625}$", "final_answer": "\\frac{2112}{9765625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "43d6a5e55b75ba27", "problem_id": "63787a17b4e7f363", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{6} x}{3} + \\frac{x}{2} - \\frac{3}{2} - \\frac{\\sqrt{6}}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{1}{2} - \\frac{\\sqrt{6}}{3}$, c=$- \\frac{3}{2} - \\frac{\\sqrt{6}}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{5 \\sqrt{6}}{3} + \\frac{83}{12}$, so √Δ = $\\sqrt{\\frac{5 \\sqrt{6}}{3} + \\frac{83}{12}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{1}{4} + \\frac{\\sqrt{6}}{6} + \\frac{\\sqrt{60 \\sqrt{6} + 249}}{12}$.\n7) x2 = $- \\frac{\\sqrt{60 \\sqrt{6} + 249}}{12} - \\frac{1}{4} + \\frac{\\sqrt{6}}{6}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{1}{4} + \\frac{\\sqrt{6}}{6} + \\frac{\\sqrt{60\\cdot \\sqrt{6} + 249}}{12}, - \\frac{\\sqrt{60\\cdot \\sqrt{6} + 249}}{12} - \\frac{1}{4} + \\frac{\\sqrt{6}}{6}\\}$", "final_answer": "x \\in \\{- \\frac{1}{4} + \\frac{\\sqrt{6}}{6} + \\frac{\\sqrt{60\\cdot \\sqrt{6} + 249}}{12}, - \\frac{\\sqrt{60\\cdot \\sqrt{6} + 249}}{12} - \\frac{1}{4} + \\frac{\\sqrt{6}}{6}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d8a79a2bf0913edd", "problem_id": "414fde83bfa8e3ed", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 2 x^{2} + x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}) = - \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} - \\frac{6 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} - \\frac{12 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} - \\frac{2 x^{2}}{2 x^{4} - 12 x^{2} + 18} + \\frac{24 x}{2 x^{4} - 12 x^{2} + 18} - \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 6 x^{2} + 9} + \\frac{9 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} + \\frac{18 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} - \\frac{6}{2 x^{4} - 12 x^{2} + 18} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 6} + \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3} - \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}) = \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - 3\\right)^{2}} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x^{2} - 3\\right)} - \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 4 x \\left(- 2 x^{2} + x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right) \\left(4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8 x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{2 \\left(x^{2} - 3\\right)^{2}}$.\nFinal answer: $\\frac{- 4 x \\left(- 2 x^{2} + x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right) \\left(4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8 x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{2 \\left(x^{2} - 3\\right)^{2}}$", "final_answer": "\\frac{- 4 x \\left(- 2 x^{2} + x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right) \\left(4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8 x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{2 \\left(x^{2} - 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "1791aefb44e92640", "problem_id": "8b73ee7f7d846704", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 11 x + 15$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$11$, c=$15$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $1$, so √Δ = $1$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{5}{2}$.\nx2 = $-3$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{5}{2}, -3\\}$", "final_answer": "x \\in \\{- \\frac{5}{2}, -3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9a5745aa18e1d9e7", "problem_id": "90b5c9913946f531", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- x^{2} - x - 2\\right) \\cos{\\left(2 x^{2} + 2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x^{2} \\cos{\\left(2 x^{2} + 2 x \\right)}) = x^{2} \\sin{\\left(2 x \\left(x + 1\\right) \\right)}$.\n3) So $f'(x) = x^{2} \\sin{\\left(2 x^{2} + 2 x \\right)} + 4 x^{2} \\sin{\\left(2 x^{2} + 2 x \\right)} + 2 x e^{x} + 10 x \\sin{\\left(2 x^{2} + 2 x \\right)} + 4 e^{x} + 4 \\sin{\\left(2 x^{2} + 2 x \\right)} - \\cos{\\left(2 x^{2} + 2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x^{2} \\cos{\\left(2 x^{2} + 2 x \\right)}) = x^{2} \\left(4 x + 2\\right) \\sin{\\left(2 x^{2} + 2 x \\right)} - 2 x \\cos{\\left(2 x^{2} + 2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(- 2 x - 1\\right) \\cos{\\left(2 x^{2} + 2 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 2 \\left(2 x + 1\\right) \\left(x \\left(- x - 1\\right) - 2\\right) \\sin{\\left(2 x^{2} + 2 x \\right)} + 2 e^{x}$.\nFinal answer: $\\left(- 2 x - 1\\right) \\cos{\\left(2 x^{2} + 2 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 2 \\left(2 x + 1\\right) \\left(x \\left(- x - 1\\right) - 2\\right) \\sin{\\left(2 x^{2} + 2 x \\right)} + 2 e^{x}$", "final_answer": "\\left(- 2 x - 1\\right) \\cos{\\left(2 x^{2} + 2 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 2 \\left(2 x + 1\\right) \\left(x \\left(- x - 1\\right) - 2\\right) \\sin{\\left(2 x^{2} + 2 x \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "953e31dd879bb896", "problem_id": "600e8cb5f73071bc", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$- 5 x + 8 y$ = $29$\n$- x + 6 y$ = $19$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $5 x - 8 y$ = $-29$.\nMultiply the second equation by $-5$: $5 x - 30 y$ = $-95$.\nSubtract to eliminate x: $22 y$ = $66$.\nSolve for y: y = $3$.\nSubstitute back to find x: x = $-1$.\nCheck: plug (x,y)=(-1,3) into both equations.\nFinal answer: $(x,y)=\\left(-1, 3\\right)$", "final_answer": "(x,y)=\\left(-1, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fe3c3ae4d0babb89", "problem_id": "600e8cb5f73071bc", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$- 5 x + 8 y$ = $29$\n$- x + 6 y$ = $19$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $5 x - 8 y$ = $-29$.\n3) Multiply the second equation by $-5$: $5 x - 30 y$ = $-95$.\n4) Subtract to eliminate x: $22 y$ = $66$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, 3\\right)$", "final_answer": "(x,y)=\\left(-1, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0fcb5793fd8fe9f4", "problem_id": "22c6a5526b357020", "style": "answer_only", "instruction": "Final answer only.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} 2 x^{4} + 2 x^{3} + 4 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}\\,dx$.", "output": "$\\frac{\\pi^{4}}{32} + \\frac{\\pi^{5}}{80} + 5 + \\frac{\\pi^{3}}{6}$", "final_answer": "\\frac{\\pi^{4}}{32} + \\frac{\\pi^{5}}{80} + 5 + \\frac{\\pi^{3}}{6}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ef68e8054c68a47c", "problem_id": "22c6a5526b357020", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} 2 x^{4} + 2 x^{3} + 4 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{4} + 2 x^{3} + 4 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\n3) Subtract: F(\\frac{\\pi}{2}) − F(0) = $\\frac{\\pi^{4}}{32} + \\frac{\\pi^{5}}{80} + 5 + \\frac{\\pi^{3}}{6}$.\n4) Simplify the result.\nFinal answer: $\\frac{\\pi^{4}}{32} + \\frac{\\pi^{5}}{80} + 5 + \\frac{\\pi^{3}}{6}$", "final_answer": "\\frac{\\pi^{4}}{32} + \\frac{\\pi^{5}}{80} + 5 + \\frac{\\pi^{3}}{6}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5f1da75e4780cd74", "problem_id": "371f838ed2bf1461", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{2} x}{3} + \\frac{11 x}{3} - \\frac{7 \\sqrt{2}}{9} + \\frac{28}{9}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{11}{3} - \\frac{\\sqrt{2}}{3}$, c=$\\frac{28}{9} - \\frac{7 \\sqrt{2}}{9}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{2 \\sqrt{2}}{3} + \\frac{11}{9}$, so √Δ = $\\sqrt{\\frac{2 \\sqrt{2}}{3} + \\frac{11}{9}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{11}{6} + \\frac{\\sqrt{2}}{6} + \\frac{\\sqrt{6 \\sqrt{2} + 11}}{6}$.\n7) x2 = $- \\frac{11}{6} - \\frac{\\sqrt{6 \\sqrt{2} + 11}}{6} + \\frac{\\sqrt{2}}{6}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{11}{6} + \\frac{\\sqrt{2}}{6} + \\frac{\\sqrt{6\\cdot \\sqrt{2} + 11}}{6}, - \\frac{11}{6} - \\frac{\\sqrt{6\\cdot \\sqrt{2} + 11}}{6} + \\frac{\\sqrt{2}}{6}\\}$", "final_answer": "x \\in \\{- \\frac{11}{6} + \\frac{\\sqrt{2}}{6} + \\frac{\\sqrt{6\\cdot \\sqrt{2} + 11}}{6}, - \\frac{11}{6} - \\frac{\\sqrt{6\\cdot \\sqrt{2} + 11}}{6} + \\frac{\\sqrt{2}}{6}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1dec1054eac5d55a", "problem_id": "371f838ed2bf1461", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{2} x}{3} + \\frac{11 x}{3} - \\frac{7 \\sqrt{2}}{9} + \\frac{28}{9}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{11}{3} - \\frac{\\sqrt{2}}{3}$, c=$\\frac{28}{9} - \\frac{7 \\sqrt{2}}{9}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{- \\frac{11}{6} + \\frac{\\sqrt{2}}{6} + \\frac{\\sqrt{6\\cdot \\sqrt{2} + 11}}{6}, - \\frac{11}{6} - \\frac{\\sqrt{6\\cdot \\sqrt{2} + 11}}{6} + \\frac{\\sqrt{2}}{6}\\}$", "final_answer": "x \\in \\{- \\frac{11}{6} + \\frac{\\sqrt{2}}{6} + \\frac{\\sqrt{6\\cdot \\sqrt{2} + 11}}{6}, - \\frac{11}{6} - \\frac{\\sqrt{6\\cdot \\sqrt{2} + 11}}{6} + \\frac{\\sqrt{2}}{6}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b1b5a5e1f41986ea", "problem_id": "6f9a0487402bd1e7", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the linear system:\n$- 5 x + y$ = $19$\n$8 x$ = $-32$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $8$: $- 40 x + 8 y$ = $152$.\nMultiply the second equation by $-5$: $- 40 x$ = $160$.\nFinal answer: $(x,y)=\\left(-4, -1\\right)$", "final_answer": "(x,y)=\\left(-4, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a2da805985abd855", "problem_id": "6f9a0487402bd1e7", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$- 5 x + y$ = $19$\n$8 x$ = $-32$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $8$: $- 40 x + 8 y$ = $152$.\n3) Multiply the second equation by $-5$: $- 40 x$ = $160$.\n4) Subtract to eliminate x: $8 y$ = $-8$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $-4$.\nFinal answer: $(x,y)=\\left(-4, -1\\right)$", "final_answer": "(x,y)=\\left(-4, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c9be603e6de083f4", "problem_id": "d263b8f0be913e6d", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - x - 4$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{2} - x + 1$ and a = 3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 2 x - 1$.\n5) Evaluate at a: p'(3) = $-7$.\nFinal answer: $-7$", "final_answer": "-7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bf50ae61e2577dd8", "problem_id": "d263b8f0be913e6d", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - x - 4$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{2} - x + 1$ and a = 3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-7$", "final_answer": "-7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "98e9be359a0beff7", "problem_id": "11a77f0e853dedb2", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} 2 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $2 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\nSubtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{3 \\pi}{2} - \\frac{3 \\pi^{2}}{8} + 2 + \\frac{\\pi^{3}}{12}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{3\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{2}}{8} + 2 + \\frac{\\pi^{3}}{12}$", "final_answer": "- \\frac{3\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{2}}{8} + 2 + \\frac{\\pi^{3}}{12}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fbcef90d0259514d", "problem_id": "11a77f0e853dedb2", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} 2 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\n3) Subtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{3 \\pi}{2} - \\frac{3 \\pi^{2}}{8} + 2 + \\frac{\\pi^{3}}{12}$.\n4) Simplify the result.\nFinal answer: $- \\frac{3\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{2}}{8} + 2 + \\frac{\\pi^{3}}{12}$", "final_answer": "- \\frac{3\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{2}}{8} + 2 + \\frac{\\pi^{3}}{12}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4d732f88bb08c7ed", "problem_id": "4c65f5e9bb278050", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_5=36$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_5=31.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=31 and k=5.\nCompute C(35,4) = 52360.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $52360$", "final_answer": "52360", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d360e15e3d92e084", "problem_id": "4c65f5e9bb278050", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_5=36$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_5=31.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=31 and k=5.\n4) Compute C(35,4) = 52360.\nFinal answer: $52360$", "final_answer": "52360", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "63ff685139455721", "problem_id": "90ce192073de0a89", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{4} - 2 x^{3} + x^{2} - 2 x$.", "output": "1) Differentiate term-by-term:\n2) d/dx(x^{2}) = 2 x.\n3) d/dx(- 2 x) = -2.\n4) d/dx(- 2 x^{3}) = - 6 x^{2}.\n5) d/dx(2 x^{4}) = 8 x^{3}.\n6) Combine: f'(x) = 2 \\left(4 x^{3} - 3 x^{2} + x - 1\\right).\nFinal answer: $8\\cdot x^{3} - 6\\cdot x^{2} + 2\\cdot x - 2$", "final_answer": "8\\cdot x^{3} - 6\\cdot x^{2} + 2\\cdot x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7bf9ab91f42f8589", "problem_id": "90ce192073de0a89", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{4} - 2 x^{3} + x^{2} - 2 x$.", "output": "Differentiate term-by-term:\nd/dx(x^{2}) = 2 x.\nd/dx(- 2 x) = -2.\nd/dx(- 2 x^{3}) = - 6 x^{2}.\nd/dx(2 x^{4}) = 8 x^{3}.\nCombine: f'(x) = 2 \\left(4 x^{3} - 3 x^{2} + x - 1\\right).\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $8\\cdot x^{3} - 6\\cdot x^{2} + 2\\cdot x - 2$", "final_answer": "8\\cdot x^{3} - 6\\cdot x^{2} + 2\\cdot x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bf715f6d70ba611c", "problem_id": "8539dc75b22b44f4", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 3 x^{3} - 6 x^{2} + 21 x - 63$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $3 x^{4} + 3 x^{3} + 3 x^{2} - 1$ and a = -3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $12 x^{3} + 9 x^{2} + 6 x$.\nEvaluate at a: p'(-3) = $-261$.\nCheck: the identity gives $-261$.\nFinal answer: $-261$", "final_answer": "-261", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9e5e2e75245bc882", "problem_id": "8539dc75b22b44f4", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 3 x^{3} - 6 x^{2} + 21 x - 63$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{4} + 3 x^{3} + 3 x^{2} - 1$ and a = -3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $12 x^{3} + 9 x^{2} + 6 x$.\n5) Evaluate at a: p'(-3) = $-261$.\nFinal answer: $-261$", "final_answer": "-261", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bcd6057bd31a7e9d", "problem_id": "9597b2b53d8e8b81", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Two friends split $1540 in the ratio 8:2. How much does each receive?", "output": "1) Total ratio parts = 8+2 = 10.\n2) Each part = 1540 ÷ 10 = 154.\n3) First share = 8 × 154 = 1232.\n4) Second share = 2 × 154 = 308.\nFinal answer: $308$", "final_answer": "308", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "44e165160c1155fb", "problem_id": "9597b2b53d8e8b81", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Two friends split $1540 in the ratio 8:2. How much does each receive?", "output": "Total ratio parts = 8+2 = 10.\nEach part = 1540 ÷ 10 = 154.\nFirst share = 8 × 154 = 1232.\nFinal answer: $308$", "final_answer": "308", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "971b610824af4529", "problem_id": "04d75f9653431b55", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 14 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=14, k=8, p=$\\frac{2}{5}$ and simplify.\nFinal answer: $\\frac{560431872}{6103515625}$", "final_answer": "\\frac{560431872}{6103515625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2706b1f086af7735", "problem_id": "04d75f9653431b55", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 14 times, what is the probability of exactly 8 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=14, k=8, p=$\\frac{2}{5}$ and simplify.\n4) Here C(14,8) = 3003.\nFinal answer: $\\frac{560431872}{6103515625}$", "final_answer": "\\frac{560431872}{6103515625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "75a60dcf7f538e81", "problem_id": "596920dcc1b388d3", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Two friends split $528 in the ratio 7:4. How much does each receive?", "output": "1) Total ratio parts = 7+4 = 11.\n2) Each part = 528 ÷ 11 = 48.\n3) First share = 7 × 48 = 336.\n4) Second share = 4 × 48 = 192.\nFinal answer: $192$", "final_answer": "192", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5bb2d39816231c3b", "problem_id": "596920dcc1b388d3", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Two friends split $528 in the ratio 7:4. How much does each receive?", "output": "Total ratio parts = 7+4 = 11.\nEach part = 528 ÷ 11 = 48.\nFirst share = 7 × 48 = 336.\nSecond share = 4 × 48 = 192.\nCheck: 336+192=528.\nFinal answer: $192$", "final_answer": "192", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3043a6079ea6cb68", "problem_id": "7f946c81c713b88d", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A cyclist rides at 79 miles per hour for 2 hours, then at 65 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 79 × 2 = 158 miles.\n2) Second leg distance = 65 × 2 = 130 miles.\n3) Total distance = 158 + 130 = 288 miles.\nFinal answer: $288$", "final_answer": "288", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "15efa486084c4506", "problem_id": "7f946c81c713b88d", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A cyclist rides at 79 miles per hour for 2 hours, then at 65 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 79 × 2 = 158 miles.\nSecond leg distance = 65 × 2 = 130 miles.\nTotal distance = 158 + 130 = 288 miles.\nCheck: adding both legs gives 288.\nFinal answer: $288$", "final_answer": "288", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "97d1b512e5d0ed3c", "problem_id": "b5d48ea5295d992d", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡1 (mod 9), x≡1 (mod 5). Give the solution modulo 45.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 45.\nCompute the CRT solution to get x ≡ 1 (mod 45).\nCheck: 1 mod 9=1 and mod 5=1.\nFinal answer: $x\\equiv 1\\pmod{45}$", "final_answer": "x\\equiv 1\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "79285d80b317573e", "problem_id": "e9ea8694d5d5651d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - x^{3} + 2 x^{2} - 3 x + 4$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to - x^{3} + 2 x^{2} - 4 x + 4.\n3) Then the limit is 28.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 26.\nFinal answer: $26$", "final_answer": "26", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6a49f88fa333e142", "problem_id": "17885421928afba8", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 13 red balls and 9 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(22,7).\nFavorable samples: choose k reds and n−k blues: C(13,3)·C(9,4).\nCompute totals: C(22,7) = 170544, and favorable = 36036.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{273}{1292}$.\nFinal answer: $\\frac{273}{1292}$", "final_answer": "\\frac{273}{1292}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "66e729a7011a63c6", "problem_id": "17885421928afba8", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "An urn has 13 red balls and 9 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(22,7).\n3) Favorable samples: choose k reds and n−k blues: C(13,3)·C(9,4).\n4) Compute totals: C(22,7) = 170544, and favorable = 36036.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{273}{1292}$", "final_answer": "\\frac{273}{1292}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "afcf8c90799bce07", "problem_id": "f0f532a3dba0d260", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Two friends split $870 in the ratio 7:8. How much does each receive?", "output": "1) Total ratio parts = 7+8 = 15.\n2) Each part = 870 ÷ 15 = 58.\n3) First share = 7 × 58 = 406.\n4) Second share = 8 × 58 = 464.\nFinal answer: $464$", "final_answer": "464", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5fb7576d0e7542ba", "problem_id": "f0edbf272a0e2609", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} + x^{2} - 2 x + 4$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x) = - 2 x$.\n3) So $f'(x) = - 6 x^{2}$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x) = -2$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- 3 x^{2} + x - 1\\right)$.\nFinal answer: $2 \\left(- 3 x^{2} + x - 1\\right)$", "final_answer": "2 \\left(- 3 x^{2} + x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "bd55361cd1f6fab8", "problem_id": "782b486e54673e16", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Worker A finishes a job in 12 hours and Worker B finishes it in 15 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/12 job/hour. B's rate = 1/15 job/hour.\n2) Combined rate = 1/12 + 1/15 = 3/20 job/hour.\n3) Time = 1 ÷ (combined rate) = 20/3 hours.\nFinal answer: $\\frac{20}{3}$", "final_answer": "\\frac{20}{3}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5ae7173591687580", "problem_id": "782b486e54673e16", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Worker A finishes a job in 12 hours and Worker B finishes it in 15 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/12 job/hour. B's rate = 1/15 job/hour.\nCombined rate = 1/12 + 1/15 = 3/20 job/hour.\nTime = 1 ÷ (combined rate) = 20/3 hours.\nCheck: (combined rate)×(time) = 3/20×20/3 = 1.\nFinal answer: $\\frac{20}{3}$", "final_answer": "\\frac{20}{3}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "46eb9b51303153b1", "problem_id": "43dff622bce8ae38", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the multiplicative inverse of 44 modulo 53.", "output": "1) We want x such that 44x ≡ 1 (mod 53).\n2) Use extended Euclid to find s,t with 44s + 53t = 1.\n3) One Bezout identity is: 44·(-6) + 53·(5) = 1.\n4) Reduce s modulo 53: x ≡ 47 (mod 53).\nFinal answer: $x\\equiv 47\\pmod{53}$", "final_answer": "x\\equiv 47\\pmod{53}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1a8e55becbc7d6c5", "problem_id": "43dff622bce8ae38", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Find the multiplicative inverse of 44 modulo 53.", "output": "We want x such that 44x ≡ 1 (mod 53).\nUse extended Euclid to find s,t with 44s + 53t = 1.\nOne Bezout identity is: 44·(-6) + 53·(5) = 1.\nFinal answer: $x\\equiv 47\\pmod{53}$", "final_answer": "x\\equiv 47\\pmod{53}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4ce497295e88c830", "problem_id": "5a3ecc58cf59e101", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $214. It is discounted by 40%. You buy 12 backpacks. What is the total cost?", "output": "Discounted price = 214 × (1 − 40/100) = 214 × 60/100 = 642/5.\nSubtotal = 12 × 642/5 = 7704/5.\nCheck: recomputing the formula yields 7704/5.\nFinal answer: $\\frac{7704}{5}$", "final_answer": "\\frac{7704}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cc60d5daa2baa7f5", "problem_id": "5a3ecc58cf59e101", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A store sells a backpack for $214. It is discounted by 40%. You buy 12 backpacks. What is the total cost?", "output": "1) Discounted price = 214 × (1 − 40/100) = 214 × 60/100 = 642/5.\n2) Subtotal = 12 × 642/5 = 7704/5.\nFinal answer: $\\frac{7704}{5}$", "final_answer": "\\frac{7704}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f9dc8b856b4ac25c", "problem_id": "db847c31dbedb3b0", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 18 x + 27$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} - 18 x + 27$ = $3 \\left(x - 3\\right)^{2}$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $3$.\n5) From the second factor: x = $3$.\nFinal answer: $x \\in \\{3, 3\\}$", "final_answer": "x \\in \\{3, 3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e05205a4fe6c52e1", "problem_id": "db847c31dbedb3b0", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 18 x + 27$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} - 18 x + 27$ = $3 \\left(x - 3\\right)^{2}$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $3$.\nFrom the second factor: x = $3$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{3, 3\\}$", "final_answer": "x \\in \\{3, 3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7d73ff895bd5d082", "problem_id": "ee391dd8555cbd05", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 x^{3} - 7 x^{2} + 11 x - 21$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{4} - x^{3} - 3 x^{2} + x + 2$ and a = -2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $12 x^{3} - 3 x^{2} - 6 x + 1$.\n5) Evaluate at a: p'(-2) = $-95$.\nFinal answer: $-95$", "final_answer": "-95", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e682d527eec7a23b", "problem_id": "046f2807e22ccc9d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{3} \\sin{\\left(x \\right)} - x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 x^{3} \\sin{\\left(x \\right)}) = 3 x^{3} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = 3 x^{3} \\cos{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 x^{3} \\sin{\\left(x \\right)}) = 3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$.\nFinal answer: $3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$", "final_answer": "3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "19b86fb1f77bdd88", "problem_id": "9488f536ff79d462", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A disease has prevalence $\\frac{1}{200}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{417}{4000}$.\n3) Compute P(D|+) = $\\frac{19}{417}$.\nFinal answer: $\\frac{19}{417}$", "final_answer": "\\frac{19}{417}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "97bc532e641b7507", "problem_id": "9488f536ff79d462", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A disease has prevalence $\\frac{1}{200}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{417}{4000}$.\nCompute P(D|+) = $\\frac{19}{417}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{19}{417}$", "final_answer": "\\frac{19}{417}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ef6279866df1d212", "problem_id": "f8106611341ace03", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 11 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=5, p=$\\frac{4}{5}$ and simplify.\n4) Here C(11,5) = 462.\nFinal answer: $\\frac{473088}{48828125}$", "final_answer": "\\frac{473088}{48828125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "14c74c9efefaf1df", "problem_id": "ded2825e9d1f2dcf", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 2 x + 1$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to x + 1.\n3) Then the limit is 0.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: -1.\nFinal answer: $-1$", "final_answer": "-1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "76ad01a12843783c", "problem_id": "082525e5d03de029", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the multiplicative inverse of 13 modulo 19.", "output": "1) We want x such that 13x ≡ 1 (mod 19).\n2) Use extended Euclid to find s,t with 13s + 19t = 1.\n3) One Bezout identity is: 13·(3) + 19·(-2) = 1.\n4) Reduce s modulo 19: x ≡ 3 (mod 19).\nFinal answer: $x\\equiv 3\\pmod{19}$", "final_answer": "x\\equiv 3\\pmod{19}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2810d34301e7e3fe", "problem_id": "082525e5d03de029", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the multiplicative inverse of 13 modulo 19.", "output": "We want x such that 13x ≡ 1 (mod 19).\nUse extended Euclid to find s,t with 13s + 19t = 1.\nOne Bezout identity is: 13·(3) + 19·(-2) = 1.\nReduce s modulo 19: x ≡ 3 (mod 19).\nCheck: 13·3 mod 19 = 1.\nFinal answer: $x\\equiv 3\\pmod{19}$", "final_answer": "x\\equiv 3\\pmod{19}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bfeb923e92aa15d3", "problem_id": "9bb76206f7e9ba6f", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡1 (mod 3), x≡1 (mod 11). Give the solution modulo 33.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 33.\n3) Compute the CRT solution to get x ≡ 1 (mod 33).\nFinal answer: $x\\equiv 1\\pmod{33}$", "final_answer": "x\\equiv 1\\pmod{33}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7135c991d3cb84e0", "problem_id": "9bb76206f7e9ba6f", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡1 (mod 11). Give the solution modulo 33.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 33.\nCompute the CRT solution to get x ≡ 1 (mod 33).\nFinal answer: $x\\equiv 1\\pmod{33}$", "final_answer": "x\\equiv 1\\pmod{33}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f39011386c10e0d3", "problem_id": "6b5889dc0a0af1a5", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- 2 x + 3 y$ = $9$\n$- 3 y$ = $-9$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $-2$: $6 y$ = $18$.\nSubtract to eliminate x: $- 6 y$ = $-18$.\nSolve for y: y = $3$.\nSubstitute back to find x: x = $0$.\nCheck: plug (x,y)=(0,3) into both equations.\nFinal answer: $(x,y)=\\left(0, 3\\right)$", "final_answer": "(x,y)=\\left(0, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1273e027d9ee187d", "problem_id": "6b5889dc0a0af1a5", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$- 2 x + 3 y$ = $9$\n$- 3 y$ = $-9$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $-2$: $6 y$ = $18$.\n4) Subtract to eliminate x: $- 6 y$ = $-18$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, 3\\right)$", "final_answer": "(x,y)=\\left(0, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "36dee35460429e34", "problem_id": "f94bb98cea397d4b", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{3} \\sin{\\left(x \\right)} + x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $3 x^{3} \\sin{\\left(x \\right)} + x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} + 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} + x^{2}\\cdot \\cos{\\left(x \\right)} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}$", "final_answer": "3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} + 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} + x^{2}\\cdot \\cos{\\left(x \\right)} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b39a36c55027b483", "problem_id": "f94bb98cea397d4b", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{3} \\sin{\\left(x \\right)} + x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $3 x^{3} \\sin{\\left(x \\right)} + x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} + 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} + x^{2}\\cdot \\cos{\\left(x \\right)} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}$", "final_answer": "3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} + 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} + x^{2}\\cdot \\cos{\\left(x \\right)} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cd8a74b32fc7e43a", "problem_id": "2b28f58588e468d8", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - x^{3} - x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- x^{3} - x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2$.\n3) Compute an antiderivative: F(x) = $- \\frac{x^{4}}{4} - \\frac{x^{3}}{3} - \\frac{3 x^{2}}{2} + 2 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{x^{4}}{4} - \\frac{x^{3}}{3} - \\frac{3\\cdot x^{2}}{2} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{x^{4}}{4} - \\frac{x^{3}}{3} - \\frac{3\\cdot x^{2}}{2} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e900deffa5f2b6f1", "problem_id": "2b28f58588e468d8", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - x^{3} - x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- x^{3} - x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2$.\nCompute an antiderivative: F(x) = $- \\frac{x^{4}}{4} - \\frac{x^{3}}{3} - \\frac{3 x^{2}}{2} + 2 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - \\frac{x^{4}}{4} - \\frac{x^{3}}{3} - \\frac{3\\cdot x^{2}}{2} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{x^{4}}{4} - \\frac{x^{3}}{3} - \\frac{3\\cdot x^{2}}{2} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d82801838f34f441", "problem_id": "88cacb7ea037b76c", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 11 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=3, p=$\\frac{2}{5}$ and simplify.\n4) Here C(11,3) = 165.\nFinal answer: $\\frac{1732104}{9765625}$", "final_answer": "\\frac{1732104}{9765625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b777314352296187", "problem_id": "88cacb7ea037b76c", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 11 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=11, k=3, p=$\\frac{2}{5}$ and simplify.\nHere C(11,3) = 165.\nCheck: expression simplifies to $\\frac{1732104}{9765625}$.\nFinal answer: $\\frac{1732104}{9765625}$", "final_answer": "\\frac{1732104}{9765625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8cea31214a609189", "problem_id": "383ad08a2280152f", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 2 \\sqrt{5} x + \\frac{x}{2} - \\frac{1}{2} + \\sqrt{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{1}{2} - 2 \\sqrt{5}$, c=$- \\frac{1}{2} + \\sqrt{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{89}{4} - 6 \\sqrt{5}$, so √Δ = $\\sqrt{\\frac{89}{4} - 6 \\sqrt{5}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{1}{4} + \\frac{\\sqrt{89 - 24 \\sqrt{5}}}{4} + \\sqrt{5}$.\n7) x2 = $- \\frac{\\sqrt{89 - 24 \\sqrt{5}}}{4} - \\frac{1}{4} + \\sqrt{5}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{1}{4} + \\frac{\\sqrt{89 - 24\\cdot \\sqrt{5}}}{4} + \\sqrt{5}, - \\frac{\\sqrt{89 - 24\\cdot \\sqrt{5}}}{4} - \\frac{1}{4} + \\sqrt{5}\\}$", "final_answer": "x \\in \\{- \\frac{1}{4} + \\frac{\\sqrt{89 - 24\\cdot \\sqrt{5}}}{4} + \\sqrt{5}, - \\frac{\\sqrt{89 - 24\\cdot \\sqrt{5}}}{4} - \\frac{1}{4} + \\sqrt{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3d33104cd009c718", "problem_id": "383ad08a2280152f", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 2 \\sqrt{5} x + \\frac{x}{2} - \\frac{1}{2} + \\sqrt{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{1}{2} - 2 \\sqrt{5}$, c=$- \\frac{1}{2} + \\sqrt{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{89}{4} - 6 \\sqrt{5}$, so √Δ = $\\sqrt{\\frac{89}{4} - 6 \\sqrt{5}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{1}{4} + \\frac{\\sqrt{89 - 24 \\sqrt{5}}}{4} + \\sqrt{5}$.\nx2 = $- \\frac{\\sqrt{89 - 24 \\sqrt{5}}}{4} - \\frac{1}{4} + \\sqrt{5}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{1}{4} + \\frac{\\sqrt{89 - 24\\cdot \\sqrt{5}}}{4} + \\sqrt{5}, - \\frac{\\sqrt{89 - 24\\cdot \\sqrt{5}}}{4} - \\frac{1}{4} + \\sqrt{5}\\}$", "final_answer": "x \\in \\{- \\frac{1}{4} + \\frac{\\sqrt{89 - 24\\cdot \\sqrt{5}}}{4} + \\sqrt{5}, - \\frac{\\sqrt{89 - 24\\cdot \\sqrt{5}}}{4} - \\frac{1}{4} + \\sqrt{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ac1c8e64788cfa1e", "problem_id": "8f1eaacb1e695c23", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the linear system:\n$- 2 x - 2 y$ = $0$\n$- 8 x$ = $72$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-8$: $16 x + 16 y$ = $0$.\n3) Multiply the second equation by $-2$: $16 x$ = $-144$.\n4) Subtract to eliminate x: $16 y$ = $144$.\n5) Solve for y: y = $9$.\n6) Substitute back to find x: x = $-9$.\nFinal answer: $(x,y)=\\left(-9, 9\\right)$", "final_answer": "(x,y)=\\left(-9, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "057cae0d76b03526", "problem_id": "8f1eaacb1e695c23", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$- 2 x - 2 y$ = $0$\n$- 8 x$ = $72$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-8$: $16 x + 16 y$ = $0$.\nMultiply the second equation by $-2$: $16 x$ = $-144$.\nSubtract to eliminate x: $16 y$ = $144$.\nSolve for y: y = $9$.\nSubstitute back to find x: x = $-9$.\nCheck: plug (x,y)=(-9,9) into both equations.\nFinal answer: $(x,y)=\\left(-9, 9\\right)$", "final_answer": "(x,y)=\\left(-9, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "166b687446b84fde", "problem_id": "af76294830020138", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Find the multiplicative inverse of 2 modulo 19.", "output": "1) We want x such that 2x ≡ 1 (mod 19).\n2) Use extended Euclid to find s,t with 2s + 19t = 1.\n3) One Bezout identity is: 2·(-9) + 19·(1) = 1.\n4) Reduce s modulo 19: x ≡ 10 (mod 19).\nFinal answer: $x\\equiv 10\\pmod{19}$", "final_answer": "x\\equiv 10\\pmod{19}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2bd1f5908d24e558", "problem_id": "af76294830020138", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Find the multiplicative inverse of 2 modulo 19.", "output": "$x\\equiv 10\\pmod{19}$", "final_answer": "x\\equiv 10\\pmod{19}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0f4a5151b458fc91", "problem_id": "cf90a095a929c3ec", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 14 red balls and 8 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(22,7).\nFavorable samples: choose k reds and n−k blues: C(14,2)·C(8,5).\nCompute totals: C(22,7) = 170544, and favorable = 5096.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{637}{21318}$.\nFinal answer: $\\frac{637}{21318}$", "final_answer": "\\frac{637}{21318}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d540a7c9783d7e8", "problem_id": "cf90a095a929c3ec", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "An urn has 14 red balls and 8 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(22,7).\nFavorable samples: choose k reds and n−k blues: C(14,2)·C(8,5).\nFinal answer: $\\frac{637}{21318}$", "final_answer": "\\frac{637}{21318}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "417b50c3b7c242b8", "problem_id": "6f073458c78b7f19", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{7} x + \\frac{5 x}{3} - 4 + \\frac{4 \\sqrt{7}}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{5}{3} - \\sqrt{7}$, c=$-4 + \\frac{4 \\sqrt{7}}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{232}{9} - \\frac{26 \\sqrt{7}}{3}$, so √Δ = $\\sqrt{\\frac{232}{9} - \\frac{26 \\sqrt{7}}{3}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{5}{6} + \\frac{\\sqrt{232 - 78 \\sqrt{7}}}{6} + \\frac{\\sqrt{7}}{2}$.\n7) x2 = $- \\frac{\\sqrt{232 - 78 \\sqrt{7}}}{6} - \\frac{5}{6} + \\frac{\\sqrt{7}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{5}{6} + \\frac{\\sqrt{232 - 78\\cdot \\sqrt{7}}}{6} + \\frac{\\sqrt{7}}{2}, - \\frac{\\sqrt{232 - 78\\cdot \\sqrt{7}}}{6} - \\frac{5}{6} + \\frac{\\sqrt{7}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{5}{6} + \\frac{\\sqrt{232 - 78\\cdot \\sqrt{7}}}{6} + \\frac{\\sqrt{7}}{2}, - \\frac{\\sqrt{232 - 78\\cdot \\sqrt{7}}}{6} - \\frac{5}{6} + \\frac{\\sqrt{7}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db1c56936b9d9ff0", "problem_id": "6f073458c78b7f19", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{7} x + \\frac{5 x}{3} - 4 + \\frac{4 \\sqrt{7}}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{5}{3} - \\sqrt{7}$, c=$-4 + \\frac{4 \\sqrt{7}}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{232}{9} - \\frac{26 \\sqrt{7}}{3}$, so √Δ = $\\sqrt{\\frac{232}{9} - \\frac{26 \\sqrt{7}}{3}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{5}{6} + \\frac{\\sqrt{232 - 78 \\sqrt{7}}}{6} + \\frac{\\sqrt{7}}{2}$.\nx2 = $- \\frac{\\sqrt{232 - 78 \\sqrt{7}}}{6} - \\frac{5}{6} + \\frac{\\sqrt{7}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{5}{6} + \\frac{\\sqrt{232 - 78\\cdot \\sqrt{7}}}{6} + \\frac{\\sqrt{7}}{2}, - \\frac{\\sqrt{232 - 78\\cdot \\sqrt{7}}}{6} - \\frac{5}{6} + \\frac{\\sqrt{7}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{5}{6} + \\frac{\\sqrt{232 - 78\\cdot \\sqrt{7}}}{6} + \\frac{\\sqrt{7}}{2}, - \\frac{\\sqrt{232 - 78\\cdot \\sqrt{7}}}{6} - \\frac{5}{6} + \\frac{\\sqrt{7}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "724b24035d29eda5", "problem_id": "d03d63d2ce6875d9", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 48 x + 180$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} - 48 x + 180$ = $3 \\left(x - 10\\right) \\left(x - 6\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $10$.\n5) From the second factor: x = $6$.\nFinal answer: $x \\in \\{10, 6\\}$", "final_answer": "x \\in \\{10, 6\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5532ec4add58449d", "problem_id": "d03d63d2ce6875d9", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 48 x + 180$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} - 48 x + 180$ = $3 \\left(x - 10\\right) \\left(x - 6\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $10$.\nFrom the second factor: x = $6$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{10, 6\\}$", "final_answer": "x \\in \\{10, 6\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "353a15abd10499fe", "problem_id": "113cf6a3e6d8ad60", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$- 4 y$ = $-24$\n$- 2 x + 4 y$ = $12$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $8 y$ = $48$.\n3) Multiply the second equation by $0$: $0$ = $0$.\n4) Subtract to eliminate x: $8 y$ = $48$.\n5) Solve for y: y = $6$.\n6) Substitute back to find x: x = $6$.\nFinal answer: $(x,y)=\\left(6, 6\\right)$", "final_answer": "(x,y)=\\left(6, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "df9a6e285c553104", "problem_id": "113cf6a3e6d8ad60", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- 4 y$ = $-24$\n$- 2 x + 4 y$ = $12$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $8 y$ = $48$.\nMultiply the second equation by $0$: $0$ = $0$.\nSubtract to eliminate x: $8 y$ = $48$.\nSolve for y: y = $6$.\nSubstitute back to find x: x = $6$.\nCheck: plug (x,y)=(6,6) into both equations.\nFinal answer: $(x,y)=\\left(6, 6\\right)$", "final_answer": "(x,y)=\\left(6, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e88e4612a2b32d38", "problem_id": "354e8a17254341ab", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A disease has prevalence $\\frac{1}{25}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{13}{100}$.\n3) Compute P(D|+) = $\\frac{17}{65}$.\nFinal answer: $\\frac{17}{65}$", "final_answer": "\\frac{17}{65}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "49c359c76044c77c", "problem_id": "354e8a17254341ab", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A disease has prevalence $\\frac{1}{25}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{13}{100}$.\nCompute P(D|+) = $\\frac{17}{65}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{17}{65}$", "final_answer": "\\frac{17}{65}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4b25f7f8c7bd8bf4", "problem_id": "7338c586c5eb8c02", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- 6 x - 5 y$ = $-3$\n$7 x + 2 y$ = $38$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $7$: $- 42 x - 35 y$ = $-21$.\nMultiply the second equation by $-6$: $- 42 x - 12 y$ = $-228$.\nSubtract to eliminate x: $- 23 y$ = $207$.\nSolve for y: y = $-9$.\nSubstitute back to find x: x = $8$.\nCheck: plug (x,y)=(8,-9) into both equations.\nFinal answer: $(x,y)=\\left(8, -9\\right)$", "final_answer": "(x,y)=\\left(8, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "897f0b37c1cf84a4", "problem_id": "7338c586c5eb8c02", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$- 6 x - 5 y$ = $-3$\n$7 x + 2 y$ = $38$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $7$: $- 42 x - 35 y$ = $-21$.\n3) Multiply the second equation by $-6$: $- 42 x - 12 y$ = $-228$.\n4) Subtract to eliminate x: $- 23 y$ = $207$.\n5) Solve for y: y = $-9$.\n6) Substitute back to find x: x = $8$.\nFinal answer: $(x,y)=\\left(8, -9\\right)$", "final_answer": "(x,y)=\\left(8, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aac71b2a08d9fadf", "problem_id": "aba8aa584564eef1", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 16 times, what is the probability of exactly 9 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=16, k=9, p=$\\frac{1}{6}$ and simplify.\n4) Here C(16,9) = 11440.\nFinal answer: $\\frac{55859375}{176319369216}$", "final_answer": "\\frac{55859375}{176319369216}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "02fdaa0207cb53ad", "problem_id": "aba8aa584564eef1", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 16 times, what is the probability of exactly 9 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=16, k=9, p=$\\frac{1}{6}$ and simplify.\nHere C(16,9) = 11440.\nCheck: expression simplifies to $\\frac{55859375}{176319369216}$.\nFinal answer: $\\frac{55859375}{176319369216}$", "final_answer": "\\frac{55859375}{176319369216}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6fec9505c728f023", "problem_id": "3e91011f9f796948", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 21 miles per hour for 3/2 hours, then at 39 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 21 × 3/2 = 63/2 miles.\nSecond leg distance = 39 × 2 = 78 miles.\nTotal distance = 63/2 + 78 = 219/2 miles.\nCheck: adding both legs gives 219/2.\nFinal answer: $\\frac{219}{2}$", "final_answer": "\\frac{219}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "01a7056ddad95856", "problem_id": "3e91011f9f796948", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A cyclist rides at 21 miles per hour for 3/2 hours, then at 39 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 21 × 3/2 = 63/2 miles.\n2) Second leg distance = 39 × 2 = 78 miles.\n3) Total distance = 63/2 + 78 = 219/2 miles.\nFinal answer: $\\frac{219}{2}$", "final_answer": "\\frac{219}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d469b09c5dea21b1", "problem_id": "6baed82c882284b5", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$8 x - 6 y$ = $-64$\n$- 5 x - 8 y$ = $40$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-5$: $- 40 x + 30 y$ = $320$.\n3) Multiply the second equation by $8$: $- 40 x - 64 y$ = $320$.\n4) Subtract to eliminate x: $94 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $-8$.\nFinal answer: $(x,y)=\\left(-8, 0\\right)$", "final_answer": "(x,y)=\\left(-8, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "22f63b9cb96ad896", "problem_id": "6baed82c882284b5", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$8 x - 6 y$ = $-64$\n$- 5 x - 8 y$ = $40$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-5$: $- 40 x + 30 y$ = $320$.\nMultiply the second equation by $8$: $- 40 x - 64 y$ = $320$.\nSubtract to eliminate x: $94 y$ = $0$.\nSolve for y: y = $0$.\nSubstitute back to find x: x = $-8$.\nCheck: plug (x,y)=(-8,0) into both equations.\nFinal answer: $(x,y)=\\left(-8, 0\\right)$", "final_answer": "(x,y)=\\left(-8, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bd6754a6cf51a826", "problem_id": "c7639cbee79b57ee", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡8 (mod 11), x≡5 (mod 13). Give the solution modulo 143.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 143.\nCompute the CRT solution to get x ≡ 96 (mod 143).\nCheck: 96 mod 11=8 and mod 13=5.\nFinal answer: $x\\equiv 96\\pmod{143}$", "final_answer": "x\\equiv 96\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "89e00854f3537bc9", "problem_id": "c7639cbee79b57ee", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡8 (mod 11), x≡5 (mod 13). Give the solution modulo 143.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 143.\n3) Compute the CRT solution to get x ≡ 96 (mod 143).\nFinal answer: $x\\equiv 96\\pmod{143}$", "final_answer": "x\\equiv 96\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9a9625a539a1b8b5", "problem_id": "87783971dad38077", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A store sells a backpack for $930. It is discounted by 30%. You buy 8 backpacks. What is the total cost?", "output": "Discounted price = 930 × (1 − 30/100) = 930 × 70/100 = 651.\nSubtotal = 8 × 651 = 5208.\nCheck: recomputing the formula yields 5208.\nFinal answer: $5208$", "final_answer": "5208", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "605ab798c9d25a9a", "problem_id": "87783971dad38077", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A store sells a backpack for $930. It is discounted by 30%. You buy 8 backpacks. What is the total cost?", "output": "1) Discounted price = 930 × (1 − 30/100) = 930 × 70/100 = 651.\n2) Subtotal = 8 × 651 = 5208.\nFinal answer: $5208$", "final_answer": "5208", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d717d3b170a49890", "problem_id": "773909b394ed0124", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{3} x + \\frac{29 x}{12} - \\frac{7 \\sqrt{3}}{4} + \\frac{7}{6}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{29}{12} - \\sqrt{3}$, c=$\\frac{7}{6} - \\frac{7 \\sqrt{3}}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{13 \\sqrt{3}}{6} + \\frac{601}{144}$, so √Δ = $\\sqrt{\\frac{13 \\sqrt{3}}{6} + \\frac{601}{144}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{29}{24} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{312 \\sqrt{3} + 601}}{24}$.\n7) x2 = $- \\frac{\\sqrt{312 \\sqrt{3} + 601}}{24} - \\frac{29}{24} + \\frac{\\sqrt{3}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{29}{24} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{312\\cdot \\sqrt{3} + 601}}{24}, - \\frac{\\sqrt{312\\cdot \\sqrt{3} + 601}}{24} - \\frac{29}{24} + \\frac{\\sqrt{3}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{29}{24} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{312\\cdot \\sqrt{3} + 601}}{24}, - \\frac{\\sqrt{312\\cdot \\sqrt{3} + 601}}{24} - \\frac{29}{24} + \\frac{\\sqrt{3}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b0a5b95faba64d74", "problem_id": "773909b394ed0124", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{3} x + \\frac{29 x}{12} - \\frac{7 \\sqrt{3}}{4} + \\frac{7}{6}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{29}{12} - \\sqrt{3}$, c=$\\frac{7}{6} - \\frac{7 \\sqrt{3}}{4}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{- \\frac{29}{24} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{312\\cdot \\sqrt{3} + 601}}{24}, - \\frac{\\sqrt{312\\cdot \\sqrt{3} + 601}}{24} - \\frac{29}{24} + \\frac{\\sqrt{3}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{29}{24} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{312\\cdot \\sqrt{3} + 601}}{24}, - \\frac{\\sqrt{312\\cdot \\sqrt{3} + 601}}{24} - \\frac{29}{24} + \\frac{\\sqrt{3}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b456d2421ded4019", "problem_id": "54d4bda99e0443cd", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$- 4 x + 6 y$ = $50$\n$- 6 x - 6 y$ = $0$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-6$: $24 x - 36 y$ = $-300$.\n3) Multiply the second equation by $-4$: $24 x + 24 y$ = $0$.\n4) Subtract to eliminate x: $- 60 y$ = $-300$.\n5) Solve for y: y = $5$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, 5\\right)$", "final_answer": "(x,y)=\\left(-5, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3575e70f757eb31c", "problem_id": "54d4bda99e0443cd", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$- 4 x + 6 y$ = $50$\n$- 6 x - 6 y$ = $0$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-6$: $24 x - 36 y$ = $-300$.\nMultiply the second equation by $-4$: $24 x + 24 y$ = $0$.\nSubtract to eliminate x: $- 60 y$ = $-300$.\nSolve for y: y = $5$.\nSubstitute back to find x: x = $-5$.\nCheck: plug (x,y)=(-5,5) into both equations.\nFinal answer: $(x,y)=\\left(-5, 5\\right)$", "final_answer": "(x,y)=\\left(-5, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff94ebcefc90c75d", "problem_id": "37c1fa38fb04f96b", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $- 2 x^{3} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $2 \\left(- x^{3} \\cos{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 3 \\sin{\\left(2 x \\right)}\\right)$.\nFinal answer: $- 2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x\\cdot \\cos{\\left(x \\right)} - 2\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)}$", "final_answer": "- 2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x\\cdot \\cos{\\left(x \\right)} - 2\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "137e753d0911e48c", "problem_id": "37c1fa38fb04f96b", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $- 2 x^{3} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $2 \\left(- x^{3} \\cos{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 3 \\sin{\\left(2 x \\right)}\\right)$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x\\cdot \\cos{\\left(x \\right)} - 2\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)}$", "final_answer": "- 2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x\\cdot \\cos{\\left(x \\right)} - 2\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b1526784927fbe2f", "problem_id": "9368ca5af380de55", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 4 x^{3} - 4 x^{2} - 3 x\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 4 x^{3} - 4 x^{2} - 3 x$.\n3) Compute an antiderivative: F(x) = $- x^{4} - \\frac{4 x^{3}}{3} - \\frac{3 x^{2}}{2}$.\n4) Include the constant of integration C.\nFinal answer: $C - x^{4} - \\frac{4\\cdot x^{3}}{3} - \\frac{3\\cdot x^{2}}{2}$", "final_answer": "C - x^{4} - \\frac{4\\cdot x^{3}}{3} - \\frac{3\\cdot x^{2}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e1d8267360542d6", "problem_id": "127845561a322a32", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{3} + 4 x^{2} + 7 x + 11$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} + 4 x^{2} + 6 x + 11.\n3) Then the limit is 55.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 57.\nFinal answer: $57$", "final_answer": "57", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3da9fb6d7bdb44cf", "problem_id": "929499a07de2f8df", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the multiplicative inverse of 19 modulo 23.", "output": "1) We want x such that 19x ≡ 1 (mod 23).\n2) Use extended Euclid to find s,t with 19s + 23t = 1.\n3) One Bezout identity is: 19·(-6) + 23·(5) = 1.\n4) Reduce s modulo 23: x ≡ 17 (mod 23).\nFinal answer: $x\\equiv 17\\pmod{23}$", "final_answer": "x\\equiv 17\\pmod{23}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6398c408a00a1bce", "problem_id": "929499a07de2f8df", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the multiplicative inverse of 19 modulo 23.", "output": "We want x such that 19x ≡ 1 (mod 23).\nUse extended Euclid to find s,t with 19s + 23t = 1.\nOne Bezout identity is: 19·(-6) + 23·(5) = 1.\nReduce s modulo 23: x ≡ 17 (mod 23).\nCheck: 19·17 mod 23 = 1.\nFinal answer: $x\\equiv 17\\pmod{23}$", "final_answer": "x\\equiv 17\\pmod{23}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "369ae7ae21556d76", "problem_id": "ece95a7cd994dc04", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 21 times, what is the probability of exactly 16 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=21, k=16, p=$\\frac{2}{3}$ and simplify.\n4) Here C(21,16) = 20349.\nFinal answer: $\\frac{148176896}{1162261467}$", "final_answer": "\\frac{148176896}{1162261467}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e8d91e911b85e69", "problem_id": "d8ba5ecd6ddbe0bf", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 3 x^{3} - 6 x^{2} - 9 x - 15$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{3} - 6 x^{2} - 10 x - 15.\n3) Then the limit is -83.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -81.\nFinal answer: $-81$", "final_answer": "-81", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d4cacd00c4a55a00", "problem_id": "3b4f27a23d30cfd9", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$- 4 x - 4 y$ = $8$\n$- 6 x$ = $48$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-6$: $24 x + 24 y$ = $-48$.\n3) Multiply the second equation by $-4$: $24 x$ = $-192$.\n4) Subtract to eliminate x: $24 y$ = $144$.\n5) Solve for y: y = $6$.\n6) Substitute back to find x: x = $-8$.\nFinal answer: $(x,y)=\\left(-8, 6\\right)$", "final_answer": "(x,y)=\\left(-8, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "630ad144d08e25eb", "problem_id": "3b4f27a23d30cfd9", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$- 4 x - 4 y$ = $8$\n$- 6 x$ = $48$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-6$: $24 x + 24 y$ = $-48$.\nMultiply the second equation by $-4$: $24 x$ = $-192$.\nSubtract to eliminate x: $24 y$ = $144$.\nSolve for y: y = $6$.\nSubstitute back to find x: x = $-8$.\nCheck: plug (x,y)=(-8,6) into both equations.\nFinal answer: $(x,y)=\\left(-8, 6\\right)$", "final_answer": "(x,y)=\\left(-8, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a824330f86872b1c", "problem_id": "d3b896883ce251b6", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$4 x + 4 y$ = $68$\n$5 x - 7 y$ = $-11$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $5$: $20 x + 20 y$ = $340$.\n3) Multiply the second equation by $4$: $20 x - 28 y$ = $-44$.\n4) Subtract to eliminate x: $48 y$ = $384$.\n5) Solve for y: y = $8$.\n6) Substitute back to find x: x = $9$.\nFinal answer: $(x,y)=\\left(9, 8\\right)$", "final_answer": "(x,y)=\\left(9, 8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a18c90e7fac82f1", "problem_id": "d3b896883ce251b6", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the linear system:\n$4 x + 4 y$ = $68$\n$5 x - 7 y$ = $-11$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $5$: $20 x + 20 y$ = $340$.\nMultiply the second equation by $4$: $20 x - 28 y$ = $-44$.\nFinal answer: $(x,y)=\\left(9, 8\\right)$", "final_answer": "(x,y)=\\left(9, 8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "829382e7d0db119f", "problem_id": "e128671d2aac1cd5", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A disease has prevalence $\\frac{3}{50}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{49}{500}$.\nCompute P(D|+) = $\\frac{51}{98}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{51}{98}$", "final_answer": "\\frac{51}{98}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a1725c728f773080", "problem_id": "e128671d2aac1cd5", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A disease has prevalence $\\frac{3}{50}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{49}{500}$.\n3) Compute P(D|+) = $\\frac{51}{98}$.\nFinal answer: $\\frac{51}{98}$", "final_answer": "\\frac{51}{98}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0dd02c85e6355fbd", "problem_id": "83f32936f10a9e56", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} x^{3} + 2 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $x^{3} + 2 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{15 \\pi^{4}}{64} - 3 \\pi - 2 + \\frac{3 \\pi^{2}}{4} + \\frac{3 \\pi^{3}}{4}$.\n4) Simplify the result.\nFinal answer: $- \\frac{15\\cdot \\pi^{4}}{64} - 3\\cdot \\pi - 2 + \\frac{3\\cdot \\pi^{2}}{4} + \\frac{3\\cdot \\pi^{3}}{4}$", "final_answer": "- \\frac{15\\cdot \\pi^{4}}{64} - 3\\cdot \\pi - 2 + \\frac{3\\cdot \\pi^{2}}{4} + \\frac{3\\cdot \\pi^{3}}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ca324d3c18c4c9fe", "problem_id": "83f32936f10a9e56", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} x^{3} + 2 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $x^{3} + 2 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{15 \\pi^{4}}{64} - 3 \\pi - 2 + \\frac{3 \\pi^{2}}{4} + \\frac{3 \\pi^{3}}{4}$.\nFinal answer: $- \\frac{15\\cdot \\pi^{4}}{64} - 3\\cdot \\pi - 2 + \\frac{3\\cdot \\pi^{2}}{4} + \\frac{3\\cdot \\pi^{3}}{4}$", "final_answer": "- \\frac{15\\cdot \\pi^{4}}{64} - 3\\cdot \\pi - 2 + \\frac{3\\cdot \\pi^{2}}{4} + \\frac{3\\cdot \\pi^{3}}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "357e056a457b8ba4", "problem_id": "df768cff59b9db9c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(3 x^{2} - x + 1\\right) \\cos{\\left(x^{2} + x + 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x e^{x}) = x e^{x}$.\n3) So $f'(x) = - 6 x^{3} \\sin{\\left(\\left(x^{2} + x\\right) + 2 \\right)} - x^{2} \\sin{\\left(\\left(x^{2} + x\\right) + 2 \\right)} + x e^{x} - x \\sin{\\left(\\left(x^{2} + x\\right) + 2 \\right)} + 6 x \\cos{\\left(\\left(x^{2} + x\\right) + 2 \\right)} - 3 e^{x} - \\sin{\\left(\\left(x^{2} + x\\right) + 2 \\right)} - \\cos{\\left(\\left(x^{2} + x\\right) + 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x e^{x}) = x e^{x} + e^{x}$.\nTherefore the correct derivative is $f'(x) = \\left(x - 3\\right) e^{x} - \\left(2 x + 1\\right) \\left(x \\left(3 x - 1\\right) + 1\\right) \\sin{\\left(\\left(x^{2} + x\\right) + 2 \\right)} + \\left(6 x - 1\\right) \\cos{\\left(\\left(x^{2} + x\\right) + 2 \\right)} + e^{x}$.\nFinal answer: $\\left(x - 3\\right) e^{x} - \\left(2 x + 1\\right) \\left(x \\left(3 x - 1\\right) + 1\\right) \\sin{\\left(\\left(x^{2} + x\\right) + 2 \\right)} + \\left(6 x - 1\\right) \\cos{\\left(\\left(x^{2} + x\\right) + 2 \\right)} + e^{x}$", "final_answer": "\\left(x - 3\\right) e^{x} - \\left(2 x + 1\\right) \\left(x \\left(3 x - 1\\right) + 1\\right) \\sin{\\left(\\left(x^{2} + x\\right) + 2 \\right)} + \\left(6 x - 1\\right) \\cos{\\left(\\left(x^{2} + x\\right) + 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8e2a27a820972fd5", "problem_id": "646a3b6e4a3be68d", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Find the equation of the tangent line to $y=3 x^{2} - x + 2 \\sin{\\left(x \\right)}$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $6 x + 2 \\cos{\\left(x \\right)} - 1$.\n2) Slope at x=-2: m = f'(-2) = $-13 + 2 \\cos{\\left(2 \\right)}$.\n3) Point on curve: (-2, f(-2)) = (-2, $14 - 2 \\sin{\\left(2 \\right)}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 14 - 2 \\sin{\\left(2 \\right)} = -13 + 2 \\cos{\\left(2 \\right)}(x − -2).\n6) Optionally expand: y = $- 13 x + 2 x \\cos{\\left(2 \\right)} - 12 - 2 \\sin{\\left(2 \\right)} + 4 \\cos{\\left(2 \\right)}$.\nFinal answer: $y = - 13\\cdot x + 2\\cdot x\\cdot \\cos{\\left(2 \\right)} - 12 - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)}$", "final_answer": "y = - 13\\cdot x + 2\\cdot x\\cdot \\cos{\\left(2 \\right)} - 12 - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ee9cfa429082c14d", "problem_id": "08a40ac65250480d", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{38 x}{5} - \\frac{8}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$- \\frac{38}{5}$, c=$- \\frac{8}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{4, - \\frac{1}{5}\\}$", "final_answer": "x \\in \\{4, - \\frac{1}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4d84b250281c14de", "problem_id": "08a40ac65250480d", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{38 x}{5} - \\frac{8}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$- \\frac{38}{5}$, c=$- \\frac{8}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{1764}{25}$, so √Δ = $\\frac{42}{5}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $4$.\n7) x2 = $- \\frac{1}{5}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{4, - \\frac{1}{5}\\}$", "final_answer": "x \\in \\{4, - \\frac{1}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ffbdc4d8348191b7", "problem_id": "2e477fd7cf3acd46", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} - x^{3} + x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{3} + x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $-2 + \\frac{3 \\pi^{2}}{4} + \\frac{3 \\pi^{3}}{8} + \\frac{15 \\pi^{4}}{64}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $-2 + \\frac{3\\cdot \\pi^{2}}{4} + \\frac{3\\cdot \\pi^{3}}{8} + \\frac{15\\cdot \\pi^{4}}{64}$", "final_answer": "-2 + \\frac{3\\cdot \\pi^{2}}{4} + \\frac{3\\cdot \\pi^{3}}{8} + \\frac{15\\cdot \\pi^{4}}{64}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4be0336597053de9", "problem_id": "29f1f9e398c80276", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 3 x^{3} + 9 x^{2} + 17 x + 31$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} + 9 x^{2} + 16 x + 31.\n3) Then the limit is 123.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 125.\nFinal answer: $125$", "final_answer": "125", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "36360ce50f5c45dc", "problem_id": "c9c46c9f83cd1f6f", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A cyclist rides at 75 miles per hour for 1 hours, then at 49 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 75 × 1 = 75 miles.\nSecond leg distance = 49 × 1 = 49 miles.\nTotal distance = 75 + 49 = 124 miles.\nCheck: adding both legs gives 124.\nFinal answer: $124$", "final_answer": "124", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b537afb60edffbf3", "problem_id": "c9c46c9f83cd1f6f", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A cyclist rides at 75 miles per hour for 1 hours, then at 49 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 75 × 1 = 75 miles.\n2) Second leg distance = 49 × 1 = 49 miles.\n3) Total distance = 75 + 49 = 124 miles.\nFinal answer: $124$", "final_answer": "124", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dc4cf4e1aa8828e1", "problem_id": "647464f7511ece8a", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$- 6 x - 4 y$ = $6$\n$6 x + 3 y$ = $-9$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $6$: $- 36 x - 24 y$ = $36$.\n3) Multiply the second equation by $-6$: $- 36 x - 18 y$ = $54$.\n4) Subtract to eliminate x: $- 6 y$ = $-18$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, 3\\right)$", "final_answer": "(x,y)=\\left(-3, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a97bfd773d98673b", "problem_id": "8ca33e415b9c53cf", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} + 3 x + 4$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x^{2}) = - 4 x^{2}$.\n3) So $f'(x) = 3 - 4 x^{2}$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x^{2}) = - 4 x$.\nTherefore the correct derivative is $f'(x) = 3 - 4 x$.\nFinal answer: $3 - 4 x$", "final_answer": "3 - 4 x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8d026144f9da3d66", "problem_id": "1e22d24a68ea403b", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$4 y$ = $-20$\n$4 x - y$ = $5$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $16 y$ = $-80$.\n3) Multiply the second equation by $0$: $0$ = $0$.\n4) Subtract to eliminate x: $16 y$ = $-80$.\n5) Solve for y: y = $-5$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, -5\\right)$", "final_answer": "(x,y)=\\left(0, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6f420b12c9d80be5", "problem_id": "1e22d24a68ea403b", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$4 y$ = $-20$\n$4 x - y$ = $5$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $16 y$ = $-80$.\nMultiply the second equation by $0$: $0$ = $0$.\nSubtract to eliminate x: $16 y$ = $-80$.\nSolve for y: y = $-5$.\nSubstitute back to find x: x = $0$.\nCheck: plug (x,y)=(0,-5) into both equations.\nFinal answer: $(x,y)=\\left(0, -5\\right)$", "final_answer": "(x,y)=\\left(0, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b10eaa5ae9db8879", "problem_id": "e4bc3407d28c0cba", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 13 x + 20$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$-13$, c=$20$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $9$, so √Δ = $3$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $4$.\n7) x2 = $\\frac{5}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{4, \\frac{5}{2}\\}$", "final_answer": "x \\in \\{4, \\frac{5}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0439a716a84a5c28", "problem_id": "192e44a45e2bafd4", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} - x + 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x^{3}) = 3 x^{3}$.\n3) So $f'(x) = 3 x^{3} - 1$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x^{3}) = 3 x^{2}$.\nTherefore the correct derivative is $f'(x) = 3 x^{2} - 1$.\nFinal answer: $3 x^{2} - 1$", "final_answer": "3 x^{2} - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "653c1a0af014ed6a", "problem_id": "34c921c0ba47d56e", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_6=18$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_6=6.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=6 and k=6.\n4) Compute C(11,5) = 462.\nFinal answer: $462$", "final_answer": "462", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ee84a23a9144b3be", "problem_id": "34c921c0ba47d56e", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_6=18$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_6=6.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=6 and k=6.\nCompute C(11,5) = 462.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $462$", "final_answer": "462", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4349aa3bc165b306", "problem_id": "f476f20cd4fa4a2e", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_7=39$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_7=25.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=25 and k=7.\n4) Compute C(31,6) = 736281.\nFinal answer: $736281$", "final_answer": "736281", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "46866b40529fd963", "problem_id": "f476f20cd4fa4a2e", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_7=39$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_7=25.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=25 and k=7.\nCompute C(31,6) = 736281.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $736281$", "final_answer": "736281", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72cd09f2c02b1dae", "problem_id": "6ded8149f7046695", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 24 miles per hour for 3/2 hours, then at 86 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 24 × 3/2 = 36 miles.\nSecond leg distance = 86 × 2 = 172 miles.\nTotal distance = 36 + 172 = 208 miles.\nCheck: adding both legs gives 208.\nFinal answer: $208$", "final_answer": "208", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "02228d4fc14a81a8", "problem_id": "6ded8149f7046695", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A cyclist rides at 24 miles per hour for 3/2 hours, then at 86 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 24 × 3/2 = 36 miles.\n2) Second leg distance = 86 × 2 = 172 miles.\n3) Total distance = 36 + 172 = 208 miles.\nFinal answer: $208$", "final_answer": "208", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "055c721cfb8d0685", "problem_id": "9c603f79a61202df", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "You want 42 liters of a 40% salt solution. You have a 30% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 42−x liters is 70% solution.\nSalt equation: 30x + 70(42−x) = 40·42.\n(30-70)x = (40-70)·42 ⇒ x = 63/2.\nFinal answer: $63/2 L of 30% and 21/2 L of 70%$", "final_answer": "63/2 L of 30% and 21/2 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "50d978b2811b363c", "problem_id": "9c603f79a61202df", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "You want 42 liters of a 40% salt solution. You have a 30% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 42−x liters is 70% solution.\n2) Salt equation: 30x + 70(42−x) = 40·42.\n3) (30-70)x = (40-70)·42 ⇒ x = 63/2.\n4) Then 42−x = 21/2.\nFinal answer: $63/2 L of 30% and 21/2 L of 70%$", "final_answer": "63/2 L of 30% and 21/2 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dd8955a27bbb4ae0", "problem_id": "d65674277b7464b1", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{2 x}{5} - \\frac{8}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$\\frac{2}{5}$, c=$- \\frac{8}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{324}{25}$, so √Δ = $\\frac{18}{5}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{4}{5}$.\n7) x2 = $-1$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{4}{5}, -1\\}$", "final_answer": "x \\in \\{\\frac{4}{5}, -1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fd7541ec6eb85f57", "problem_id": "d65674277b7464b1", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{2 x}{5} - \\frac{8}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$\\frac{2}{5}$, c=$- \\frac{8}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{4}{5}, -1\\}$", "final_answer": "x \\in \\{\\frac{4}{5}, -1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3274d096db4220cc", "problem_id": "a351e3480b9fa015", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} 2 x^{3} + 6 x^{2} + 15 x + 42$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $2 x^{4} - 3 x^{2} - 3 x - 1$ and a = 3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $8 x^{3} - 6 x - 3$.\n5) Evaluate at a: p'(3) = $195$.\nFinal answer: $195$", "final_answer": "195", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72b255d9d1d91b26", "problem_id": "a351e3480b9fa015", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} 2 x^{3} + 6 x^{2} + 15 x + 42$.", "output": "$195$", "final_answer": "195", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "36011b04f23dce83", "problem_id": "3d3a3832d17b97b3", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A disease has prevalence $\\frac{3}{100}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{31}{250}$.\n3) Compute P(D|+) = $\\frac{27}{124}$.\nFinal answer: $\\frac{27}{124}$", "final_answer": "\\frac{27}{124}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "af3df727fa1317ed", "problem_id": "3d3a3832d17b97b3", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A disease has prevalence $\\frac{3}{100}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{31}{250}$.\nCompute P(D|+) = $\\frac{27}{124}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{27}{124}$", "final_answer": "\\frac{27}{124}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "28d0f66349174f16", "problem_id": "910fd6fef821479d", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$6 x - 5 y$ = $33$\n$- 4 x + 8 y$ = $-64$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $- 24 x + 20 y$ = $-132$.\n3) Multiply the second equation by $6$: $- 24 x + 48 y$ = $-384$.\n4) Subtract to eliminate x: $- 28 y$ = $252$.\n5) Solve for y: y = $-9$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, -9\\right)$", "final_answer": "(x,y)=\\left(-2, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "30eeda227b8e3345", "problem_id": "910fd6fef821479d", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$6 x - 5 y$ = $33$\n$- 4 x + 8 y$ = $-64$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $- 24 x + 20 y$ = $-132$.\nMultiply the second equation by $6$: $- 24 x + 48 y$ = $-384$.\nSubtract to eliminate x: $- 28 y$ = $252$.\nSolve for y: y = $-9$.\nSubstitute back to find x: x = $-2$.\nCheck: plug (x,y)=(-2,-9) into both equations.\nFinal answer: $(x,y)=\\left(-2, -9\\right)$", "final_answer": "(x,y)=\\left(-2, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2e4a62a6b8b2a28e", "problem_id": "ea1789cff106c30e", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "You want 30 liters of a 50% salt solution. You have a 30% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 30−x liters is 70% solution.\n2) Salt equation: 30x + 70(30−x) = 50·30.\n3) (30-70)x = (50-70)·30 ⇒ x = 15.\n4) Then 30−x = 15.\nFinal answer: $15 L of 30% and 15 L of 70%$", "final_answer": "15 L of 30% and 15 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c00a3f070598b06f", "problem_id": "ea1789cff106c30e", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "You want 30 liters of a 50% salt solution. You have a 30% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 30−x liters is 70% solution.\nSalt equation: 30x + 70(30−x) = 50·30.\n(30-70)x = (50-70)·30 ⇒ x = 15.\nFinal answer: $15 L of 30% and 15 L of 70%$", "final_answer": "15 L of 30% and 15 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8cac86ee25ce78ae", "problem_id": "d5dc854944b9506b", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "You want 41 liters of a 25% salt solution. You have a 20% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 41−x liters is 60% solution.\nSalt equation: 20x + 60(41−x) = 25·41.\n(20-60)x = (25-60)·41 ⇒ x = 287/8.\nThen 41−x = 41/8.\nCheck: 20·287/8 + 60·41/8 = 25·41.\nFinal answer: $287/8 L of 20% and 41/8 L of 60%$", "final_answer": "287/8 L of 20% and 41/8 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d1dfabab1ef42ff8", "problem_id": "d5dc854944b9506b", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "You want 41 liters of a 25% salt solution. You have a 20% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 41−x liters is 60% solution.\n2) Salt equation: 20x + 60(41−x) = 25·41.\n3) (20-60)x = (25-60)·41 ⇒ x = 287/8.\n4) Then 41−x = 41/8.\nFinal answer: $287/8 L of 20% and 41/8 L of 60%$", "final_answer": "287/8 L of 20% and 41/8 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "65ef78162ccd6c50", "problem_id": "d9a0f61d269f57de", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(3 x^{3} - x^{2} - x - 3\\right) \\cos{\\left(2 x - 3 \\right)}$.", "output": "1) Let f(x) = $2 \\left(x + 2\\right) e^{x} + \\left(3 x^{3} - x^{2} - x - 3\\right) \\cos{\\left(2 x - 3 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $2 \\left(x + 2\\right) e^{x} + \\left(9 x^{2} - 2 x - 1\\right) \\cos{\\left(2 x - 3 \\right)} - 2 \\left(3 x^{3} - x^{2} - x - 3\\right) \\sin{\\left(2 x - 3 \\right)} + 2 e^{x}$.\nFinal answer: $\\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(9\\cdot x^{2} - 2\\cdot x - 1\\right)\\cdot \\cos{\\left(2\\cdot x - 3 \\right)} - \\left(6\\cdot x^{3} - 2\\cdot x^{2} - 2\\cdot x - 6\\right)\\cdot \\sin{\\left(2\\cdot x - 3 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(9\\cdot x^{2} - 2\\cdot x - 1\\right)\\cdot \\cos{\\left(2\\cdot x - 3 \\right)} - \\left(6\\cdot x^{3} - 2\\cdot x^{2} - 2\\cdot x - 6\\right)\\cdot \\sin{\\left(2\\cdot x - 3 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a5e34e694a5b0b2a", "problem_id": "d9a0f61d269f57de", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(3 x^{3} - x^{2} - x - 3\\right) \\cos{\\left(2 x - 3 \\right)}$.", "output": "Let f(x) = $2 \\left(x + 2\\right) e^{x} + \\left(3 x^{3} - x^{2} - x - 3\\right) \\cos{\\left(2 x - 3 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $2 \\left(x + 2\\right) e^{x} + \\left(9 x^{2} - 2 x - 1\\right) \\cos{\\left(2 x - 3 \\right)} - 2 \\left(3 x^{3} - x^{2} - x - 3\\right) \\sin{\\left(2 x - 3 \\right)} + 2 e^{x}$.\nFinal answer: $\\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(9\\cdot x^{2} - 2\\cdot x - 1\\right)\\cdot \\cos{\\left(2\\cdot x - 3 \\right)} - \\left(6\\cdot x^{3} - 2\\cdot x^{2} - 2\\cdot x - 6\\right)\\cdot \\sin{\\left(2\\cdot x - 3 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(9\\cdot x^{2} - 2\\cdot x - 1\\right)\\cdot \\cos{\\left(2\\cdot x - 3 \\right)} - \\left(6\\cdot x^{3} - 2\\cdot x^{2} - 2\\cdot x - 6\\right)\\cdot \\sin{\\left(2\\cdot x - 3 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ad53e9e1b1956942", "problem_id": "1d145a9c44f1532a", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Two friends split $680 in the ratio 6:2. How much does each receive?", "output": "1) Total ratio parts = 6+2 = 8.\n2) Each part = 680 ÷ 8 = 85.\n3) First share = 6 × 85 = 510.\n4) Second share = 2 × 85 = 170.\nFinal answer: $170$", "final_answer": "170", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5c4edd9b3a86501d", "problem_id": "1d145a9c44f1532a", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $680 in the ratio 6:2. How much does each receive?", "output": "Total ratio parts = 6+2 = 8.\nEach part = 680 ÷ 8 = 85.\nFirst share = 6 × 85 = 510.\nSecond share = 2 × 85 = 170.\nCheck: 510+170=680.\nFinal answer: $170$", "final_answer": "170", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5090b9af02e6be48", "problem_id": "34956e149d5b0fcf", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{11} x + \\frac{11 x}{6} - \\frac{5 \\sqrt{11}}{2} - \\frac{5}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{11}{6} - \\sqrt{11}$, c=$- \\frac{5 \\sqrt{11}}{2} - \\frac{5}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{19 \\sqrt{11}}{3} + \\frac{757}{36}$, so √Δ = $\\sqrt{\\frac{19 \\sqrt{11}}{3} + \\frac{757}{36}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{11}{12} + \\frac{\\sqrt{11}}{2} + \\frac{\\sqrt{228 \\sqrt{11} + 757}}{12}$.\n7) x2 = $- \\frac{\\sqrt{228 \\sqrt{11} + 757}}{12} - \\frac{11}{12} + \\frac{\\sqrt{11}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{11}{12} + \\frac{\\sqrt{11}}{2} + \\frac{\\sqrt{228\\cdot \\sqrt{11} + 757}}{12}, - \\frac{\\sqrt{228\\cdot \\sqrt{11} + 757}}{12} - \\frac{11}{12} + \\frac{\\sqrt{11}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{11}{12} + \\frac{\\sqrt{11}}{2} + \\frac{\\sqrt{228\\cdot \\sqrt{11} + 757}}{12}, - \\frac{\\sqrt{228\\cdot \\sqrt{11} + 757}}{12} - \\frac{11}{12} + \\frac{\\sqrt{11}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6543ef76dfd695cf", "problem_id": "34956e149d5b0fcf", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{11} x + \\frac{11 x}{6} - \\frac{5 \\sqrt{11}}{2} - \\frac{5}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{11}{6} - \\sqrt{11}$, c=$- \\frac{5 \\sqrt{11}}{2} - \\frac{5}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{19 \\sqrt{11}}{3} + \\frac{757}{36}$, so √Δ = $\\sqrt{\\frac{19 \\sqrt{11}}{3} + \\frac{757}{36}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{11}{12} + \\frac{\\sqrt{11}}{2} + \\frac{\\sqrt{228 \\sqrt{11} + 757}}{12}$.\nx2 = $- \\frac{\\sqrt{228 \\sqrt{11} + 757}}{12} - \\frac{11}{12} + \\frac{\\sqrt{11}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{11}{12} + \\frac{\\sqrt{11}}{2} + \\frac{\\sqrt{228\\cdot \\sqrt{11} + 757}}{12}, - \\frac{\\sqrt{228\\cdot \\sqrt{11} + 757}}{12} - \\frac{11}{12} + \\frac{\\sqrt{11}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{11}{12} + \\frac{\\sqrt{11}}{2} + \\frac{\\sqrt{228\\cdot \\sqrt{11} + 757}}{12}, - \\frac{\\sqrt{228\\cdot \\sqrt{11} + 757}}{12} - \\frac{11}{12} + \\frac{\\sqrt{11}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "802a7de3e636c207", "problem_id": "dc65040bfa353515", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 6 x - 36$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} + 6 x - 36$ = $2 \\left(x - 3\\right) \\left(x + 6\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $3$.\n5) From the second factor: x = $-6$.\nFinal answer: $x \\in \\{3, -6\\}$", "final_answer": "x \\in \\{3, -6\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8b9a6da65685ecab", "problem_id": "dc65040bfa353515", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 6 x - 36$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} + 6 x - 36$ = $2 \\left(x - 3\\right) \\left(x + 6\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $3$.\nFrom the second factor: x = $-6$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{3, -6\\}$", "final_answer": "x \\in \\{3, -6\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d80dbedbb95085af", "problem_id": "ae0ce6d4c81b5ec1", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 3 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2$.\n3) Compute an antiderivative: F(x) = $- x^{3} - x^{2} + 2 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C - x^{3} - x^{2} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - x^{3} - x^{2} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f844106daba09ee2", "problem_id": "ae0ce6d4c81b5ec1", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 3 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2$.\nCompute an antiderivative: F(x) = $- x^{3} - x^{2} + 2 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - x^{3} - x^{2} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - x^{3} - x^{2} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f9e0b4ef58709ccc", "problem_id": "a3fbebf907925eda", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_6=44$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_6=38.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=38 and k=6.\nCompute C(43,5) = 962598.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $962598$", "final_answer": "962598", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9bf3a3099957bb88", "problem_id": "a3fbebf907925eda", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_6=44$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_6=38.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=38 and k=6.\n4) Compute C(43,5) = 962598.\nFinal answer: $962598$", "final_answer": "962598", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "55dd638898b83dde", "problem_id": "033c64d843e511f1", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{5} x - x - 6 - 2 \\sqrt{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\sqrt{5} - 1$, c=$-6 - 2 \\sqrt{5}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{1}{2} + \\frac{\\sqrt{5}}{2} + \\frac{\\sqrt{10\\cdot \\sqrt{5} + 30}}{2}, - \\frac{\\sqrt{10\\cdot \\sqrt{5} + 30}}{2} + \\frac{1}{2} + \\frac{\\sqrt{5}}{2}\\}$", "final_answer": "x \\in \\{\\frac{1}{2} + \\frac{\\sqrt{5}}{2} + \\frac{\\sqrt{10\\cdot \\sqrt{5} + 30}}{2}, - \\frac{\\sqrt{10\\cdot \\sqrt{5} + 30}}{2} + \\frac{1}{2} + \\frac{\\sqrt{5}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "81c8a09e096695e5", "problem_id": "033c64d843e511f1", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{5} x - x - 6 - 2 \\sqrt{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\sqrt{5} - 1$, c=$-6 - 2 \\sqrt{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $10 \\sqrt{5} + 30$, so √Δ = $\\sqrt{10 \\sqrt{5} + 30}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{1}{2} + \\frac{\\sqrt{5}}{2} + \\frac{\\sqrt{10 \\sqrt{5} + 30}}{2}$.\nx2 = $- \\frac{\\sqrt{10 \\sqrt{5} + 30}}{2} + \\frac{1}{2} + \\frac{\\sqrt{5}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{1}{2} + \\frac{\\sqrt{5}}{2} + \\frac{\\sqrt{10\\cdot \\sqrt{5} + 30}}{2}, - \\frac{\\sqrt{10\\cdot \\sqrt{5} + 30}}{2} + \\frac{1}{2} + \\frac{\\sqrt{5}}{2}\\}$", "final_answer": "x \\in \\{\\frac{1}{2} + \\frac{\\sqrt{5}}{2} + \\frac{\\sqrt{10\\cdot \\sqrt{5} + 30}}{2}, - \\frac{\\sqrt{10\\cdot \\sqrt{5} + 30}}{2} + \\frac{1}{2} + \\frac{\\sqrt{5}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e1e23dc003764a88", "problem_id": "87b9883f6cadafd0", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Worker A finishes a job in 8 hours and Worker B finishes it in 17 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/8 job/hour. B's rate = 1/17 job/hour.\nCombined rate = 1/8 + 1/17 = 25/136 job/hour.\nTime = 1 ÷ (combined rate) = 136/25 hours.\nFinal answer: $\\frac{136}{25}$", "final_answer": "\\frac{136}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6835143deff530d0", "problem_id": "87b9883f6cadafd0", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Worker A finishes a job in 8 hours and Worker B finishes it in 17 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/8 job/hour. B's rate = 1/17 job/hour.\n2) Combined rate = 1/8 + 1/17 = 25/136 job/hour.\n3) Time = 1 ÷ (combined rate) = 136/25 hours.\nFinal answer: $\\frac{136}{25}$", "final_answer": "\\frac{136}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "110243b03b2c8cf8", "problem_id": "139649f73c7b27d9", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡1 (mod 8). Give the solution modulo 24.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 24.\n3) Compute the CRT solution to get x ≡ 1 (mod 24).\nFinal answer: $x\\equiv 1\\pmod{24}$", "final_answer": "x\\equiv 1\\pmod{24}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ba267ba0fee67ef7", "problem_id": "139649f73c7b27d9", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡1 (mod 8). Give the solution modulo 24.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 24.\nCompute the CRT solution to get x ≡ 1 (mod 24).\nCheck: 1 mod 3=1 and mod 8=1.\nFinal answer: $x\\equiv 1\\pmod{24}$", "final_answer": "x\\equiv 1\\pmod{24}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d36e275037864a1a", "problem_id": "12b13366b0ee2647", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "An urn has 8 red balls and 16 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(24,8).\n3) Favorable samples: choose k reds and n−k blues: C(8,1)·C(16,7).\n4) Compute totals: C(24,8) = 735471, and favorable = 91520.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{8320}{66861}$", "final_answer": "\\frac{8320}{66861}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "87432b04a4b2c44b", "problem_id": "12b13366b0ee2647", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "An urn has 8 red balls and 16 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(24,8).\nFavorable samples: choose k reds and n−k blues: C(8,1)·C(16,7).\nFinal answer: $\\frac{8320}{66861}$", "final_answer": "\\frac{8320}{66861}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "32fa5639e1d416b6", "problem_id": "90ab1a68841320a7", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "You want 37 liters of a 25% salt solution. You have a 20% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 37−x liters is 60% solution.\nSalt equation: 20x + 60(37−x) = 25·37.\n(20-60)x = (25-60)·37 ⇒ x = 259/8.\nFinal answer: $259/8 L of 20% and 37/8 L of 60%$", "final_answer": "259/8 L of 20% and 37/8 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e253527612deb073", "problem_id": "90ab1a68841320a7", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "You want 37 liters of a 25% salt solution. You have a 20% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 37−x liters is 60% solution.\n2) Salt equation: 20x + 60(37−x) = 25·37.\n3) (20-60)x = (25-60)·37 ⇒ x = 259/8.\n4) Then 37−x = 37/8.\nFinal answer: $259/8 L of 20% and 37/8 L of 60%$", "final_answer": "259/8 L of 20% and 37/8 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2daa4ac47f5a04ba", "problem_id": "762f7c1affce33bf", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "You want 59 liters of a 30% salt solution. You have a 20% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 59−x liters is 60% solution.\n2) Salt equation: 20x + 60(59−x) = 30·59.\n3) (20-60)x = (30-60)·59 ⇒ x = 177/4.\n4) Then 59−x = 59/4.\nFinal answer: $177/4 L of 20% and 59/4 L of 60%$", "final_answer": "177/4 L of 20% and 59/4 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "305626fd82da47e8", "problem_id": "a3343dda6c927aaf", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} - x + \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{x + 6}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 6}) = \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 6}$.\n3) So $f'(x) = \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 24 x + 72} + \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + 24 x + 72} - \\frac{2 x^{2}}{2 x^{2} + 24 x + 72} + \\frac{12 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 24 x + 72} + \\frac{24 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + 24 x + 72} - \\frac{24 x}{2 x^{2} + 24 x + 72} + \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 12 x + 36} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x + 12} - \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 6} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 6} + \\frac{36 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 24 x + 72} + \\frac{72 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + 24 x + 72} - \\frac{6}{2 x^{2} + 24 x + 72} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 6}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 6}) = \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x + 6\\right)} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 6} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x + 6\\right)^{2}} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 6}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 x^{2} + 2 x + \\left(x + 6\\right) \\left(- 4 x + \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 2 \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6}{2 \\left(x + 6\\right)^{2}}$.\nFinal answer: $\\frac{2 x^{2} + 2 x + \\left(x + 6\\right) \\left(- 4 x + \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 2 \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6}{2 \\left(x + 6\\right)^{2}}$", "final_answer": "\\frac{2 x^{2} + 2 x + \\left(x + 6\\right) \\left(- 4 x + \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 2 \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6}{2 \\left(x + 6\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e54cc1088d3f1dfb", "problem_id": "02d3f48b48038cc0", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Worker A finishes a job in 18 hours and Worker B finishes it in 12 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/18 job/hour. B's rate = 1/12 job/hour.\nCombined rate = 1/18 + 1/12 = 5/36 job/hour.\nTime = 1 ÷ (combined rate) = 36/5 hours.\nCheck: (combined rate)×(time) = 5/36×36/5 = 1.\nFinal answer: $\\frac{36}{5}$", "final_answer": "\\frac{36}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e2ed5757c8f7273a", "problem_id": "1bea1c540a0aa966", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(- 2 x^{3} + x^{2} - x - 3\\right) \\cos{\\left(x + 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x e^{x}) = x e^{x}$.\n3) So $f'(x) = 2 x^{3} \\sin{\\left(x + 2 \\right)} - x^{2} \\sin{\\left(x + 2 \\right)} - 6 x^{2} \\cos{\\left(x + 2 \\right)} + x e^{x} + x \\sin{\\left(x + 2 \\right)} + 2 x \\cos{\\left(x + 2 \\right)} + e^{x} + 3 \\sin{\\left(x + 2 \\right)} - \\cos{\\left(x + 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x e^{x}) = x e^{x} + e^{x}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 1\\right) e^{x} - \\left(x \\left(x \\left(1 - 2 x\\right) - 1\\right) - 3\\right) \\sin{\\left(x + 2 \\right)} + \\left(- 6 x^{2} + 2 x - 1\\right) \\cos{\\left(x + 2 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 1\\right) e^{x} - \\left(x \\left(x \\left(1 - 2 x\\right) - 1\\right) - 3\\right) \\sin{\\left(x + 2 \\right)} + \\left(- 6 x^{2} + 2 x - 1\\right) \\cos{\\left(x + 2 \\right)} + e^{x}$", "final_answer": "\\left(x + 1\\right) e^{x} - \\left(x \\left(x \\left(1 - 2 x\\right) - 1\\right) - 3\\right) \\sin{\\left(x + 2 \\right)} + \\left(- 6 x^{2} + 2 x - 1\\right) \\cos{\\left(x + 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "9dd0e96a465bf0d4", "problem_id": "cf92d694d2fe7316", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 16 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=16, k=5, p=$\\frac{3}{7}$ and simplify.\n4) Here C(16,5) = 4368.\nFinal answer: $\\frac{635990704128}{4747561509943}$", "final_answer": "\\frac{635990704128}{4747561509943}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "25ad93a37f2e8fc8", "problem_id": "cf92d694d2fe7316", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 16 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=16, k=5, p=$\\frac{3}{7}$ and simplify.\nHere C(16,5) = 4368.\nCheck: expression simplifies to $\\frac{635990704128}{4747561509943}$.\nFinal answer: $\\frac{635990704128}{4747561509943}$", "final_answer": "\\frac{635990704128}{4747561509943}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7526c8fe18b570dc", "problem_id": "014b643c8ea0f237", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{4} + 4 x^{3} - 3 x^{2} + 1$.", "output": "1) Differentiate term-by-term:\n2) d/dx(1) = 0.\n3) d/dx(- 3 x^{2}) = - 6 x.\n4) d/dx(4 x^{3}) = 12 x^{2}.\n5) d/dx(4 x^{4}) = 16 x^{3}.\n6) Combine: f'(x) = 2 x \\left(8 x^{2} + 6 x - 3\\right).\nFinal answer: $2\\cdot x\\cdot \\left(8\\cdot x^{2} + 6\\cdot x - 3\\right)$", "final_answer": "2\\cdot x\\cdot \\left(8\\cdot x^{2} + 6\\cdot x - 3\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3afaee4b59f923d4", "problem_id": "014b643c8ea0f237", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{4} + 4 x^{3} - 3 x^{2} + 1$.", "output": "$2\\cdot x\\cdot \\left(8\\cdot x^{2} + 6\\cdot x - 3\\right)$", "final_answer": "2\\cdot x\\cdot \\left(8\\cdot x^{2} + 6\\cdot x - 3\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "100984e2aa283fe8", "problem_id": "fd894b170f808b49", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} + 2 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 3$ at $x=-1$.", "output": "Differentiate: f'(x) = $- 6 x^{2} + 4 x + 2 \\cos{\\left(x \\right)} - 2$.\nSlope at x=-1: m = f'(-1) = $-12 + 2 \\cos{\\left(1 \\right)}$.\nPoint on curve: (-1, f(-1)) = (-1, $3 - 2 \\sin{\\left(1 \\right)}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 3 - 2 \\sin{\\left(1 \\right)} = -12 + 2 \\cos{\\left(1 \\right)}(x − -1).\nOptionally expand: y = $- 12 x + 2 x \\cos{\\left(1 \\right)} - 9 - 2 \\sin{\\left(1 \\right)} + 2 \\cos{\\left(1 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 12\\cdot x + 2\\cdot x\\cdot \\cos{\\left(1 \\right)} - 9 - 2\\cdot \\sin{\\left(1 \\right)} + 2\\cdot \\cos{\\left(1 \\right)}$", "final_answer": "y = - 12\\cdot x + 2\\cdot x\\cdot \\cos{\\left(1 \\right)} - 9 - 2\\cdot \\sin{\\left(1 \\right)} + 2\\cdot \\cos{\\left(1 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6abf5ffc6f345942", "problem_id": "fd894b170f808b49", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} + 2 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 3$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $- 6 x^{2} + 4 x + 2 \\cos{\\left(x \\right)} - 2$.\n2) Slope at x=-1: m = f'(-1) = $-12 + 2 \\cos{\\left(1 \\right)}$.\n3) Point on curve: (-1, f(-1)) = (-1, $3 - 2 \\sin{\\left(1 \\right)}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 3 - 2 \\sin{\\left(1 \\right)} = -12 + 2 \\cos{\\left(1 \\right)}(x − -1).\n6) Optionally expand: y = $- 12 x + 2 x \\cos{\\left(1 \\right)} - 9 - 2 \\sin{\\left(1 \\right)} + 2 \\cos{\\left(1 \\right)}$.\nFinal answer: $y = - 12\\cdot x + 2\\cdot x\\cdot \\cos{\\left(1 \\right)} - 9 - 2\\cdot \\sin{\\left(1 \\right)} + 2\\cdot \\cos{\\left(1 \\right)}$", "final_answer": "y = - 12\\cdot x + 2\\cdot x\\cdot \\cos{\\left(1 \\right)} - 9 - 2\\cdot \\sin{\\left(1 \\right)} + 2\\cdot \\cos{\\left(1 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9b7a20c7a84875a1", "problem_id": "74e4d58d58099a1a", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(x^{3} + x^{2} + 2\\right) \\cos{\\left(3 x + 1 \\right)}$.", "output": "1) Let f(x) = $\\left(x + 1\\right) e^{x} + \\left(x^{3} + x^{2} + 2\\right) \\cos{\\left(3 x + 1 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $x \\left(3 x + 2\\right) \\cos{\\left(3 x + 1 \\right)} + \\left(x + 1\\right) e^{x} - 3 \\left(x^{3} + x^{2} + 2\\right) \\sin{\\left(3 x + 1 \\right)} + e^{x}$.\nFinal answer: $x\\cdot \\left(3\\cdot x + 2\\right)\\cdot \\cos{\\left(3\\cdot x + 1 \\right)} + \\left(x + 1\\right)\\cdot e^{x} - \\left(3\\cdot x^{3} + 3\\cdot x^{2} + 6\\right)\\cdot \\sin{\\left(3\\cdot x + 1 \\right)} + e^{x}$", "final_answer": "x\\cdot \\left(3\\cdot x + 2\\right)\\cdot \\cos{\\left(3\\cdot x + 1 \\right)} + \\left(x + 1\\right)\\cdot e^{x} - \\left(3\\cdot x^{3} + 3\\cdot x^{2} + 6\\right)\\cdot \\sin{\\left(3\\cdot x + 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0415ddb315387f04", "problem_id": "74e4d58d58099a1a", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(x^{3} + x^{2} + 2\\right) \\cos{\\left(3 x + 1 \\right)}$.", "output": "Let f(x) = $\\left(x + 1\\right) e^{x} + \\left(x^{3} + x^{2} + 2\\right) \\cos{\\left(3 x + 1 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $x \\left(3 x + 2\\right) \\cos{\\left(3 x + 1 \\right)} + \\left(x + 1\\right) e^{x} - 3 \\left(x^{3} + x^{2} + 2\\right) \\sin{\\left(3 x + 1 \\right)} + e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $x\\cdot \\left(3\\cdot x + 2\\right)\\cdot \\cos{\\left(3\\cdot x + 1 \\right)} + \\left(x + 1\\right)\\cdot e^{x} - \\left(3\\cdot x^{3} + 3\\cdot x^{2} + 6\\right)\\cdot \\sin{\\left(3\\cdot x + 1 \\right)} + e^{x}$", "final_answer": "x\\cdot \\left(3\\cdot x + 2\\right)\\cdot \\cos{\\left(3\\cdot x + 1 \\right)} + \\left(x + 1\\right)\\cdot e^{x} - \\left(3\\cdot x^{3} + 3\\cdot x^{2} + 6\\right)\\cdot \\sin{\\left(3\\cdot x + 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "29961c50c11b4509", "problem_id": "c845a4d972868a75", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 x}{2} - \\frac{\\sqrt{6} x}{2} - \\frac{5}{2} - \\frac{\\sqrt{6}}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{3}{2} - \\frac{\\sqrt{6}}{2}$, c=$- \\frac{5}{2} - \\frac{\\sqrt{6}}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{7 \\sqrt{6}}{2} + \\frac{55}{4}$, so √Δ = $\\sqrt{\\frac{7 \\sqrt{6}}{2} + \\frac{55}{4}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{6}}{4} + \\frac{3}{4} + \\frac{\\sqrt{14 \\sqrt{6} + 55}}{4}$.\n7) x2 = $- \\frac{\\sqrt{14 \\sqrt{6} + 55}}{4} + \\frac{\\sqrt{6}}{4} + \\frac{3}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{6}}{4} + \\frac{3}{4} + \\frac{\\sqrt{14\\cdot \\sqrt{6} + 55}}{4}, - \\frac{\\sqrt{14\\cdot \\sqrt{6} + 55}}{4} + \\frac{\\sqrt{6}}{4} + \\frac{3}{4}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{6}}{4} + \\frac{3}{4} + \\frac{\\sqrt{14\\cdot \\sqrt{6} + 55}}{4}, - \\frac{\\sqrt{14\\cdot \\sqrt{6} + 55}}{4} + \\frac{\\sqrt{6}}{4} + \\frac{3}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec9ff43afc123a0b", "problem_id": "3ef96dfabadc66d8", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 2 x^{3} + 5 x^{2} - 6 x + 8$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{3} + 5 x^{2} - 7 x + 8.\n3) Then the limit is 22.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: 21.\nFinal answer: $21$", "final_answer": "21", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5fceab097a065b58", "problem_id": "0cae86ff7b54c0b6", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "You want 29 liters of a 40% salt solution. You have a 30% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 29−x liters is 50% solution.\nSalt equation: 30x + 50(29−x) = 40·29.\n(30-50)x = (40-50)·29 ⇒ x = 29/2.\nThen 29−x = 29/2.\nCheck: 30·29/2 + 50·29/2 = 40·29.\nFinal answer: $29/2 L of 30% and 29/2 L of 50%$", "final_answer": "29/2 L of 30% and 29/2 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f6e19ac97794c35d", "problem_id": "0cae86ff7b54c0b6", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "You want 29 liters of a 40% salt solution. You have a 30% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 29−x liters is 50% solution.\n2) Salt equation: 30x + 50(29−x) = 40·29.\n3) (30-50)x = (40-50)·29 ⇒ x = 29/2.\n4) Then 29−x = 29/2.\nFinal answer: $29/2 L of 30% and 29/2 L of 50%$", "final_answer": "29/2 L of 30% and 29/2 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "af3f827f5824879b", "problem_id": "9ac5656afd08dbf1", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Two friends split $228 in the ratio 8:4. How much does each receive?", "output": "Total ratio parts = 8+4 = 12.\nEach part = 228 ÷ 12 = 19.\nFirst share = 8 × 19 = 152.\nFinal answer: $76$", "final_answer": "76", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6ab32fd99b280a93", "problem_id": "9ac5656afd08dbf1", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Two friends split $228 in the ratio 8:4. How much does each receive?", "output": "1) Total ratio parts = 8+4 = 12.\n2) Each part = 228 ÷ 12 = 19.\n3) First share = 8 × 19 = 152.\n4) Second share = 4 × 19 = 76.\nFinal answer: $76$", "final_answer": "76", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ae100e343465c30e", "problem_id": "3451b6d8ff262c62", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x^{3} \\sin{\\left(x \\right)}) = - x^{3} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - x^{3} \\cos{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x^{3} \\sin{\\left(x \\right)}) = - x^{3} \\cos{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - x^{3} \\cos{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.\nFinal answer: $- x^{3} \\cos{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$", "final_answer": "- x^{3} \\cos{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "846803997d11ad64", "problem_id": "cbcef28301f79ed4", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "You want 42 liters of a 45% salt solution. You have a 15% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 42−x liters is 60% solution.\n2) Salt equation: 15x + 60(42−x) = 45·42.\n3) (15-60)x = (45-60)·42 ⇒ x = 14.\n4) Then 42−x = 28.\nFinal answer: $14 L of 15% and 28 L of 60%$", "final_answer": "14 L of 15% and 28 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c6a6388bf955fc73", "problem_id": "b1b01e769504ff59", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 14 red balls and 26 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(40,8).\n3) Favorable samples: choose k reds and n−k blues: C(14,3)·C(26,5).\n4) Compute totals: C(40,8) = 76904685, and favorable = 23943920.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{33488}{107559}$", "final_answer": "\\frac{33488}{107559}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b2231970cf592c72", "problem_id": "b1b01e769504ff59", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 14 red balls and 26 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(40,8).\nFavorable samples: choose k reds and n−k blues: C(14,3)·C(26,5).\nCompute totals: C(40,8) = 76904685, and favorable = 23943920.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{33488}{107559}$.\nFinal answer: $\\frac{33488}{107559}$", "final_answer": "\\frac{33488}{107559}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ebd432d1d1cae4c4", "problem_id": "8235f35bda04d3e4", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} 3 x^{3} + 4 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{3} + 4 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2$.\nEvaluate at the bounds: F(2 \\pi) and F(0).\nSubtract: F(2 \\pi) − F(0) = $\\frac{2 \\pi \\left(6 + 9 \\pi + 16 \\pi^{2} + 18 \\pi^{3}\\right)}{3}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(6 + 9\\cdot \\pi + 16\\cdot \\pi^{2} + 18\\cdot \\pi^{3}\\right)}{3}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(6 + 9\\cdot \\pi + 16\\cdot \\pi^{2} + 18\\cdot \\pi^{3}\\right)}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "14810f8fbee17014", "problem_id": "8235f35bda04d3e4", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} 3 x^{3} + 4 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{3} + 4 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(0).\n3) Subtract: F(2 \\pi) − F(0) = $\\frac{2 \\pi \\left(6 + 9 \\pi + 16 \\pi^{2} + 18 \\pi^{3}\\right)}{3}$.\n4) Simplify the result.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(6 + 9\\cdot \\pi + 16\\cdot \\pi^{2} + 18\\cdot \\pi^{3}\\right)}{3}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(6 + 9\\cdot \\pi + 16\\cdot \\pi^{2} + 18\\cdot \\pi^{3}\\right)}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6c77c245894cd037", "problem_id": "8a3b964e9fa62cb1", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 2 \\sqrt{6} x - \\frac{23 x}{5} + \\frac{12}{5} + \\frac{6 \\sqrt{6}}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- 2 \\sqrt{6} - \\frac{23}{5}$, c=$\\frac{12}{5} + \\frac{6 \\sqrt{6}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{68 \\sqrt{6}}{5} + \\frac{889}{25}$, so √Δ = $\\sqrt{\\frac{68 \\sqrt{6}}{5} + \\frac{889}{25}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{23}{10} + \\sqrt{6} + \\frac{\\sqrt{340 \\sqrt{6} + 889}}{10}$.\n7) x2 = $- \\frac{\\sqrt{340 \\sqrt{6} + 889}}{10} + \\frac{23}{10} + \\sqrt{6}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{23}{10} + \\sqrt{6} + \\frac{\\sqrt{340\\cdot \\sqrt{6} + 889}}{10}, - \\frac{\\sqrt{340\\cdot \\sqrt{6} + 889}}{10} + \\frac{23}{10} + \\sqrt{6}\\}$", "final_answer": "x \\in \\{\\frac{23}{10} + \\sqrt{6} + \\frac{\\sqrt{340\\cdot \\sqrt{6} + 889}}{10}, - \\frac{\\sqrt{340\\cdot \\sqrt{6} + 889}}{10} + \\frac{23}{10} + \\sqrt{6}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a9ea509c4c24cfb9", "problem_id": "e725b9b3a9f46c53", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(2 x^{2} + 2 x + 1\\right) \\cos{\\left(2 x^{2} + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\cos{\\left(2 x^{2} + 3 \\right)}) = - \\sin{\\left(2 x^{2} + 3 \\right)}$.\n3) So $f'(x) = - 8 x^{3} \\sin{\\left(2 x^{2} + 3 \\right)} - 8 x^{2} \\sin{\\left(2 x^{2} + 3 \\right)} + x e^{x} + 4 x \\cos{\\left(2 x^{2} + 3 \\right)} + 3 e^{x} - \\sin{\\left(2 x^{2} + 3 \\right)} + 2 \\cos{\\left(2 x^{2} + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\cos{\\left(2 x^{2} + 3 \\right)}) = - 4 x \\sin{\\left(2 x^{2} + 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = - 4 x \\left(2 x \\left(x + 1\\right) + 1\\right) \\sin{\\left(2 x^{2} + 3 \\right)} + \\left(x + 2\\right) e^{x} + 2 \\left(2 x + 1\\right) \\cos{\\left(2 x^{2} + 3 \\right)} + e^{x}$.\nFinal answer: $- 4 x \\left(2 x \\left(x + 1\\right) + 1\\right) \\sin{\\left(2 x^{2} + 3 \\right)} + \\left(x + 2\\right) e^{x} + 2 \\left(2 x + 1\\right) \\cos{\\left(2 x^{2} + 3 \\right)} + e^{x}$", "final_answer": "- 4 x \\left(2 x \\left(x + 1\\right) + 1\\right) \\sin{\\left(2 x^{2} + 3 \\right)} + \\left(x + 2\\right) e^{x} + 2 \\left(2 x + 1\\right) \\cos{\\left(2 x^{2} + 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ef7a78f1b83b1d3b", "problem_id": "a93d71bead376c93", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the linear system:\n$4 x - 3 y$ = $7$\n$- 3 x + 3 y$ = $-6$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $- 12 x + 9 y$ = $-21$.\n3) Multiply the second equation by $4$: $- 12 x + 12 y$ = $-24$.\n4) Subtract to eliminate x: $- 3 y$ = $3$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $1$.\nFinal answer: $(x,y)=\\left(1, -1\\right)$", "final_answer": "(x,y)=\\left(1, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "79c3794b1e3b835a", "problem_id": "4d555e3f0bec517a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 2 x^{2} - 2 x - 1$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{2} - 3 x - 1.\n3) Then the limit is 4.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: 3.\nFinal answer: $3$", "final_answer": "3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0cf5d3ccbfd551a0", "problem_id": "ae4e94b3d7938c56", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A cyclist rides at 38 miles per hour for 2 hours, then at 20 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 38 × 2 = 76 miles.\n2) Second leg distance = 20 × 5/2 = 50 miles.\n3) Total distance = 76 + 50 = 126 miles.\nFinal answer: $126$", "final_answer": "126", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b1f0cab1353246ca", "problem_id": "ae4e94b3d7938c56", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A cyclist rides at 38 miles per hour for 2 hours, then at 20 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 38 × 2 = 76 miles.\nSecond leg distance = 20 × 5/2 = 50 miles.\nTotal distance = 76 + 50 = 126 miles.\nFinal answer: $126$", "final_answer": "126", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6dfa209b4bbf3f02", "problem_id": "d8d35b6d9871079c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- \\frac{3 x^{2}}{2} - x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1}{x - 4}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}) = \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4}$.\n3) So $f'(x) = \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{3 x^{2}}{2 x^{2} - 16 x + 32} - \\frac{8 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{16 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{24 x}{2 x^{2} - 16 x + 32} + \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 8 x + 16} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 8} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} - \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} + \\frac{16 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{32 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{10}{2 x^{2} - 16 x + 32} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}) = \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x - 4\\right)} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x - 4\\right)^{2}} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}$.\nTherefore the correct derivative is $f'(x) = \\frac{3 x^{2} + 2 x + \\left(x - 4\\right) \\left(- 6 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{2 \\left(x - 4\\right)^{2}}$.\nFinal answer: $\\frac{3 x^{2} + 2 x + \\left(x - 4\\right) \\left(- 6 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{2 \\left(x - 4\\right)^{2}}$", "final_answer": "\\frac{3 x^{2} + 2 x + \\left(x - 4\\right) \\left(- 6 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{2 \\left(x - 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0d4ea347c1ef40ae", "problem_id": "dc5750e97cf2078d", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} 3 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\n3) Subtract: F(\\frac{\\pi}{2}) − F(0) = $\\frac{\\pi^{2}}{8} + \\frac{\\pi}{2} + 2 + \\frac{\\pi^{3}}{8}$.\n4) Simplify the result.\nFinal answer: $\\frac{\\pi^{2}}{8} + \\frac{\\pi}{2} + 2 + \\frac{\\pi^{3}}{8}$", "final_answer": "\\frac{\\pi^{2}}{8} + \\frac{\\pi}{2} + 2 + \\frac{\\pi^{3}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4bc7bf112c6c82ed", "problem_id": "ad9c6672da8ca64a", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 3 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $- 6 \\pi - \\frac{3 \\pi^{2}}{2} - 4 + 9 \\pi^{3}$.\n4) Simplify the result.\nFinal answer: $- 6\\cdot \\pi - \\frac{3\\cdot \\pi^{2}}{2} - 4 + 9\\cdot \\pi^{3}$", "final_answer": "- 6\\cdot \\pi - \\frac{3\\cdot \\pi^{2}}{2} - 4 + 9\\cdot \\pi^{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db8da358982dd1b1", "problem_id": "f159eca792120ea4", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} 2 x^{3} + 11 x^{2} + 41 x + 162$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} + 11 x^{2} + 40 x + 162.\n3) Then the limit is 626.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: 630.\nFinal answer: $630$", "final_answer": "630", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "481ef709145e7a19", "problem_id": "ae5ebb9642461a0e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} 3 x^{3} - 2 x - 2$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} - 3 x - 2.\n3) Then the limit is -2.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: -1.\nFinal answer: $-1$", "final_answer": "-1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e512c17cc8f674b4", "problem_id": "6da872766fc81c07", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 13 red balls and 15 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(28,8).\nFavorable samples: choose k reds and n−k blues: C(13,4)·C(15,4).\nCompute totals: C(28,8) = 3108105, and favorable = 975975.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{65}{207}$.\nFinal answer: $\\frac{65}{207}$", "final_answer": "\\frac{65}{207}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ad35c3b7427fd8b3", "problem_id": "42ffaf92e63db81e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - 3 x - 11$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to - 4 x - 11.\n3) Then the limit is -23.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: -20.\nFinal answer: $-20$", "final_answer": "-20", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6dca42f5b094f0ca", "problem_id": "d8d5b92ad283da00", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Two friends split $1190 in the ratio 4:6. How much does each receive?", "output": "Total ratio parts = 4+6 = 10.\nEach part = 1190 ÷ 10 = 119.\nFirst share = 4 × 119 = 476.\nSecond share = 6 × 119 = 714.\nCheck: 476+714=1190.\nFinal answer: $714$", "final_answer": "714", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a380622ed7a26b35", "problem_id": "f12c049d600435b6", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $594 in the ratio 9:9. How much does each receive?", "output": "Total ratio parts = 9+9 = 18.\nEach part = 594 ÷ 18 = 33.\nFirst share = 9 × 33 = 297.\nSecond share = 9 × 33 = 297.\nCheck: 297+297=594.\nFinal answer: $297$", "final_answer": "297", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "79868568c7b5e6f3", "problem_id": "f12c049d600435b6", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Two friends split $594 in the ratio 9:9. How much does each receive?", "output": "Total ratio parts = 9+9 = 18.\nEach part = 594 ÷ 18 = 33.\nFirst share = 9 × 33 = 297.\nFinal answer: $297$", "final_answer": "297", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "541325150e1330e6", "problem_id": "b514321070f45156", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A store sells a backpack for $78. It is discounted by 40%. You buy 6 backpacks. What is the total cost?", "output": "Discounted price = 78 × (1 − 40/100) = 78 × 60/100 = 234/5.\nSubtotal = 6 × 234/5 = 1404/5.\nFinal answer: $\\frac{1404}{5}$", "final_answer": "\\frac{1404}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ad833ed45b2ad482", "problem_id": "b514321070f45156", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $78. It is discounted by 40%. You buy 6 backpacks. What is the total cost?", "output": "Discounted price = 78 × (1 − 40/100) = 78 × 60/100 = 234/5.\nSubtotal = 6 × 234/5 = 1404/5.\nCheck: recomputing the formula yields 1404/5.\nFinal answer: $\\frac{1404}{5}$", "final_answer": "\\frac{1404}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a13f0ee9d7621514", "problem_id": "d903d488291fd67c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 2 x^{2} + 2 x + \\left(x^{2} + 3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{x^{2} + 3 x - 5}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 5}) = \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3 x - 5}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{6 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{12 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{16 x^{2}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{6 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 6 x^{3} - x^{2} - 30 x + 25} - \\frac{30 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{60 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{48 x}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{9 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 6 x^{3} - x^{2} - 30 x + 25} - \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 6 x - 10} + \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3 x - 5} - \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 5} + \\frac{25 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{50 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{8}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 5}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 5}) = \\frac{3 x \\left(- 2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(x^{2} + 3 x\\right) - 5\\right)^{2}} + \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(x^{2} + 3 x\\right) - 5\\right)} + \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 5} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 5}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(- 2 x - 3\\right) \\left(- 2 x^{2} + 2 x + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(x \\left(x + 3\\right) - 5\\right) \\left(- 8 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x + 3\\right) - 5\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(- 2 x - 3\\right) \\left(- 2 x^{2} + 2 x + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(x \\left(x + 3\\right) - 5\\right) \\left(- 8 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x + 3\\right) - 5\\right)^{2}}$", "final_answer": "\\frac{2 \\left(- 2 x - 3\\right) \\left(- 2 x^{2} + 2 x + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(x \\left(x + 3\\right) - 5\\right) \\left(- 8 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x + 3\\right) - 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5dded8ac97b6fdf2", "problem_id": "39c50ddcf2498768", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x - 1$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{2} - 3 x - 3$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $2 x - 3$.\n5) Evaluate at a: p'(2) = $1$.\nFinal answer: $1$", "final_answer": "1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ba97929b7e3a7e9", "problem_id": "f290be13e9781e3a", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A store sells a backpack for $420. It is discounted by 5%. You buy 13 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 420 × (1 − 5/100) = 420 × 95/100 = 399.\n2) Subtotal = 13 × 399 = 5187.\n3) Total with tax = 5187 × 109/100 = 565383/100.\nFinal answer: $\\frac{565383}{100}$", "final_answer": "\\frac{565383}{100}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7455f22e738eb753", "problem_id": "f290be13e9781e3a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A store sells a backpack for $420. It is discounted by 5%. You buy 13 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 420 × (1 − 5/100) = 420 × 95/100 = 399.\nSubtotal = 13 × 399 = 5187.\nTotal with tax = 5187 × 109/100 = 565383/100.\nCheck: recomputing the formula yields 565383/100.\nFinal answer: $\\frac{565383}{100}$", "final_answer": "\\frac{565383}{100}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8a524846e4f06df8", "problem_id": "7be7a12a45734b17", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡2 (mod 4), x≡6 (mod 7). Give the solution modulo 28.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 28.\n3) Compute the CRT solution to get x ≡ 6 (mod 28).\nFinal answer: $x\\equiv 6\\pmod{28}$", "final_answer": "x\\equiv 6\\pmod{28}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b46a81e9ea68dbe9", "problem_id": "7be7a12a45734b17", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡2 (mod 4), x≡6 (mod 7). Give the solution modulo 28.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 28.\nCompute the CRT solution to get x ≡ 6 (mod 28).\nCheck: 6 mod 4=2 and mod 7=6.\nFinal answer: $x\\equiv 6\\pmod{28}$", "final_answer": "x\\equiv 6\\pmod{28}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9e264ff1ff8c7907", "problem_id": "7a5f13547cb329da", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A disease has prevalence $\\frac{3}{200}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{861}{20000}$.\nCompute P(D|+) = $\\frac{90}{287}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{90}{287}$", "final_answer": "\\frac{90}{287}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "334bb99829609b6a", "problem_id": "9d3d72eaf058d2a7", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} 3 x^{3} + 6 x^{2} + 15 x + 48$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} + 6 x^{2} + 14 x + 48.\n3) Then the limit is 225.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: 228.\nFinal answer: $228$", "final_answer": "228", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f9efc90a58316e2d", "problem_id": "45382144d46e78a5", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_10=32$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_10=2.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=2 and k=10.\n4) Compute C(11,9) = 55.\nFinal answer: $55$", "final_answer": "55", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c57547ab08a305a5", "problem_id": "45382144d46e78a5", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_10=32$?", "output": "Let y_i=x_i-3, then y_i≥0 and y_1+…+y_10=2.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=2 and k=10.\nCompute C(11,9) = 55.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $55$", "final_answer": "55", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3a80101e731fcbac", "problem_id": "d425f2e063466925", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 2 x^{3} - 3 x^{2} + 8 x - 23$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} - 3 x^{2} + 7 x - 23.\n3) Then the limit is -125.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -128.\nFinal answer: $-128$", "final_answer": "-128", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b3bfa4ac146cf3f7", "problem_id": "418709f777900570", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$- 2 x + 7 y$ = $45$\n$- 9 x + 7 y$ = $31$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-9$: $18 x - 63 y$ = $-405$.\n3) Multiply the second equation by $-2$: $18 x - 14 y$ = $-62$.\n4) Subtract to eliminate x: $- 49 y$ = $-343$.\n5) Solve for y: y = $7$.\n6) Substitute back to find x: x = $2$.\nFinal answer: $(x,y)=\\left(2, 7\\right)$", "final_answer": "(x,y)=\\left(2, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "89ac334d6d237029", "problem_id": "01b6f0131f2438e2", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Worker A finishes a job in 17 hours and Worker B finishes it in 5 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/17 job/hour. B's rate = 1/5 job/hour.\n2) Combined rate = 1/17 + 1/5 = 22/85 job/hour.\n3) Time = 1 ÷ (combined rate) = 85/22 hours.\nFinal answer: $\\frac{85}{22}$", "final_answer": "\\frac{85}{22}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8bdc46d49e3b394a", "problem_id": "232140c6d24e0324", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} x^{2} - 6 x + 22$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to x^{2} - 7 x + 22.\n3) Then the limit is 66.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: 62.\nFinal answer: $62$", "final_answer": "62", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "11b338c2d846e7ae", "problem_id": "c4c739fc624f1161", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the multiplicative inverse of 9 modulo 29.", "output": "1) We want x such that 9x ≡ 1 (mod 29).\n2) Use extended Euclid to find s,t with 9s + 29t = 1.\n3) One Bezout identity is: 9·(13) + 29·(-4) = 1.\n4) Reduce s modulo 29: x ≡ 13 (mod 29).\nFinal answer: $x\\equiv 13\\pmod{29}$", "final_answer": "x\\equiv 13\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4e5bcd491cf2587a", "problem_id": "1973eb501f26a2f2", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} + 2 x + \\left(x^{2} - x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{x^{2} - x + 7}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7}) = - \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 7}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{4 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{15 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{30 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{2 x^{2}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{2 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 15 x^{2} - 14 x + 49} - \\frac{14 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{28 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{40 x}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 15 x^{2} - 14 x + 49} + \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 2 x + 14} - \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 7} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7} + \\frac{49 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{98 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{34}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7}) = - \\frac{x \\left(1 - 2 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(x^{2} - x\\right) + 7\\right)^{2}} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(x^{2} - x\\right) + 7\\right)} - \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(x - 1\\right) + 7\\right) \\left(- 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x - 1\\right) + 7\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(x - 1\\right) + 7\\right) \\left(- 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x - 1\\right) + 7\\right)^{2}}$", "final_answer": "\\frac{2 \\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(x - 1\\right) + 7\\right) \\left(- 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x - 1\\right) + 7\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "80e6bbb87db8c216", "problem_id": "1f05e186a5486864", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Worker A finishes a job in 18 hours and Worker B finishes it in 7 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/18 job/hour. B's rate = 1/7 job/hour.\n2) Combined rate = 1/18 + 1/7 = 25/126 job/hour.\n3) Time = 1 ÷ (combined rate) = 126/25 hours.\nFinal answer: $\\frac{126}{25}$", "final_answer": "\\frac{126}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ba705f834b2bd70", "problem_id": "1f05e186a5486864", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Worker A finishes a job in 18 hours and Worker B finishes it in 7 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/18 job/hour. B's rate = 1/7 job/hour.\nCombined rate = 1/18 + 1/7 = 25/126 job/hour.\nTime = 1 ÷ (combined rate) = 126/25 hours.\nCheck: (combined rate)×(time) = 25/126×126/25 = 1.\nFinal answer: $\\frac{126}{25}$", "final_answer": "\\frac{126}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ab07197ac897066f", "problem_id": "6bb667c67166215c", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "How many binary strings of length 24 contain exactly 10 ones with no two ones adjacent?", "output": "Reserve 9 mandatory zeros between the 10 ones.\nThis leaves n−k+1 = 15 valid slots for the ones.\nChoose the slots: C(15,10) = 3003.\nFinal answer: $3003$", "final_answer": "3003", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "75cdd42bcc92f2a0", "problem_id": "6bb667c67166215c", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "How many binary strings of length 24 contain exactly 10 ones with no two ones adjacent?", "output": "$3003$", "final_answer": "3003", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e00374ebcd1942f", "problem_id": "3c38a63278b9eeee", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} - 2 x^{2} - 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $- 6 x^{2} - 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 3$.\n2) Slope at x=-2: m = f'(-2) = $-19 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}$.\n3) Point on curve: (-2, f(-2)) = (-2, $\\cos{\\left(4 \\right)} + e^{-1} + 15$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(4 \\right)} + e^{-1} + 15 = -19 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}(x − -2).\n6) Optionally expand: y = $- 19 x + 2 x \\sin{\\left(4 \\right)} + \\frac{x}{2 e} - 23 + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$.\nFinal answer: $y = - 19\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 23 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$", "final_answer": "y = - 19\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 23 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9cfd7d4ed12cec41", "problem_id": "3c38a63278b9eeee", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} - 2 x^{2} - 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=-2$.", "output": "Differentiate: f'(x) = $- 6 x^{2} - 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 3$.\nSlope at x=-2: m = f'(-2) = $-19 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}$.\nPoint on curve: (-2, f(-2)) = (-2, $\\cos{\\left(4 \\right)} + e^{-1} + 15$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − \\cos{\\left(4 \\right)} + e^{-1} + 15 = -19 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}(x − -2).\nOptionally expand: y = $- 19 x + 2 x \\sin{\\left(4 \\right)} + \\frac{x}{2 e} - 23 + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 19\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 23 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$", "final_answer": "y = - 19\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 23 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "73e5a775dd5a0ef7", "problem_id": "0cbc4a6f177a0b58", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "You want 30 liters of a 20% salt solution. You have a 10% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 30−x liters is 40% solution.\n2) Salt equation: 10x + 40(30−x) = 20·30.\n3) (10-40)x = (20-40)·30 ⇒ x = 20.\n4) Then 30−x = 10.\nFinal answer: $20 L of 10% and 10 L of 40%$", "final_answer": "20 L of 10% and 10 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bb1751ffd13b62c3", "problem_id": "0cbc4a6f177a0b58", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "You want 30 liters of a 20% salt solution. You have a 10% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 30−x liters is 40% solution.\nSalt equation: 10x + 40(30−x) = 20·30.\n(10-40)x = (20-40)·30 ⇒ x = 20.\nThen 30−x = 10.\nCheck: 10·20 + 40·10 = 20·30.\nFinal answer: $20 L of 10% and 10 L of 40%$", "final_answer": "20 L of 10% and 10 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3d91741923aef266", "problem_id": "37ce152e01870031", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A store sells a backpack for $400. It is discounted by 15%. You buy 7 backpacks. What is the total cost?", "output": "1) Discounted price = 400 × (1 − 15/100) = 400 × 85/100 = 340.\n2) Subtotal = 7 × 340 = 2380.\nFinal answer: $2380$", "final_answer": "2380", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec6b654656fba75a", "problem_id": "37ce152e01870031", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $400. It is discounted by 15%. You buy 7 backpacks. What is the total cost?", "output": "Discounted price = 400 × (1 − 15/100) = 400 × 85/100 = 340.\nSubtotal = 7 × 340 = 2380.\nCheck: recomputing the formula yields 2380.\nFinal answer: $2380$", "final_answer": "2380", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "191f72256be08255", "problem_id": "89e2773a701b5b11", "style": "answer_only", "instruction": "Final answer only.", "input": "Solve the linear system:\n$2 x$ = $-6$\n$4 x + y$ = $-11$", "output": "$(x,y)=\\left(-3, 1\\right)$", "final_answer": "(x,y)=\\left(-3, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "635c80c5584807a8", "problem_id": "d903d488291fd67c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 2 x^{2} + 2 x + \\left(x^{2} + 3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{x^{2} + 3 x - 5}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 5}) = - \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3 x - 5}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{6 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{12 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{16 x^{2}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{30 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{60 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{48 x}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{10 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 6 x^{3} - x^{2} - 30 x + 25} + \\frac{25 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{50 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{8}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{15 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 6 x^{3} - x^{2} - 30 x + 25} + \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 6 x - 10} - \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3 x - 5} + \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 5}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 5}) = - \\frac{5 \\left(- 2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(x^{2} + 3 x\\right) - 5\\right)^{2}} - \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(x^{2} + 3 x\\right) - 5\\right)} - \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 5}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(- 2 x - 3\\right) \\left(- 2 x^{2} + 2 x + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(x \\left(x + 3\\right) - 5\\right) \\left(- 8 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x + 3\\right) - 5\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(- 2 x - 3\\right) \\left(- 2 x^{2} + 2 x + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(x \\left(x + 3\\right) - 5\\right) \\left(- 8 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x + 3\\right) - 5\\right)^{2}}$", "final_answer": "\\frac{2 \\left(- 2 x - 3\\right) \\left(- 2 x^{2} + 2 x + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(x \\left(x + 3\\right) - 5\\right) \\left(- 8 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x + 3\\right) - 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "1dccd3df14b7fe43", "problem_id": "46c8b3095b3b3f80", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡4 (mod 7). Give the solution modulo 21.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 21.\nCompute the CRT solution to get x ≡ 4 (mod 21).\nCheck: 4 mod 3=1 and mod 7=4.\nFinal answer: $x\\equiv 4\\pmod{21}$", "final_answer": "x\\equiv 4\\pmod{21}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a7d424ee3791f4c9", "problem_id": "46c8b3095b3b3f80", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡4 (mod 7). Give the solution modulo 21.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 21.\n3) Compute the CRT solution to get x ≡ 4 (mod 21).\nFinal answer: $x\\equiv 4\\pmod{21}$", "final_answer": "x\\equiv 4\\pmod{21}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "96a38c73aab3e654", "problem_id": "ceca83b7662c4734", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡6 (mod 7), x≡7 (mod 9). Give the solution modulo 63.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 63.\n3) Compute the CRT solution to get x ≡ 34 (mod 63).\nFinal answer: $x\\equiv 34\\pmod{63}$", "final_answer": "x\\equiv 34\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c2e0a5a0b73403f0", "problem_id": "ceca83b7662c4734", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡6 (mod 7), x≡7 (mod 9). Give the solution modulo 63.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 63.\nCompute the CRT solution to get x ≡ 34 (mod 63).\nCheck: 34 mod 7=6 and mod 9=7.\nFinal answer: $x\\equiv 34\\pmod{63}$", "final_answer": "x\\equiv 34\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ea3f7b7c53d77318", "problem_id": "97b5f2f0cefbb82f", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$- 8 x + 4 y$ = $-16$\n$x - 4 y$ = $-12$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $- 8 x + 4 y$ = $-16$.\n3) Multiply the second equation by $-8$: $- 8 x + 32 y$ = $96$.\n4) Subtract to eliminate x: $- 28 y$ = $-112$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, 4\\right)$", "final_answer": "(x,y)=\\left(4, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a7c206f7fbde1001", "problem_id": "97b5f2f0cefbb82f", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- 8 x + 4 y$ = $-16$\n$x - 4 y$ = $-12$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $- 8 x + 4 y$ = $-16$.\nMultiply the second equation by $-8$: $- 8 x + 32 y$ = $96$.\nSubtract to eliminate x: $- 28 y$ = $-112$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $4$.\nCheck: plug (x,y)=(4,4) into both equations.\nFinal answer: $(x,y)=\\left(4, 4\\right)$", "final_answer": "(x,y)=\\left(4, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f4f2634b42c351c3", "problem_id": "070b8d8e920c171e", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- x^{2} + 3 x - 1\\right) \\cos{\\left(3 x \\right)}$.", "output": "1) Let f(x) = $2 \\left(x + 1\\right) e^{x} + \\left(- x^{2} + 3 x - 1\\right) \\cos{\\left(3 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\left(3 - 2 x\\right) \\cos{\\left(3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 3 \\left(- x^{2} + 3 x - 1\\right) \\sin{\\left(3 x \\right)} + 2 e^{x}$.\nFinal answer: $\\left(3 - 2\\cdot x\\right)\\cdot \\cos{\\left(3\\cdot x \\right)} + \\left(2\\cdot x + 2\\right)\\cdot e^{x} - \\left(- 3\\cdot x^{2} + 9\\cdot x - 3\\right)\\cdot \\sin{\\left(3\\cdot x \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(3 - 2\\cdot x\\right)\\cdot \\cos{\\left(3\\cdot x \\right)} + \\left(2\\cdot x + 2\\right)\\cdot e^{x} - \\left(- 3\\cdot x^{2} + 9\\cdot x - 3\\right)\\cdot \\sin{\\left(3\\cdot x \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "39182ec59f8d7f64", "problem_id": "070b8d8e920c171e", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- x^{2} + 3 x - 1\\right) \\cos{\\left(3 x \\right)}$.", "output": "Let f(x) = $2 \\left(x + 1\\right) e^{x} + \\left(- x^{2} + 3 x - 1\\right) \\cos{\\left(3 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\left(3 - 2 x\\right) \\cos{\\left(3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 3 \\left(- x^{2} + 3 x - 1\\right) \\sin{\\left(3 x \\right)} + 2 e^{x}$.\nFinal answer: $\\left(3 - 2\\cdot x\\right)\\cdot \\cos{\\left(3\\cdot x \\right)} + \\left(2\\cdot x + 2\\right)\\cdot e^{x} - \\left(- 3\\cdot x^{2} + 9\\cdot x - 3\\right)\\cdot \\sin{\\left(3\\cdot x \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(3 - 2\\cdot x\\right)\\cdot \\cos{\\left(3\\cdot x \\right)} + \\left(2\\cdot x + 2\\right)\\cdot e^{x} - \\left(- 3\\cdot x^{2} + 9\\cdot x - 3\\right)\\cdot \\sin{\\left(3\\cdot x \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e8df35a7ed4ef701", "problem_id": "5a02fb3ca642a960", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the congruences: x≡2 (mod 8), x≡3 (mod 7). Give the solution modulo 56.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 56.\n3) Compute the CRT solution to get x ≡ 10 (mod 56).\nFinal answer: $x\\equiv 10\\pmod{56}$", "final_answer": "x\\equiv 10\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c6ac22d5b0613e7b", "problem_id": "5a02fb3ca642a960", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡2 (mod 8), x≡3 (mod 7). Give the solution modulo 56.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 56.\nCompute the CRT solution to get x ≡ 10 (mod 56).\nCheck: 10 mod 8=2 and mod 7=3.\nFinal answer: $x\\equiv 10\\pmod{56}$", "final_answer": "x\\equiv 10\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "454363699e6e44c0", "problem_id": "16be25eafa06faff", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "You want 30 liters of a 25% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 30−x liters is 70% solution.\n2) Salt equation: 20x + 70(30−x) = 25·30.\n3) (20-70)x = (25-70)·30 ⇒ x = 27.\n4) Then 30−x = 3.\nFinal answer: $27 L of 20% and 3 L of 70%$", "final_answer": "27 L of 20% and 3 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a21343e4cc7e8550", "problem_id": "16be25eafa06faff", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 30 liters of a 25% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 30−x liters is 70% solution.\nSalt equation: 20x + 70(30−x) = 25·30.\n(20-70)x = (25-70)·30 ⇒ x = 27.\nThen 30−x = 3.\nCheck: 20·27 + 70·3 = 25·30.\nFinal answer: $27 L of 20% and 3 L of 70%$", "final_answer": "27 L of 20% and 3 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b9ee29a007beeb67", "problem_id": "d09f4a86774a3157", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 7 red balls and 16 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(23,6).\n3) Favorable samples: choose k reds and n−k blues: C(7,2)·C(16,4).\n4) Compute totals: C(23,6) = 100947, and favorable = 38220.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1820}{4807}$", "final_answer": "\\frac{1820}{4807}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "963a69cead25cd9a", "problem_id": "d09f4a86774a3157", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 7 red balls and 16 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(23,6).\nFavorable samples: choose k reds and n−k blues: C(7,2)·C(16,4).\nCompute totals: C(23,6) = 100947, and favorable = 38220.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1820}{4807}$.\nFinal answer: $\\frac{1820}{4807}$", "final_answer": "\\frac{1820}{4807}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f4360f7090202ce5", "problem_id": "7a1f0e318e19a3c0", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{4} + x^{3} + 2 x^{2} - 2 x + 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(2 x^{2}) = 4 x^{2}$.\n3) So $f'(x) = - 8 x^{3} + 7 x^{2} - 2$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(2 x^{2}) = 4 x$.\nTherefore the correct derivative is $f'(x) = - 8 x^{3} + 3 x^{2} + 4 x - 2$.\nFinal answer: $- 8 x^{3} + 3 x^{2} + 4 x - 2$", "final_answer": "- 8 x^{3} + 3 x^{2} + 4 x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ec50707a743bef16", "problem_id": "882d258a5a6a2639", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the linear system:\n$4 x + 3 y$ = $2$\n$- x - y$ = $-1$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $- 4 x - 3 y$ = $-2$.\n3) Multiply the second equation by $4$: $- 4 x - 4 y$ = $-4$.\n4) Subtract to eliminate x: $y$ = $2$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, 2\\right)$", "final_answer": "(x,y)=\\left(-1, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "619ec0151ce13ba5", "problem_id": "f3d3b4e037485f8c", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(x^{2} - 3 x - 2\\right) \\cos{\\left(- 3 x^{2} + 3 x + 3 \\right)}$.", "output": "Let f(x) = $\\left(x + 2\\right) e^{x} + \\left(x^{2} - 3 x - 2\\right) \\cos{\\left(- 3 x^{2} + 3 x + 3 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- 3 \\left(1 - 2 x\\right) \\left(x^{2} - 3 x - 2\\right) \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} + \\left(x + 2\\right) e^{x} + \\left(2 x - 3\\right) \\cos{\\left(- 3 x^{2} + 3 x + 3 \\right)} + e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\left(x + 2\\right)\\cdot e^{x} + \\left(2\\cdot x - 3\\right)\\cdot \\cos{\\left(- 3\\cdot x^{2} + 3\\cdot x + 3 \\right)} + \\left(6\\cdot x - 3\\right)\\cdot \\left(x^{2} - 3\\cdot x - 2\\right)\\cdot \\sin{\\left(- 3\\cdot x^{2} + 3\\cdot x + 3 \\right)} + e^{x}$", "final_answer": "\\left(x + 2\\right)\\cdot e^{x} + \\left(2\\cdot x - 3\\right)\\cdot \\cos{\\left(- 3\\cdot x^{2} + 3\\cdot x + 3 \\right)} + \\left(6\\cdot x - 3\\right)\\cdot \\left(x^{2} - 3\\cdot x - 2\\right)\\cdot \\sin{\\left(- 3\\cdot x^{2} + 3\\cdot x + 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8fbe33bfd439e3e0", "problem_id": "f3d3b4e037485f8c", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(x^{2} - 3 x - 2\\right) \\cos{\\left(- 3 x^{2} + 3 x + 3 \\right)}$.", "output": "1) Let f(x) = $\\left(x + 2\\right) e^{x} + \\left(x^{2} - 3 x - 2\\right) \\cos{\\left(- 3 x^{2} + 3 x + 3 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- 3 \\left(1 - 2 x\\right) \\left(x^{2} - 3 x - 2\\right) \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} + \\left(x + 2\\right) e^{x} + \\left(2 x - 3\\right) \\cos{\\left(- 3 x^{2} + 3 x + 3 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 2\\right)\\cdot e^{x} + \\left(2\\cdot x - 3\\right)\\cdot \\cos{\\left(- 3\\cdot x^{2} + 3\\cdot x + 3 \\right)} + \\left(6\\cdot x - 3\\right)\\cdot \\left(x^{2} - 3\\cdot x - 2\\right)\\cdot \\sin{\\left(- 3\\cdot x^{2} + 3\\cdot x + 3 \\right)} + e^{x}$", "final_answer": "\\left(x + 2\\right)\\cdot e^{x} + \\left(2\\cdot x - 3\\right)\\cdot \\cos{\\left(- 3\\cdot x^{2} + 3\\cdot x + 3 \\right)} + \\left(6\\cdot x - 3\\right)\\cdot \\left(x^{2} - 3\\cdot x - 2\\right)\\cdot \\sin{\\left(- 3\\cdot x^{2} + 3\\cdot x + 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "408100240be4d269", "problem_id": "5377178775cdd578", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{2} - 4 x$.", "output": "1) Differentiate term-by-term:\n2) d/dx(- x^{2}) = - 2 x.\n3) d/dx(- 4 x) = -4.\n4) Combine: f'(x) = 2 \\left(- x - 2\\right).\nFinal answer: $- 2\\cdot x - 4$", "final_answer": "- 2\\cdot x - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fdd363264093e008", "problem_id": "5377178775cdd578", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{2} - 4 x$.", "output": "Differentiate term-by-term:\nd/dx(- x^{2}) = - 2 x.\nd/dx(- 4 x) = -4.\nCombine: f'(x) = 2 \\left(- x - 2\\right).\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 2\\cdot x - 4$", "final_answer": "- 2\\cdot x - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6fdb0068ed50b4d0", "problem_id": "9e7e804921d6d527", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 79 miles per hour for 1/2 hours, then at 34 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 79 × 1/2 = 79/2 miles.\nSecond leg distance = 34 × 3/2 = 51 miles.\nTotal distance = 79/2 + 51 = 181/2 miles.\nCheck: adding both legs gives 181/2.\nFinal answer: $\\frac{181}{2}$", "final_answer": "\\frac{181}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9c1381dcac5a1e19", "problem_id": "8a6217f9111d785d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 - 3 x$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 3 - 4 x.\n3) Then the limit is 11.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 9.\nFinal answer: $9$", "final_answer": "9", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7e59a3f800d41754", "problem_id": "4ef51e440f9db9f1", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{7} x}{2} + 2 x - \\frac{9 \\sqrt{7}}{2} - 3$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$2 - \\frac{3 \\sqrt{7}}{2}$, c=$- \\frac{9 \\sqrt{7}}{2} - 3$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{-1 + \\frac{3\\cdot \\sqrt{7}}{4} + \\frac{\\sqrt{48\\cdot \\sqrt{7} + 127}}{4}, - \\frac{\\sqrt{48\\cdot \\sqrt{7} + 127}}{4} - 1 + \\frac{3\\cdot \\sqrt{7}}{4}\\}$", "final_answer": "x \\in \\{-1 + \\frac{3\\cdot \\sqrt{7}}{4} + \\frac{\\sqrt{48\\cdot \\sqrt{7} + 127}}{4}, - \\frac{\\sqrt{48\\cdot \\sqrt{7} + 127}}{4} - 1 + \\frac{3\\cdot \\sqrt{7}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e80ad6420ad54c8f", "problem_id": "4ef51e440f9db9f1", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{7} x}{2} + 2 x - \\frac{9 \\sqrt{7}}{2} - 3$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$2 - \\frac{3 \\sqrt{7}}{2}$, c=$- \\frac{9 \\sqrt{7}}{2} - 3$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $12 \\sqrt{7} + \\frac{127}{4}$, so √Δ = $\\sqrt{12 \\sqrt{7} + \\frac{127}{4}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $-1 + \\frac{3 \\sqrt{7}}{4} + \\frac{\\sqrt{48 \\sqrt{7} + 127}}{4}$.\n7) x2 = $- \\frac{\\sqrt{48 \\sqrt{7} + 127}}{4} - 1 + \\frac{3 \\sqrt{7}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{-1 + \\frac{3\\cdot \\sqrt{7}}{4} + \\frac{\\sqrt{48\\cdot \\sqrt{7} + 127}}{4}, - \\frac{\\sqrt{48\\cdot \\sqrt{7} + 127}}{4} - 1 + \\frac{3\\cdot \\sqrt{7}}{4}\\}$", "final_answer": "x \\in \\{-1 + \\frac{3\\cdot \\sqrt{7}}{4} + \\frac{\\sqrt{48\\cdot \\sqrt{7} + 127}}{4}, - \\frac{\\sqrt{48\\cdot \\sqrt{7} + 127}}{4} - 1 + \\frac{3\\cdot \\sqrt{7}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2444c2143928c5b2", "problem_id": "7a95093493e9331b", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "You want 23 liters of a 30% salt solution. You have a 25% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 23−x liters is 60% solution.\n2) Salt equation: 25x + 60(23−x) = 30·23.\n3) (25-60)x = (30-60)·23 ⇒ x = 138/7.\n4) Then 23−x = 23/7.\nFinal answer: $138/7 L of 25% and 23/7 L of 60%$", "final_answer": "138/7 L of 25% and 23/7 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ba435f434e1457cc", "problem_id": "7a95093493e9331b", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "You want 23 liters of a 30% salt solution. You have a 25% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 23−x liters is 60% solution.\nSalt equation: 25x + 60(23−x) = 30·23.\n(25-60)x = (30-60)·23 ⇒ x = 138/7.\nThen 23−x = 23/7.\nCheck: 25·138/7 + 60·23/7 = 30·23.\nFinal answer: $138/7 L of 25% and 23/7 L of 60%$", "final_answer": "138/7 L of 25% and 23/7 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3389fde4efdcc5ab", "problem_id": "7c38fdfd2070353b", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Two friends split $1611 in the ratio 1:8. How much does each receive?", "output": "1) Total ratio parts = 1+8 = 9.\n2) Each part = 1611 ÷ 9 = 179.\n3) First share = 1 × 179 = 179.\n4) Second share = 8 × 179 = 1432.\nFinal answer: $1432$", "final_answer": "1432", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2fa7c4e362093f07", "problem_id": "b708677091c5e5b4", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(2 x^{3} - x^{2} + x - 2\\right) \\cos{\\left(2 x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x^{3} \\cos{\\left(2 x + 3 \\right)}) = - 2 x^{3} \\sin{\\left(2 x + 3 \\right)}$.\n3) So $f'(x) = - 2 x^{3} \\sin{\\left(2 x + 3 \\right)} + 2 x^{2} \\sin{\\left(2 x + 3 \\right)} + 2 x e^{x} - 2 x \\sin{\\left(2 x + 3 \\right)} - 2 x \\cos{\\left(2 x + 3 \\right)} + 6 e^{x} + 4 \\sin{\\left(2 x + 3 \\right)} + \\cos{\\left(2 x + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x^{3} \\cos{\\left(2 x + 3 \\right)}) = - 4 x^{3} \\sin{\\left(2 x + 3 \\right)} + 6 x^{2} \\cos{\\left(2 x + 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x + 2\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) + 1\\right) - 2\\right) \\sin{\\left(2 x + 3 \\right)} + \\left(6 x^{2} - 2 x + 1\\right) \\cos{\\left(2 x + 3 \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(x + 2\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) + 1\\right) - 2\\right) \\sin{\\left(2 x + 3 \\right)} + \\left(6 x^{2} - 2 x + 1\\right) \\cos{\\left(2 x + 3 \\right)} + 2 e^{x}$", "final_answer": "2 \\left(x + 2\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) + 1\\right) - 2\\right) \\sin{\\left(2 x + 3 \\right)} + \\left(6 x^{2} - 2 x + 1\\right) \\cos{\\left(2 x + 3 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "57877042860e1895", "problem_id": "b82e99176332d679", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} x \\left(- x - 2\\right)$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{3} - 3 x^{2} - 2 x + 2$ and a = -1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 3 x^{2} - 6 x - 2$.\n5) Evaluate at a: p'(-1) = $1$.\nFinal answer: $1$", "final_answer": "1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "50f350d20934bd72", "problem_id": "b82e99176332d679", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} x \\left(- x - 2\\right)$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{3} - 3 x^{2} - 2 x + 2$ and a = -1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 3 x^{2} - 6 x - 2$.\nEvaluate at a: p'(-1) = $1$.\nCheck: the identity gives $1$.\nFinal answer: $1$", "final_answer": "1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ffe9e64f9bc9b926", "problem_id": "24a35d7ce058c53f", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "An urn has 29 red balls and 10 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(39,8).\nFavorable samples: choose k reds and n−k blues: C(29,4)·C(10,4).\nFinal answer: $\\frac{21315}{262922}$", "final_answer": "\\frac{21315}{262922}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d0448e4be7475f30", "problem_id": "24a35d7ce058c53f", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 29 red balls and 10 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(39,8).\nFavorable samples: choose k reds and n−k blues: C(29,4)·C(10,4).\nCompute totals: C(39,8) = 61523748, and favorable = 4987710.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{21315}{262922}$.\nFinal answer: $\\frac{21315}{262922}$", "final_answer": "\\frac{21315}{262922}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b6629ddee5598556", "problem_id": "08427d3cfc08945a", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Worker A finishes a job in 6 hours and Worker B finishes it in 16 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/6 job/hour. B's rate = 1/16 job/hour.\n2) Combined rate = 1/6 + 1/16 = 11/48 job/hour.\n3) Time = 1 ÷ (combined rate) = 48/11 hours.\nFinal answer: $\\frac{48}{11}$", "final_answer": "\\frac{48}{11}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a53f06e9a493db50", "problem_id": "08427d3cfc08945a", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Worker A finishes a job in 6 hours and Worker B finishes it in 16 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "$\\frac{48}{11}$", "final_answer": "\\frac{48}{11}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c1fcfbd43768f1b3", "problem_id": "9edd22cec7d638d9", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_6=45$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_6=27.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=27 and k=6.\n4) Compute C(32,5) = 201376.\nFinal answer: $201376$", "final_answer": "201376", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d5c0355f5966a9d", "problem_id": "1973eb501f26a2f2", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} + 2 x + \\left(x^{2} - x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{x^{2} - x + 7}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7}) = \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 7}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{4 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 15 x^{2} - 14 x + 49} + \\frac{15 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{30 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{2 x^{2}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 15 x^{2} - 14 x + 49} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 2 x + 14} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 7} - \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7} - \\frac{14 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{28 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{40 x}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7} + \\frac{49 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{98 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{34}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7}) = \\frac{x^{2} \\left(1 - 2 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(x^{2} - x\\right) + 7\\right)^{2}} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(x^{2} - x\\right) + 7\\right)} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7} + \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(x - 1\\right) + 7\\right) \\left(- 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x - 1\\right) + 7\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(x - 1\\right) + 7\\right) \\left(- 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x - 1\\right) + 7\\right)^{2}}$", "final_answer": "\\frac{2 \\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(x - 1\\right) + 7\\right) \\left(- 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x - 1\\right) + 7\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "88f14cb71d76423f", "problem_id": "939ede2e1a317e20", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 \\sin{\\left(x \\right)}) = 2 \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 \\sin{\\left(x \\right)}) = 2 \\cos{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $- 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$", "final_answer": "- 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5f6338592cf13ab5", "problem_id": "9683a5728ece0dd1", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - x^{3} + 2 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{3} + 2 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{255 \\pi^{4}}{64} - 1 + \\frac{5 \\pi}{2} + \\frac{15 \\pi^{2}}{8} + \\frac{65 \\pi^{3}}{12}$.\n4) Simplify the result.\nFinal answer: $- \\frac{255\\cdot \\pi^{4}}{64} - 1 + \\frac{5\\cdot \\pi}{2} + \\frac{15\\cdot \\pi^{2}}{8} + \\frac{65\\cdot \\pi^{3}}{12}$", "final_answer": "- \\frac{255\\cdot \\pi^{4}}{64} - 1 + \\frac{5\\cdot \\pi}{2} + \\frac{15\\cdot \\pi^{2}}{8} + \\frac{65\\cdot \\pi^{3}}{12}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c562fec9136117b1", "problem_id": "143c86e4aa90dba1", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(- 3 x^{3} - x^{2} - 2\\right) \\cos{\\left(3 x + 1 \\right)}$.", "output": "1) Let f(x) = $2 \\left(x - 3\\right) e^{x} + \\left(- 3 x^{3} - x^{2} - 2\\right) \\cos{\\left(3 x + 1 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $x \\left(- 9 x - 2\\right) \\cos{\\left(3 x + 1 \\right)} + 2 \\left(x - 3\\right) e^{x} - 3 \\left(- 3 x^{3} - x^{2} - 2\\right) \\sin{\\left(3 x + 1 \\right)} + 2 e^{x}$.\nFinal answer: $x\\cdot \\left(- 9\\cdot x - 2\\right)\\cdot \\cos{\\left(3\\cdot x + 1 \\right)} + \\left(2\\cdot x - 6\\right)\\cdot e^{x} - \\left(- 9\\cdot x^{3} - 3\\cdot x^{2} - 6\\right)\\cdot \\sin{\\left(3\\cdot x + 1 \\right)} + 2\\cdot e^{x}$", "final_answer": "x\\cdot \\left(- 9\\cdot x - 2\\right)\\cdot \\cos{\\left(3\\cdot x + 1 \\right)} + \\left(2\\cdot x - 6\\right)\\cdot e^{x} - \\left(- 9\\cdot x^{3} - 3\\cdot x^{2} - 6\\right)\\cdot \\sin{\\left(3\\cdot x + 1 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1fc7a6131cc0ed59", "problem_id": "4303602b72d4f841", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} 3 x^{3} - 11 x^{2} + 47 x - 191$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} - 11 x^{2} + 46 x - 191.\n3) Then the limit is -743.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: -747.\nFinal answer: $-747$", "final_answer": "-747", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "68d995ea683fb161", "problem_id": "1338e2ceb2eba40b", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "An urn has 12 red balls and 14 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(26,7).\nFavorable samples: choose k reds and n−k blues: C(12,2)·C(14,5).\nFinal answer: $\\frac{231}{1150}$", "final_answer": "\\frac{231}{1150}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "74fdc62b43c6f5a6", "problem_id": "1338e2ceb2eba40b", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 12 red balls and 14 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(26,7).\n3) Favorable samples: choose k reds and n−k blues: C(12,2)·C(14,5).\n4) Compute totals: C(26,7) = 657800, and favorable = 132132.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{231}{1150}$", "final_answer": "\\frac{231}{1150}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9fec5102e2f3e3a8", "problem_id": "9793109064b373b0", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 8 x - 42$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 8 x - 42$ = $2 \\left(x - 7\\right) \\left(x + 3\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $7$.\n5) From the second factor: x = $-3$.\nFinal answer: $x \\in \\{7, -3\\}$", "final_answer": "x \\in \\{7, -3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5c926cecb6eae367", "problem_id": "9793109064b373b0", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 8 x - 42$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} - 8 x - 42$ = $2 \\left(x - 7\\right) \\left(x + 3\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFinal answer: $x \\in \\{7, -3\\}$", "final_answer": "x \\in \\{7, -3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1f01e42ef0ad3270", "problem_id": "dc2923b800951abb", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $226. It is discounted by 20%. You buy 13 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 226 × (1 − 20/100) = 226 × 80/100 = 904/5.\nSubtotal = 13 × 904/5 = 11752/5.\nTotal with tax = 11752/5 × 108/100 = 317304/125.\nCheck: recomputing the formula yields 317304/125.\nFinal answer: $\\frac{317304}{125}$", "final_answer": "\\frac{317304}{125}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "40983e9b0858b8a5", "problem_id": "dc2923b800951abb", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A store sells a backpack for $226. It is discounted by 20%. You buy 13 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 226 × (1 − 20/100) = 226 × 80/100 = 904/5.\n2) Subtotal = 13 × 904/5 = 11752/5.\n3) Total with tax = 11752/5 × 108/100 = 317304/125.\nFinal answer: $\\frac{317304}{125}$", "final_answer": "\\frac{317304}{125}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9f4e849fae311876", "problem_id": "4b4ef155686c6bfd", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A store sells a backpack for $116. It is discounted by 10%. You buy 8 backpacks. What is the total cost?", "output": "Discounted price = 116 × (1 − 10/100) = 116 × 90/100 = 522/5.\nSubtotal = 8 × 522/5 = 4176/5.\nFinal answer: $\\frac{4176}{5}$", "final_answer": "\\frac{4176}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bf852e40b58515e5", "problem_id": "4b4ef155686c6bfd", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A store sells a backpack for $116. It is discounted by 10%. You buy 8 backpacks. What is the total cost?", "output": "1) Discounted price = 116 × (1 − 10/100) = 116 × 90/100 = 522/5.\n2) Subtotal = 8 × 522/5 = 4176/5.\nFinal answer: $\\frac{4176}{5}$", "final_answer": "\\frac{4176}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3e1813d06382b9fb", "problem_id": "ca884422e7f124a3", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$3 x + 4 y$ = $11$\n$- 4 y$ = $-20$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $3$: $- 12 y$ = $-60$.\n4) Subtract to eliminate x: $12 y$ = $60$.\n5) Solve for y: y = $5$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, 5\\right)$", "final_answer": "(x,y)=\\left(-3, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "16a4c6939f10db95", "problem_id": "ca884422e7f124a3", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the linear system:\n$3 x + 4 y$ = $11$\n$- 4 y$ = $-20$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $3$: $- 12 y$ = $-60$.\nFinal answer: $(x,y)=\\left(-3, 5\\right)$", "final_answer": "(x,y)=\\left(-3, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ea1a34cd7458f5a3", "problem_id": "9f03831cca1f19fe", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 16 red balls and 29 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(45,11).\nFavorable samples: choose k reds and n−k blues: C(16,2)·C(29,9).\nCompute totals: C(45,11) = 10150595910, and favorable = 1201800600.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{146740}{1239389}$.\nFinal answer: $\\frac{146740}{1239389}$", "final_answer": "\\frac{146740}{1239389}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c2f4b29d3fcdc8a6", "problem_id": "9f03831cca1f19fe", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "An urn has 16 red balls and 29 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(45,11).\n3) Favorable samples: choose k reds and n−k blues: C(16,2)·C(29,9).\n4) Compute totals: C(45,11) = 10150595910, and favorable = 1201800600.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{146740}{1239389}$", "final_answer": "\\frac{146740}{1239389}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "60bea1e55a06001c", "problem_id": "186969d3492f660d", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A cyclist rides at 38 miles per hour for 1/2 hours, then at 51 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 38 × 1/2 = 19 miles.\n2) Second leg distance = 51 × 5/2 = 255/2 miles.\n3) Total distance = 19 + 255/2 = 293/2 miles.\nFinal answer: $\\frac{293}{2}$", "final_answer": "\\frac{293}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b24635ea027be65b", "problem_id": "186969d3492f660d", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A cyclist rides at 38 miles per hour for 1/2 hours, then at 51 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 38 × 1/2 = 19 miles.\nSecond leg distance = 51 × 5/2 = 255/2 miles.\nTotal distance = 19 + 255/2 = 293/2 miles.\nCheck: adding both legs gives 293/2.\nFinal answer: $\\frac{293}{2}$", "final_answer": "\\frac{293}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "60ae742de56f7cce", "problem_id": "1fc9e8a8e2c1ca93", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$- x$ = $0$\n$4 x + 3 y$ = $9$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $- 4 x$ = $0$.\n3) Multiply the second equation by $-1$: $- 4 x - 3 y$ = $-9$.\n4) Subtract to eliminate x: $3 y$ = $9$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, 3\\right)$", "final_answer": "(x,y)=\\left(0, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b51885d1d916ce7c", "problem_id": "1fc9e8a8e2c1ca93", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$- x$ = $0$\n$4 x + 3 y$ = $9$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $- 4 x$ = $0$.\nMultiply the second equation by $-1$: $- 4 x - 3 y$ = $-9$.\nSubtract to eliminate x: $3 y$ = $9$.\nSolve for y: y = $3$.\nSubstitute back to find x: x = $0$.\nCheck: plug (x,y)=(0,3) into both equations.\nFinal answer: $(x,y)=\\left(0, 3\\right)$", "final_answer": "(x,y)=\\left(0, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "71391a643bb0fa14", "problem_id": "d494abb8c5ad15c9", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Two friends split $324 in the ratio 4:8. How much does each receive?", "output": "1) Total ratio parts = 4+8 = 12.\n2) Each part = 324 ÷ 12 = 27.\n3) First share = 4 × 27 = 108.\n4) Second share = 8 × 27 = 216.\nFinal answer: $216$", "final_answer": "216", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cb73f1aa80261d39", "problem_id": "b2ea521ab8090dcc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(2 x^{3} + x^{2} + 3 x - 3\\right) \\cos{\\left(2 x + 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x^{3} \\cos{\\left(2 x + 2 \\right)}) = - 2 x^{3} \\sin{\\left(2 x + 2 \\right)}$.\n3) So $f'(x) = - 2 x^{3} \\sin{\\left(2 x + 2 \\right)} - 2 x^{2} \\sin{\\left(2 x + 2 \\right)} + 2 x e^{x} - 6 x \\sin{\\left(2 x + 2 \\right)} + 2 x \\cos{\\left(2 x + 2 \\right)} - 4 e^{x} + 6 \\sin{\\left(2 x + 2 \\right)} + 3 \\cos{\\left(2 x + 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x^{3} \\cos{\\left(2 x + 2 \\right)}) = - 4 x^{3} \\sin{\\left(2 x + 2 \\right)} + 6 x^{2} \\cos{\\left(2 x + 2 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x - 3\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x + 1\\right) + 3\\right) - 3\\right) \\sin{\\left(2 x + 2 \\right)} + \\left(6 x^{2} + 2 x + 3\\right) \\cos{\\left(2 x + 2 \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(x - 3\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x + 1\\right) + 3\\right) - 3\\right) \\sin{\\left(2 x + 2 \\right)} + \\left(6 x^{2} + 2 x + 3\\right) \\cos{\\left(2 x + 2 \\right)} + 2 e^{x}$", "final_answer": "2 \\left(x - 3\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x + 1\\right) + 3\\right) - 3\\right) \\sin{\\left(2 x + 2 \\right)} + \\left(6 x^{2} + 2 x + 3\\right) \\cos{\\left(2 x + 2 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7f381114cabd6798", "problem_id": "61597971a8e26afc", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - x^{2} - 2 x - 3$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{3} + x^{2} + 3 x - 3$ and a = 3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-18$", "final_answer": "-18", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "000408657fe9030a", "problem_id": "61597971a8e26afc", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - x^{2} - 2 x - 3$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{3} + x^{2} + 3 x - 3$ and a = 3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 3 x^{2} + 2 x + 3$.\n5) Evaluate at a: p'(3) = $-18$.\nFinal answer: $-18$", "final_answer": "-18", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eb48855083914c9c", "problem_id": "705b04a7667749cf", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "You want 20 liters of a 25% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 20−x liters is 70% solution.\n2) Salt equation: 20x + 70(20−x) = 25·20.\n3) (20-70)x = (25-70)·20 ⇒ x = 18.\n4) Then 20−x = 2.\nFinal answer: $18 L of 20% and 2 L of 70%$", "final_answer": "18 L of 20% and 2 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3d6d0ddc2aa394a8", "problem_id": "705b04a7667749cf", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "You want 20 liters of a 25% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 20−x liters is 70% solution.\nSalt equation: 20x + 70(20−x) = 25·20.\n(20-70)x = (25-70)·20 ⇒ x = 18.\nThen 20−x = 2.\nCheck: 20·18 + 70·2 = 25·20.\nFinal answer: $18 L of 20% and 2 L of 70%$", "final_answer": "18 L of 20% and 2 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cd7a034d4a43a48e", "problem_id": "aae1fb5eaa1aecc6", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} - 3 x^{2} + 11 x - 41$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 3 x^{3} - x^{2} + 3 x - 2$ and a = -4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 9 x^{2} - 2 x + 3$.\n5) Evaluate at a: p'(-4) = $-133$.\nFinal answer: $-133$", "final_answer": "-133", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1e0f45f0bf13db66", "problem_id": "aae1fb5eaa1aecc6", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} - 3 x^{2} + 11 x - 41$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 3 x^{3} - x^{2} + 3 x - 2$ and a = -4.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 9 x^{2} - 2 x + 3$.\nEvaluate at a: p'(-4) = $-133$.\nCheck: the identity gives $-133$.\nFinal answer: $-133$", "final_answer": "-133", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "861c7be190b3470a", "problem_id": "9c34718406ea5ba7", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$4 x^{2} - 9 x - \\frac{4 \\sqrt{5} x}{3} + 3 \\sqrt{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$4$, b=$-9 - \\frac{4 \\sqrt{5}}{3}$, c=$3 \\sqrt{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{809}{9} - 24 \\sqrt{5}$, so √Δ = $\\sqrt{\\frac{809}{9} - 24 \\sqrt{5}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{5}}{6} + \\frac{\\sqrt{809 - 216 \\sqrt{5}}}{24} + \\frac{9}{8}$.\n7) x2 = $- \\frac{\\sqrt{809 - 216 \\sqrt{5}}}{24} + \\frac{\\sqrt{5}}{6} + \\frac{9}{8}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{5}}{6} + \\frac{\\sqrt{809 - 216\\cdot \\sqrt{5}}}{24} + \\frac{9}{8}, - \\frac{\\sqrt{809 - 216\\cdot \\sqrt{5}}}{24} + \\frac{\\sqrt{5}}{6} + \\frac{9}{8}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{5}}{6} + \\frac{\\sqrt{809 - 216\\cdot \\sqrt{5}}}{24} + \\frac{9}{8}, - \\frac{\\sqrt{809 - 216\\cdot \\sqrt{5}}}{24} + \\frac{\\sqrt{5}}{6} + \\frac{9}{8}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ee144dd116012b55", "problem_id": "9c34718406ea5ba7", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$4 x^{2} - 9 x - \\frac{4 \\sqrt{5} x}{3} + 3 \\sqrt{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$4$, b=$-9 - \\frac{4 \\sqrt{5}}{3}$, c=$3 \\sqrt{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{809}{9} - 24 \\sqrt{5}$, so √Δ = $\\sqrt{\\frac{809}{9} - 24 \\sqrt{5}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{5}}{6} + \\frac{\\sqrt{809 - 216 \\sqrt{5}}}{24} + \\frac{9}{8}$.\nx2 = $- \\frac{\\sqrt{809 - 216 \\sqrt{5}}}{24} + \\frac{\\sqrt{5}}{6} + \\frac{9}{8}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{5}}{6} + \\frac{\\sqrt{809 - 216\\cdot \\sqrt{5}}}{24} + \\frac{9}{8}, - \\frac{\\sqrt{809 - 216\\cdot \\sqrt{5}}}{24} + \\frac{\\sqrt{5}}{6} + \\frac{9}{8}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{5}}{6} + \\frac{\\sqrt{809 - 216\\cdot \\sqrt{5}}}{24} + \\frac{9}{8}, - \\frac{\\sqrt{809 - 216\\cdot \\sqrt{5}}}{24} + \\frac{\\sqrt{5}}{6} + \\frac{9}{8}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "748ecc273cf0cb29", "problem_id": "8977d33d66bc046b", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{5} x}{2} + \\frac{3 x}{2} - \\frac{5}{2} + \\frac{3 \\sqrt{5}}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{3}{2} - \\frac{3 \\sqrt{5}}{2}$, c=$- \\frac{5}{2} + \\frac{3 \\sqrt{5}}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{47}{2} - \\frac{21 \\sqrt{5}}{2}$, so √Δ = $\\sqrt{\\frac{47}{2} - \\frac{21 \\sqrt{5}}{2}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{3}{4} + \\frac{\\sqrt{94 - 42 \\sqrt{5}}}{4} + \\frac{3 \\sqrt{5}}{4}$.\nx2 = $- \\frac{3}{4} - \\frac{\\sqrt{94 - 42 \\sqrt{5}}}{4} + \\frac{3 \\sqrt{5}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{3}{4} + \\frac{\\sqrt{94 - 42\\cdot \\sqrt{5}}}{4} + \\frac{3\\cdot \\sqrt{5}}{4}, - \\frac{3}{4} - \\frac{\\sqrt{94 - 42\\cdot \\sqrt{5}}}{4} + \\frac{3\\cdot \\sqrt{5}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{3}{4} + \\frac{\\sqrt{94 - 42\\cdot \\sqrt{5}}}{4} + \\frac{3\\cdot \\sqrt{5}}{4}, - \\frac{3}{4} - \\frac{\\sqrt{94 - 42\\cdot \\sqrt{5}}}{4} + \\frac{3\\cdot \\sqrt{5}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1b2e85f0ab44887a", "problem_id": "8977d33d66bc046b", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{5} x}{2} + \\frac{3 x}{2} - \\frac{5}{2} + \\frac{3 \\sqrt{5}}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{3}{2} - \\frac{3 \\sqrt{5}}{2}$, c=$- \\frac{5}{2} + \\frac{3 \\sqrt{5}}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{47}{2} - \\frac{21 \\sqrt{5}}{2}$, so √Δ = $\\sqrt{\\frac{47}{2} - \\frac{21 \\sqrt{5}}{2}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{3}{4} + \\frac{\\sqrt{94 - 42 \\sqrt{5}}}{4} + \\frac{3 \\sqrt{5}}{4}$.\n7) x2 = $- \\frac{3}{4} - \\frac{\\sqrt{94 - 42 \\sqrt{5}}}{4} + \\frac{3 \\sqrt{5}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{3}{4} + \\frac{\\sqrt{94 - 42\\cdot \\sqrt{5}}}{4} + \\frac{3\\cdot \\sqrt{5}}{4}, - \\frac{3}{4} - \\frac{\\sqrt{94 - 42\\cdot \\sqrt{5}}}{4} + \\frac{3\\cdot \\sqrt{5}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{3}{4} + \\frac{\\sqrt{94 - 42\\cdot \\sqrt{5}}}{4} + \\frac{3\\cdot \\sqrt{5}}{4}, - \\frac{3}{4} - \\frac{\\sqrt{94 - 42\\cdot \\sqrt{5}}}{4} + \\frac{3\\cdot \\sqrt{5}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3058bb4e707ed3af", "problem_id": "f1f772a9847fc661", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Two friends split $272 in the ratio 8:8. How much does each receive?", "output": "1) Total ratio parts = 8+8 = 16.\n2) Each part = 272 ÷ 16 = 17.\n3) First share = 8 × 17 = 136.\n4) Second share = 8 × 17 = 136.\nFinal answer: $136$", "final_answer": "136", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e0d481fc55950b27", "problem_id": "f1f772a9847fc661", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $272 in the ratio 8:8. How much does each receive?", "output": "Total ratio parts = 8+8 = 16.\nEach part = 272 ÷ 16 = 17.\nFirst share = 8 × 17 = 136.\nSecond share = 8 × 17 = 136.\nCheck: 136+136=272.\nFinal answer: $136$", "final_answer": "136", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "93f5100a8d64a3dd", "problem_id": "9b265e83c27dc684", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{7}$. If it is flipped 10 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=4, p=$\\frac{1}{7}$ and simplify.\n4) Here C(10,4) = 210.\nFinal answer: $\\frac{1399680}{40353607}$", "final_answer": "\\frac{1399680}{40353607}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d7f1a1777eb466ca", "problem_id": "9b265e83c27dc684", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{7}$. If it is flipped 10 times, what is the probability of exactly 4 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=4, p=$\\frac{1}{7}$ and simplify.\nHere C(10,4) = 210.\nCheck: expression simplifies to $\\frac{1399680}{40353607}$.\nFinal answer: $\\frac{1399680}{40353607}$", "final_answer": "\\frac{1399680}{40353607}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ac84a83f3bf0985b", "problem_id": "d8a0546b963e2602", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 16 red balls and 7 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(23,7).\n3) Favorable samples: choose k reds and n−k blues: C(16,1)·C(7,6).\n4) Compute totals: C(23,7) = 245157, and favorable = 112.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{112}{245157}$", "final_answer": "\\frac{112}{245157}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f283b64bb476c675", "problem_id": "a9836c54ff5c5377", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{3} + 3 x^{2} + 2 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{2 x - 3}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 3}) = - \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 3}$.\n3) So $f'(x) = \\frac{8 x^{3}}{8 x^{2} - 24 x + 18} + \\frac{4 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{2} - 24 x + 18} + \\frac{8 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{2} - 24 x + 18} - \\frac{6 x^{2}}{8 x^{2} - 24 x + 18} - \\frac{12 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{2} - 24 x + 18} - \\frac{24 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{2} - 24 x + 18} - \\frac{36 x}{8 x^{2} - 24 x + 18} + \\frac{9 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{2} - 24 x + 18} + \\frac{18 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{2} - 24 x + 18} - \\frac{20}{8 x^{2} - 24 x + 18} - \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{4 x^{2} - 12 x + 9} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{4 x - 6} - \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 3} + \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 3}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 3}) = - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(2 x - 3\\right)} - \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 3} + \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x - 3\\right)^{2}}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 4 x^{3} - 12 x^{2} - 8 x + \\left(2 x - 3\\right) \\left(6 x^{2} + 12 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 4 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8}{2 \\left(2 x - 3\\right)^{2}}$.\nFinal answer: $\\frac{- 4 x^{3} - 12 x^{2} - 8 x + \\left(2 x - 3\\right) \\left(6 x^{2} + 12 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 4 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8}{2 \\left(2 x - 3\\right)^{2}}$", "final_answer": "\\frac{- 4 x^{3} - 12 x^{2} - 8 x + \\left(2 x - 3\\right) \\left(6 x^{2} + 12 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 4 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8}{2 \\left(2 x - 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "1c6695fc8534b485", "problem_id": "492f095e9cb5ae7e", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} - 3 x^{3} + x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{3} + x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\n2) Evaluate at the bounds: F(\\pi) and F(0).\n3) Subtract: F(\\pi) − F(0) = $- \\frac{3 \\pi^{4}}{4} + 4 + 2 \\pi + \\pi^{2} + \\frac{\\pi^{3}}{3}$.\n4) Simplify the result.\nFinal answer: $- \\frac{3\\cdot \\pi^{4}}{4} + 4 + 2\\cdot \\pi + \\pi^{2} + \\frac{\\pi^{3}}{3}$", "final_answer": "- \\frac{3\\cdot \\pi^{4}}{4} + 4 + 2\\cdot \\pi + \\pi^{2} + \\frac{\\pi^{3}}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e257c10b41d788c1", "problem_id": "492f095e9cb5ae7e", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} - 3 x^{3} + x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{3} + x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\nEvaluate at the bounds: F(\\pi) and F(0).\nSubtract: F(\\pi) − F(0) = $- \\frac{3 \\pi^{4}}{4} + 4 + 2 \\pi + \\pi^{2} + \\frac{\\pi^{3}}{3}$.\nFinal answer: $- \\frac{3\\cdot \\pi^{4}}{4} + 4 + 2\\cdot \\pi + \\pi^{2} + \\frac{\\pi^{3}}{3}$", "final_answer": "- \\frac{3\\cdot \\pi^{4}}{4} + 4 + 2\\cdot \\pi + \\pi^{2} + \\frac{\\pi^{3}}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f419770759f9770e", "problem_id": "95eb60d16dd49a77", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the linear system:\n$10 x + 10 y$ = $120$\n$- 12 x + 12 y$ = $24$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-12$: $- 120 x - 120 y$ = $-1440$.\n3) Multiply the second equation by $10$: $- 120 x + 120 y$ = $240$.\n4) Subtract to eliminate x: $- 240 y$ = $-1680$.\n5) Solve for y: y = $7$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, 7\\right)$", "final_answer": "(x,y)=\\left(5, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a6ae99324a57e0c1", "problem_id": "95eb60d16dd49a77", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the linear system:\n$10 x + 10 y$ = $120$\n$- 12 x + 12 y$ = $24$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-12$: $- 120 x - 120 y$ = $-1440$.\nMultiply the second equation by $10$: $- 120 x + 120 y$ = $240$.\nFinal answer: $(x,y)=\\left(5, 7\\right)$", "final_answer": "(x,y)=\\left(5, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a2c467839a3f12ba", "problem_id": "a4280c3d163e25dc", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 28 red balls and 16 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(44,9).\n3) Favorable samples: choose k reds and n−k blues: C(28,3)·C(16,6).\n4) Compute totals: C(44,9) = 708930508, and favorable = 26234208.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{45864}{1239389}$", "final_answer": "\\frac{45864}{1239389}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "faf284b39afb1b6d", "problem_id": "a4280c3d163e25dc", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 28 red balls and 16 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(44,9).\nFavorable samples: choose k reds and n−k blues: C(28,3)·C(16,6).\nCompute totals: C(44,9) = 708930508, and favorable = 26234208.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{45864}{1239389}$.\nFinal answer: $\\frac{45864}{1239389}$", "final_answer": "\\frac{45864}{1239389}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0d63c1a65a61d6c8", "problem_id": "57d117776d96c884", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A cyclist rides at 36 miles per hour for 3/2 hours, then at 39 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 36 × 3/2 = 54 miles.\nSecond leg distance = 39 × 5/2 = 195/2 miles.\nTotal distance = 54 + 195/2 = 303/2 miles.\nFinal answer: $\\frac{303}{2}$", "final_answer": "\\frac{303}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "863aa1232eaf9c96", "problem_id": "57d117776d96c884", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A cyclist rides at 36 miles per hour for 3/2 hours, then at 39 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 36 × 3/2 = 54 miles.\nSecond leg distance = 39 × 5/2 = 195/2 miles.\nTotal distance = 54 + 195/2 = 303/2 miles.\nCheck: adding both legs gives 303/2.\nFinal answer: $\\frac{303}{2}$", "final_answer": "\\frac{303}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c24d679af12ce15b", "problem_id": "befc982755f3b965", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{34 x}{5} - 2 \\sqrt{11} x + \\frac{24}{5} + \\frac{8 \\sqrt{11}}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{34}{5} - 2 \\sqrt{11}$, c=$\\frac{24}{5} + \\frac{8 \\sqrt{11}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{104 \\sqrt{11}}{5} + \\frac{1776}{25}$, so √Δ = $\\sqrt{\\frac{104 \\sqrt{11}}{5} + \\frac{1776}{25}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\sqrt{11} + \\frac{17}{5} + \\frac{\\sqrt{130 \\sqrt{11} + 444}}{5}$.\n7) x2 = $- \\frac{\\sqrt{130 \\sqrt{11} + 444}}{5} + \\sqrt{11} + \\frac{17}{5}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\sqrt{11} + \\frac{17}{5} + \\frac{\\sqrt{130\\cdot \\sqrt{11} + 444}}{5}, - \\frac{\\sqrt{130\\cdot \\sqrt{11} + 444}}{5} + \\sqrt{11} + \\frac{17}{5}\\}$", "final_answer": "x \\in \\{\\sqrt{11} + \\frac{17}{5} + \\frac{\\sqrt{130\\cdot \\sqrt{11} + 444}}{5}, - \\frac{\\sqrt{130\\cdot \\sqrt{11} + 444}}{5} + \\sqrt{11} + \\frac{17}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "10314098652f310f", "problem_id": "befc982755f3b965", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{34 x}{5} - 2 \\sqrt{11} x + \\frac{24}{5} + \\frac{8 \\sqrt{11}}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{34}{5} - 2 \\sqrt{11}$, c=$\\frac{24}{5} + \\frac{8 \\sqrt{11}}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{104 \\sqrt{11}}{5} + \\frac{1776}{25}$, so √Δ = $\\sqrt{\\frac{104 \\sqrt{11}}{5} + \\frac{1776}{25}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\sqrt{11} + \\frac{17}{5} + \\frac{\\sqrt{130 \\sqrt{11} + 444}}{5}$.\nx2 = $- \\frac{\\sqrt{130 \\sqrt{11} + 444}}{5} + \\sqrt{11} + \\frac{17}{5}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\sqrt{11} + \\frac{17}{5} + \\frac{\\sqrt{130\\cdot \\sqrt{11} + 444}}{5}, - \\frac{\\sqrt{130\\cdot \\sqrt{11} + 444}}{5} + \\sqrt{11} + \\frac{17}{5}\\}$", "final_answer": "x \\in \\{\\sqrt{11} + \\frac{17}{5} + \\frac{\\sqrt{130\\cdot \\sqrt{11} + 444}}{5}, - \\frac{\\sqrt{130\\cdot \\sqrt{11} + 444}}{5} + \\sqrt{11} + \\frac{17}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d1dd5ccabe9bf3e4", "problem_id": "2dab6a074c97e1a1", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{2} - 3 x - 2\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 3 x^{2} - 3 x - 2$.\n3) Compute an antiderivative: F(x) = $- x^{3} - \\frac{3 x^{2}}{2} - 2 x$.\n4) Include the constant of integration C.\nFinal answer: $C - x^{3} - \\frac{3\\cdot x^{2}}{2} - 2\\cdot x$", "final_answer": "C - x^{3} - \\frac{3\\cdot x^{2}}{2} - 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d467dd64d51802a", "problem_id": "77059cff0e0e9a9a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} x^{2} + 2$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to x^{2} - x + 2.\n3) Then the limit is 2.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: 3.\nFinal answer: $3$", "final_answer": "3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e295207c6f0b428a", "problem_id": "bb5b2c74cf509f0e", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 15 times, what is the probability of exactly 4 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=15, k=4, p=$\\frac{2}{5}$ and simplify.\nHere C(15,4) = 1365.\nCheck: expression simplifies to $\\frac{773778096}{6103515625}$.\nFinal answer: $\\frac{773778096}{6103515625}$", "final_answer": "\\frac{773778096}{6103515625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "273bee818661addf", "problem_id": "bb5b2c74cf509f0e", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 15 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=15, k=4, p=$\\frac{2}{5}$ and simplify.\n4) Here C(15,4) = 1365.\nFinal answer: $\\frac{773778096}{6103515625}$", "final_answer": "\\frac{773778096}{6103515625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5f827eec15f317c8", "problem_id": "b71bb7c7959cdfd8", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} - 2 x^{2} + 4 x - 4$.", "output": "Differentiate term-by-term:\nd/dx(-4) = 0.\nd/dx(- 2 x^{2}) = - 4 x.\nd/dx(- 2 x^{3}) = - 6 x^{2}.\nd/dx(4 x) = 4.\nCombine: f'(x) = 2 \\left(- 3 x^{2} - 2 x + 2\\right).\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 6\\cdot x^{2} - 4\\cdot x + 4$", "final_answer": "- 6\\cdot x^{2} - 4\\cdot x + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "77debb9437a5b168", "problem_id": "b71bb7c7959cdfd8", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} - 2 x^{2} + 4 x - 4$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-4) = 0.\n3) d/dx(- 2 x^{2}) = - 4 x.\n4) d/dx(- 2 x^{3}) = - 6 x^{2}.\n5) d/dx(4 x) = 4.\n6) Combine: f'(x) = 2 \\left(- 3 x^{2} - 2 x + 2\\right).\nFinal answer: $- 6\\cdot x^{2} - 4\\cdot x + 4$", "final_answer": "- 6\\cdot x^{2} - 4\\cdot x + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1d71720b884f08be", "problem_id": "05a9f380c25694fe", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A disease has prevalence $\\frac{3}{100}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{273}{5000}$.\nCompute P(D|+) = $\\frac{85}{182}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{85}{182}$", "final_answer": "\\frac{85}{182}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9d05204b16f04cb1", "problem_id": "05a9f380c25694fe", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A disease has prevalence $\\frac{3}{100}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{273}{5000}$.\n3) Compute P(D|+) = $\\frac{85}{182}$.\nFinal answer: $\\frac{85}{182}$", "final_answer": "\\frac{85}{182}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "13ce286a78ea927d", "problem_id": "b7b2598cae1756f2", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "You want 28 liters of a 25% salt solution. You have a 20% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 28−x liters is 40% solution.\n2) Salt equation: 20x + 40(28−x) = 25·28.\n3) (20-40)x = (25-40)·28 ⇒ x = 21.\n4) Then 28−x = 7.\nFinal answer: $21 L of 20% and 7 L of 40%$", "final_answer": "21 L of 20% and 7 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e018da23c40a0af0", "problem_id": "b708677091c5e5b4", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(2 x^{3} - x^{2} + x - 2\\right) \\cos{\\left(2 x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x \\cos{\\left(2 x + 3 \\right)}) = - x \\sin{\\left(2 x + 3 \\right)}$.\n3) So $f'(x) = - 4 x^{3} \\sin{\\left(2 x + 3 \\right)} + 2 x^{2} \\sin{\\left(2 x + 3 \\right)} + 6 x^{2} \\cos{\\left(2 x + 3 \\right)} + 2 x e^{x} - x \\sin{\\left(2 x + 3 \\right)} - 2 x \\cos{\\left(2 x + 3 \\right)} + 6 e^{x} + 4 \\sin{\\left(2 x + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x \\cos{\\left(2 x + 3 \\right)}) = - 2 x \\sin{\\left(2 x + 3 \\right)} + \\cos{\\left(2 x + 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x + 2\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) + 1\\right) - 2\\right) \\sin{\\left(2 x + 3 \\right)} + \\left(6 x^{2} - 2 x + 1\\right) \\cos{\\left(2 x + 3 \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(x + 2\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) + 1\\right) - 2\\right) \\sin{\\left(2 x + 3 \\right)} + \\left(6 x^{2} - 2 x + 1\\right) \\cos{\\left(2 x + 3 \\right)} + 2 e^{x}$", "final_answer": "2 \\left(x + 2\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) + 1\\right) - 2\\right) \\sin{\\left(2 x + 3 \\right)} + \\left(6 x^{2} - 2 x + 1\\right) \\cos{\\left(2 x + 3 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "25dafc5649b324b6", "problem_id": "c6c6b83692c14cc3", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$x + 4 y$ = $6$\n$2 x + 2 y$ = $6$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $2 x + 8 y$ = $12$.\n3) Multiply the second equation by $1$: $2 x + 2 y$ = $6$.\n4) Subtract to eliminate x: $6 y$ = $6$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $2$.\nFinal answer: $(x,y)=\\left(2, 1\\right)$", "final_answer": "(x,y)=\\left(2, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c0ddd8efa02e4600", "problem_id": "4130f56657689103", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(2 x^{2} - 3 x - 3\\right) \\cos{\\left(3 x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x e^{x}) = x e^{x}$.\n3) So $f'(x) = - 6 x^{2} \\sin{\\left(3 x + 3 \\right)} + x e^{x} + 9 x \\sin{\\left(3 x + 3 \\right)} + 4 x \\cos{\\left(3 x + 3 \\right)} - 3 e^{x} + 9 \\sin{\\left(3 x + 3 \\right)} - 3 \\cos{\\left(3 x + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x e^{x}) = x e^{x} + e^{x}$.\nTherefore the correct derivative is $f'(x) = \\left(x - 3\\right) e^{x} + \\left(4 x - 3\\right) \\cos{\\left(3 x + 3 \\right)} - 3 \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x + 3 \\right)} + e^{x}$.\nFinal answer: $\\left(x - 3\\right) e^{x} + \\left(4 x - 3\\right) \\cos{\\left(3 x + 3 \\right)} - 3 \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x + 3 \\right)} + e^{x}$", "final_answer": "\\left(x - 3\\right) e^{x} + \\left(4 x - 3\\right) \\cos{\\left(3 x + 3 \\right)} - 3 \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x + 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "fd954e151cd0cb2d", "problem_id": "acf2864b30440e70", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡1 (mod 4), x≡1 (mod 9). Give the solution modulo 36.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 36.\nCompute the CRT solution to get x ≡ 1 (mod 36).\nCheck: 1 mod 4=1 and mod 9=1.\nFinal answer: $x\\equiv 1\\pmod{36}$", "final_answer": "x\\equiv 1\\pmod{36}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4e13695c37b1a746", "problem_id": "acf2864b30440e70", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡1 (mod 4), x≡1 (mod 9). Give the solution modulo 36.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 36.\n3) Compute the CRT solution to get x ≡ 1 (mod 36).\nFinal answer: $x\\equiv 1\\pmod{36}$", "final_answer": "x\\equiv 1\\pmod{36}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b87cb2c4ff732930", "problem_id": "686bb2946a7217ea", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} + 2 x - 1$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-1) = 0.\n3) d/dx(2 x) = 2.\n4) d/dx(2 x^{2}) = 4 x.\n5) Combine: f'(x) = 2 \\left(2 x + 1\\right).\nFinal answer: $4\\cdot x + 2$", "final_answer": "4\\cdot x + 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ad1fedc9b5a0a997", "problem_id": "686bb2946a7217ea", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} + 2 x - 1$.", "output": "Differentiate term-by-term:\nd/dx(-1) = 0.\nd/dx(2 x) = 2.\nd/dx(2 x^{2}) = 4 x.\nCombine: f'(x) = 2 \\left(2 x + 1\\right).\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $4\\cdot x + 2$", "final_answer": "4\\cdot x + 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a34c72988892cee4", "problem_id": "92be65c1f4601be4", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$x - y$ = $0$\n$- x$ = $4$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $- x + y$ = $0$.\n3) Multiply the second equation by $1$: $- x$ = $4$.\n4) Subtract to eliminate x: $y$ = $-4$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $-4$.\nFinal answer: $(x,y)=\\left(-4, -4\\right)$", "final_answer": "(x,y)=\\left(-4, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a6308219156f568c", "problem_id": "f988366da0f25484", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{3} - x^{2} - 3 x + \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{3 x - 2}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 2}) = - \\frac{2 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 2}$.\n3) So $f'(x) = - \\frac{36 x^{3}}{18 x^{2} - 24 x + 8} + \\frac{9 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} - 24 x + 8} + \\frac{18 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} - 24 x + 8} + \\frac{30 x^{2}}{18 x^{2} - 24 x + 8} - \\frac{12 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} - 24 x + 8} - \\frac{24 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} - 24 x + 8} + \\frac{8 x}{18 x^{2} - 24 x + 8} + \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} - 24 x + 8} + \\frac{8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} - 24 x + 8} + \\frac{24}{18 x^{2} - 24 x + 8} - \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{2} - 12 x + 4} - \\frac{2 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 2} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 2} + \\frac{2 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 2}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 2}) = - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 2} - \\frac{2 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 2} + \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(3 x - 2\\right)^{2}}$.\nTherefore the correct derivative is $f'(x) = \\frac{18 x^{3} + 6 x^{2} + 18 x + \\left(3 x - 2\\right) \\left(- 18 x^{2} - 4 x + \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6\\right) - 6 \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 12}{2 \\left(3 x - 2\\right)^{2}}$.\nFinal answer: $\\frac{18 x^{3} + 6 x^{2} + 18 x + \\left(3 x - 2\\right) \\left(- 18 x^{2} - 4 x + \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6\\right) - 6 \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 12}{2 \\left(3 x - 2\\right)^{2}}$", "final_answer": "\\frac{18 x^{3} + 6 x^{2} + 18 x + \\left(3 x - 2\\right) \\left(- 18 x^{2} - 4 x + \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6\\right) - 6 \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 12}{2 \\left(3 x - 2\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c40a132b6f352cfc", "problem_id": "ee7d3934fe73248c", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$- 3 x + 3 y$ = $15$\n$4 x + 7 y$ = $13$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $- 12 x + 12 y$ = $60$.\n3) Multiply the second equation by $-3$: $- 12 x - 21 y$ = $-39$.\n4) Subtract to eliminate x: $33 y$ = $99$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, 3\\right)$", "final_answer": "(x,y)=\\left(-2, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6cd940cf54150df9", "problem_id": "ee7d3934fe73248c", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- 3 x + 3 y$ = $15$\n$4 x + 7 y$ = $13$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $- 12 x + 12 y$ = $60$.\nMultiply the second equation by $-3$: $- 12 x - 21 y$ = $-39$.\nSubtract to eliminate x: $33 y$ = $99$.\nSolve for y: y = $3$.\nSubstitute back to find x: x = $-2$.\nCheck: plug (x,y)=(-2,3) into both equations.\nFinal answer: $(x,y)=\\left(-2, 3\\right)$", "final_answer": "(x,y)=\\left(-2, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "15e416367cd924f5", "problem_id": "ccc1ddb463aabc7d", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} - 2 x^{3} - x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{3} - x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{9 \\pi}{2} - \\frac{3 \\pi^{3}}{8} - 2 + \\frac{3 \\pi^{2}}{4} + \\frac{15 \\pi^{4}}{32}$.\nFinal answer: $- \\frac{9\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{3}}{8} - 2 + \\frac{3\\cdot \\pi^{2}}{4} + \\frac{15\\cdot \\pi^{4}}{32}$", "final_answer": "- \\frac{9\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{3}}{8} - 2 + \\frac{3\\cdot \\pi^{2}}{4} + \\frac{15\\cdot \\pi^{4}}{32}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "97447157679233f1", "problem_id": "ccc1ddb463aabc7d", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} - 2 x^{3} - x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{3} - x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{9 \\pi}{2} - \\frac{3 \\pi^{3}}{8} - 2 + \\frac{3 \\pi^{2}}{4} + \\frac{15 \\pi^{4}}{32}$.\n4) Simplify the result.\nFinal answer: $- \\frac{9\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{3}}{8} - 2 + \\frac{3\\cdot \\pi^{2}}{4} + \\frac{15\\cdot \\pi^{4}}{32}$", "final_answer": "- \\frac{9\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{3}}{8} - 2 + \\frac{3\\cdot \\pi^{2}}{4} + \\frac{15\\cdot \\pi^{4}}{32}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3a2bbc4aafa38c5f", "problem_id": "491fdd667a875fbe", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A cyclist rides at 52 miles per hour for 2 hours, then at 77 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 52 × 2 = 104 miles.\nSecond leg distance = 77 × 2 = 154 miles.\nTotal distance = 104 + 154 = 258 miles.\nFinal answer: $258$", "final_answer": "258", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "48c77b3a293f5de9", "problem_id": "491fdd667a875fbe", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A cyclist rides at 52 miles per hour for 2 hours, then at 77 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 52 × 2 = 104 miles.\n2) Second leg distance = 77 × 2 = 154 miles.\n3) Total distance = 104 + 154 = 258 miles.\nFinal answer: $258$", "final_answer": "258", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "44c103f9e474e367", "problem_id": "8eed75e4d8a3037c", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{7} x}{2} - \\frac{3 x}{2} + \\frac{1}{2} + \\frac{3 \\sqrt{7}}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{3 \\sqrt{7}}{2} - \\frac{3}{2}$, c=$\\frac{1}{2} + \\frac{3 \\sqrt{7}}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $16 - \\frac{3 \\sqrt{7}}{2}$, so √Δ = $\\sqrt{16 - \\frac{3 \\sqrt{7}}{2}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{3}{4} + \\frac{\\sqrt{64 - 6 \\sqrt{7}}}{4} + \\frac{3 \\sqrt{7}}{4}$.\n7) x2 = $- \\frac{\\sqrt{64 - 6 \\sqrt{7}}}{4} + \\frac{3}{4} + \\frac{3 \\sqrt{7}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{3}{4} + \\frac{\\sqrt{64 - 6\\cdot \\sqrt{7}}}{4} + \\frac{3\\cdot \\sqrt{7}}{4}, - \\frac{\\sqrt{64 - 6\\cdot \\sqrt{7}}}{4} + \\frac{3}{4} + \\frac{3\\cdot \\sqrt{7}}{4}\\}$", "final_answer": "x \\in \\{\\frac{3}{4} + \\frac{\\sqrt{64 - 6\\cdot \\sqrt{7}}}{4} + \\frac{3\\cdot \\sqrt{7}}{4}, - \\frac{\\sqrt{64 - 6\\cdot \\sqrt{7}}}{4} + \\frac{3}{4} + \\frac{3\\cdot \\sqrt{7}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3e5f4b0909d037b9", "problem_id": "d723cd3fbf671a71", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{3} + 2 x^{2} - x + \\left(2 x^{2} + x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{2 x^{2} + x - 6}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x^{2} + x\\right) - 6}) = - \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + x - 6}$.\n3) So $f'(x) = \\frac{4 x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} + \\frac{8 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} - \\frac{4 x^{4}}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} + \\frac{4 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} + \\frac{8 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} - \\frac{4 x^{3}}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} - \\frac{23 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} - \\frac{46 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} + \\frac{44 x^{2}}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} - \\frac{12 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} - \\frac{24 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} - \\frac{24 x}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} - \\frac{24 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{4 x^{4} + 4 x^{3} - 23 x^{2} - 12 x + 36} + \\frac{36 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} + \\frac{72 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} + \\frac{18}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} - \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{4 x^{4} + 4 x^{3} - 23 x^{2} - 12 x + 36} - \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + x - 6} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x^{2} + x\\right) - 6} + \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(2 x^{2} + x\\right) - 6}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x^{2} + x\\right) - 6}) = - \\frac{6 \\left(- 4 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(2 x^{2} + x\\right) - 6\\right)^{2}} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x^{2} + x\\right) - 6} - \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(2 x^{2} + x\\right) - 6}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(- 4 x - 1\\right) \\left(- x^{3} + 2 x^{2} - x + \\left(x \\left(2 x + 1\\right) - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x \\left(2 x + 1\\right) - 6\\right) \\left(- 6 x^{2} + 8 x + 2 \\left(4 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 x + 1\\right) - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 x + 1\\right) - 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x \\left(2 x + 1\\right) - 6\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(- 4 x - 1\\right) \\left(- x^{3} + 2 x^{2} - x + \\left(x \\left(2 x + 1\\right) - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x \\left(2 x + 1\\right) - 6\\right) \\left(- 6 x^{2} + 8 x + 2 \\left(4 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 x + 1\\right) - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 x + 1\\right) - 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x \\left(2 x + 1\\right) - 6\\right)^{2}}$", "final_answer": "\\frac{2 \\left(- 4 x - 1\\right) \\left(- x^{3} + 2 x^{2} - x + \\left(x \\left(2 x + 1\\right) - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x \\left(2 x + 1\\right) - 6\\right) \\left(- 6 x^{2} + 8 x + 2 \\left(4 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 x + 1\\right) - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 x + 1\\right) - 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x \\left(2 x + 1\\right) - 6\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "002502905f839f1a", "problem_id": "ba7a3dc10cbd6e18", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "An urn has 7 red balls and 13 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(20,6).\nFavorable samples: choose k reds and n−k blues: C(7,3)·C(13,3).\nFinal answer: $\\frac{1001}{3876}$", "final_answer": "\\frac{1001}{3876}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "691b6d95d899b4ad", "problem_id": "ba7a3dc10cbd6e18", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "An urn has 7 red balls and 13 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(20,6).\n3) Favorable samples: choose k reds and n−k blues: C(7,3)·C(13,3).\n4) Compute totals: C(20,6) = 38760, and favorable = 10010.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1001}{3876}$", "final_answer": "\\frac{1001}{3876}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "74e9f7962e992b76", "problem_id": "e8e9872fa4d9bd6c", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the multiplicative inverse of 26 modulo 31.", "output": "1) We want x such that 26x ≡ 1 (mod 31).\n2) Use extended Euclid to find s,t with 26s + 31t = 1.\n3) One Bezout identity is: 26·(6) + 31·(-5) = 1.\n4) Reduce s modulo 31: x ≡ 6 (mod 31).\nFinal answer: $x\\equiv 6\\pmod{31}$", "final_answer": "x\\equiv 6\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "693f909aff9d0d19", "problem_id": "e8e9872fa4d9bd6c", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the multiplicative inverse of 26 modulo 31.", "output": "We want x such that 26x ≡ 1 (mod 31).\nUse extended Euclid to find s,t with 26s + 31t = 1.\nOne Bezout identity is: 26·(6) + 31·(-5) = 1.\nReduce s modulo 31: x ≡ 6 (mod 31).\nCheck: 26·6 mod 31 = 1.\nFinal answer: $x\\equiv 6\\pmod{31}$", "final_answer": "x\\equiv 6\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "02cfba2c1547474a", "problem_id": "871ad565c3e9d995", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{20 x}{3} + \\frac{16}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$\\frac{20}{3}$, c=$\\frac{16}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{16}{9}$, so √Δ = $\\frac{4}{3}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{4}{3}$.\n7) x2 = $-2$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{4}{3}, -2\\}$", "final_answer": "x \\in \\{- \\frac{4}{3}, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cd8edb068ceeea08", "problem_id": "871ad565c3e9d995", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{20 x}{3} + \\frac{16}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$\\frac{20}{3}$, c=$\\frac{16}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{16}{9}$, so √Δ = $\\frac{4}{3}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{4}{3}$.\nx2 = $-2$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{4}{3}, -2\\}$", "final_answer": "x \\in \\{- \\frac{4}{3}, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0bb26ccac4ff178a", "problem_id": "8151f168017c9a34", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} x^{2} + 3 x + 11$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to x^{2} + 2 x + 11.\n3) Then the limit is 26.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: 29.\nFinal answer: $29$", "final_answer": "29", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "af4dc68e986c0911", "problem_id": "027b8ccd18e6c05e", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{4} - 2 x^{3} - 2 x^{2} - 3 x - 2\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $3 x^{4} - 2 x^{3} - 2 x^{2} - 3 x - 2$.\nCompute an antiderivative: F(x) = $\\frac{3 x^{5}}{5} - \\frac{x^{4}}{2} - \\frac{2 x^{3}}{3} - \\frac{3 x^{2}}{2} - 2 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{3\\cdot x^{5}}{5} - \\frac{x^{4}}{2} - \\frac{2\\cdot x^{3}}{3} - \\frac{3\\cdot x^{2}}{2} - 2\\cdot x$", "final_answer": "C + \\frac{3\\cdot x^{5}}{5} - \\frac{x^{4}}{2} - \\frac{2\\cdot x^{3}}{3} - \\frac{3\\cdot x^{2}}{2} - 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "22fb52a70b0b0f53", "problem_id": "e7b8d490ac4bc9cf", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many binary strings of length 25 contain exactly 6 ones with no two ones adjacent?", "output": "1) Reserve 5 mandatory zeros between the 6 ones.\n2) This leaves n−k+1 = 20 valid slots for the ones.\n3) Choose the slots: C(20,6) = 38760.\nFinal answer: $38760$", "final_answer": "38760", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5847a8674015bf28", "problem_id": "e7b8d490ac4bc9cf", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many binary strings of length 25 contain exactly 6 ones with no two ones adjacent?", "output": "Reserve 5 mandatory zeros between the 6 ones.\nThis leaves n−k+1 = 20 valid slots for the ones.\nChoose the slots: C(20,6) = 38760.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $38760$", "final_answer": "38760", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6d6fa63465dd56e5", "problem_id": "1788da1060e6c4bf", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Worker A finishes a job in 16 hours and Worker B finishes it in 20 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/16 job/hour. B's rate = 1/20 job/hour.\n2) Combined rate = 1/16 + 1/20 = 9/80 job/hour.\n3) Time = 1 ÷ (combined rate) = 80/9 hours.\nFinal answer: $\\frac{80}{9}$", "final_answer": "\\frac{80}{9}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "07ee84d03a3ce424", "problem_id": "1788da1060e6c4bf", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Worker A finishes a job in 16 hours and Worker B finishes it in 20 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/16 job/hour. B's rate = 1/20 job/hour.\nCombined rate = 1/16 + 1/20 = 9/80 job/hour.\nTime = 1 ÷ (combined rate) = 80/9 hours.\nCheck: (combined rate)×(time) = 9/80×80/9 = 1.\nFinal answer: $\\frac{80}{9}$", "final_answer": "\\frac{80}{9}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "21257cc06da8e532", "problem_id": "0c20ef26520015c6", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the multiplicative inverse of 37 modulo 41.", "output": "We want x such that 37x ≡ 1 (mod 41).\nUse extended Euclid to find s,t with 37s + 41t = 1.\nOne Bezout identity is: 37·(10) + 41·(-9) = 1.\nReduce s modulo 41: x ≡ 10 (mod 41).\nCheck: 37·10 mod 41 = 1.\nFinal answer: $x\\equiv 10\\pmod{41}$", "final_answer": "x\\equiv 10\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a7ed6aab15eec42", "problem_id": "0c20ef26520015c6", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the multiplicative inverse of 37 modulo 41.", "output": "1) We want x such that 37x ≡ 1 (mod 41).\n2) Use extended Euclid to find s,t with 37s + 41t = 1.\n3) One Bezout identity is: 37·(10) + 41·(-9) = 1.\n4) Reduce s modulo 41: x ≡ 10 (mod 41).\nFinal answer: $x\\equiv 10\\pmod{41}$", "final_answer": "x\\equiv 10\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "29314844a5e8db5b", "problem_id": "01bd4fd4e5855b64", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Worker A finishes a job in 7 hours and Worker B finishes it in 19 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/7 job/hour. B's rate = 1/19 job/hour.\n2) Combined rate = 1/7 + 1/19 = 26/133 job/hour.\n3) Time = 1 ÷ (combined rate) = 133/26 hours.\nFinal answer: $\\frac{133}{26}$", "final_answer": "\\frac{133}{26}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a11cac4f8e660bf0", "problem_id": "01bd4fd4e5855b64", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Worker A finishes a job in 7 hours and Worker B finishes it in 19 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/7 job/hour. B's rate = 1/19 job/hour.\nCombined rate = 1/7 + 1/19 = 26/133 job/hour.\nTime = 1 ÷ (combined rate) = 133/26 hours.\nCheck: (combined rate)×(time) = 26/133×133/26 = 1.\nFinal answer: $\\frac{133}{26}$", "final_answer": "\\frac{133}{26}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "13f4df0d9400420d", "problem_id": "bd5b39f80c58d122", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - 2 x^{4} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{4} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{205 \\pi^{5}}{16} - \\frac{45 \\pi^{2}}{8} - 1 + 5 \\pi$.\n4) Simplify the result.\nFinal answer: $- \\frac{205\\cdot \\pi^{5}}{16} - \\frac{45\\cdot \\pi^{2}}{8} - 1 + 5\\cdot \\pi$", "final_answer": "- \\frac{205\\cdot \\pi^{5}}{16} - \\frac{45\\cdot \\pi^{2}}{8} - 1 + 5\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c5f083b94801abed", "problem_id": "bd5b39f80c58d122", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - 2 x^{4} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{4} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{205 \\pi^{5}}{16} - \\frac{45 \\pi^{2}}{8} - 1 + 5 \\pi$.\nFinal answer: $- \\frac{205\\cdot \\pi^{5}}{16} - \\frac{45\\cdot \\pi^{2}}{8} - 1 + 5\\cdot \\pi$", "final_answer": "- \\frac{205\\cdot \\pi^{5}}{16} - \\frac{45\\cdot \\pi^{2}}{8} - 1 + 5\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d08a146b190dae09", "problem_id": "b770b4bc25360e1a", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 27 red balls and 27 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(54,7).\nFavorable samples: choose k reds and n−k blues: C(27,3)·C(27,4).\nCompute totals: C(54,7) = 177100560, and favorable = 51333750.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{14625}{50456}$.\nFinal answer: $\\frac{14625}{50456}$", "final_answer": "\\frac{14625}{50456}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3455abfa8114b6bf", "problem_id": "b770b4bc25360e1a", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 27 red balls and 27 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(54,7).\n3) Favorable samples: choose k reds and n−k blues: C(27,3)·C(27,4).\n4) Compute totals: C(54,7) = 177100560, and favorable = 51333750.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{14625}{50456}$", "final_answer": "\\frac{14625}{50456}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "22b91683acb577c6", "problem_id": "2f2490502ad4b2dc", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_8=35$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_8=27.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=27 and k=8.\n4) Compute C(34,7) = 5379616.\nFinal answer: $5379616$", "final_answer": "5379616", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d9496c75ab9e0e0f", "problem_id": "f63f7a44d91b8951", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 2 x^{2} + 3 x + \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 6}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 6}) = \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 6}$.\n3) So $f'(x) = \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 24 x + 72} + \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 24 x + 72} - \\frac{4 x^{2}}{2 x^{2} - 24 x + 72} - \\frac{12 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 24 x + 72} - \\frac{24 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 24 x + 72} + \\frac{48 x}{2 x^{2} - 24 x + 72} + \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 12 x + 36} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 12} - \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 6} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 6} + \\frac{36 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 24 x + 72} + \\frac{72 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 24 x + 72} - \\frac{36}{2 x^{2} - 24 x + 72} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 6}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 6}) = \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x - 6\\right)} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 6} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x - 6\\right)^{2}} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 6}$.\nTherefore the correct derivative is $f'(x) = \\frac{4 x^{2} - 6 x + \\left(x - 6\\right) \\left(- 8 x + \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 2 \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x - 6\\right)^{2}}$.\nFinal answer: $\\frac{4 x^{2} - 6 x + \\left(x - 6\\right) \\left(- 8 x + \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 2 \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x - 6\\right)^{2}}$", "final_answer": "\\frac{4 x^{2} - 6 x + \\left(x - 6\\right) \\left(- 8 x + \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 2 \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x - 6\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "22be2410cb22077f", "problem_id": "9bffa29733583575", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Find the multiplicative inverse of 24 modulo 31.", "output": "We want x such that 24x ≡ 1 (mod 31).\nUse extended Euclid to find s,t with 24s + 31t = 1.\nOne Bezout identity is: 24·(-9) + 31·(7) = 1.\nFinal answer: $x\\equiv 22\\pmod{31}$", "final_answer": "x\\equiv 22\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b726924a1b3f7345", "problem_id": "9bffa29733583575", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the multiplicative inverse of 24 modulo 31.", "output": "1) We want x such that 24x ≡ 1 (mod 31).\n2) Use extended Euclid to find s,t with 24s + 31t = 1.\n3) One Bezout identity is: 24·(-9) + 31·(7) = 1.\n4) Reduce s modulo 31: x ≡ 22 (mod 31).\nFinal answer: $x\\equiv 22\\pmod{31}$", "final_answer": "x\\equiv 22\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "184db0d7c6008296", "problem_id": "56aa3efb44d81dc4", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{31 x}{6} - \\frac{2 \\sqrt{10} x}{3} + 3 + 3 \\sqrt{10}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{31}{6} - \\frac{2 \\sqrt{10}}{3}$, c=$3 + 3 \\sqrt{10}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{689}{36} - \\frac{46 \\sqrt{10}}{9}$, so √Δ = $\\sqrt{\\frac{689}{36} - \\frac{46 \\sqrt{10}}{9}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{689 - 184 \\sqrt{10}}}{12} + \\frac{\\sqrt{10}}{3} + \\frac{31}{12}$.\nx2 = $- \\frac{\\sqrt{689 - 184 \\sqrt{10}}}{12} + \\frac{\\sqrt{10}}{3} + \\frac{31}{12}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{689 - 184\\cdot \\sqrt{10}}}{12} + \\frac{\\sqrt{10}}{3} + \\frac{31}{12}, - \\frac{\\sqrt{689 - 184\\cdot \\sqrt{10}}}{12} + \\frac{\\sqrt{10}}{3} + \\frac{31}{12}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{689 - 184\\cdot \\sqrt{10}}}{12} + \\frac{\\sqrt{10}}{3} + \\frac{31}{12}, - \\frac{\\sqrt{689 - 184\\cdot \\sqrt{10}}}{12} + \\frac{\\sqrt{10}}{3} + \\frac{31}{12}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7b1c64ab8e1d9432", "problem_id": "56aa3efb44d81dc4", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{31 x}{6} - \\frac{2 \\sqrt{10} x}{3} + 3 + 3 \\sqrt{10}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{31}{6} - \\frac{2 \\sqrt{10}}{3}$, c=$3 + 3 \\sqrt{10}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{689}{36} - \\frac{46 \\sqrt{10}}{9}$, so √Δ = $\\sqrt{\\frac{689}{36} - \\frac{46 \\sqrt{10}}{9}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{689 - 184 \\sqrt{10}}}{12} + \\frac{\\sqrt{10}}{3} + \\frac{31}{12}$.\n7) x2 = $- \\frac{\\sqrt{689 - 184 \\sqrt{10}}}{12} + \\frac{\\sqrt{10}}{3} + \\frac{31}{12}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{689 - 184\\cdot \\sqrt{10}}}{12} + \\frac{\\sqrt{10}}{3} + \\frac{31}{12}, - \\frac{\\sqrt{689 - 184\\cdot \\sqrt{10}}}{12} + \\frac{\\sqrt{10}}{3} + \\frac{31}{12}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{689 - 184\\cdot \\sqrt{10}}}{12} + \\frac{\\sqrt{10}}{3} + \\frac{31}{12}, - \\frac{\\sqrt{689 - 184\\cdot \\sqrt{10}}}{12} + \\frac{\\sqrt{10}}{3} + \\frac{31}{12}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0996728e77e716fa", "problem_id": "85cfc6a3c93f40a0", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "You want 18 liters of a 30% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 18−x liters is 70% solution.\n2) Salt equation: 25x + 70(18−x) = 30·18.\n3) (25-70)x = (30-70)·18 ⇒ x = 16.\n4) Then 18−x = 2.\nFinal answer: $16 L of 25% and 2 L of 70%$", "final_answer": "16 L of 25% and 2 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c6438e46f6530793", "problem_id": "b834c3003c0c1aab", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - x^{2} + 4 x - 9$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} + 3 x - 9.\n3) Then the limit is -19.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -21.\nFinal answer: $-21$", "final_answer": "-21", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "928e447af06eb514", "problem_id": "86b258ce2d1631ea", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(- x^{3} - 2 x^{2} + 3\\right) \\cos{\\left(x + 1 \\right)}$.", "output": "1) Let f(x) = $\\left(x - 3\\right) e^{x} + \\left(- x^{3} - 2 x^{2} + 3\\right) \\cos{\\left(x + 1 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $x \\left(- 3 x - 4\\right) \\cos{\\left(x + 1 \\right)} + \\left(x - 3\\right) e^{x} - \\left(- x^{3} - 2 x^{2} + 3\\right) \\sin{\\left(x + 1 \\right)} + e^{x}$.\nFinal answer: $x\\cdot \\left(- 3\\cdot x - 4\\right)\\cdot \\cos{\\left(x + 1 \\right)} + \\left(x - 3\\right)\\cdot e^{x} - \\left(- x^{3} - 2\\cdot x^{2} + 3\\right)\\cdot \\sin{\\left(x + 1 \\right)} + e^{x}$", "final_answer": "x\\cdot \\left(- 3\\cdot x - 4\\right)\\cdot \\cos{\\left(x + 1 \\right)} + \\left(x - 3\\right)\\cdot e^{x} - \\left(- x^{3} - 2\\cdot x^{2} + 3\\right)\\cdot \\sin{\\left(x + 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5c79b7a459978ff0", "problem_id": "cbfc2742c4c3ebdf", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- \\frac{x^{2}}{2} + x + \\left(x^{2} - x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\frac{1}{2}}{x^{2} - x + 1}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 1}) = \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 1}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} - \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} - \\frac{4 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} + \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 1} + \\frac{3 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} + \\frac{6 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} - \\frac{x^{2}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 1} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 2 x + 2} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 1} - \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 1} - \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} - \\frac{4 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} - \\frac{4 x}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} - \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 1} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} + \\frac{2 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} + \\frac{3}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 1}) = \\frac{x^{2} \\left(1 - 2 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(x^{2} - x\\right) + 1\\right)^{2}} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(x^{2} - x\\right) + 1\\right)} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 1} + \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 1}$.\nTherefore the correct derivative is $f'(x) = \\frac{\\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + 2 \\left(x \\left(x - 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(x \\left(x - 1\\right) + 1\\right) \\left(- 2 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{2 \\left(x \\left(x - 1\\right) + 1\\right)^{2}}$.\nFinal answer: $\\frac{\\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + 2 \\left(x \\left(x - 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(x \\left(x - 1\\right) + 1\\right) \\left(- 2 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{2 \\left(x \\left(x - 1\\right) + 1\\right)^{2}}$", "final_answer": "\\frac{\\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + 2 \\left(x \\left(x - 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(x \\left(x - 1\\right) + 1\\right) \\left(- 2 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{2 \\left(x \\left(x - 1\\right) + 1\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "fbe35f5c4fd19200", "problem_id": "2f460b9885f27eca", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x^{2} - 4 x - 5$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{3} - 2 x^{2} + 3 x + 2$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 3 x^{2} - 4 x + 3$.\n5) Evaluate at a: p'(2) = $-17$.\nFinal answer: $-17$", "final_answer": "-17", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "498ca161779c274b", "problem_id": "2f460b9885f27eca", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x^{2} - 4 x - 5$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{3} - 2 x^{2} + 3 x + 2$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-17$", "final_answer": "-17", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ac63a8536aff50b1", "problem_id": "242fa246c2e4234f", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Two friends split $288 in the ratio 3:3. How much does each receive?", "output": "Total ratio parts = 3+3 = 6.\nEach part = 288 ÷ 6 = 48.\nFirst share = 3 × 48 = 144.\nSecond share = 3 × 48 = 144.\nCheck: 144+144=288.\nFinal answer: $144$", "final_answer": "144", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3e132640af1ef776", "problem_id": "d14866f0ac52d8c3", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(3 x^{2} + x - 2\\right) \\cos{\\left(3 x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\n3) So $f'(x) = - 9 x^{2} \\sin{\\left(3 x + 3 \\right)} + 2 x e^{x} - 3 x \\sin{\\left(3 x + 3 \\right)} + 6 x \\cos{\\left(3 x + 3 \\right)} + 4 e^{x} + 6 \\sin{\\left(3 x + 3 \\right)} + \\cos{\\left(3 x + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x + 1\\right) e^{x} + \\left(6 x + 1\\right) \\cos{\\left(3 x + 3 \\right)} - 3 \\left(x \\left(3 x + 1\\right) - 2\\right) \\sin{\\left(3 x + 3 \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(x + 1\\right) e^{x} + \\left(6 x + 1\\right) \\cos{\\left(3 x + 3 \\right)} - 3 \\left(x \\left(3 x + 1\\right) - 2\\right) \\sin{\\left(3 x + 3 \\right)} + 2 e^{x}$", "final_answer": "2 \\left(x + 1\\right) e^{x} + \\left(6 x + 1\\right) \\cos{\\left(3 x + 3 \\right)} - 3 \\left(x \\left(3 x + 1\\right) - 2\\right) \\sin{\\left(3 x + 3 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a636b0e9d067ac3e", "problem_id": "f95f6e1640acefb3", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=41$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=41 and k=10.\n4) Compute C(50,9) = 2505433700.\nFinal answer: $2505433700$", "final_answer": "2505433700", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4e1e2b1e01594148", "problem_id": "f95f6e1640acefb3", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=41$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=41 and k=10.\nCompute C(50,9) = 2505433700.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $2505433700$", "final_answer": "2505433700", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "de477d9aeca7a6c4", "problem_id": "01e6d860993453e4", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡0 (mod 4), x≡4 (mod 11). Give the solution modulo 44.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 44.\n3) Compute the CRT solution to get x ≡ 4 (mod 44).\nFinal answer: $x\\equiv 4\\pmod{44}$", "final_answer": "x\\equiv 4\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4d395db1c2ea6a83", "problem_id": "01e6d860993453e4", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the congruences: x≡0 (mod 4), x≡4 (mod 11). Give the solution modulo 44.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 44.\nCompute the CRT solution to get x ≡ 4 (mod 44).\nFinal answer: $x\\equiv 4\\pmod{44}$", "final_answer": "x\\equiv 4\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ee086ae2b0fa3a2", "problem_id": "298ad5efa97dcad2", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{7} x}{2} + \\frac{3 x}{4} - \\frac{9}{8} + \\frac{9 \\sqrt{7}}{8}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{3}{4} - \\frac{3 \\sqrt{7}}{2}$, c=$- \\frac{9}{8} + \\frac{9 \\sqrt{7}}{8}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{333}{16} - \\frac{27 \\sqrt{7}}{4}$, so √Δ = $\\sqrt{\\frac{333}{16} - \\frac{27 \\sqrt{7}}{4}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{3}{8} + \\frac{3 \\sqrt{37 - 12 \\sqrt{7}}}{8} + \\frac{3 \\sqrt{7}}{4}$.\nx2 = $- \\frac{3 \\sqrt{37 - 12 \\sqrt{7}}}{8} - \\frac{3}{8} + \\frac{3 \\sqrt{7}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{3}{8} + \\frac{3\\cdot \\sqrt{37 - 12\\cdot \\sqrt{7}}}{8} + \\frac{3\\cdot \\sqrt{7}}{4}, - \\frac{3\\cdot \\sqrt{37 - 12\\cdot \\sqrt{7}}}{8} - \\frac{3}{8} + \\frac{3\\cdot \\sqrt{7}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{3}{8} + \\frac{3\\cdot \\sqrt{37 - 12\\cdot \\sqrt{7}}}{8} + \\frac{3\\cdot \\sqrt{7}}{4}, - \\frac{3\\cdot \\sqrt{37 - 12\\cdot \\sqrt{7}}}{8} - \\frac{3}{8} + \\frac{3\\cdot \\sqrt{7}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "235ed39cb960f4f2", "problem_id": "298ad5efa97dcad2", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{7} x}{2} + \\frac{3 x}{4} - \\frac{9}{8} + \\frac{9 \\sqrt{7}}{8}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{3}{4} - \\frac{3 \\sqrt{7}}{2}$, c=$- \\frac{9}{8} + \\frac{9 \\sqrt{7}}{8}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{333}{16} - \\frac{27 \\sqrt{7}}{4}$, so √Δ = $\\sqrt{\\frac{333}{16} - \\frac{27 \\sqrt{7}}{4}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{3}{8} + \\frac{3 \\sqrt{37 - 12 \\sqrt{7}}}{8} + \\frac{3 \\sqrt{7}}{4}$.\n7) x2 = $- \\frac{3 \\sqrt{37 - 12 \\sqrt{7}}}{8} - \\frac{3}{8} + \\frac{3 \\sqrt{7}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{3}{8} + \\frac{3\\cdot \\sqrt{37 - 12\\cdot \\sqrt{7}}}{8} + \\frac{3\\cdot \\sqrt{7}}{4}, - \\frac{3\\cdot \\sqrt{37 - 12\\cdot \\sqrt{7}}}{8} - \\frac{3}{8} + \\frac{3\\cdot \\sqrt{7}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{3}{8} + \\frac{3\\cdot \\sqrt{37 - 12\\cdot \\sqrt{7}}}{8} + \\frac{3\\cdot \\sqrt{7}}{4}, - \\frac{3\\cdot \\sqrt{37 - 12\\cdot \\sqrt{7}}}{8} - \\frac{3}{8} + \\frac{3\\cdot \\sqrt{7}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "da82e4aa8ebd0439", "problem_id": "c451f5d79ccf9b3a", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡3 (mod 5), x≡9 (mod 11). Give the solution modulo 55.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 55.\n3) Compute the CRT solution to get x ≡ 53 (mod 55).\nFinal answer: $x\\equiv 53\\pmod{55}$", "final_answer": "x\\equiv 53\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c62edc7ec834886f", "problem_id": "c451f5d79ccf9b3a", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡3 (mod 5), x≡9 (mod 11). Give the solution modulo 55.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 55.\nCompute the CRT solution to get x ≡ 53 (mod 55).\nCheck: 53 mod 5=3 and mod 11=9.\nFinal answer: $x\\equiv 53\\pmod{55}$", "final_answer": "x\\equiv 53\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2807f7f623d4ff37", "problem_id": "11debd86f5d588d0", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the equation of the tangent line to $y=x^{2} + 4 x - 4$ at $x=-2$.", "output": "Differentiate: f'(x) = $2 x + 4$.\nSlope at x=-2: m = f'(-2) = $0$.\nPoint on curve: (-2, f(-2)) = (-2, $-8$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -8 = 0(x − -2).\nOptionally expand: y = $-8$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = -8$", "final_answer": "y = -8", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "993c9495c36258f2", "problem_id": "11debd86f5d588d0", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Find the equation of the tangent line to $y=x^{2} + 4 x - 4$ at $x=-2$.", "output": "Differentiate: f'(x) = $2 x + 4$.\nSlope at x=-2: m = f'(-2) = $0$.\nPoint on curve: (-2, f(-2)) = (-2, $-8$).\nFinal answer: $y = -8$", "final_answer": "y = -8", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "136343fa325b476b", "problem_id": "f449087ab0993e22", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} + 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $- 3 x^{2} + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 2$.\n2) Slope at x=-1: m = f'(-1) = $-1 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\n3) Point on curve: (-1, f(-1)) = (-1, $\\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} = -1 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}(x − -1).\n6) Optionally expand: y = $- x + \\frac{x}{2 e^{\\frac{1}{2}}} + 2 x \\sin{\\left(2 \\right)} - 1 + \\cos{\\left(2 \\right)} + \\frac{3}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = - x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 1 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 1 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e74c3a93efed54e0", "problem_id": "f449087ab0993e22", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} + 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=-1$.", "output": "Differentiate: f'(x) = $- 3 x^{2} + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 2$.\nSlope at x=-1: m = f'(-1) = $-1 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nPoint on curve: (-1, f(-1)) = (-1, $\\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}}$).\nFinal answer: $y = - x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 1 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 1 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c46b282570a8374c", "problem_id": "3c689bc18b2e55d9", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the multiplicative inverse of 3 modulo 59.", "output": "1) We want x such that 3x ≡ 1 (mod 59).\n2) Use extended Euclid to find s,t with 3s + 59t = 1.\n3) One Bezout identity is: 3·(20) + 59·(-1) = 1.\n4) Reduce s modulo 59: x ≡ 20 (mod 59).\nFinal answer: $x\\equiv 20\\pmod{59}$", "final_answer": "x\\equiv 20\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "94131f7c169ad38b", "problem_id": "192cc84e7fa41147", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} x^{3} + 4 x^{2} + 17 x + 71$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to x^{3} + 4 x^{2} + 16 x + 71.\n3) Then the limit is 263.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: 267.\nFinal answer: $267$", "final_answer": "267", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5bf64430534fdcce", "problem_id": "92c55353878aa3c8", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Two friends split $882 in the ratio 5:1. How much does each receive?", "output": "1) Total ratio parts = 5+1 = 6.\n2) Each part = 882 ÷ 6 = 147.\n3) First share = 5 × 147 = 735.\n4) Second share = 1 × 147 = 147.\nFinal answer: $147$", "final_answer": "147", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "17c4ac405462d177", "problem_id": "92c55353878aa3c8", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Two friends split $882 in the ratio 5:1. How much does each receive?", "output": "Total ratio parts = 5+1 = 6.\nEach part = 882 ÷ 6 = 147.\nFirst share = 5 × 147 = 735.\nSecond share = 1 × 147 = 147.\nCheck: 735+147=882.\nFinal answer: $147$", "final_answer": "147", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7bfcb51017ba0078", "problem_id": "77920310844ea92d", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the multiplicative inverse of 51 modulo 59.", "output": "1) We want x such that 51x ≡ 1 (mod 59).\n2) Use extended Euclid to find s,t with 51s + 59t = 1.\n3) One Bezout identity is: 51·(22) + 59·(-19) = 1.\n4) Reduce s modulo 59: x ≡ 22 (mod 59).\nFinal answer: $x\\equiv 22\\pmod{59}$", "final_answer": "x\\equiv 22\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c75e86d842c1b6c6", "problem_id": "77920310844ea92d", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Find the multiplicative inverse of 51 modulo 59.", "output": "We want x such that 51x ≡ 1 (mod 59).\nUse extended Euclid to find s,t with 51s + 59t = 1.\nOne Bezout identity is: 51·(22) + 59·(-19) = 1.\nFinal answer: $x\\equiv 22\\pmod{59}$", "final_answer": "x\\equiv 22\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9fa8c74ff6a4eaf8", "problem_id": "1193087165f856b2", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} 2 x^{4} - 3 x^{3} + 4 x^{2} + 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{4} - 3 x^{3} + 4 x^{2} + 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4$.\n2) Evaluate at the bounds: F(\\pi) and F(0).\n3) Subtract: F(\\pi) − F(0) = $\\frac{\\pi \\left(- 45 \\pi^{3} - 240 + 60 \\pi + 80 \\pi^{2} + 24 \\pi^{4}\\right)}{60}$.\n4) Simplify the result.\nFinal answer: $\\frac{\\pi\\cdot \\left(- 45\\cdot \\pi^{3} - 240 + 60\\cdot \\pi + 80\\cdot \\pi^{2} + 24\\cdot \\pi^{4}\\right)}{60}$", "final_answer": "\\frac{\\pi\\cdot \\left(- 45\\cdot \\pi^{3} - 240 + 60\\cdot \\pi + 80\\cdot \\pi^{2} + 24\\cdot \\pi^{4}\\right)}{60}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c6fb3ee21a6b6f1f", "problem_id": "1193087165f856b2", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} 2 x^{4} - 3 x^{3} + 4 x^{2} + 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $2 x^{4} - 3 x^{3} + 4 x^{2} + 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4$.\nEvaluate at the bounds: F(\\pi) and F(0).\nSubtract: F(\\pi) − F(0) = $\\frac{\\pi \\left(- 45 \\pi^{3} - 240 + 60 \\pi + 80 \\pi^{2} + 24 \\pi^{4}\\right)}{60}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{\\pi\\cdot \\left(- 45\\cdot \\pi^{3} - 240 + 60\\cdot \\pi + 80\\cdot \\pi^{2} + 24\\cdot \\pi^{4}\\right)}{60}$", "final_answer": "\\frac{\\pi\\cdot \\left(- 45\\cdot \\pi^{3} - 240 + 60\\cdot \\pi + 80\\cdot \\pi^{2} + 24\\cdot \\pi^{4}\\right)}{60}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "025f4be98a987661", "problem_id": "ad926e265edac09e", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $585 in the ratio 2:3. How much does each receive?", "output": "Total ratio parts = 2+3 = 5.\nEach part = 585 ÷ 5 = 117.\nFirst share = 2 × 117 = 234.\nSecond share = 3 × 117 = 351.\nCheck: 234+351=585.\nFinal answer: $351$", "final_answer": "351", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "57ae62450c9e2d35", "problem_id": "ad926e265edac09e", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Two friends split $585 in the ratio 2:3. How much does each receive?", "output": "1) Total ratio parts = 2+3 = 5.\n2) Each part = 585 ÷ 5 = 117.\n3) First share = 2 × 117 = 234.\n4) Second share = 3 × 117 = 351.\nFinal answer: $351$", "final_answer": "351", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4d48ef0da1ca5582", "problem_id": "fe18a94228871668", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} - 3 x^{3} + x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{3} + x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1$.\n2) Evaluate at the bounds: F(\\pi) and F(0).\n3) Subtract: F(\\pi) − F(0) = $\\pi \\left(- \\frac{3 \\pi^{3}}{4} - \\pi - 1 + \\frac{\\pi^{2}}{3}\\right)$.\n4) Simplify the result.\nFinal answer: $\\pi\\cdot \\left(- \\frac{3\\cdot \\pi^{3}}{4} - \\pi - 1 + \\frac{\\pi^{2}}{3}\\right)$", "final_answer": "\\pi\\cdot \\left(- \\frac{3\\cdot \\pi^{3}}{4} - \\pi - 1 + \\frac{\\pi^{2}}{3}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eeb804ada3bd22a7", "problem_id": "fe18a94228871668", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} - 3 x^{3} + x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{3} + x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1$.\nEvaluate at the bounds: F(\\pi) and F(0).\nSubtract: F(\\pi) − F(0) = $\\pi \\left(- \\frac{3 \\pi^{3}}{4} - \\pi - 1 + \\frac{\\pi^{2}}{3}\\right)$.\nFinal answer: $\\pi\\cdot \\left(- \\frac{3\\cdot \\pi^{3}}{4} - \\pi - 1 + \\frac{\\pi^{2}}{3}\\right)$", "final_answer": "\\pi\\cdot \\left(- \\frac{3\\cdot \\pi^{3}}{4} - \\pi - 1 + \\frac{\\pi^{2}}{3}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bf6dc1ca38cb8f38", "problem_id": "eb5fa22de9f7c136", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 3 x + 8$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{2} + 2 x + 1$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $6 x + 2$.\n5) Evaluate at a: p'(2) = $14$.\nFinal answer: $14$", "final_answer": "14", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "62dbfe18d741ad4d", "problem_id": "eb5fa22de9f7c136", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 3 x + 8$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $3 x^{2} + 2 x + 1$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $6 x + 2$.\nEvaluate at a: p'(2) = $14$.\nCheck: the identity gives $14$.\nFinal answer: $14$", "final_answer": "14", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "779cf89b64557cf0", "problem_id": "d1634042c042ac2f", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$6 y$ = $-54$\n$- 2 x + 8 y$ = $-62$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $- 12 y$ = $108$.\n3) Multiply the second equation by $0$: $0$ = $0$.\n4) Subtract to eliminate x: $- 12 y$ = $108$.\n5) Solve for y: y = $-9$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, -9\\right)$", "final_answer": "(x,y)=\\left(-5, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f16c8f58ddf9205a", "problem_id": "dbe34a8b224711d0", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 x^{3} - 4 x^{2} + 11 x - 19$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} - 4 x^{2} + 10 x - 19.\n3) Then the limit is -79.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -81.\nFinal answer: $-81$", "final_answer": "-81", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d4030e11846f9b0b", "problem_id": "34ac3cd7048f37b1", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A store sells a backpack for $308. It is discounted by 40%. You buy 13 backpacks. What is the total cost?", "output": "1) Discounted price = 308 × (1 − 40/100) = 308 × 60/100 = 924/5.\n2) Subtotal = 13 × 924/5 = 12012/5.\nFinal answer: $\\frac{12012}{5}$", "final_answer": "\\frac{12012}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "11df4c49eca14df6", "problem_id": "34ac3cd7048f37b1", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $308. It is discounted by 40%. You buy 13 backpacks. What is the total cost?", "output": "Discounted price = 308 × (1 − 40/100) = 308 × 60/100 = 924/5.\nSubtotal = 13 × 924/5 = 12012/5.\nCheck: recomputing the formula yields 12012/5.\nFinal answer: $\\frac{12012}{5}$", "final_answer": "\\frac{12012}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "145c264ff95d883a", "problem_id": "acc938d7490b5358", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "An urn has 11 red balls and 6 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(17,6).\n3) Favorable samples: choose k reds and n−k blues: C(11,3)·C(6,3).\n4) Compute totals: C(17,6) = 12376, and favorable = 3300.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{825}{3094}$", "final_answer": "\\frac{825}{3094}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3d0acd48510209d4", "problem_id": "acc938d7490b5358", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 11 red balls and 6 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(17,6).\nFavorable samples: choose k reds and n−k blues: C(11,3)·C(6,3).\nCompute totals: C(17,6) = 12376, and favorable = 3300.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{825}{3094}$.\nFinal answer: $\\frac{825}{3094}$", "final_answer": "\\frac{825}{3094}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "749daf4ad9f4ca35", "problem_id": "2e92acf1804900ca", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A cyclist rides at 24 miles per hour for 2 hours, then at 59 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 24 × 2 = 48 miles.\n2) Second leg distance = 59 × 3 = 177 miles.\n3) Total distance = 48 + 177 = 225 miles.\nFinal answer: $225$", "final_answer": "225", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3be9496da3a0540c", "problem_id": "2e92acf1804900ca", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A cyclist rides at 24 miles per hour for 2 hours, then at 59 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 24 × 2 = 48 miles.\nSecond leg distance = 59 × 3 = 177 miles.\nTotal distance = 48 + 177 = 225 miles.\nCheck: adding both legs gives 225.\nFinal answer: $225$", "final_answer": "225", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "33b60ccc56fe84bb", "problem_id": "fadfba899225a32e", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A store sells a backpack for $160. It is discounted by 30%. You buy 5 backpacks. What is the total cost?", "output": "1) Discounted price = 160 × (1 − 30/100) = 160 × 70/100 = 112.\n2) Subtotal = 5 × 112 = 560.\nFinal answer: $560$", "final_answer": "560", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4f949880bad89da6", "problem_id": "fadfba899225a32e", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A store sells a backpack for $160. It is discounted by 30%. You buy 5 backpacks. What is the total cost?", "output": "Discounted price = 160 × (1 − 30/100) = 160 × 70/100 = 112.\nSubtotal = 5 × 112 = 560.\nFinal answer: $560$", "final_answer": "560", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "86e4e5e3c23afb2d", "problem_id": "03df527187c09f76", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡0 (mod 3), x≡2 (mod 5). Give the solution modulo 15.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 15.\nCompute the CRT solution to get x ≡ 12 (mod 15).\nCheck: 12 mod 3=0 and mod 5=2.\nFinal answer: $x\\equiv 12\\pmod{15}$", "final_answer": "x\\equiv 12\\pmod{15}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6533505c4d736f4b", "problem_id": "03df527187c09f76", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the congruences: x≡0 (mod 3), x≡2 (mod 5). Give the solution modulo 15.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 15.\n3) Compute the CRT solution to get x ≡ 12 (mod 15).\nFinal answer: $x\\equiv 12\\pmod{15}$", "final_answer": "x\\equiv 12\\pmod{15}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "762bed244c7b4871", "problem_id": "02bffa89a6bc2a19", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} 3 x^{4} + 2 x^{3} - x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{4} + 2 x^{3} - x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{65 \\pi^{3}}{24} - \\frac{15 \\pi^{2}}{8} - 1 + \\frac{5 \\pi}{2} + \\frac{255 \\pi^{4}}{32} + \\frac{615 \\pi^{5}}{32}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{65\\cdot \\pi^{3}}{24} - \\frac{15\\cdot \\pi^{2}}{8} - 1 + \\frac{5\\cdot \\pi}{2} + \\frac{255\\cdot \\pi^{4}}{32} + \\frac{615\\cdot \\pi^{5}}{32}$", "final_answer": "- \\frac{65\\cdot \\pi^{3}}{24} - \\frac{15\\cdot \\pi^{2}}{8} - 1 + \\frac{5\\cdot \\pi}{2} + \\frac{255\\cdot \\pi^{4}}{32} + \\frac{615\\cdot \\pi^{5}}{32}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4934564c6f620db7", "problem_id": "02bffa89a6bc2a19", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} 3 x^{4} + 2 x^{3} - x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{4} + 2 x^{3} - x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{65 \\pi^{3}}{24} - \\frac{15 \\pi^{2}}{8} - 1 + \\frac{5 \\pi}{2} + \\frac{255 \\pi^{4}}{32} + \\frac{615 \\pi^{5}}{32}$.\n4) Simplify the result.\nFinal answer: $- \\frac{65\\cdot \\pi^{3}}{24} - \\frac{15\\cdot \\pi^{2}}{8} - 1 + \\frac{5\\cdot \\pi}{2} + \\frac{255\\cdot \\pi^{4}}{32} + \\frac{615\\cdot \\pi^{5}}{32}$", "final_answer": "- \\frac{65\\cdot \\pi^{3}}{24} - \\frac{15\\cdot \\pi^{2}}{8} - 1 + \\frac{5\\cdot \\pi}{2} + \\frac{255\\cdot \\pi^{4}}{32} + \\frac{615\\cdot \\pi^{5}}{32}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c3aa38821b0f3ad2", "problem_id": "28dfa09c78d29520", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} - 3 x^{2} - 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=0$.", "output": "1) Differentiate: f'(x) = $9 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 3$.\n2) Slope at x=0: m = f'(0) = $- \\frac{5}{2}$.\n3) Point on curve: (0, f(0)) = (0, $1$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 1 = - \\frac{5}{2}(x − 0).\n6) Optionally expand: y = $1 - \\frac{5 x}{2}$.\nFinal answer: $y = 1 - \\frac{5\\cdot x}{2}$", "final_answer": "y = 1 - \\frac{5\\cdot x}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aa0b8197247ff50b", "problem_id": "28dfa09c78d29520", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} - 3 x^{2} - 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=0$.", "output": "Differentiate: f'(x) = $9 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 3$.\nSlope at x=0: m = f'(0) = $- \\frac{5}{2}$.\nPoint on curve: (0, f(0)) = (0, $1$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 1 = - \\frac{5}{2}(x − 0).\nOptionally expand: y = $1 - \\frac{5 x}{2}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 1 - \\frac{5\\cdot x}{2}$", "final_answer": "y = 1 - \\frac{5\\cdot x}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "65a3fb6408f2058a", "problem_id": "42e3112797ea5109", "style": "answer_only", "instruction": "Final answer only.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_6=29$?", "output": "$26334$", "final_answer": "26334", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "703df05e531cae67", "problem_id": "42e3112797ea5109", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_6=29$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_6=17.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=17 and k=6.\n4) Compute C(22,5) = 26334.\nFinal answer: $26334$", "final_answer": "26334", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "042df149cbcba205", "problem_id": "3957c5889c0020b5", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 18 times, what is the probability of exactly 7 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=18, k=7, p=$\\frac{1}{4}$ and simplify.\nHere C(18,7) = 31824.\nCheck: expression simplifies to $\\frac{352345383}{4294967296}$.\nFinal answer: $\\frac{352345383}{4294967296}$", "final_answer": "\\frac{352345383}{4294967296}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9589b25dca767681", "problem_id": "3957c5889c0020b5", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 18 times, what is the probability of exactly 7 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=18, k=7, p=$\\frac{1}{4}$ and simplify.\n4) Here C(18,7) = 31824.\nFinal answer: $\\frac{352345383}{4294967296}$", "final_answer": "\\frac{352345383}{4294967296}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a310de6087c01c36", "problem_id": "8eb129c8ffc74457", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 x^{2} - 5 x + 8$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{3} + x^{2} - 2 x + 2$ and a = -2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $9 x^{2} + 2 x - 2$.\n5) Evaluate at a: p'(-2) = $30$.\nFinal answer: $30$", "final_answer": "30", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "82538bcecbf18cfa", "problem_id": "8eb129c8ffc74457", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 x^{2} - 5 x + 8$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $3 x^{3} + x^{2} - 2 x + 2$ and a = -2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $30$", "final_answer": "30", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5f232bea61f358da", "problem_id": "4e150a079cfdc667", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "You want 51 liters of a 45% salt solution. You have a 20% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 51−x liters is 50% solution.\n2) Salt equation: 20x + 50(51−x) = 45·51.\n3) (20-50)x = (45-50)·51 ⇒ x = 17/2.\n4) Then 51−x = 85/2.\nFinal answer: $17/2 L of 20% and 85/2 L of 50%$", "final_answer": "17/2 L of 20% and 85/2 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "862d3ac0a118870f", "problem_id": "4e150a079cfdc667", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "You want 51 liters of a 45% salt solution. You have a 20% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 51−x liters is 50% solution.\nSalt equation: 20x + 50(51−x) = 45·51.\n(20-50)x = (45-50)·51 ⇒ x = 17/2.\nFinal answer: $17/2 L of 20% and 85/2 L of 50%$", "final_answer": "17/2 L of 20% and 85/2 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "45182715eb078f97", "problem_id": "928c844d461859f4", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} + 2 x^{2} - x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 3$ at $x=2$.", "output": "1) Differentiate: f'(x) = $6 x^{2} + 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 1$.\n2) Slope at x=2: m = f'(2) = $\\frac{e}{2} - 2 \\sin{\\left(4 \\right)} + 31$.\n3) Point on curve: (2, f(2)) = (2, $\\cos{\\left(4 \\right)} + e + 19$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(4 \\right)} + e + 19 = \\frac{e}{2} - 2 \\sin{\\left(4 \\right)} + 31(x − 2).\n6) Optionally expand: y = $\\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 31 x - 43 + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$.\nFinal answer: $y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 31\\cdot x - 43 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$", "final_answer": "y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 31\\cdot x - 43 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "55ab80319287481f", "problem_id": "928c844d461859f4", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} + 2 x^{2} - x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 3$ at $x=2$.", "output": "Differentiate: f'(x) = $6 x^{2} + 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 1$.\nSlope at x=2: m = f'(2) = $\\frac{e}{2} - 2 \\sin{\\left(4 \\right)} + 31$.\nPoint on curve: (2, f(2)) = (2, $\\cos{\\left(4 \\right)} + e + 19$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − \\cos{\\left(4 \\right)} + e + 19 = \\frac{e}{2} - 2 \\sin{\\left(4 \\right)} + 31(x − 2).\nOptionally expand: y = $\\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 31 x - 43 + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 31\\cdot x - 43 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$", "final_answer": "y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 31\\cdot x - 43 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c9d02e27340e5b0a", "problem_id": "c925178170b5b1c9", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 14 times, what is the probability of exactly 9 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=14, k=9, p=$\\frac{1}{4}$ and simplify.\nFinal answer: $\\frac{243243}{134217728}$", "final_answer": "\\frac{243243}{134217728}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "651ea47760a2afb9", "problem_id": "c925178170b5b1c9", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 14 times, what is the probability of exactly 9 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=14, k=9, p=$\\frac{1}{4}$ and simplify.\nHere C(14,9) = 2002.\nCheck: expression simplifies to $\\frac{243243}{134217728}$.\nFinal answer: $\\frac{243243}{134217728}$", "final_answer": "\\frac{243243}{134217728}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c2d443632c2cf02b", "problem_id": "d679b07c160e2d77", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_3=19$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=19 and k=3.\nCompute C(21,2) = 210.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $210$", "final_answer": "210", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c169d172a587a5bb", "problem_id": "d679b07c160e2d77", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_3=19$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=19 and k=3.\n4) Compute C(21,2) = 210.\nFinal answer: $210$", "final_answer": "210", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "68e739051fa7655e", "problem_id": "7976e7f19e5b5273", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$8 x - 8 y$ = $-56$\n$5 x - 8 y$ = $-59$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $5$: $40 x - 40 y$ = $-280$.\n3) Multiply the second equation by $8$: $40 x - 64 y$ = $-472$.\n4) Subtract to eliminate x: $24 y$ = $192$.\n5) Solve for y: y = $8$.\n6) Substitute back to find x: x = $1$.\nFinal answer: $(x,y)=\\left(1, 8\\right)$", "final_answer": "(x,y)=\\left(1, 8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0bb26ccac4ff178a", "problem_id": "8151f168017c9a34", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} x^{2} + 3 x + 11$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to x^{2} + 2 x + 11.\n3) Then the limit is 26.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: 29.\nFinal answer: $29$", "final_answer": "29", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d5e414bd682b610c", "problem_id": "4a9467a6c7b84f9d", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A cyclist rides at 60 miles per hour for 3/2 hours, then at 70 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 60 × 3/2 = 90 miles.\n2) Second leg distance = 70 × 1 = 70 miles.\n3) Total distance = 90 + 70 = 160 miles.\nFinal answer: $160$", "final_answer": "160", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2b4c6bec3fab5070", "problem_id": "4a9467a6c7b84f9d", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A cyclist rides at 60 miles per hour for 3/2 hours, then at 70 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 60 × 3/2 = 90 miles.\nSecond leg distance = 70 × 1 = 70 miles.\nTotal distance = 90 + 70 = 160 miles.\nCheck: adding both legs gives 160.\nFinal answer: $160$", "final_answer": "160", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fdd4571add2d68c8", "problem_id": "d38563a5a0d6e311", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Two friends split $125 in the ratio 3:2. How much does each receive?", "output": "1) Total ratio parts = 3+2 = 5.\n2) Each part = 125 ÷ 5 = 25.\n3) First share = 3 × 25 = 75.\n4) Second share = 2 × 25 = 50.\nFinal answer: $50$", "final_answer": "50", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "df202bb723a04fcc", "problem_id": "3787fad3e50f603c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} + 4$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(3 x^{2}) = 6 x^{2}$.\n3) So $f'(x) = 6 x^{2}$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(3 x^{2}) = 6 x$.\nTherefore the correct derivative is $f'(x) = 6 x$.\nFinal answer: $6 x$", "final_answer": "6 x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "1876ad7fbdcf0051", "problem_id": "171507cacba2478d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 2 x^{3} + x^{2} + 2 x + \\left(- 3 x^{2} + x + 8\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{- 3 x^{2} + x + 8}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{8 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- 3 x^{2} + x\\right) + 8}) = \\frac{8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{- 3 x^{2} + x + 8}$.\n3) So $f'(x) = \\frac{9 x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} + \\frac{18 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} + \\frac{12 x^{4}}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} - \\frac{6 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} - \\frac{12 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} - \\frac{8 x^{3}}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} - \\frac{47 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} - \\frac{94 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} - \\frac{82 x^{2}}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} + \\frac{16 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} + \\frac{32 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} - \\frac{4 x}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} - \\frac{48 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{4} - 6 x^{3} - 47 x^{2} + 16 x + 64} + \\frac{64 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} + \\frac{128 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} + \\frac{38}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} + \\frac{8 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{4} - 6 x^{3} - 47 x^{2} + 16 x + 64} + \\frac{8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{- 3 x^{2} + x + 8} - \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- 3 x^{2} + x\\right) + 8} - \\frac{8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(- 3 x^{2} + x\\right) + 8}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{8 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- 3 x^{2} + x\\right) + 8}) = \\frac{8 \\left(6 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(- 3 x^{2} + x\\right) + 8\\right)^{2}} + \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- 3 x^{2} + x\\right) + 8} + \\frac{8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(- 3 x^{2} + x\\right) + 8}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(6 x - 1\\right) \\left(- 2 x^{3} + x^{2} + 2 x + \\left(x \\left(1 - 3 x\\right) + 8\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x \\left(1 - 3 x\\right) + 8\\right) \\left(- 12 x^{2} + 4 x + 2 \\left(1 - 6 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(1 - 3 x\\right) + 8\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(1 - 3 x\\right) + 8\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(1 - 3 x\\right) + 8\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(6 x - 1\\right) \\left(- 2 x^{3} + x^{2} + 2 x + \\left(x \\left(1 - 3 x\\right) + 8\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x \\left(1 - 3 x\\right) + 8\\right) \\left(- 12 x^{2} + 4 x + 2 \\left(1 - 6 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(1 - 3 x\\right) + 8\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(1 - 3 x\\right) + 8\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(1 - 3 x\\right) + 8\\right)^{2}}$", "final_answer": "\\frac{2 \\left(6 x - 1\\right) \\left(- 2 x^{3} + x^{2} + 2 x + \\left(x \\left(1 - 3 x\\right) + 8\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x \\left(1 - 3 x\\right) + 8\\right) \\left(- 12 x^{2} + 4 x + 2 \\left(1 - 6 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(1 - 3 x\\right) + 8\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(1 - 3 x\\right) + 8\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(1 - 3 x\\right) + 8\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f733eb79adca83a2", "problem_id": "3874bb846d99cd65", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 9 x + 8$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} + 9 x + 8$ = $\\left(x + 1\\right) \\left(x + 8\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-1$.\n5) From the second factor: x = $-8$.\nFinal answer: $x \\in \\{-1, -8\\}$", "final_answer": "x \\in \\{-1, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b023c9b385f904e1", "problem_id": "4014a631239f2622", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡1 (mod 4), x≡12 (mod 13). Give the solution modulo 52.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 52.\n3) Compute the CRT solution to get x ≡ 25 (mod 52).\nFinal answer: $x\\equiv 25\\pmod{52}$", "final_answer": "x\\equiv 25\\pmod{52}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "87b50a9b2e4ba638", "problem_id": "4014a631239f2622", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡1 (mod 4), x≡12 (mod 13). Give the solution modulo 52.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 52.\nCompute the CRT solution to get x ≡ 25 (mod 52).\nCheck: 25 mod 4=1 and mod 13=12.\nFinal answer: $x\\equiv 25\\pmod{52}$", "final_answer": "x\\equiv 25\\pmod{52}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c43ea49bd8796728", "problem_id": "c6dbbdf0256eb22e", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A cyclist rides at 84 miles per hour for 3/2 hours, then at 83 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 84 × 3/2 = 126 miles.\n2) Second leg distance = 83 × 2 = 166 miles.\n3) Total distance = 126 + 166 = 292 miles.\nFinal answer: $292$", "final_answer": "292", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bc85f9979a8b81bc", "problem_id": "c6dbbdf0256eb22e", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 84 miles per hour for 3/2 hours, then at 83 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 84 × 3/2 = 126 miles.\nSecond leg distance = 83 × 2 = 166 miles.\nTotal distance = 126 + 166 = 292 miles.\nCheck: adding both legs gives 292.\nFinal answer: $292$", "final_answer": "292", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "79f24aa865b8b991", "problem_id": "c71c126cc36d6e29", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A cyclist rides at 61 miles per hour for 2 hours, then at 21 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 61 × 2 = 122 miles.\nSecond leg distance = 21 × 1 = 21 miles.\nTotal distance = 122 + 21 = 143 miles.\nCheck: adding both legs gives 143.\nFinal answer: $143$", "final_answer": "143", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b47bebd5efbce33d", "problem_id": "c71c126cc36d6e29", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A cyclist rides at 61 miles per hour for 2 hours, then at 21 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 61 × 2 = 122 miles.\n2) Second leg distance = 21 × 1 = 21 miles.\n3) Total distance = 122 + 21 = 143 miles.\nFinal answer: $143$", "final_answer": "143", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "723fa86c3be28354", "problem_id": "c0fe0fde2aa2a058", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "You want 25 liters of a 25% salt solution. You have a 20% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 25−x liters is 60% solution.\n2) Salt equation: 20x + 60(25−x) = 25·25.\n3) (20-60)x = (25-60)·25 ⇒ x = 175/8.\n4) Then 25−x = 25/8.\nFinal answer: $175/8 L of 20% and 25/8 L of 60%$", "final_answer": "175/8 L of 20% and 25/8 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e4581b622b08b325", "problem_id": "c0fe0fde2aa2a058", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "You want 25 liters of a 25% salt solution. You have a 20% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 25−x liters is 60% solution.\nSalt equation: 20x + 60(25−x) = 25·25.\n(20-60)x = (25-60)·25 ⇒ x = 175/8.\nThen 25−x = 25/8.\nCheck: 20·175/8 + 60·25/8 = 25·25.\nFinal answer: $175/8 L of 20% and 25/8 L of 60%$", "final_answer": "175/8 L of 20% and 25/8 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec651227f297c536", "problem_id": "fc06541925808c59", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the multiplicative inverse of 7 modulo 29.", "output": "1) We want x such that 7x ≡ 1 (mod 29).\n2) Use extended Euclid to find s,t with 7s + 29t = 1.\n3) One Bezout identity is: 7·(-4) + 29·(1) = 1.\n4) Reduce s modulo 29: x ≡ 25 (mod 29).\nFinal answer: $x\\equiv 25\\pmod{29}$", "final_answer": "x\\equiv 25\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "91f83046547460df", "problem_id": "fc06541925808c59", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the multiplicative inverse of 7 modulo 29.", "output": "We want x such that 7x ≡ 1 (mod 29).\nUse extended Euclid to find s,t with 7s + 29t = 1.\nOne Bezout identity is: 7·(-4) + 29·(1) = 1.\nReduce s modulo 29: x ≡ 25 (mod 29).\nCheck: 7·25 mod 29 = 1.\nFinal answer: $x\\equiv 25\\pmod{29}$", "final_answer": "x\\equiv 25\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "68bf9d01717db994", "problem_id": "012099cd915bd478", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} - 2 x^{3} - x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{3} - x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $- \\frac{15 \\pi^{4}}{2} - 3 \\pi^{3} - 9 \\pi - 4 + \\frac{3 \\pi^{2}}{2}$.\n4) Simplify the result.\nFinal answer: $- \\frac{15\\cdot \\pi^{4}}{2} - 3\\cdot \\pi^{3} - 9\\cdot \\pi - 4 + \\frac{3\\cdot \\pi^{2}}{2}$", "final_answer": "- \\frac{15\\cdot \\pi^{4}}{2} - 3\\cdot \\pi^{3} - 9\\cdot \\pi - 4 + \\frac{3\\cdot \\pi^{2}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b5385d35eefedb5f", "problem_id": "012099cd915bd478", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} - 2 x^{3} - x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{3} - x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $- \\frac{15 \\pi^{4}}{2} - 3 \\pi^{3} - 9 \\pi - 4 + \\frac{3 \\pi^{2}}{2}$.\nFinal answer: $- \\frac{15\\cdot \\pi^{4}}{2} - 3\\cdot \\pi^{3} - 9\\cdot \\pi - 4 + \\frac{3\\cdot \\pi^{2}}{2}$", "final_answer": "- \\frac{15\\cdot \\pi^{4}}{2} - 3\\cdot \\pi^{3} - 9\\cdot \\pi - 4 + \\frac{3\\cdot \\pi^{2}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e1085b1fc4ed4811", "problem_id": "21baddfaea8d05cd", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "An urn has 15 red balls and 16 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(31,12).\n3) Favorable samples: choose k reds and n−k blues: C(15,2)·C(16,10).\n4) Compute totals: C(31,12) = 141120525, and favorable = 840840.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{616}{103385}$", "final_answer": "\\frac{616}{103385}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5799909632e29b66", "problem_id": "21baddfaea8d05cd", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "An urn has 15 red balls and 16 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(31,12).\nFavorable samples: choose k reds and n−k blues: C(15,2)·C(16,10).\nFinal answer: $\\frac{616}{103385}$", "final_answer": "\\frac{616}{103385}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "44c4480f63fec0ac", "problem_id": "a0cbae6b41c1d781", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡7 (mod 8), x≡5 (mod 7). Give the solution modulo 56.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 56.\nCompute the CRT solution to get x ≡ 47 (mod 56).\nCheck: 47 mod 8=7 and mod 7=5.\nFinal answer: $x\\equiv 47\\pmod{56}$", "final_answer": "x\\equiv 47\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d9b873970e119684", "problem_id": "a0cbae6b41c1d781", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡7 (mod 8), x≡5 (mod 7). Give the solution modulo 56.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 56.\n3) Compute the CRT solution to get x ≡ 47 (mod 56).\nFinal answer: $x\\equiv 47\\pmod{56}$", "final_answer": "x\\equiv 47\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8e64712f0b58e0b8", "problem_id": "eabdb9071f0884da", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$- 4 x - 4 y$ = $72$\n$6 x + 2 y$ = $-72$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $6$: $- 24 x - 24 y$ = $432$.\nMultiply the second equation by $-4$: $- 24 x - 8 y$ = $288$.\nSubtract to eliminate x: $- 16 y$ = $144$.\nSolve for y: y = $-9$.\nSubstitute back to find x: x = $-9$.\nCheck: plug (x,y)=(-9,-9) into both equations.\nFinal answer: $(x,y)=\\left(-9, -9\\right)$", "final_answer": "(x,y)=\\left(-9, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1445b5749814becd", "problem_id": "eabdb9071f0884da", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$- 4 x - 4 y$ = $72$\n$6 x + 2 y$ = $-72$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $6$: $- 24 x - 24 y$ = $432$.\n3) Multiply the second equation by $-4$: $- 24 x - 8 y$ = $288$.\n4) Subtract to eliminate x: $- 16 y$ = $144$.\n5) Solve for y: y = $-9$.\n6) Substitute back to find x: x = $-9$.\nFinal answer: $(x,y)=\\left(-9, -9\\right)$", "final_answer": "(x,y)=\\left(-9, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "225469be4a9784b8", "problem_id": "f183f12c710c1989", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "An urn has 27 red balls and 30 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(57,11).\nFavorable samples: choose k reds and n−k blues: C(27,3)·C(30,8).\nFinal answer: $\\frac{29264625}{315400456}$", "final_answer": "\\frac{29264625}{315400456}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "93235452b0c0b092", "problem_id": "f183f12c710c1989", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 27 red balls and 30 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(57,11).\n3) Favorable samples: choose k reds and n−k blues: C(27,3)·C(30,8).\n4) Compute totals: C(57,11) = 184509266760, and favorable = 17119805625.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{29264625}{315400456}$", "final_answer": "\\frac{29264625}{315400456}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a1402494f607254d", "problem_id": "3ec4cbb32148a4fc", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 17 x + 72$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} - 17 x + 72$ = $\\left(x - 9\\right) \\left(x - 8\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $9$.\n5) From the second factor: x = $8$.\nFinal answer: $x \\in \\{9, 8\\}$", "final_answer": "x \\in \\{9, 8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d34a9462a4aaf68", "problem_id": "3ec4cbb32148a4fc", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 17 x + 72$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} - 17 x + 72$ = $\\left(x - 9\\right) \\left(x - 8\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $9$.\nFrom the second factor: x = $8$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{9, 8\\}$", "final_answer": "x \\in \\{9, 8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a4d4153d4b16f5f4", "problem_id": "1c2bd529d20a3a2b", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 29 red balls and 28 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(57,12).\n3) Favorable samples: choose k reds and n−k blues: C(29,4)·C(28,8).\n4) Compute totals: C(57,12) = 707285522580, and favorable = 73820601855.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{335907}{3218372}$", "final_answer": "\\frac{335907}{3218372}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "92248d6755569ade", "problem_id": "855a8eedf256ec62", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x^{2} + 5 x + 7$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to x^{2} + 4 x + 7.\n3) Then the limit is 19.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 21.\nFinal answer: $21$", "final_answer": "21", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3638f7172da48df6", "problem_id": "d56bd844bc413b03", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 2 x^{3} + x^{2} + x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 2 x^{3} + x^{2} + x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 3$.\n3) Compute an antiderivative: F(x) = $- \\frac{x^{4}}{2} + \\frac{x^{3}}{3} + \\frac{x^{2}}{2} + 3 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{x^{4}}{2} + \\frac{x^{3}}{3} + \\frac{x^{2}}{2} + 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{x^{4}}{2} + \\frac{x^{3}}{3} + \\frac{x^{2}}{2} + 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c2e1e81d2cb34331", "problem_id": "7ad602b7d6879177", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- 3 x^{2} - 3 x - 3\\right) \\cos{\\left(2 x + 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 3 x \\cos{\\left(2 x + 1 \\right)}) = 3 x \\sin{\\left(2 x + 1 \\right)}$.\n3) So $f'(x) = 6 x^{2} \\sin{\\left(2 x + 1 \\right)} + 2 x e^{x} + 3 x \\sin{\\left(2 x + 1 \\right)} - 6 x \\cos{\\left(2 x + 1 \\right)} + 4 e^{x} + 6 \\sin{\\left(2 x + 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 3 x \\cos{\\left(2 x + 1 \\right)}) = 6 x \\sin{\\left(2 x + 1 \\right)} - 3 \\cos{\\left(2 x + 1 \\right)}$.\nTherefore the correct derivative is $f'(x) = 3 \\left(- 2 x - 1\\right) \\cos{\\left(2 x + 1 \\right)} + 2 \\left(x + 1\\right) e^{x} - 6 \\left(x \\left(- x - 1\\right) - 1\\right) \\sin{\\left(2 x + 1 \\right)} + 2 e^{x}$.\nFinal answer: $3 \\left(- 2 x - 1\\right) \\cos{\\left(2 x + 1 \\right)} + 2 \\left(x + 1\\right) e^{x} - 6 \\left(x \\left(- x - 1\\right) - 1\\right) \\sin{\\left(2 x + 1 \\right)} + 2 e^{x}$", "final_answer": "3 \\left(- 2 x - 1\\right) \\cos{\\left(2 x + 1 \\right)} + 2 \\left(x + 1\\right) e^{x} - 6 \\left(x \\left(- x - 1\\right) - 1\\right) \\sin{\\left(2 x + 1 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ca27f03b51d4ec7a", "problem_id": "f04363ae8e8f0741", "style": "answer_only", "instruction": "Final answer only.", "input": "Solve the linear system:\n$3 x + 3 y$ = $0$\n$- 3 x + 3 y$ = $54$", "output": "$(x,y)=\\left(-9, 9\\right)$", "final_answer": "(x,y)=\\left(-9, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dbe97b1703c6e37a", "problem_id": "f04363ae8e8f0741", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$3 x + 3 y$ = $0$\n$- 3 x + 3 y$ = $54$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $- 9 x - 9 y$ = $0$.\n3) Multiply the second equation by $3$: $- 9 x + 9 y$ = $162$.\n4) Subtract to eliminate x: $- 18 y$ = $-162$.\n5) Solve for y: y = $9$.\n6) Substitute back to find x: x = $-9$.\nFinal answer: $(x,y)=\\left(-9, 9\\right)$", "final_answer": "(x,y)=\\left(-9, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d9787c35cbe04a55", "problem_id": "6ada227493c72117", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "An urn has 13 red balls and 22 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(35,11).\nFavorable samples: choose k reds and n−k blues: C(13,1)·C(22,10).\nFinal answer: $\\frac{2717}{134850}$", "final_answer": "\\frac{2717}{134850}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bec3d999471baf3a", "problem_id": "6ada227493c72117", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "An urn has 13 red balls and 22 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(35,11).\n3) Favorable samples: choose k reds and n−k blues: C(13,1)·C(22,10).\n4) Compute totals: C(35,11) = 417225900, and favorable = 8406398.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{2717}{134850}$", "final_answer": "\\frac{2717}{134850}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4178b778dfdb43ff", "problem_id": "ffe13513d6f2a422", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Two friends split $840 in the ratio 5:9. How much does each receive?", "output": "1) Total ratio parts = 5+9 = 14.\n2) Each part = 840 ÷ 14 = 60.\n3) First share = 5 × 60 = 300.\n4) Second share = 9 × 60 = 540.\nFinal answer: $540$", "final_answer": "540", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "38c601dfaa5641e4", "problem_id": "ffe13513d6f2a422", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $840 in the ratio 5:9. How much does each receive?", "output": "Total ratio parts = 5+9 = 14.\nEach part = 840 ÷ 14 = 60.\nFirst share = 5 × 60 = 300.\nSecond share = 9 × 60 = 540.\nCheck: 300+540=840.\nFinal answer: $540$", "final_answer": "540", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bd0226ce3704236a", "problem_id": "54571c3109c02cf8", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A disease has prevalence $\\frac{1}{25}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{33}{250}$.\n3) Compute P(D|+) = $\\frac{3}{11}$.\nFinal answer: $\\frac{3}{11}$", "final_answer": "\\frac{3}{11}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff579adf25ae6568", "problem_id": "54571c3109c02cf8", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A disease has prevalence $\\frac{1}{25}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{33}{250}$.\nCompute P(D|+) = $\\frac{3}{11}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{3}{11}$", "final_answer": "\\frac{3}{11}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b5adb6d98efa3697", "problem_id": "63257aef8f99b133", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 21 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=21, k=5, p=$\\frac{3}{8}$ and simplify.\n4) Here C(21,5) = 20349.\nFinal answer: $\\frac{754517669677734375}{9223372036854775808}$", "final_answer": "\\frac{754517669677734375}{9223372036854775808}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "da7362f2bd419aaf", "problem_id": "63257aef8f99b133", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 21 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=21, k=5, p=$\\frac{3}{8}$ and simplify.\nHere C(21,5) = 20349.\nCheck: expression simplifies to $\\frac{754517669677734375}{9223372036854775808}$.\nFinal answer: $\\frac{754517669677734375}{9223372036854775808}$", "final_answer": "\\frac{754517669677734375}{9223372036854775808}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e76e62c69836f9b4", "problem_id": "06c221c7dc60c39f", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{10}$. If it is flipped 8 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=8, k=6, p=$\\frac{3}{10}$ and simplify.\n4) Here C(8,6) = 28.\nFinal answer: $\\frac{250047}{25000000}$", "final_answer": "\\frac{250047}{25000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "47f58679fc0005e5", "problem_id": "06c221c7dc60c39f", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{10}$. If it is flipped 8 times, what is the probability of exactly 6 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=8, k=6, p=$\\frac{3}{10}$ and simplify.\nHere C(8,6) = 28.\nCheck: expression simplifies to $\\frac{250047}{25000000}$.\nFinal answer: $\\frac{250047}{25000000}$", "final_answer": "\\frac{250047}{25000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b4eb509168b9fbc5", "problem_id": "8b04e34990ee3066", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A disease has prevalence $\\frac{3}{200}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{31}{500}$.\nCompute P(D|+) = $\\frac{51}{248}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{51}{248}$", "final_answer": "\\frac{51}{248}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d0378bd2168077c3", "problem_id": "8b04e34990ee3066", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A disease has prevalence $\\frac{3}{200}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{31}{500}$.\n3) Compute P(D|+) = $\\frac{51}{248}$.\nFinal answer: $\\frac{51}{248}$", "final_answer": "\\frac{51}{248}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "20d33c0ecf49f6de", "problem_id": "ffb4bbaf950eb930", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡1 (mod 5), x≡4 (mod 9). Give the solution modulo 45.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 45.\nCompute the CRT solution to get x ≡ 31 (mod 45).\nCheck: 31 mod 5=1 and mod 9=4.\nFinal answer: $x\\equiv 31\\pmod{45}$", "final_answer": "x\\equiv 31\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "11bb8fced14daec6", "problem_id": "ffb4bbaf950eb930", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the congruences: x≡1 (mod 5), x≡4 (mod 9). Give the solution modulo 45.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 45.\n3) Compute the CRT solution to get x ≡ 31 (mod 45).\nFinal answer: $x\\equiv 31\\pmod{45}$", "final_answer": "x\\equiv 31\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cd30c53f9151ae40", "problem_id": "f7ae2436bf54fd28", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A cyclist rides at 27 miles per hour for 1/2 hours, then at 33 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 27 × 1/2 = 27/2 miles.\n2) Second leg distance = 33 × 5/2 = 165/2 miles.\n3) Total distance = 27/2 + 165/2 = 96 miles.\nFinal answer: $96$", "final_answer": "96", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5ad3d22b68ebc41d", "problem_id": "f7ae2436bf54fd28", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 27 miles per hour for 1/2 hours, then at 33 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 27 × 1/2 = 27/2 miles.\nSecond leg distance = 33 × 5/2 = 165/2 miles.\nTotal distance = 27/2 + 165/2 = 96 miles.\nCheck: adding both legs gives 96.\nFinal answer: $96$", "final_answer": "96", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "57e2459b92af0c82", "problem_id": "8168d0b62efc3c85", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A store sells a backpack for $745. It is discounted by 40%. You buy 7 backpacks. What is the total cost?", "output": "1) Discounted price = 745 × (1 − 40/100) = 745 × 60/100 = 447.\n2) Subtotal = 7 × 447 = 3129.\nFinal answer: $3129$", "final_answer": "3129", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff0d4bfadeba67a0", "problem_id": "8168d0b62efc3c85", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A store sells a backpack for $745. It is discounted by 40%. You buy 7 backpacks. What is the total cost?", "output": "Discounted price = 745 × (1 − 40/100) = 745 × 60/100 = 447.\nSubtotal = 7 × 447 = 3129.\nCheck: recomputing the formula yields 3129.\nFinal answer: $3129$", "final_answer": "3129", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "41ef8a09ff28111f", "problem_id": "5ee675c3b0f00cc7", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 3 x^{3} + 9 x^{2} - 24 x + 70$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 3 x^{4} + 3 x^{2} - 2 x + 1$ and a = -3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 12 x^{3} + 6 x - 2$.\nEvaluate at a: p'(-3) = $304$.\nCheck: the identity gives $304$.\nFinal answer: $304$", "final_answer": "304", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4d1977e693aeb26d", "problem_id": "5ee675c3b0f00cc7", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 3 x^{3} + 9 x^{2} - 24 x + 70$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 3 x^{4} + 3 x^{2} - 2 x + 1$ and a = -3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 12 x^{3} + 6 x - 2$.\n5) Evaluate at a: p'(-3) = $304$.\nFinal answer: $304$", "final_answer": "304", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec17e3eccfc2a514", "problem_id": "e971a75f11a2fe09", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$- 2 x + 7 y$ = $-66$\n$- 3 x + 7 y$ = $-71$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $6 x - 21 y$ = $198$.\nMultiply the second equation by $-2$: $6 x - 14 y$ = $142$.\nSubtract to eliminate x: $- 7 y$ = $56$.\nSolve for y: y = $-8$.\nSubstitute back to find x: x = $5$.\nCheck: plug (x,y)=(5,-8) into both equations.\nFinal answer: $(x,y)=\\left(5, -8\\right)$", "final_answer": "(x,y)=\\left(5, -8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3e58fdf261ca72a0", "problem_id": "e971a75f11a2fe09", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$- 2 x + 7 y$ = $-66$\n$- 3 x + 7 y$ = $-71$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $6 x - 21 y$ = $198$.\n3) Multiply the second equation by $-2$: $6 x - 14 y$ = $142$.\n4) Subtract to eliminate x: $- 7 y$ = $56$.\n5) Solve for y: y = $-8$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, -8\\right)$", "final_answer": "(x,y)=\\left(5, -8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e1fd94dba02feaa0", "problem_id": "24fda73f02fbfb71", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Worker A finishes a job in 5 hours and Worker B finishes it in 6 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/5 job/hour. B's rate = 1/6 job/hour.\nCombined rate = 1/5 + 1/6 = 11/30 job/hour.\nTime = 1 ÷ (combined rate) = 30/11 hours.\nFinal answer: $\\frac{30}{11}$", "final_answer": "\\frac{30}{11}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "00b788c9d86730da", "problem_id": "24fda73f02fbfb71", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Worker A finishes a job in 5 hours and Worker B finishes it in 6 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/5 job/hour. B's rate = 1/6 job/hour.\n2) Combined rate = 1/5 + 1/6 = 11/30 job/hour.\n3) Time = 1 ÷ (combined rate) = 30/11 hours.\nFinal answer: $\\frac{30}{11}$", "final_answer": "\\frac{30}{11}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7b939ae18a30e9f1", "problem_id": "347ef49faece6c15", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 6 x - 105$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} + 6 x - 105$ = $3 \\left(x - 5\\right) \\left(x + 7\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $5$.\nFrom the second factor: x = $-7$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{5, -7\\}$", "final_answer": "x \\in \\{5, -7\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ab03d9ba982db995", "problem_id": "347ef49faece6c15", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 6 x - 105$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} + 6 x - 105$ = $3 \\left(x - 5\\right) \\left(x + 7\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $5$.\n5) From the second factor: x = $-7$.\nFinal answer: $x \\in \\{5, -7\\}$", "final_answer": "x \\in \\{5, -7\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a11d206a09a8380c", "problem_id": "7a8f5e4e0af43983", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{13 x}{3} - \\frac{28}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$- \\frac{13}{3}$, c=$- \\frac{28}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{841}{9}$, so √Δ = $\\frac{29}{3}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{7}{2}$.\n7) x2 = $- \\frac{4}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{7}{2}, - \\frac{4}{3}\\}$", "final_answer": "x \\in \\{\\frac{7}{2}, - \\frac{4}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "276707b88c308780", "problem_id": "7a8f5e4e0af43983", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{13 x}{3} - \\frac{28}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$- \\frac{13}{3}$, c=$- \\frac{28}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{7}{2}, - \\frac{4}{3}\\}$", "final_answer": "x \\in \\{\\frac{7}{2}, - \\frac{4}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c65f608499775fb5", "problem_id": "1457c0ab9483595a", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A disease has prevalence $\\frac{3}{200}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{219}{5000}$.\n3) Compute P(D|+) = $\\frac{95}{292}$.\nFinal answer: $\\frac{95}{292}$", "final_answer": "\\frac{95}{292}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "890f71e05928cf78", "problem_id": "1457c0ab9483595a", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A disease has prevalence $\\frac{3}{200}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{219}{5000}$.\nCompute P(D|+) = $\\frac{95}{292}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{95}{292}$", "final_answer": "\\frac{95}{292}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c0ddd8efa02e4600", "problem_id": "4130f56657689103", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(2 x^{2} - 3 x - 3\\right) \\cos{\\left(3 x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x e^{x}) = x e^{x}$.\n3) So $f'(x) = - 6 x^{2} \\sin{\\left(3 x + 3 \\right)} + x e^{x} + 9 x \\sin{\\left(3 x + 3 \\right)} + 4 x \\cos{\\left(3 x + 3 \\right)} - 3 e^{x} + 9 \\sin{\\left(3 x + 3 \\right)} - 3 \\cos{\\left(3 x + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x e^{x}) = x e^{x} + e^{x}$.\nTherefore the correct derivative is $f'(x) = \\left(x - 3\\right) e^{x} + \\left(4 x - 3\\right) \\cos{\\left(3 x + 3 \\right)} - 3 \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x + 3 \\right)} + e^{x}$.\nFinal answer: $\\left(x - 3\\right) e^{x} + \\left(4 x - 3\\right) \\cos{\\left(3 x + 3 \\right)} - 3 \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x + 3 \\right)} + e^{x}$", "final_answer": "\\left(x - 3\\right) e^{x} + \\left(4 x - 3\\right) \\cos{\\left(3 x + 3 \\right)} - 3 \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x + 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "60a71c0c04c156e6", "problem_id": "1c6b20ea6bc49805", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 16 red balls and 30 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(46,12).\n3) Favorable samples: choose k reds and n−k blues: C(16,5)·C(30,7).\n4) Compute totals: C(46,12) = 38910617655, and favorable = 8892374400.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{6514560}{28505947}$", "final_answer": "\\frac{6514560}{28505947}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "009241a4492868f3", "problem_id": "ebbc207139e4f84e", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 19 red balls and 17 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(36,8).\n3) Favorable samples: choose k reds and n−k blues: C(19,1)·C(17,7).\n4) Compute totals: C(36,8) = 30260340, and favorable = 369512.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{494}{40455}$", "final_answer": "\\frac{494}{40455}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "857c50117f5d4f25", "problem_id": "ebbc207139e4f84e", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 19 red balls and 17 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(36,8).\nFavorable samples: choose k reds and n−k blues: C(19,1)·C(17,7).\nCompute totals: C(36,8) = 30260340, and favorable = 369512.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{494}{40455}$.\nFinal answer: $\\frac{494}{40455}$", "final_answer": "\\frac{494}{40455}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e7f4a19999fb85ef", "problem_id": "9bc7ef2d1792229b", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the multiplicative inverse of 14 modulo 29.", "output": "1) We want x such that 14x ≡ 1 (mod 29).\n2) Use extended Euclid to find s,t with 14s + 29t = 1.\n3) One Bezout identity is: 14·(-2) + 29·(1) = 1.\n4) Reduce s modulo 29: x ≡ 27 (mod 29).\nFinal answer: $x\\equiv 27\\pmod{29}$", "final_answer": "x\\equiv 27\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "97992bd92ffd9055", "problem_id": "9bc7ef2d1792229b", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the multiplicative inverse of 14 modulo 29.", "output": "We want x such that 14x ≡ 1 (mod 29).\nUse extended Euclid to find s,t with 14s + 29t = 1.\nOne Bezout identity is: 14·(-2) + 29·(1) = 1.\nReduce s modulo 29: x ≡ 27 (mod 29).\nCheck: 14·27 mod 29 = 1.\nFinal answer: $x\\equiv 27\\pmod{29}$", "final_answer": "x\\equiv 27\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "167d8bafe0e7414a", "problem_id": "f4dcc49fa60ea487", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A store sells a backpack for $510. It is discounted by 15%. You buy 8 backpacks. What is the total cost?", "output": "Discounted price = 510 × (1 − 15/100) = 510 × 85/100 = 867/2.\nSubtotal = 8 × 867/2 = 3468.\nFinal answer: $3468$", "final_answer": "3468", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a4bb362eade39f2", "problem_id": "f4dcc49fa60ea487", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A store sells a backpack for $510. It is discounted by 15%. You buy 8 backpacks. What is the total cost?", "output": "1) Discounted price = 510 × (1 − 15/100) = 510 × 85/100 = 867/2.\n2) Subtotal = 8 × 867/2 = 3468.\nFinal answer: $3468$", "final_answer": "3468", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8990efaa3c8b32ca", "problem_id": "de0620872ca109aa", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} - x^{3} + 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{3} + 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $\\frac{\\pi \\left(\\pi^{2} + 12\\right)}{4}$.\n4) Simplify the result.\nFinal answer: $\\frac{\\pi\\cdot \\left(\\pi^{2} + 12\\right)}{4}$", "final_answer": "\\frac{\\pi\\cdot \\left(\\pi^{2} + 12\\right)}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3bbbf10eaa4453b9", "problem_id": "de0620872ca109aa", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} - x^{3} + 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{3} + 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $\\frac{\\pi \\left(\\pi^{2} + 12\\right)}{4}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{\\pi\\cdot \\left(\\pi^{2} + 12\\right)}{4}$", "final_answer": "\\frac{\\pi\\cdot \\left(\\pi^{2} + 12\\right)}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9f47e8eace021ee5", "problem_id": "1f5c7817fd153bec", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{4 x}{3} - \\frac{2}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$- \\frac{4}{3}$, c=$- \\frac{2}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{64}{9}$, so √Δ = $\\frac{8}{3}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $1$.\nx2 = $- \\frac{1}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{1, - \\frac{1}{3}\\}$", "final_answer": "x \\in \\{1, - \\frac{1}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c75fa6696d246330", "problem_id": "1f5c7817fd153bec", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{4 x}{3} - \\frac{2}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$- \\frac{4}{3}$, c=$- \\frac{2}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{64}{9}$, so √Δ = $\\frac{8}{3}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $1$.\n7) x2 = $- \\frac{1}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{1, - \\frac{1}{3}\\}$", "final_answer": "x \\in \\{1, - \\frac{1}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d4204d76624af9cd", "problem_id": "ca8f0d005fb535b0", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the linear system:\n$8 x + 3 y$ = $75$\n$10 x - 4 y$ = $86$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $10$: $80 x + 30 y$ = $750$.\n3) Multiply the second equation by $8$: $80 x - 32 y$ = $688$.\n4) Subtract to eliminate x: $62 y$ = $62$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $9$.\nFinal answer: $(x,y)=\\left(9, 1\\right)$", "final_answer": "(x,y)=\\left(9, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "57c27893288eab34", "problem_id": "9324b99eb1983050", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(3 x^{2} - 2 x - 3\\right) \\cos{\\left(2 x^{2} + x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 3 \\cos{\\left(2 x^{2} + x \\right)}) = 3 \\sin{\\left(x \\left(2 x + 1\\right) \\right)}$.\n3) So $f'(x) = - 12 x^{3} \\sin{\\left(2 x^{2} + x \\right)} + 5 x^{2} \\sin{\\left(2 x^{2} + x \\right)} + x e^{x} + 2 x \\sin{\\left(2 x^{2} + x \\right)} + 6 x \\cos{\\left(2 x^{2} + x \\right)} + 3 e^{x} + 3 \\sin{\\left(2 x^{2} + x \\right)} - 2 \\cos{\\left(2 x^{2} + x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 3 \\cos{\\left(2 x^{2} + x \\right)}) = 3 \\left(4 x + 1\\right) \\sin{\\left(2 x^{2} + x \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 2\\right) e^{x} + 2 \\left(3 x - 1\\right) \\cos{\\left(2 x^{2} + x \\right)} - \\left(4 x + 1\\right) \\left(x \\left(3 x - 2\\right) - 3\\right) \\sin{\\left(2 x^{2} + x \\right)} + e^{x}$.\nFinal answer: $\\left(x + 2\\right) e^{x} + 2 \\left(3 x - 1\\right) \\cos{\\left(2 x^{2} + x \\right)} - \\left(4 x + 1\\right) \\left(x \\left(3 x - 2\\right) - 3\\right) \\sin{\\left(2 x^{2} + x \\right)} + e^{x}$", "final_answer": "\\left(x + 2\\right) e^{x} + 2 \\left(3 x - 1\\right) \\cos{\\left(2 x^{2} + x \\right)} - \\left(4 x + 1\\right) \\left(x \\left(3 x - 2\\right) - 3\\right) \\sin{\\left(2 x^{2} + x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7110e72e4aede21b", "problem_id": "7fe2a74d1b59da3a", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{2} x - \\frac{x}{3} - \\frac{2}{3} + \\sqrt{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\sqrt{2} - \\frac{1}{3}$, c=$- \\frac{2}{3} + \\sqrt{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{43}{9} - \\frac{10 \\sqrt{2}}{3}$, so √Δ = $\\sqrt{\\frac{43}{9} - \\frac{10 \\sqrt{2}}{3}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{43 - 30 \\sqrt{2}}}{6} + \\frac{1}{6} + \\frac{\\sqrt{2}}{2}$.\n7) x2 = $- \\frac{\\sqrt{43 - 30 \\sqrt{2}}}{6} + \\frac{1}{6} + \\frac{\\sqrt{2}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{43 - 30\\cdot \\sqrt{2}}}{6} + \\frac{1}{6} + \\frac{\\sqrt{2}}{2}, - \\frac{\\sqrt{43 - 30\\cdot \\sqrt{2}}}{6} + \\frac{1}{6} + \\frac{\\sqrt{2}}{2}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{43 - 30\\cdot \\sqrt{2}}}{6} + \\frac{1}{6} + \\frac{\\sqrt{2}}{2}, - \\frac{\\sqrt{43 - 30\\cdot \\sqrt{2}}}{6} + \\frac{1}{6} + \\frac{\\sqrt{2}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f74abd01ab72cfdf", "problem_id": "7fe2a74d1b59da3a", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{2} x - \\frac{x}{3} - \\frac{2}{3} + \\sqrt{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\sqrt{2} - \\frac{1}{3}$, c=$- \\frac{2}{3} + \\sqrt{2}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{43}{9} - \\frac{10 \\sqrt{2}}{3}$, so √Δ = $\\sqrt{\\frac{43}{9} - \\frac{10 \\sqrt{2}}{3}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{43 - 30 \\sqrt{2}}}{6} + \\frac{1}{6} + \\frac{\\sqrt{2}}{2}$.\nx2 = $- \\frac{\\sqrt{43 - 30 \\sqrt{2}}}{6} + \\frac{1}{6} + \\frac{\\sqrt{2}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{43 - 30\\cdot \\sqrt{2}}}{6} + \\frac{1}{6} + \\frac{\\sqrt{2}}{2}, - \\frac{\\sqrt{43 - 30\\cdot \\sqrt{2}}}{6} + \\frac{1}{6} + \\frac{\\sqrt{2}}{2}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{43 - 30\\cdot \\sqrt{2}}}{6} + \\frac{1}{6} + \\frac{\\sqrt{2}}{2}, - \\frac{\\sqrt{43 - 30\\cdot \\sqrt{2}}}{6} + \\frac{1}{6} + \\frac{\\sqrt{2}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "157f9a54688fc638", "problem_id": "61107e90434b4098", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{3} + x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 3 x^{3} + x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3$.\n3) Compute an antiderivative: F(x) = $- \\frac{3 x^{4}}{4} + \\frac{x^{3}}{3} + x^{2} - 3 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{3\\cdot x^{4}}{4} + \\frac{x^{3}}{3} + x^{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{3\\cdot x^{4}}{4} + \\frac{x^{3}}{3} + x^{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b74b845bb22bdf41", "problem_id": "61107e90434b4098", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{3} + x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 3 x^{3} + x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3$.\nCompute an antiderivative: F(x) = $- \\frac{3 x^{4}}{4} + \\frac{x^{3}}{3} + x^{2} - 3 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - \\frac{3\\cdot x^{4}}{4} + \\frac{x^{3}}{3} + x^{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{3\\cdot x^{4}}{4} + \\frac{x^{3}}{3} + x^{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "94c2b00efe09d1cc", "problem_id": "e4bdbe5d6712457b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{3 x^{2} - x + \\left(- 3 x^{2} + 2 x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{- 3 x^{2} + 2 x + 5}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- 3 x^{2} + 2 x\\right) + 5}) = \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{- 3 x^{2} + 2 x + 5}$.\n3) So $f'(x) = \\frac{9 x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} + \\frac{18 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} - \\frac{12 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} - \\frac{24 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} - \\frac{26 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} - \\frac{52 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} + \\frac{6 x^{2}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} + \\frac{20 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} + \\frac{40 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} + \\frac{96 x}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} - \\frac{30 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{4} - 12 x^{3} - 26 x^{2} + 20 x + 25} + \\frac{25 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} + \\frac{50 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} - \\frac{22}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} + \\frac{10 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{4} - 12 x^{3} - 26 x^{2} + 20 x + 25} + \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{- 3 x^{2} + 2 x + 5} - \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{- 6 x^{2} + 4 x + 10} - \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(- 3 x^{2} + 2 x\\right) + 5}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- 3 x^{2} + 2 x\\right) + 5}) = \\frac{5 \\left(6 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(- 3 x^{2} + 2 x\\right) + 5\\right)^{2}} + \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(- 3 x^{2} + 2 x\\right) + 5\\right)} + \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(- 3 x^{2} + 2 x\\right) + 5}$.\nTherefore the correct derivative is $f'(x) = \\frac{4 \\left(3 x - 1\\right) \\left(3 x^{2} - x + \\left(x \\left(2 - 3 x\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(2 - 3 x\\right) + 5\\right) \\left(12 x + 4 \\left(1 - 3 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 - 3 x\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 - 3 x\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x \\left(2 - 3 x\\right) + 5\\right)^{2}}$.\nFinal answer: $\\frac{4 \\left(3 x - 1\\right) \\left(3 x^{2} - x + \\left(x \\left(2 - 3 x\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(2 - 3 x\\right) + 5\\right) \\left(12 x + 4 \\left(1 - 3 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 - 3 x\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 - 3 x\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x \\left(2 - 3 x\\right) + 5\\right)^{2}}$", "final_answer": "\\frac{4 \\left(3 x - 1\\right) \\left(3 x^{2} - x + \\left(x \\left(2 - 3 x\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(2 - 3 x\\right) + 5\\right) \\left(12 x + 4 \\left(1 - 3 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 - 3 x\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 - 3 x\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x \\left(2 - 3 x\\right) + 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6f1e83ece00284aa", "problem_id": "faf0d2dd830dc4fc", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡7 (mod 8), x≡10 (mod 11). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 87 (mod 88).\nFinal answer: $x\\equiv 87\\pmod{88}$", "final_answer": "x\\equiv 87\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "309b2c4aafae1195", "problem_id": "6a31e15816e79371", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "A store sells a backpack for $240. It is discounted by 15%. You buy 9 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 240 × (1 − 15/100) = 240 × 85/100 = 204.\nSubtotal = 9 × 204 = 1836.\nTotal with tax = 1836 × 105/100 = 9639/5.\nFinal answer: $\\frac{9639}{5}$", "final_answer": "\\frac{9639}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1f134bd73aaaecc0", "problem_id": "6a31e15816e79371", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A store sells a backpack for $240. It is discounted by 15%. You buy 9 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 240 × (1 − 15/100) = 240 × 85/100 = 204.\n2) Subtotal = 9 × 204 = 1836.\n3) Total with tax = 1836 × 105/100 = 9639/5.\nFinal answer: $\\frac{9639}{5}$", "final_answer": "\\frac{9639}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "34c6467d51be04fd", "problem_id": "b834c3003c0c1aab", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - x^{2} + 4 x - 9$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{3} + 2 x^{2} - x - 1$ and a = -2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 3 x^{2} + 4 x - 1$.\nEvaluate at a: p'(-2) = $-21$.\nCheck: the identity gives $-21$.\nFinal answer: $-21$", "final_answer": "-21", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6f958d4345d415d4", "problem_id": "fe88d3ab437c78a9", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} - x^{2} - 3 x - 3$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} - 4 x - 3.\n3) Then the limit is -8.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: -7.\nFinal answer: $-7$", "final_answer": "-7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "eee141d43c013285", "problem_id": "6236a20ff3755e82", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A cyclist rides at 85 miles per hour for 1 hours, then at 60 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 85 × 1 = 85 miles.\n2) Second leg distance = 60 × 3/2 = 90 miles.\n3) Total distance = 85 + 90 = 175 miles.\nFinal answer: $175$", "final_answer": "175", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f978e93079fc7c3d", "problem_id": "6236a20ff3755e82", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 85 miles per hour for 1 hours, then at 60 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 85 × 1 = 85 miles.\nSecond leg distance = 60 × 3/2 = 90 miles.\nTotal distance = 85 + 90 = 175 miles.\nCheck: adding both legs gives 175.\nFinal answer: $175$", "final_answer": "175", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "390de3ccd9250c57", "problem_id": "31509bb532fdf481", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$3 y$ = $-6$\n$4 x + y$ = $18$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $12 y$ = $-24$.\n3) Multiply the second equation by $0$: $0$ = $0$.\n4) Subtract to eliminate x: $12 y$ = $-24$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, -2\\right)$", "final_answer": "(x,y)=\\left(5, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "69d0ce8fc2b2dcd5", "problem_id": "31509bb532fdf481", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$3 y$ = $-6$\n$4 x + y$ = $18$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $12 y$ = $-24$.\nMultiply the second equation by $0$: $0$ = $0$.\nSubtract to eliminate x: $12 y$ = $-24$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $5$.\nCheck: plug (x,y)=(5,-2) into both equations.\nFinal answer: $(x,y)=\\left(5, -2\\right)$", "final_answer": "(x,y)=\\left(5, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ce6676d009607da9", "problem_id": "1d9a1723f3f36f0c", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Worker A finishes a job in 17 hours and Worker B finishes it in 16 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/17 job/hour. B's rate = 1/16 job/hour.\n2) Combined rate = 1/17 + 1/16 = 33/272 job/hour.\n3) Time = 1 ÷ (combined rate) = 272/33 hours.\nFinal answer: $\\frac{272}{33}$", "final_answer": "\\frac{272}{33}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4bfe8e37cc773e6d", "problem_id": "6ba99aa955fab2e0", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} + 4 x$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(4 x) = 4 x$.\n3) So $f'(x) = - 6 x^{2} + 4 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(4 x) = 4$.\nTherefore the correct derivative is $f'(x) = 2 \\left(2 - 3 x^{2}\\right)$.\nFinal answer: $2 \\left(2 - 3 x^{2}\\right)$", "final_answer": "2 \\left(2 - 3 x^{2}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "faccf687ad4737ba", "problem_id": "4515d91ae660cecc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - 2 x^{2} + 3 x - 8$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{2} + 2 x - 8.\n3) Then the limit is -20.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -22.\nFinal answer: $-22$", "final_answer": "-22", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d4523cfbfbf0ce88", "problem_id": "d591fa8df36f4bda", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 18 red balls and 27 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(45,12).\nFavorable samples: choose k reds and n−k blues: C(18,3)·C(27,9).\nCompute totals: C(45,12) = 28760021745, and favorable = 3824449200.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{60720}{456617}$.\nFinal answer: $\\frac{60720}{456617}$", "final_answer": "\\frac{60720}{456617}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f87469ff316d8cb6", "problem_id": "1bc42323064be4aa", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the multiplicative inverse of 39 modulo 43.", "output": "We want x such that 39x ≡ 1 (mod 43).\nUse extended Euclid to find s,t with 39s + 43t = 1.\nOne Bezout identity is: 39·(-11) + 43·(10) = 1.\nReduce s modulo 43: x ≡ 32 (mod 43).\nCheck: 39·32 mod 43 = 1.\nFinal answer: $x\\equiv 32\\pmod{43}$", "final_answer": "x\\equiv 32\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e03a98f037d84068", "problem_id": "1bc42323064be4aa", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Find the multiplicative inverse of 39 modulo 43.", "output": "1) We want x such that 39x ≡ 1 (mod 43).\n2) Use extended Euclid to find s,t with 39s + 43t = 1.\n3) One Bezout identity is: 39·(-11) + 43·(10) = 1.\n4) Reduce s modulo 43: x ≡ 32 (mod 43).\nFinal answer: $x\\equiv 32\\pmod{43}$", "final_answer": "x\\equiv 32\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fb7444024377e08e", "problem_id": "a55aff6d90e9d856", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many binary strings of length 22 contain exactly 8 ones with no two ones adjacent?", "output": "1) Reserve 7 mandatory zeros between the 8 ones.\n2) This leaves n−k+1 = 15 valid slots for the ones.\n3) Choose the slots: C(15,8) = 6435.\nFinal answer: $6435$", "final_answer": "6435", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b9e861675233c84", "problem_id": "a55aff6d90e9d856", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "How many binary strings of length 22 contain exactly 8 ones with no two ones adjacent?", "output": "Reserve 7 mandatory zeros between the 8 ones.\nThis leaves n−k+1 = 15 valid slots for the ones.\nChoose the slots: C(15,8) = 6435.\nFinal answer: $6435$", "final_answer": "6435", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "beaea092eb692946", "problem_id": "d41f7e73fd3602b8", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 14 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=14, k=4, p=$\\frac{1}{5}$ and simplify.\n4) Here C(14,4) = 1001.\nFinal answer: $\\frac{1049624576}{6103515625}$", "final_answer": "\\frac{1049624576}{6103515625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b17595e83665dd4a", "problem_id": "d41f7e73fd3602b8", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 14 times, what is the probability of exactly 4 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=14, k=4, p=$\\frac{1}{5}$ and simplify.\nFinal answer: $\\frac{1049624576}{6103515625}$", "final_answer": "\\frac{1049624576}{6103515625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f9b81b50fe0ba5f2", "problem_id": "b361e3a369a1ce8d", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} 2 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1$.\n2) Evaluate at the bounds: F(\\pi) and F(0).\n3) Subtract: F(\\pi) − F(0) = $- \\frac{\\pi^{2}}{2} + \\pi + 4 + \\frac{2 \\pi^{3}}{3}$.\n4) Simplify the result.\nFinal answer: $- \\frac{\\pi^{2}}{2} + \\pi + 4 + \\frac{2\\cdot \\pi^{3}}{3}$", "final_answer": "- \\frac{\\pi^{2}}{2} + \\pi + 4 + \\frac{2\\cdot \\pi^{3}}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e89939ebf635332c", "problem_id": "303abfbf507cb79d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{2} \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 3 x^{2} \\sin{\\left(x \\right)}) = - 3 x^{2} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 3 x^{2} \\cos{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 3 x^{2} \\sin{\\left(x \\right)}) = - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $- 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$", "final_answer": "- 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "9744a1a996b2208b", "problem_id": "d842d67a4870ba88", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "An urn has 11 red balls and 11 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(22,12).\nFavorable samples: choose k reds and n−k blues: C(11,1)·C(11,11).\nFinal answer: $\\frac{1}{58786}$", "final_answer": "\\frac{1}{58786}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f9711f891957dd31", "problem_id": "d842d67a4870ba88", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 11 red balls and 11 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(22,12).\nFavorable samples: choose k reds and n−k blues: C(11,1)·C(11,11).\nCompute totals: C(22,12) = 646646, and favorable = 11.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1}{58786}$.\nFinal answer: $\\frac{1}{58786}$", "final_answer": "\\frac{1}{58786}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d28b9f36f3df896f", "problem_id": "28e190d99fafb88a", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} - 2 x^{4} - x^{3} + 4 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{4} - x^{3} + 4 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{33 \\pi^{5}}{80} - 6 \\pi + 5 + \\frac{3 \\pi^{2}}{4} + \\frac{15 \\pi^{4}}{64} + \\frac{3 \\pi^{3}}{2}$.\n4) Simplify the result.\nFinal answer: $- \\frac{33\\cdot \\pi^{5}}{80} - 6\\cdot \\pi + 5 + \\frac{3\\cdot \\pi^{2}}{4} + \\frac{15\\cdot \\pi^{4}}{64} + \\frac{3\\cdot \\pi^{3}}{2}$", "final_answer": "- \\frac{33\\cdot \\pi^{5}}{80} - 6\\cdot \\pi + 5 + \\frac{3\\cdot \\pi^{2}}{4} + \\frac{15\\cdot \\pi^{4}}{64} + \\frac{3\\cdot \\pi^{3}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e1d5ca279a2cacf7", "problem_id": "28e190d99fafb88a", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} - 2 x^{4} - x^{3} + 4 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{4} - x^{3} + 4 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{33 \\pi^{5}}{80} - 6 \\pi + 5 + \\frac{3 \\pi^{2}}{4} + \\frac{15 \\pi^{4}}{64} + \\frac{3 \\pi^{3}}{2}$.\nFinal answer: $- \\frac{33\\cdot \\pi^{5}}{80} - 6\\cdot \\pi + 5 + \\frac{3\\cdot \\pi^{2}}{4} + \\frac{15\\cdot \\pi^{4}}{64} + \\frac{3\\cdot \\pi^{3}}{2}$", "final_answer": "- \\frac{33\\cdot \\pi^{5}}{80} - 6\\cdot \\pi + 5 + \\frac{3\\cdot \\pi^{2}}{4} + \\frac{15\\cdot \\pi^{4}}{64} + \\frac{3\\cdot \\pi^{3}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4b85bd9e1c6e05ac", "problem_id": "a11ce872093009a1", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 10 times, what is the probability of exactly 6 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=6, p=$\\frac{3}{7}$ and simplify.\nHere C(10,6) = 210.\nCheck: expression simplifies to $\\frac{5598720}{40353607}$.\nFinal answer: $\\frac{5598720}{40353607}$", "final_answer": "\\frac{5598720}{40353607}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0ca7739810828acc", "problem_id": "a11ce872093009a1", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 10 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=6, p=$\\frac{3}{7}$ and simplify.\n4) Here C(10,6) = 210.\nFinal answer: $\\frac{5598720}{40353607}$", "final_answer": "\\frac{5598720}{40353607}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8468a2bf3a3b336e", "problem_id": "b52e4baadd9d1202", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{3} \\sin{\\left(x \\right)} - x^{2} \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $- 3 x^{3} \\sin{\\left(x \\right)} - x^{2} \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$.\nFinal answer: $- 3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - x^{2}\\cdot \\cos{\\left(x \\right)} - 2\\cdot x\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)}$", "final_answer": "- 3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - x^{2}\\cdot \\cos{\\left(x \\right)} - 2\\cdot x\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4bea520d1da2cea0", "problem_id": "1408609af3a0ebfc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - x^{3} - x^{2} - 5 x - 17$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to - x^{3} - x^{2} - 6 x - 17.\n3) Then the limit is -121.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: -117.\nFinal answer: $-117$", "final_answer": "-117", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b7340d38fc947f7e", "problem_id": "d3226169838040b6", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 21 times, what is the probability of exactly 2 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=21, k=2, p=$\\frac{3}{8}$ and simplify.\nHere C(21,2) = 210.\nCheck: expression simplifies to $\\frac{18024444580078125}{4611686018427387904}$.\nFinal answer: $\\frac{18024444580078125}{4611686018427387904}$", "final_answer": "\\frac{18024444580078125}{4611686018427387904}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2606a3050a24bcf3", "problem_id": "d3226169838040b6", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 21 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=21, k=2, p=$\\frac{3}{8}$ and simplify.\n4) Here C(21,2) = 210.\nFinal answer: $\\frac{18024444580078125}{4611686018427387904}$", "final_answer": "\\frac{18024444580078125}{4611686018427387904}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5da6404fdf2d6ba8", "problem_id": "c89111db2eb520a5", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A store sells a backpack for $111. It is discounted by 10%. You buy 13 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 111 × (1 − 10/100) = 111 × 90/100 = 999/10.\n2) Subtotal = 13 × 999/10 = 12987/10.\n3) Total with tax = 12987/10 × 107/100 = 1389609/1000.\nFinal answer: $\\frac{1389609}{1000}$", "final_answer": "\\frac{1389609}{1000}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "74dad4a9b524e319", "problem_id": "c89111db2eb520a5", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A store sells a backpack for $111. It is discounted by 10%. You buy 13 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 111 × (1 − 10/100) = 111 × 90/100 = 999/10.\nSubtotal = 13 × 999/10 = 12987/10.\nTotal with tax = 12987/10 × 107/100 = 1389609/1000.\nCheck: recomputing the formula yields 1389609/1000.\nFinal answer: $\\frac{1389609}{1000}$", "final_answer": "\\frac{1389609}{1000}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8d26c380df589d1f", "problem_id": "3ce11da3240b3d7a", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- 8 x + 2 y$ = $44$\n$- 7 x - y$ = $44$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-7$: $56 x - 14 y$ = $-308$.\nMultiply the second equation by $-8$: $56 x + 8 y$ = $-352$.\nSubtract to eliminate x: $- 22 y$ = $44$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $-6$.\nCheck: plug (x,y)=(-6,-2) into both equations.\nFinal answer: $(x,y)=\\left(-6, -2\\right)$", "final_answer": "(x,y)=\\left(-6, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "364ddde2294766de", "problem_id": "3ce11da3240b3d7a", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$- 8 x + 2 y$ = $44$\n$- 7 x - y$ = $44$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-7$: $56 x - 14 y$ = $-308$.\n3) Multiply the second equation by $-8$: $56 x + 8 y$ = $-352$.\n4) Subtract to eliminate x: $- 22 y$ = $44$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $-6$.\nFinal answer: $(x,y)=\\left(-6, -2\\right)$", "final_answer": "(x,y)=\\left(-6, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6154559b766fe518", "problem_id": "1db2711cda43d96a", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡4 (mod 8), x≡4 (mod 9). Give the solution modulo 72.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 72.\n3) Compute the CRT solution to get x ≡ 4 (mod 72).\nFinal answer: $x\\equiv 4\\pmod{72}$", "final_answer": "x\\equiv 4\\pmod{72}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a6d3c7e67a39bedd", "problem_id": "1db2711cda43d96a", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡4 (mod 8), x≡4 (mod 9). Give the solution modulo 72.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 72.\nCompute the CRT solution to get x ≡ 4 (mod 72).\nCheck: 4 mod 8=4 and mod 9=4.\nFinal answer: $x\\equiv 4\\pmod{72}$", "final_answer": "x\\equiv 4\\pmod{72}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a6bd98705ed99a2e", "problem_id": "d9235ecf3be31b39", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Worker A finishes a job in 14 hours and Worker B finishes it in 16 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/14 job/hour. B's rate = 1/16 job/hour.\nCombined rate = 1/14 + 1/16 = 15/112 job/hour.\nTime = 1 ÷ (combined rate) = 112/15 hours.\nCheck: (combined rate)×(time) = 15/112×112/15 = 1.\nFinal answer: $\\frac{112}{15}$", "final_answer": "\\frac{112}{15}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2759a4214637c0e3", "problem_id": "d9235ecf3be31b39", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Worker A finishes a job in 14 hours and Worker B finishes it in 16 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/14 job/hour. B's rate = 1/16 job/hour.\n2) Combined rate = 1/14 + 1/16 = 15/112 job/hour.\n3) Time = 1 ÷ (combined rate) = 112/15 hours.\nFinal answer: $\\frac{112}{15}$", "final_answer": "\\frac{112}{15}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d02a80a551538823", "problem_id": "0bba58c043601b57", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the multiplicative inverse of 7 modulo 13.", "output": "1) We want x such that 7x ≡ 1 (mod 13).\n2) Use extended Euclid to find s,t with 7s + 13t = 1.\n3) One Bezout identity is: 7·(2) + 13·(-1) = 1.\n4) Reduce s modulo 13: x ≡ 2 (mod 13).\nFinal answer: $x\\equiv 2\\pmod{13}$", "final_answer": "x\\equiv 2\\pmod{13}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "71aa8f89eaa05b83", "problem_id": "0bba58c043601b57", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the multiplicative inverse of 7 modulo 13.", "output": "We want x such that 7x ≡ 1 (mod 13).\nUse extended Euclid to find s,t with 7s + 13t = 1.\nOne Bezout identity is: 7·(2) + 13·(-1) = 1.\nReduce s modulo 13: x ≡ 2 (mod 13).\nCheck: 7·2 mod 13 = 1.\nFinal answer: $x\\equiv 2\\pmod{13}$", "final_answer": "x\\equiv 2\\pmod{13}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b0b27c0d562ca01a", "problem_id": "530c48e3b0828728", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Find the multiplicative inverse of 33 modulo 37.", "output": "1) We want x such that 33x ≡ 1 (mod 37).\n2) Use extended Euclid to find s,t with 33s + 37t = 1.\n3) One Bezout identity is: 33·(9) + 37·(-8) = 1.\n4) Reduce s modulo 37: x ≡ 9 (mod 37).\nFinal answer: $x\\equiv 9\\pmod{37}$", "final_answer": "x\\equiv 9\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b42916884596f65", "problem_id": "20e6e34ac6bb1843", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x - 1$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 2 x - 1.\n3) Then the limit is -5.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -3.\nFinal answer: $-3$", "final_answer": "-3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7ead69fe05fc403e", "problem_id": "720a18381f9aa55c", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A cyclist rides at 75 miles per hour for 1 hours, then at 47 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 75 × 1 = 75 miles.\nSecond leg distance = 47 × 2 = 94 miles.\nTotal distance = 75 + 94 = 169 miles.\nCheck: adding both legs gives 169.\nFinal answer: $169$", "final_answer": "169", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d2022a61fc657e5", "problem_id": "720a18381f9aa55c", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A cyclist rides at 75 miles per hour for 1 hours, then at 47 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 75 × 1 = 75 miles.\n2) Second leg distance = 47 × 2 = 94 miles.\n3) Total distance = 75 + 94 = 169 miles.\nFinal answer: $169$", "final_answer": "169", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "42b29696e32bd0c1", "problem_id": "92735ccc356d1772", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{3} + 3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $2 x^{3} + 3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3$.\n3) Compute an antiderivative: F(x) = $\\frac{x^{4}}{2} + x^{3} - \\frac{3 x^{2}}{2} - 3 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{x^{4}}{2} + x^{3} - \\frac{3\\cdot x^{2}}{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{4}}{2} + x^{3} - \\frac{3\\cdot x^{2}}{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "884a80268763f41b", "problem_id": "92735ccc356d1772", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{3} + 3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $2 x^{3} + 3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3$.\nCompute an antiderivative: F(x) = $\\frac{x^{4}}{2} + x^{3} - \\frac{3 x^{2}}{2} - 3 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{x^{4}}{2} + x^{3} - \\frac{3\\cdot x^{2}}{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{4}}{2} + x^{3} - \\frac{3\\cdot x^{2}}{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "34a6b3df6161ded2", "problem_id": "ac952133d9c2b911", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$- x - y$ = $-10$\n$- 3 x$ = $-15$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $3 x + 3 y$ = $30$.\nMultiply the second equation by $-1$: $3 x$ = $15$.\nSubtract to eliminate x: $3 y$ = $15$.\nSolve for y: y = $5$.\nSubstitute back to find x: x = $5$.\nCheck: plug (x,y)=(5,5) into both equations.\nFinal answer: $(x,y)=\\left(5, 5\\right)$", "final_answer": "(x,y)=\\left(5, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e0cb38655c2a63a", "problem_id": "ac952133d9c2b911", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$- x - y$ = $-10$\n$- 3 x$ = $-15$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $3 x + 3 y$ = $30$.\n3) Multiply the second equation by $-1$: $3 x$ = $15$.\n4) Subtract to eliminate x: $3 y$ = $15$.\n5) Solve for y: y = $5$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, 5\\right)$", "final_answer": "(x,y)=\\left(5, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8deb981abd54b143", "problem_id": "89a6a595ae40609a", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} 2 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $2 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- 3 \\pi - 2 + \\frac{3 \\pi^{2}}{4} + \\frac{3 \\pi^{3}}{4}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- 3\\cdot \\pi - 2 + \\frac{3\\cdot \\pi^{2}}{4} + \\frac{3\\cdot \\pi^{3}}{4}$", "final_answer": "- 3\\cdot \\pi - 2 + \\frac{3\\cdot \\pi^{2}}{4} + \\frac{3\\cdot \\pi^{3}}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a8e9ae9d0f9e5c87", "problem_id": "89a6a595ae40609a", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} 2 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- 3 \\pi - 2 + \\frac{3 \\pi^{2}}{4} + \\frac{3 \\pi^{3}}{4}$.\n4) Simplify the result.\nFinal answer: $- 3\\cdot \\pi - 2 + \\frac{3\\cdot \\pi^{2}}{4} + \\frac{3\\cdot \\pi^{3}}{4}$", "final_answer": "- 3\\cdot \\pi - 2 + \\frac{3\\cdot \\pi^{2}}{4} + \\frac{3\\cdot \\pi^{3}}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "62a5aa3b657b7522", "problem_id": "954dc9b1336df2a1", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(2 x^{2} - 1\\right) \\cos{\\left(3 x^{2} - 3 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x e^{x}) = x e^{x}$.\n3) So $f'(x) = - 12 x^{3} \\sin{\\left(3 x^{2} - 3 x \\right)} + 6 x^{2} \\sin{\\left(3 x^{2} - 3 x \\right)} + x e^{x} + 6 x \\sin{\\left(3 x^{2} - 3 x \\right)} + 4 x \\cos{\\left(3 x^{2} - 3 x \\right)} + 2 e^{x} - 3 \\sin{\\left(3 x^{2} - 3 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x e^{x}) = x e^{x} + e^{x}$.\nTherefore the correct derivative is $f'(x) = 4 x \\cos{\\left(3 x^{2} - 3 x \\right)} + \\left(x + 2\\right) e^{x} - 3 \\left(2 x - 1\\right) \\left(2 x^{2} - 1\\right) \\sin{\\left(3 x^{2} - 3 x \\right)} + e^{x}$.\nFinal answer: $4 x \\cos{\\left(3 x^{2} - 3 x \\right)} + \\left(x + 2\\right) e^{x} - 3 \\left(2 x - 1\\right) \\left(2 x^{2} - 1\\right) \\sin{\\left(3 x^{2} - 3 x \\right)} + e^{x}$", "final_answer": "4 x \\cos{\\left(3 x^{2} - 3 x \\right)} + \\left(x + 2\\right) e^{x} - 3 \\left(2 x - 1\\right) \\left(2 x^{2} - 1\\right) \\sin{\\left(3 x^{2} - 3 x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "27daccfdbcdc9f5c", "problem_id": "dbb48751c0f9c485", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - 3 x^{3} + 7 x^{2} - 12 x + 22$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 3 x^{4} + x^{3} + 2 x^{2} - 2 x - 3$ and a = -2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 12 x^{3} + 3 x^{2} + 4 x - 2$.\n5) Evaluate at a: p'(-2) = $98$.\nFinal answer: $98$", "final_answer": "98", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b9076e5500b3f379", "problem_id": "dc42cd99772c0e8b", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(x^{2} - x + 2\\right) \\cos{\\left(3 x \\right)}$.", "output": "1) Let f(x) = $\\left(x - 3\\right) e^{x} + \\left(x^{2} - x + 2\\right) \\cos{\\left(3 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\left(x - 3\\right) e^{x} + \\left(2 x - 1\\right) \\cos{\\left(3 x \\right)} - 3 \\left(x^{2} - x + 2\\right) \\sin{\\left(3 x \\right)} + e^{x}$.\nFinal answer: $\\left(x - 3\\right)\\cdot e^{x} + \\left(2\\cdot x - 1\\right)\\cdot \\cos{\\left(3\\cdot x \\right)} - \\left(3\\cdot x^{2} - 3\\cdot x + 6\\right)\\cdot \\sin{\\left(3\\cdot x \\right)} + e^{x}$", "final_answer": "\\left(x - 3\\right)\\cdot e^{x} + \\left(2\\cdot x - 1\\right)\\cdot \\cos{\\left(3\\cdot x \\right)} - \\left(3\\cdot x^{2} - 3\\cdot x + 6\\right)\\cdot \\sin{\\left(3\\cdot x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bd6d0050348ac87f", "problem_id": "dc42cd99772c0e8b", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(x^{2} - x + 2\\right) \\cos{\\left(3 x \\right)}$.", "output": "Let f(x) = $\\left(x - 3\\right) e^{x} + \\left(x^{2} - x + 2\\right) \\cos{\\left(3 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\left(x - 3\\right) e^{x} + \\left(2 x - 1\\right) \\cos{\\left(3 x \\right)} - 3 \\left(x^{2} - x + 2\\right) \\sin{\\left(3 x \\right)} + e^{x}$.\nFinal answer: $\\left(x - 3\\right)\\cdot e^{x} + \\left(2\\cdot x - 1\\right)\\cdot \\cos{\\left(3\\cdot x \\right)} - \\left(3\\cdot x^{2} - 3\\cdot x + 6\\right)\\cdot \\sin{\\left(3\\cdot x \\right)} + e^{x}$", "final_answer": "\\left(x - 3\\right)\\cdot e^{x} + \\left(2\\cdot x - 1\\right)\\cdot \\cos{\\left(3\\cdot x \\right)} - \\left(3\\cdot x^{2} - 3\\cdot x + 6\\right)\\cdot \\sin{\\left(3\\cdot x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "86e561d64b75aac1", "problem_id": "62f6961244fee3f1", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $1026 in the ratio 2:4. How much does each receive?", "output": "Total ratio parts = 2+4 = 6.\nEach part = 1026 ÷ 6 = 171.\nFirst share = 2 × 171 = 342.\nSecond share = 4 × 171 = 684.\nCheck: 342+684=1026.\nFinal answer: $684$", "final_answer": "684", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "088f91190d45eb78", "problem_id": "62f6961244fee3f1", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Two friends split $1026 in the ratio 2:4. How much does each receive?", "output": "1) Total ratio parts = 2+4 = 6.\n2) Each part = 1026 ÷ 6 = 171.\n3) First share = 2 × 171 = 342.\n4) Second share = 4 × 171 = 684.\nFinal answer: $684$", "final_answer": "684", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "13911703538b61ea", "problem_id": "a3343dda6c927aaf", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} - x + \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{x + 6}$.", "output": "Let f(x) = $\\frac{- x^{2} - x + \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{x + 6}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{2 x^{2} + 2 x + \\left(x + 6\\right) \\left(- 4 x + \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 2 \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6}{2 \\left(x + 6\\right)^{2}}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\frac{2\\cdot x^{2} + 2\\cdot x + \\left(x + 6\\right)\\cdot \\left(- 4\\cdot x + \\left(x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x + 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right) - \\left(2\\cdot x + 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6}{2\\cdot \\left(x + 6\\right)^{2}}$", "final_answer": "\\frac{2\\cdot x^{2} + 2\\cdot x + \\left(x + 6\\right)\\cdot \\left(- 4\\cdot x + \\left(x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x + 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right) - \\left(2\\cdot x + 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6}{2\\cdot \\left(x + 6\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8ab2e196cfc3353a", "problem_id": "e87817263420430d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(x^{3} + 2 x^{2} + 2 x + 3\\right) \\cos{\\left(3 x - 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x^{2} \\cos{\\left(3 x - 2 \\right)}) = - 2 x^{2} \\sin{\\left(3 x - 2 \\right)}$.\n3) So $f'(x) = - 3 x^{3} \\sin{\\left(3 x - 2 \\right)} - 2 x^{2} \\sin{\\left(3 x - 2 \\right)} + 3 x^{2} \\cos{\\left(3 x - 2 \\right)} + x e^{x} - 6 x \\sin{\\left(3 x - 2 \\right)} + 2 e^{x} - 9 \\sin{\\left(3 x - 2 \\right)} + 2 \\cos{\\left(3 x - 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x^{2} \\cos{\\left(3 x - 2 \\right)}) = - 6 x^{2} \\sin{\\left(3 x - 2 \\right)} + 4 x \\cos{\\left(3 x - 2 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 1\\right) e^{x} - 3 \\left(x \\left(x \\left(x + 2\\right) + 2\\right) + 3\\right) \\sin{\\left(3 x - 2 \\right)} + \\left(3 x^{2} + 4 x + 2\\right) \\cos{\\left(3 x - 2 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 1\\right) e^{x} - 3 \\left(x \\left(x \\left(x + 2\\right) + 2\\right) + 3\\right) \\sin{\\left(3 x - 2 \\right)} + \\left(3 x^{2} + 4 x + 2\\right) \\cos{\\left(3 x - 2 \\right)} + e^{x}$", "final_answer": "\\left(x + 1\\right) e^{x} - 3 \\left(x \\left(x \\left(x + 2\\right) + 2\\right) + 3\\right) \\sin{\\left(3 x - 2 \\right)} + \\left(3 x^{2} + 4 x + 2\\right) \\cos{\\left(3 x - 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e85a11d04faa6ea4", "problem_id": "edb67aad81feb229", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡2 (mod 11), x≡11 (mod 13). Give the solution modulo 143.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 143.\n3) Compute the CRT solution to get x ≡ 24 (mod 143).\nFinal answer: $x\\equiv 24\\pmod{143}$", "final_answer": "x\\equiv 24\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "61c306c59d53e8a7", "problem_id": "edb67aad81feb229", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 11), x≡11 (mod 13). Give the solution modulo 143.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 143.\nCompute the CRT solution to get x ≡ 24 (mod 143).\nCheck: 24 mod 11=2 and mod 13=11.\nFinal answer: $x\\equiv 24\\pmod{143}$", "final_answer": "x\\equiv 24\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ef8e15b3169aef55", "problem_id": "64786dacc2afc18a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 21 times, what is the probability of exactly 13 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=21, k=13, p=$\\frac{1}{8}$ and simplify.\nHere C(21,13) = 203490.\nCheck: expression simplifies to $\\frac{586539677745}{4611686018427387904}$.\nFinal answer: $\\frac{586539677745}{4611686018427387904}$", "final_answer": "\\frac{586539677745}{4611686018427387904}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f01c4d7dcc99451b", "problem_id": "64786dacc2afc18a", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 21 times, what is the probability of exactly 13 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=21, k=13, p=$\\frac{1}{8}$ and simplify.\n4) Here C(21,13) = 203490.\nFinal answer: $\\frac{586539677745}{4611686018427387904}$", "final_answer": "\\frac{586539677745}{4611686018427387904}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "89599898aa9454c5", "problem_id": "0daf9f6d3352a1f5", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- 2 x + 10 y$ = $-22$\n$- 11 x + 7 y$ = $71$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-11$: $22 x - 110 y$ = $242$.\nMultiply the second equation by $-2$: $22 x - 14 y$ = $-142$.\nSubtract to eliminate x: $- 96 y$ = $384$.\nSolve for y: y = $-4$.\nSubstitute back to find x: x = $-9$.\nCheck: plug (x,y)=(-9,-4) into both equations.\nFinal answer: $(x,y)=\\left(-9, -4\\right)$", "final_answer": "(x,y)=\\left(-9, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "550a1c716ecde647", "problem_id": "0daf9f6d3352a1f5", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the linear system:\n$- 2 x + 10 y$ = $-22$\n$- 11 x + 7 y$ = $71$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-11$: $22 x - 110 y$ = $242$.\nMultiply the second equation by $-2$: $22 x - 14 y$ = $-142$.\nFinal answer: $(x,y)=\\left(-9, -4\\right)$", "final_answer": "(x,y)=\\left(-9, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3b864ba0ba74beb3", "problem_id": "29a057d297cd0e27", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Two friends split $320 in the ratio 1:7. How much does each receive?", "output": "Total ratio parts = 1+7 = 8.\nEach part = 320 ÷ 8 = 40.\nFirst share = 1 × 40 = 40.\nSecond share = 7 × 40 = 280.\nCheck: 40+280=320.\nFinal answer: $280$", "final_answer": "280", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a77fe071740a7312", "problem_id": "29a057d297cd0e27", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Two friends split $320 in the ratio 1:7. How much does each receive?", "output": "Total ratio parts = 1+7 = 8.\nEach part = 320 ÷ 8 = 40.\nFirst share = 1 × 40 = 40.\nFinal answer: $280$", "final_answer": "280", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "405321e6c02a2a66", "problem_id": "0a9034d6bc05fb6a", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{2} + 3 x + \\left(3 x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{3 x^{2} - x + 5}$.", "output": "Let f(x) = $\\frac{- 3 x^{2} + 3 x + \\left(3 x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{3 x^{2} - x + 5}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{2 \\left(1 - 6 x\\right) \\left(- 3 x^{2} + 3 x + \\left(3 x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(3 x^{2} - x + 5\\right) \\left(- 12 x + 2 \\left(6 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(3 x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6\\right)}{2 \\left(3 x^{2} - x + 5\\right)^{2}}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\frac{\\left(2 - 12\\cdot x\\right)\\cdot \\left(- 3\\cdot x^{2} + 3\\cdot x + \\left(3\\cdot x^{2} - x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 1\\right) + \\left(3\\cdot x^{2} - x + 5\\right)\\cdot \\left(- 12\\cdot x + \\left(12\\cdot x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(3\\cdot x^{2} - x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x^{2} - 2\\cdot x + 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\right)}{2\\cdot \\left(3\\cdot x^{2} - x + 5\\right)^{2}}$", "final_answer": "\\frac{\\left(2 - 12\\cdot x\\right)\\cdot \\left(- 3\\cdot x^{2} + 3\\cdot x + \\left(3\\cdot x^{2} - x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 1\\right) + \\left(3\\cdot x^{2} - x + 5\\right)\\cdot \\left(- 12\\cdot x + \\left(12\\cdot x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(3\\cdot x^{2} - x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x^{2} - 2\\cdot x + 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\right)}{2\\cdot \\left(3\\cdot x^{2} - x + 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f36951d2ff203209", "problem_id": "0a9034d6bc05fb6a", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{2} + 3 x + \\left(3 x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{3 x^{2} - x + 5}$.", "output": "1) Let f(x) = $\\frac{- 3 x^{2} + 3 x + \\left(3 x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{3 x^{2} - x + 5}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{2 \\left(1 - 6 x\\right) \\left(- 3 x^{2} + 3 x + \\left(3 x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(3 x^{2} - x + 5\\right) \\left(- 12 x + 2 \\left(6 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(3 x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6\\right)}{2 \\left(3 x^{2} - x + 5\\right)^{2}}$.\nFinal answer: $\\frac{\\left(2 - 12\\cdot x\\right)\\cdot \\left(- 3\\cdot x^{2} + 3\\cdot x + \\left(3\\cdot x^{2} - x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 1\\right) + \\left(3\\cdot x^{2} - x + 5\\right)\\cdot \\left(- 12\\cdot x + \\left(12\\cdot x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(3\\cdot x^{2} - x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x^{2} - 2\\cdot x + 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\right)}{2\\cdot \\left(3\\cdot x^{2} - x + 5\\right)^{2}}$", "final_answer": "\\frac{\\left(2 - 12\\cdot x\\right)\\cdot \\left(- 3\\cdot x^{2} + 3\\cdot x + \\left(3\\cdot x^{2} - x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 1\\right) + \\left(3\\cdot x^{2} - x + 5\\right)\\cdot \\left(- 12\\cdot x + \\left(12\\cdot x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(3\\cdot x^{2} - x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x^{2} - 2\\cdot x + 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\right)}{2\\cdot \\left(3\\cdot x^{2} - x + 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "90ff2c57c47b9a7e", "problem_id": "219c7112060e4333", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{13} x}{2} + x - \\frac{35}{4} - \\frac{7 \\sqrt{13}}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$1 - \\frac{\\sqrt{13}}{2}$, c=$- \\frac{35}{4} - \\frac{7 \\sqrt{13}}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $6 \\sqrt{13} + \\frac{157}{4}$, so √Δ = $\\sqrt{6 \\sqrt{13} + \\frac{157}{4}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{1}{2} + \\frac{\\sqrt{13}}{4} + \\frac{\\sqrt{24 \\sqrt{13} + 157}}{4}$.\n7) x2 = $- \\frac{\\sqrt{24 \\sqrt{13} + 157}}{4} - \\frac{1}{2} + \\frac{\\sqrt{13}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{1}{2} + \\frac{\\sqrt{13}}{4} + \\frac{\\sqrt{24\\cdot \\sqrt{13} + 157}}{4}, - \\frac{\\sqrt{24\\cdot \\sqrt{13} + 157}}{4} - \\frac{1}{2} + \\frac{\\sqrt{13}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{1}{2} + \\frac{\\sqrt{13}}{4} + \\frac{\\sqrt{24\\cdot \\sqrt{13} + 157}}{4}, - \\frac{\\sqrt{24\\cdot \\sqrt{13} + 157}}{4} - \\frac{1}{2} + \\frac{\\sqrt{13}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "77b2f8a571b20793", "problem_id": "219c7112060e4333", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{13} x}{2} + x - \\frac{35}{4} - \\frac{7 \\sqrt{13}}{4}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$1 - \\frac{\\sqrt{13}}{2}$, c=$- \\frac{35}{4} - \\frac{7 \\sqrt{13}}{4}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $6 \\sqrt{13} + \\frac{157}{4}$, so √Δ = $\\sqrt{6 \\sqrt{13} + \\frac{157}{4}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{1}{2} + \\frac{\\sqrt{13}}{4} + \\frac{\\sqrt{24 \\sqrt{13} + 157}}{4}$.\nx2 = $- \\frac{\\sqrt{24 \\sqrt{13} + 157}}{4} - \\frac{1}{2} + \\frac{\\sqrt{13}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{1}{2} + \\frac{\\sqrt{13}}{4} + \\frac{\\sqrt{24\\cdot \\sqrt{13} + 157}}{4}, - \\frac{\\sqrt{24\\cdot \\sqrt{13} + 157}}{4} - \\frac{1}{2} + \\frac{\\sqrt{13}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{1}{2} + \\frac{\\sqrt{13}}{4} + \\frac{\\sqrt{24\\cdot \\sqrt{13} + 157}}{4}, - \\frac{\\sqrt{24\\cdot \\sqrt{13} + 157}}{4} - \\frac{1}{2} + \\frac{\\sqrt{13}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "44cce4d42146a48c", "problem_id": "084b08668e2649fe", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 24 red balls and 11 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(35,7).\nFavorable samples: choose k reds and n−k blues: C(24,1)·C(11,6).\nCompute totals: C(35,7) = 6724520, and favorable = 11088.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{126}{76415}$.\nFinal answer: $\\frac{126}{76415}$", "final_answer": "\\frac{126}{76415}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7c755fa453856b14", "problem_id": "084b08668e2649fe", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 24 red balls and 11 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(35,7).\n3) Favorable samples: choose k reds and n−k blues: C(24,1)·C(11,6).\n4) Compute totals: C(35,7) = 6724520, and favorable = 11088.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{126}{76415}$", "final_answer": "\\frac{126}{76415}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a022165d90d99249", "problem_id": "1c79bedd36efa53e", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 3 x^{3} - 3 x^{2} + 2 x - 5$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{4} - x^{2} - 3 x$ and a = -1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $12 x^{3} - 2 x - 3$.\n5) Evaluate at a: p'(-1) = $-13$.\nFinal answer: $-13$", "final_answer": "-13", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a59ac8d6dcbf8414", "problem_id": "2d7e3350e7114741", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{2} + 3 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{x - 4}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}) = - \\frac{4 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4}$.\n3) So $f'(x) = \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{6 x^{2}}{2 x^{2} - 16 x + 32} - \\frac{8 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{16 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{48 x}{2 x^{2} - 16 x + 32} + \\frac{16 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{32 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{20}{2 x^{2} - 16 x + 32} - \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 8 x + 16} - \\frac{4 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} + \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4} + \\frac{4 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}) = - \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4} - \\frac{4 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} + \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x - 4\\right)^{2}}$.\nTherefore the correct derivative is $f'(x) = \\frac{6 x^{2} - 6 x + \\left(x - 4\\right) \\left(- 12 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4}{2 \\left(x - 4\\right)^{2}}$.\nFinal answer: $\\frac{6 x^{2} - 6 x + \\left(x - 4\\right) \\left(- 12 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4}{2 \\left(x - 4\\right)^{2}}$", "final_answer": "\\frac{6 x^{2} - 6 x + \\left(x - 4\\right) \\left(- 12 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4}{2 \\left(x - 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "eba3be56f56b35c0", "problem_id": "cda17199fd64e870", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$- 5 x + 7 y$ = $-84$\n$6 x - 5 y$ = $77$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $6$: $- 30 x + 42 y$ = $-504$.\n3) Multiply the second equation by $-5$: $- 30 x + 25 y$ = $-385$.\n4) Subtract to eliminate x: $17 y$ = $-119$.\n5) Solve for y: y = $-7$.\n6) Substitute back to find x: x = $7$.\nFinal answer: $(x,y)=\\left(7, -7\\right)$", "final_answer": "(x,y)=\\left(7, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f86810d3be7d6815", "problem_id": "d8cc166d19e77d96", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x^{3} - 5 x^{2} - 8 x - 14$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{3} - 5 x^{2} - 9 x - 14.\n3) Then the limit is -68.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -66.\nFinal answer: $-66$", "final_answer": "-66", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "1c2d5d2b75dc4a9f", "problem_id": "5248b7aa2c2083c7", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Worker A finishes a job in 10 hours and Worker B finishes it in 6 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/10 job/hour. B's rate = 1/6 job/hour.\nCombined rate = 1/10 + 1/6 = 4/15 job/hour.\nTime = 1 ÷ (combined rate) = 15/4 hours.\nCheck: (combined rate)×(time) = 4/15×15/4 = 1.\nFinal answer: $\\frac{15}{4}$", "final_answer": "\\frac{15}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bc15413964209dd7", "problem_id": "5248b7aa2c2083c7", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Worker A finishes a job in 10 hours and Worker B finishes it in 6 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/10 job/hour. B's rate = 1/6 job/hour.\n2) Combined rate = 1/10 + 1/6 = 4/15 job/hour.\n3) Time = 1 ÷ (combined rate) = 15/4 hours.\nFinal answer: $\\frac{15}{4}$", "final_answer": "\\frac{15}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f8aa26bcad068c66", "problem_id": "62775cf5a7fc7196", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Worker A finishes a job in 5 hours and Worker B finishes it in 20 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/5 job/hour. B's rate = 1/20 job/hour.\n2) Combined rate = 1/5 + 1/20 = 1/4 job/hour.\n3) Time = 1 ÷ (combined rate) = 4 hours.\nFinal answer: $4$", "final_answer": "4", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "56144348491a37a8", "problem_id": "62775cf5a7fc7196", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Worker A finishes a job in 5 hours and Worker B finishes it in 20 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/5 job/hour. B's rate = 1/20 job/hour.\nCombined rate = 1/5 + 1/20 = 1/4 job/hour.\nTime = 1 ÷ (combined rate) = 4 hours.\nCheck: (combined rate)×(time) = 1/4×4 = 1.\nFinal answer: $4$", "final_answer": "4", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1f0949003daa4e05", "problem_id": "aae4c8c822c13cb5", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "You want 60 liters of a 25% salt solution. You have a 10% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 60−x liters is 60% solution.\nSalt equation: 10x + 60(60−x) = 25·60.\n(10-60)x = (25-60)·60 ⇒ x = 42.\nThen 60−x = 18.\nCheck: 10·42 + 60·18 = 25·60.\nFinal answer: $42 L of 10% and 18 L of 60%$", "final_answer": "42 L of 10% and 18 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7bbeb500971865ff", "problem_id": "aae4c8c822c13cb5", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "You want 60 liters of a 25% salt solution. You have a 10% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 60−x liters is 60% solution.\n2) Salt equation: 10x + 60(60−x) = 25·60.\n3) (10-60)x = (25-60)·60 ⇒ x = 42.\n4) Then 60−x = 18.\nFinal answer: $42 L of 10% and 18 L of 60%$", "final_answer": "42 L of 10% and 18 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b32ef9517960abd6", "problem_id": "28f616efab7ddfc2", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$- 2 x - y$ = $-8$\n$4 x - y$ = $16$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $- 8 x - 4 y$ = $-32$.\n3) Multiply the second equation by $-2$: $- 8 x + 2 y$ = $-32$.\n4) Subtract to eliminate x: $- 6 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, 0\\right)$", "final_answer": "(x,y)=\\left(4, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bb051bc1f559c4f6", "problem_id": "28f616efab7ddfc2", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- 2 x - y$ = $-8$\n$4 x - y$ = $16$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $- 8 x - 4 y$ = $-32$.\nMultiply the second equation by $-2$: $- 8 x + 2 y$ = $-32$.\nSubtract to eliminate x: $- 6 y$ = $0$.\nSolve for y: y = $0$.\nSubstitute back to find x: x = $4$.\nCheck: plug (x,y)=(4,0) into both equations.\nFinal answer: $(x,y)=\\left(4, 0\\right)$", "final_answer": "(x,y)=\\left(4, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8215e18d38197dfe", "problem_id": "e154a9ca592d13d7", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "You want 52 liters of a 30% salt solution. You have a 20% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 52−x liters is 40% solution.\n2) Salt equation: 20x + 40(52−x) = 30·52.\n3) (20-40)x = (30-40)·52 ⇒ x = 26.\n4) Then 52−x = 26.\nFinal answer: $26 L of 20% and 26 L of 40%$", "final_answer": "26 L of 20% and 26 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "12e8d070741c8d1d", "problem_id": "7dd28420924931e1", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{3} + \\frac{2 x^{2}}{3} + \\frac{x}{3} + \\left(x^{2} + x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - \\frac{1}{3}}{x^{2} + x + 1}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + x\\right) + 1}) = \\frac{e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + x + 1}$.\n3) So $f'(x) = \\frac{3 x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{6 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{6 x^{4}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{6 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{12 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{12 x^{3}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{9 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{18 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{20 x^{2}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{12 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{12 x}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 1} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{4}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 1} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 2 x + 2} + \\frac{e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + x + 1} - \\frac{e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} + x\\right) + 1}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + x\\right) + 1}) = \\frac{\\left(- 2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(x^{2} + x\\right) + 1\\right)^{2}} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(x^{2} + x\\right) + 1\\right)} + \\frac{e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} + x\\right) + 1}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(- 2 x - 1\\right) \\left(3 x^{3} + 2 x^{2} + x + 3 \\left(x \\left(x + 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) + \\left(x \\left(x + 1\\right) + 1\\right) \\left(18 x^{2} + 8 x + 6 \\left(2 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3 \\left(x \\left(x + 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x \\left(x + 1\\right) + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{6 \\left(x \\left(x + 1\\right) + 1\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(- 2 x - 1\\right) \\left(3 x^{3} + 2 x^{2} + x + 3 \\left(x \\left(x + 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) + \\left(x \\left(x + 1\\right) + 1\\right) \\left(18 x^{2} + 8 x + 6 \\left(2 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3 \\left(x \\left(x + 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x \\left(x + 1\\right) + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{6 \\left(x \\left(x + 1\\right) + 1\\right)^{2}}$", "final_answer": "\\frac{2 \\left(- 2 x - 1\\right) \\left(3 x^{3} + 2 x^{2} + x + 3 \\left(x \\left(x + 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) + \\left(x \\left(x + 1\\right) + 1\\right) \\left(18 x^{2} + 8 x + 6 \\left(2 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3 \\left(x \\left(x + 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x \\left(x + 1\\right) + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{6 \\left(x \\left(x + 1\\right) + 1\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2259b2a4fcaa56cb", "problem_id": "0691f597e9cfc083", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} + x^{2} + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=1$.", "output": "Differentiate: f'(x) = $- 6 x^{2} + 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)}$.\nSlope at x=1: m = f'(1) = $-4 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}$.\nPoint on curve: (1, f(1)) = (1, $\\cos{\\left(2 \\right)} + e^{\\frac{1}{2}}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} = -4 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}(x − 1).\nOptionally expand: y = $- 4 x - 2 x \\sin{\\left(2 \\right)} + \\frac{x e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2 \\sin{\\left(2 \\right)} + 4$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 4\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 4$", "final_answer": "y = - 4\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b59ff2099f13ac39", "problem_id": "0691f597e9cfc083", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} + x^{2} + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=1$.", "output": "Differentiate: f'(x) = $- 6 x^{2} + 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)}$.\nSlope at x=1: m = f'(1) = $-4 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}$.\nPoint on curve: (1, f(1)) = (1, $\\cos{\\left(2 \\right)} + e^{\\frac{1}{2}}$).\nFinal answer: $y = - 4\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 4$", "final_answer": "y = - 4\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "979f5c89d6ae193c", "problem_id": "b002d1a672dc3d96", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A cyclist rides at 46 miles per hour for 1/2 hours, then at 22 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 46 × 1/2 = 23 miles.\nSecond leg distance = 22 × 2 = 44 miles.\nTotal distance = 23 + 44 = 67 miles.\nCheck: adding both legs gives 67.\nFinal answer: $67$", "final_answer": "67", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b9b4ccbd037d83b8", "problem_id": "b002d1a672dc3d96", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A cyclist rides at 46 miles per hour for 1/2 hours, then at 22 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 46 × 1/2 = 23 miles.\n2) Second leg distance = 22 × 2 = 44 miles.\n3) Total distance = 23 + 44 = 67 miles.\nFinal answer: $67$", "final_answer": "67", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8e61937232b6ddc8", "problem_id": "4d138697ac87227d", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the multiplicative inverse of 16 modulo 23.", "output": "1) We want x such that 16x ≡ 1 (mod 23).\n2) Use extended Euclid to find s,t with 16s + 23t = 1.\n3) One Bezout identity is: 16·(-10) + 23·(7) = 1.\n4) Reduce s modulo 23: x ≡ 13 (mod 23).\nFinal answer: $x\\equiv 13\\pmod{23}$", "final_answer": "x\\equiv 13\\pmod{23}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b73a6acbe519c669", "problem_id": "93c560e9cdb2555e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{3} + 4 x^{2} + 10 x + 17$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} + 4 x^{2} + 9 x + 17.\n3) Then the limit is 67.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 69.\nFinal answer: $69$", "final_answer": "69", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c03e8eff87a810f2", "problem_id": "db689e9a2f0dcaf4", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A cyclist rides at 83 miles per hour for 1 hours, then at 84 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 83 × 1 = 83 miles.\n2) Second leg distance = 84 × 3/2 = 126 miles.\n3) Total distance = 83 + 126 = 209 miles.\nFinal answer: $209$", "final_answer": "209", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f4db02893094a853", "problem_id": "db689e9a2f0dcaf4", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A cyclist rides at 83 miles per hour for 1 hours, then at 84 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 83 × 1 = 83 miles.\nSecond leg distance = 84 × 3/2 = 126 miles.\nTotal distance = 83 + 126 = 209 miles.\nCheck: adding both legs gives 209.\nFinal answer: $209$", "final_answer": "209", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9e186c414434c9fd", "problem_id": "ba579723ea38e816", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} x^{2} + 4 x + 14$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{3} + x^{2} + 2 x$ and a = 3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $3 x^{2} + 2 x + 2$.\n5) Evaluate at a: p'(3) = $35$.\nFinal answer: $35$", "final_answer": "35", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "29ae871fbc6c52d4", "problem_id": "ba579723ea38e816", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} x^{2} + 4 x + 14$.", "output": "$35$", "final_answer": "35", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b47e6831dbb208ed", "problem_id": "ca025ae87b81be7a", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡2 (mod 8), x≡0 (mod 11). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 66 (mod 88).\nFinal answer: $x\\equiv 66\\pmod{88}$", "final_answer": "x\\equiv 66\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d8ff76f2305061ab", "problem_id": "ca025ae87b81be7a", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the congruences: x≡2 (mod 8), x≡0 (mod 11). Give the solution modulo 88.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 88.\nCompute the CRT solution to get x ≡ 66 (mod 88).\nFinal answer: $x\\equiv 66\\pmod{88}$", "final_answer": "x\\equiv 66\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b08131d6d162de7e", "problem_id": "d19d4a7916b0cfa9", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "How many binary strings of length 13 contain exactly 4 ones with no two ones adjacent?", "output": "1) Reserve 3 mandatory zeros between the 4 ones.\n2) This leaves n−k+1 = 10 valid slots for the ones.\n3) Choose the slots: C(10,4) = 210.\nFinal answer: $210$", "final_answer": "210", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1d6fd83634f7b65b", "problem_id": "d19d4a7916b0cfa9", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many binary strings of length 13 contain exactly 4 ones with no two ones adjacent?", "output": "Reserve 3 mandatory zeros between the 4 ones.\nThis leaves n−k+1 = 10 valid slots for the ones.\nChoose the slots: C(10,4) = 210.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $210$", "final_answer": "210", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a98f85cbf4990408", "problem_id": "a87ff4cdf9c43631", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{13} x}{2} - x - 2 + \\sqrt{13}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{\\sqrt{13}}{2} - 1$, c=$-2 + \\sqrt{13}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{49}{4} - 3 \\sqrt{13}$, so √Δ = $\\sqrt{\\frac{49}{4} - 3 \\sqrt{13}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{1}{2} + \\frac{\\sqrt{49 - 12 \\sqrt{13}}}{4} + \\frac{\\sqrt{13}}{4}$.\nx2 = $- \\frac{\\sqrt{49 - 12 \\sqrt{13}}}{4} + \\frac{1}{2} + \\frac{\\sqrt{13}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{1}{2} + \\frac{\\sqrt{49 - 12\\cdot \\sqrt{13}}}{4} + \\frac{\\sqrt{13}}{4}, - \\frac{\\sqrt{49 - 12\\cdot \\sqrt{13}}}{4} + \\frac{1}{2} + \\frac{\\sqrt{13}}{4}\\}$", "final_answer": "x \\in \\{\\frac{1}{2} + \\frac{\\sqrt{49 - 12\\cdot \\sqrt{13}}}{4} + \\frac{\\sqrt{13}}{4}, - \\frac{\\sqrt{49 - 12\\cdot \\sqrt{13}}}{4} + \\frac{1}{2} + \\frac{\\sqrt{13}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a4941d44aa145731", "problem_id": "a87ff4cdf9c43631", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{13} x}{2} - x - 2 + \\sqrt{13}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{\\sqrt{13}}{2} - 1$, c=$-2 + \\sqrt{13}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{1}{2} + \\frac{\\sqrt{49 - 12\\cdot \\sqrt{13}}}{4} + \\frac{\\sqrt{13}}{4}, - \\frac{\\sqrt{49 - 12\\cdot \\sqrt{13}}}{4} + \\frac{1}{2} + \\frac{\\sqrt{13}}{4}\\}$", "final_answer": "x \\in \\{\\frac{1}{2} + \\frac{\\sqrt{49 - 12\\cdot \\sqrt{13}}}{4} + \\frac{\\sqrt{13}}{4}, - \\frac{\\sqrt{49 - 12\\cdot \\sqrt{13}}}{4} + \\frac{1}{2} + \\frac{\\sqrt{13}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "11edf41931558d5d", "problem_id": "165dada9dac61996", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A disease has prevalence $\\frac{1}{25}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{157}{2500}$.\nCompute P(D|+) = $\\frac{85}{157}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{85}{157}$", "final_answer": "\\frac{85}{157}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f5ff2b913408f63", "problem_id": "165dada9dac61996", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A disease has prevalence $\\frac{1}{25}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{157}{2500}$.\n3) Compute P(D|+) = $\\frac{85}{157}$.\nFinal answer: $\\frac{85}{157}$", "final_answer": "\\frac{85}{157}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6e02e35ab5197279", "problem_id": "debddf5be840ba7f", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Worker A finishes a job in 6 hours and Worker B finishes it in 9 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/6 job/hour. B's rate = 1/9 job/hour.\nCombined rate = 1/6 + 1/9 = 5/18 job/hour.\nTime = 1 ÷ (combined rate) = 18/5 hours.\nCheck: (combined rate)×(time) = 5/18×18/5 = 1.\nFinal answer: $\\frac{18}{5}$", "final_answer": "\\frac{18}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3e9b85a5bcac4326", "problem_id": "debddf5be840ba7f", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Worker A finishes a job in 6 hours and Worker B finishes it in 9 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/6 job/hour. B's rate = 1/9 job/hour.\n2) Combined rate = 1/6 + 1/9 = 5/18 job/hour.\n3) Time = 1 ÷ (combined rate) = 18/5 hours.\nFinal answer: $\\frac{18}{5}$", "final_answer": "\\frac{18}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c94218a6cb5fbbd8", "problem_id": "1d03d918044d8874", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 15 times, what is the probability of exactly 8 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=15, k=8, p=$\\frac{1}{4}$ and simplify.\n4) Here C(15,8) = 6435.\nFinal answer: $\\frac{14073345}{1073741824}$", "final_answer": "\\frac{14073345}{1073741824}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bb5a153e0840a5ed", "problem_id": "1d03d918044d8874", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 15 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=15, k=8, p=$\\frac{1}{4}$ and simplify.\nHere C(15,8) = 6435.\nCheck: expression simplifies to $\\frac{14073345}{1073741824}$.\nFinal answer: $\\frac{14073345}{1073741824}$", "final_answer": "\\frac{14073345}{1073741824}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e2392ff06bdda864", "problem_id": "d30a0dec9f21f128", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} - 3 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\nEvaluate at the bounds: F(2 \\pi) and F(0).\nSubtract: F(2 \\pi) − F(0) = $2 \\pi \\left(- 4 \\pi^{2} - 2 + \\pi\\right)$.\nFinal answer: $2\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} - 2 + \\pi\\right)$", "final_answer": "2\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} - 2 + \\pi\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "14e5c10082a43eef", "problem_id": "d30a0dec9f21f128", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} - 3 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(0).\n3) Subtract: F(2 \\pi) − F(0) = $2 \\pi \\left(- 4 \\pi^{2} - 2 + \\pi\\right)$.\n4) Simplify the result.\nFinal answer: $2\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} - 2 + \\pi\\right)$", "final_answer": "2\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} - 2 + \\pi\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f6533185701f7957", "problem_id": "f3ede4c046370fc4", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 22 times, what is the probability of exactly 6 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=22, k=6, p=$\\frac{2}{7}$ and simplify.\nHere C(22,6) = 74613.\nCheck: expression simplifies to $\\frac{104091796875000000}{558545864083284007}$.\nFinal answer: $\\frac{104091796875000000}{558545864083284007}$", "final_answer": "\\frac{104091796875000000}{558545864083284007}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2778fdb41068411b", "problem_id": "f3ede4c046370fc4", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 22 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=22, k=6, p=$\\frac{2}{7}$ and simplify.\n4) Here C(22,6) = 74613.\nFinal answer: $\\frac{104091796875000000}{558545864083284007}$", "final_answer": "\\frac{104091796875000000}{558545864083284007}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f2abcc255b223c68", "problem_id": "5fb62f3256035004", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 6 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=6, k=4, p=$\\frac{1}{4}$ and simplify.\n4) Here C(6,4) = 15.\nFinal answer: $\\frac{135}{4096}$", "final_answer": "\\frac{135}{4096}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "546ed7bf10628974", "problem_id": "5fb62f3256035004", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 6 times, what is the probability of exactly 4 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=6, k=4, p=$\\frac{1}{4}$ and simplify.\nFinal answer: $\\frac{135}{4096}$", "final_answer": "\\frac{135}{4096}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e9bbf2be3865e101", "problem_id": "ea02fa8b30936062", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 35 liters of a 40% salt solution. You have a 20% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 35−x liters is 50% solution.\nSalt equation: 20x + 50(35−x) = 40·35.\n(20-50)x = (40-50)·35 ⇒ x = 35/3.\nThen 35−x = 70/3.\nCheck: 20·35/3 + 50·70/3 = 40·35.\nFinal answer: $35/3 L of 20% and 70/3 L of 50%$", "final_answer": "35/3 L of 20% and 70/3 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "be5c4be90ee10b56", "problem_id": "ea02fa8b30936062", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "You want 35 liters of a 40% salt solution. You have a 20% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 35−x liters is 50% solution.\n2) Salt equation: 20x + 50(35−x) = 40·35.\n3) (20-50)x = (40-50)·35 ⇒ x = 35/3.\n4) Then 35−x = 70/3.\nFinal answer: $35/3 L of 20% and 70/3 L of 50%$", "final_answer": "35/3 L of 20% and 70/3 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dfa3e373b802a352", "problem_id": "74e4d58d58099a1a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(x^{3} + x^{2} + 2\\right) \\cos{\\left(3 x + 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x^{2} \\cos{\\left(3 x + 1 \\right)}) = - x^{2} \\sin{\\left(3 x + 1 \\right)}$.\n3) So $f'(x) = - 3 x^{3} \\sin{\\left(3 x + 1 \\right)} - x^{2} \\sin{\\left(3 x + 1 \\right)} + 3 x^{2} \\cos{\\left(3 x + 1 \\right)} + x e^{x} + 2 e^{x} - 6 \\sin{\\left(3 x + 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x^{2} \\cos{\\left(3 x + 1 \\right)}) = - 3 x^{2} \\sin{\\left(3 x + 1 \\right)} + 2 x \\cos{\\left(3 x + 1 \\right)}$.\nTherefore the correct derivative is $f'(x) = x \\left(3 x + 2\\right) \\cos{\\left(3 x + 1 \\right)} + \\left(x + 1\\right) e^{x} - 3 \\left(x^{2} \\left(x + 1\\right) + 2\\right) \\sin{\\left(3 x + 1 \\right)} + e^{x}$.\nFinal answer: $x \\left(3 x + 2\\right) \\cos{\\left(3 x + 1 \\right)} + \\left(x + 1\\right) e^{x} - 3 \\left(x^{2} \\left(x + 1\\right) + 2\\right) \\sin{\\left(3 x + 1 \\right)} + e^{x}$", "final_answer": "x \\left(3 x + 2\\right) \\cos{\\left(3 x + 1 \\right)} + \\left(x + 1\\right) e^{x} - 3 \\left(x^{2} \\left(x + 1\\right) + 2\\right) \\sin{\\left(3 x + 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b2cd45c974db8a76", "problem_id": "d8686d9c80f4e4b2", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$5 x - 5 y$ = $-10$\n$6 y$ = $36$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $5$: $30 y$ = $180$.\n4) Subtract to eliminate x: $- 30 y$ = $-180$.\n5) Solve for y: y = $6$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, 6\\right)$", "final_answer": "(x,y)=\\left(4, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3e195cfd9394ff14", "problem_id": "1ad538345d0a51bf", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many binary strings of length 15 contain exactly 6 ones with no two ones adjacent?", "output": "1) Reserve 5 mandatory zeros between the 6 ones.\n2) This leaves n−k+1 = 10 valid slots for the ones.\n3) Choose the slots: C(10,6) = 210.\nFinal answer: $210$", "final_answer": "210", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8742b052fc1e06ba", "problem_id": "1ad538345d0a51bf", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "How many binary strings of length 15 contain exactly 6 ones with no two ones adjacent?", "output": "Reserve 5 mandatory zeros between the 6 ones.\nThis leaves n−k+1 = 10 valid slots for the ones.\nChoose the slots: C(10,6) = 210.\nFinal answer: $210$", "final_answer": "210", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "359748fc0db5a172", "problem_id": "c9e3f96d9296fe11", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{7} x}{2} - \\frac{3 x}{10} - \\frac{\\sqrt{7}}{10} - \\frac{1}{10}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{\\sqrt{7}}{2} - \\frac{3}{10}$, c=$- \\frac{\\sqrt{7}}{10} - \\frac{1}{10}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{7 \\sqrt{7}}{10} + \\frac{56}{25}$, so √Δ = $\\sqrt{\\frac{7 \\sqrt{7}}{10} + \\frac{56}{25}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{3}{20} + \\frac{\\sqrt{7}}{4} + \\frac{\\sqrt{70 \\sqrt{7} + 224}}{20}$.\n7) x2 = $- \\frac{\\sqrt{70 \\sqrt{7} + 224}}{20} + \\frac{3}{20} + \\frac{\\sqrt{7}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{3}{20} + \\frac{\\sqrt{7}}{4} + \\frac{\\sqrt{70\\cdot \\sqrt{7} + 224}}{20}, - \\frac{\\sqrt{70\\cdot \\sqrt{7} + 224}}{20} + \\frac{3}{20} + \\frac{\\sqrt{7}}{4}\\}$", "final_answer": "x \\in \\{\\frac{3}{20} + \\frac{\\sqrt{7}}{4} + \\frac{\\sqrt{70\\cdot \\sqrt{7} + 224}}{20}, - \\frac{\\sqrt{70\\cdot \\sqrt{7} + 224}}{20} + \\frac{3}{20} + \\frac{\\sqrt{7}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ad5804fdc4b47d49", "problem_id": "c9e3f96d9296fe11", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{7} x}{2} - \\frac{3 x}{10} - \\frac{\\sqrt{7}}{10} - \\frac{1}{10}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{\\sqrt{7}}{2} - \\frac{3}{10}$, c=$- \\frac{\\sqrt{7}}{10} - \\frac{1}{10}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{7 \\sqrt{7}}{10} + \\frac{56}{25}$, so √Δ = $\\sqrt{\\frac{7 \\sqrt{7}}{10} + \\frac{56}{25}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{3}{20} + \\frac{\\sqrt{7}}{4} + \\frac{\\sqrt{70 \\sqrt{7} + 224}}{20}$.\nx2 = $- \\frac{\\sqrt{70 \\sqrt{7} + 224}}{20} + \\frac{3}{20} + \\frac{\\sqrt{7}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{3}{20} + \\frac{\\sqrt{7}}{4} + \\frac{\\sqrt{70\\cdot \\sqrt{7} + 224}}{20}, - \\frac{\\sqrt{70\\cdot \\sqrt{7} + 224}}{20} + \\frac{3}{20} + \\frac{\\sqrt{7}}{4}\\}$", "final_answer": "x \\in \\{\\frac{3}{20} + \\frac{\\sqrt{7}}{4} + \\frac{\\sqrt{70\\cdot \\sqrt{7} + 224}}{20}, - \\frac{\\sqrt{70\\cdot \\sqrt{7} + 224}}{20} + \\frac{3}{20} + \\frac{\\sqrt{7}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "702ea81163953729", "problem_id": "beaf7b6780904fb7", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - x^{2} + 5 x - 8$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} + 4 x - 8.\n3) Then the limit is -20.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -22.\nFinal answer: $-22$", "final_answer": "-22", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "76e5d32c443b3989", "problem_id": "0e6c3be6ead05bf6", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 18 red balls and 30 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(48,8).\n3) Favorable samples: choose k reds and n−k blues: C(18,3)·C(30,5).\n4) Compute totals: C(48,8) = 377348994, and favorable = 116284896.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{6460272}{20963833}$", "final_answer": "\\frac{6460272}{20963833}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "11b744865e8d105a", "problem_id": "58f2bc92d85fd6a2", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A cyclist rides at 38 miles per hour for 3/2 hours, then at 62 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 38 × 3/2 = 57 miles.\n2) Second leg distance = 62 × 2 = 124 miles.\n3) Total distance = 57 + 124 = 181 miles.\nFinal answer: $181$", "final_answer": "181", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3c4a17d6b4a3b927", "problem_id": "58f2bc92d85fd6a2", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A cyclist rides at 38 miles per hour for 3/2 hours, then at 62 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 38 × 3/2 = 57 miles.\nSecond leg distance = 62 × 2 = 124 miles.\nTotal distance = 57 + 124 = 181 miles.\nCheck: adding both legs gives 181.\nFinal answer: $181$", "final_answer": "181", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "82604fb7a64579e9", "problem_id": "9d08efd7fd433674", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "An urn has 19 red balls and 18 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(37,12).\n3) Favorable samples: choose k reds and n−k blues: C(19,4)·C(18,8).\n4) Compute totals: C(37,12) = 1852482996, and favorable = 169606008.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{21318}{232841}$", "final_answer": "\\frac{21318}{232841}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "15209450a7d8d8b7", "problem_id": "9d08efd7fd433674", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "An urn has 19 red balls and 18 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(37,12).\nFavorable samples: choose k reds and n−k blues: C(19,4)·C(18,8).\nFinal answer: $\\frac{21318}{232841}$", "final_answer": "\\frac{21318}{232841}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eaa0bdf35ba741ec", "problem_id": "205f83ff7f162af5", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Two friends split $360 in the ratio 1:3. How much does each receive?", "output": "Total ratio parts = 1+3 = 4.\nEach part = 360 ÷ 4 = 90.\nFirst share = 1 × 90 = 90.\nFinal answer: $270$", "final_answer": "270", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c4621a241b474585", "problem_id": "205f83ff7f162af5", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Two friends split $360 in the ratio 1:3. How much does each receive?", "output": "1) Total ratio parts = 1+3 = 4.\n2) Each part = 360 ÷ 4 = 90.\n3) First share = 1 × 90 = 90.\n4) Second share = 3 × 90 = 270.\nFinal answer: $270$", "final_answer": "270", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "01f9b14124d704e7", "problem_id": "37f54105dc32b141", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{3}$. If it is flipped 10 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=5, p=$\\frac{1}{3}$ and simplify.\nHere C(10,5) = 252.\nCheck: expression simplifies to $\\frac{896}{6561}$.\nFinal answer: $\\frac{896}{6561}$", "final_answer": "\\frac{896}{6561}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "23ccc9cc10bafbbf", "problem_id": "37f54105dc32b141", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{3}$. If it is flipped 10 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=5, p=$\\frac{1}{3}$ and simplify.\n4) Here C(10,5) = 252.\nFinal answer: $\\frac{896}{6561}$", "final_answer": "\\frac{896}{6561}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "747f2a0c5b43fffc", "problem_id": "a506b6af331630f6", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 21 x + 18$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} + 21 x + 18$ = $3 \\left(x + 1\\right) \\left(x + 6\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-1$.\n5) From the second factor: x = $-6$.\nFinal answer: $x \\in \\{-1, -6\\}$", "final_answer": "x \\in \\{-1, -6\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2ed862a6e579e68a", "problem_id": "a506b6af331630f6", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 21 x + 18$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} + 21 x + 18$ = $3 \\left(x + 1\\right) \\left(x + 6\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFinal answer: $x \\in \\{-1, -6\\}$", "final_answer": "x \\in \\{-1, -6\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0c9ef4fa3aae9eb9", "problem_id": "cff1e9bcb9c02d57", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_7=19$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_7=12.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=12 and k=7.\n4) Compute C(18,6) = 18564.\nFinal answer: $18564$", "final_answer": "18564", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f86155ac0de0b9f4", "problem_id": "cff1e9bcb9c02d57", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_7=19$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_7=12.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=12 and k=7.\nCompute C(18,6) = 18564.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $18564$", "final_answer": "18564", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1854f28e1eccc221", "problem_id": "53f0ce97fc75429a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 14 red balls and 19 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(33,11).\nFavorable samples: choose k reds and n−k blues: C(14,4)·C(19,7).\nCompute totals: C(33,11) = 193536720, and favorable = 50438388.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{323323}{1240620}$.\nFinal answer: $\\frac{323323}{1240620}$", "final_answer": "\\frac{323323}{1240620}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "089f37725bbfcbd4", "problem_id": "53f0ce97fc75429a", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "An urn has 14 red balls and 19 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(33,11).\n3) Favorable samples: choose k reds and n−k blues: C(14,4)·C(19,7).\n4) Compute totals: C(33,11) = 193536720, and favorable = 50438388.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{323323}{1240620}$", "final_answer": "\\frac{323323}{1240620}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "89e8e059f5e5a5bc", "problem_id": "e42f61c177f6352f", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{65 \\pi^{3}}{24} - \\frac{5 \\pi}{2} - 2 + \\frac{45 \\pi^{2}}{8}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{65\\cdot \\pi^{3}}{24} - \\frac{5\\cdot \\pi}{2} - 2 + \\frac{45\\cdot \\pi^{2}}{8}$", "final_answer": "- \\frac{65\\cdot \\pi^{3}}{24} - \\frac{5\\cdot \\pi}{2} - 2 + \\frac{45\\cdot \\pi^{2}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e8626416eca18718", "problem_id": "e42f61c177f6352f", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{65 \\pi^{3}}{24} - \\frac{5 \\pi}{2} - 2 + \\frac{45 \\pi^{2}}{8}$.\n4) Simplify the result.\nFinal answer: $- \\frac{65\\cdot \\pi^{3}}{24} - \\frac{5\\cdot \\pi}{2} - 2 + \\frac{45\\cdot \\pi^{2}}{8}$", "final_answer": "- \\frac{65\\cdot \\pi^{3}}{24} - \\frac{5\\cdot \\pi}{2} - 2 + \\frac{45\\cdot \\pi^{2}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e58a687d8dedfad8", "problem_id": "221d604d46284338", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 21 modulo 29.", "output": "We want x such that 21x ≡ 1 (mod 29).\nUse extended Euclid to find s,t with 21s + 29t = 1.\nOne Bezout identity is: 21·(-11) + 29·(8) = 1.\nReduce s modulo 29: x ≡ 18 (mod 29).\nCheck: 21·18 mod 29 = 1.\nFinal answer: $x\\equiv 18\\pmod{29}$", "final_answer": "x\\equiv 18\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "60303c63344d3c21", "problem_id": "221d604d46284338", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Find the multiplicative inverse of 21 modulo 29.", "output": "We want x such that 21x ≡ 1 (mod 29).\nUse extended Euclid to find s,t with 21s + 29t = 1.\nOne Bezout identity is: 21·(-11) + 29·(8) = 1.\nFinal answer: $x\\equiv 18\\pmod{29}$", "final_answer": "x\\equiv 18\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "76a1ea1b5f9403b0", "problem_id": "c35ad5581f1828cb", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{2} x - \\frac{7 x}{6} + \\frac{1}{3} + \\frac{2 \\sqrt{2}}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\sqrt{2} - \\frac{7}{6}$, c=$\\frac{1}{3} + \\frac{2 \\sqrt{2}}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{73}{36} - \\frac{\\sqrt{2}}{3}$, so √Δ = $\\sqrt{\\frac{73}{36} - \\frac{\\sqrt{2}}{3}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{7}{12} + \\frac{\\sqrt{73 - 12 \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{2}$.\n7) x2 = $- \\frac{\\sqrt{73 - 12 \\sqrt{2}}}{12} + \\frac{7}{12} + \\frac{\\sqrt{2}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{7}{12} + \\frac{\\sqrt{73 - 12\\cdot \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{2}, - \\frac{\\sqrt{73 - 12\\cdot \\sqrt{2}}}{12} + \\frac{7}{12} + \\frac{\\sqrt{2}}{2}\\}$", "final_answer": "x \\in \\{\\frac{7}{12} + \\frac{\\sqrt{73 - 12\\cdot \\sqrt{2}}}{12} + \\frac{\\sqrt{2}}{2}, - \\frac{\\sqrt{73 - 12\\cdot \\sqrt{2}}}{12} + \\frac{7}{12} + \\frac{\\sqrt{2}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "68b4802067b5f0ba", "problem_id": "331742ad7366406c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 3 x^{2} - 5 x - 12$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} - 6 x - 12.\n3) Then the limit is -36.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -34.\nFinal answer: $-34$", "final_answer": "-34", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0ab32dd90c7c4b97", "problem_id": "bfd9df7582bc8879", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $-4 + 9 \\pi + \\frac{9 \\pi^{2}}{2} + 6 \\pi^{3}$.\n4) Simplify the result.\nFinal answer: $-4 + 9\\cdot \\pi + \\frac{9\\cdot \\pi^{2}}{2} + 6\\cdot \\pi^{3}$", "final_answer": "-4 + 9\\cdot \\pi + \\frac{9\\cdot \\pi^{2}}{2} + 6\\cdot \\pi^{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "90b1133aa41e6e4f", "problem_id": "bfd9df7582bc8879", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $-4 + 9 \\pi + \\frac{9 \\pi^{2}}{2} + 6 \\pi^{3}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $-4 + 9\\cdot \\pi + \\frac{9\\cdot \\pi^{2}}{2} + 6\\cdot \\pi^{3}$", "final_answer": "-4 + 9\\cdot \\pi + \\frac{9\\cdot \\pi^{2}}{2} + 6\\cdot \\pi^{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "869f727b231dbe7a", "problem_id": "f1da73a3d98b6be7", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{3} x + x - 6 \\sqrt{3} - 2$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$1 - 3 \\sqrt{3}$, c=$- 6 \\sqrt{3} - 2$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $18 \\sqrt{3} + 36$, so √Δ = $\\sqrt{18 \\sqrt{3} + 36}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{1}{2} + \\frac{3 \\sqrt{3}}{2} + \\frac{3 \\sqrt{2 \\sqrt{3} + 4}}{2}$.\n7) x2 = $- \\frac{3 \\sqrt{2 \\sqrt{3} + 4}}{2} - \\frac{1}{2} + \\frac{3 \\sqrt{3}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{1}{2} + \\frac{3\\cdot \\sqrt{3}}{2} + \\frac{3\\cdot \\sqrt{2\\cdot \\sqrt{3} + 4}}{2}, - \\frac{3\\cdot \\sqrt{2\\cdot \\sqrt{3} + 4}}{2} - \\frac{1}{2} + \\frac{3\\cdot \\sqrt{3}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{1}{2} + \\frac{3\\cdot \\sqrt{3}}{2} + \\frac{3\\cdot \\sqrt{2\\cdot \\sqrt{3} + 4}}{2}, - \\frac{3\\cdot \\sqrt{2\\cdot \\sqrt{3} + 4}}{2} - \\frac{1}{2} + \\frac{3\\cdot \\sqrt{3}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d0d089c64edf62ef", "problem_id": "f1da73a3d98b6be7", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{3} x + x - 6 \\sqrt{3} - 2$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$1 - 3 \\sqrt{3}$, c=$- 6 \\sqrt{3} - 2$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{- \\frac{1}{2} + \\frac{3\\cdot \\sqrt{3}}{2} + \\frac{3\\cdot \\sqrt{2\\cdot \\sqrt{3} + 4}}{2}, - \\frac{3\\cdot \\sqrt{2\\cdot \\sqrt{3} + 4}}{2} - \\frac{1}{2} + \\frac{3\\cdot \\sqrt{3}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{1}{2} + \\frac{3\\cdot \\sqrt{3}}{2} + \\frac{3\\cdot \\sqrt{2\\cdot \\sqrt{3} + 4}}{2}, - \\frac{3\\cdot \\sqrt{2\\cdot \\sqrt{3} + 4}}{2} - \\frac{1}{2} + \\frac{3\\cdot \\sqrt{3}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec79429cee882366", "problem_id": "06617c74386c736a", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "You want 10 liters of a 30% salt solution. You have a 20% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 10−x liters is 40% solution.\n2) Salt equation: 20x + 40(10−x) = 30·10.\n3) (20-40)x = (30-40)·10 ⇒ x = 5.\n4) Then 10−x = 5.\nFinal answer: $5 L of 20% and 5 L of 40%$", "final_answer": "5 L of 20% and 5 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e1d8267360542d6", "problem_id": "127845561a322a32", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{3} + 4 x^{2} + 7 x + 11$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} + 4 x^{2} + 6 x + 11.\n3) Then the limit is 55.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 57.\nFinal answer: $57$", "final_answer": "57", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7e36c2abbb460467", "problem_id": "f140bf46e1b00918", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A disease has prevalence $\\frac{1}{100}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{29}{500}$.\n3) Compute P(D|+) = $\\frac{17}{116}$.\nFinal answer: $\\frac{17}{116}$", "final_answer": "\\frac{17}{116}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "22a4c2c7e41be62e", "problem_id": "f140bf46e1b00918", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A disease has prevalence $\\frac{1}{100}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{29}{500}$.\nCompute P(D|+) = $\\frac{17}{116}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{17}{116}$", "final_answer": "\\frac{17}{116}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a7afabf4e24f94fe", "problem_id": "f85cb8e8a4aed11d", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 23 red balls and 17 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(40,12).\n3) Favorable samples: choose k reds and n−k blues: C(23,3)·C(17,9).\n4) Compute totals: C(40,12) = 5586853480, and favorable = 43053010.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{19481}{2527988}$", "final_answer": "\\frac{19481}{2527988}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9c3da3a69677edf9", "problem_id": "34496964d84262b4", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(3 x^{3} - 3 x^{2} + x + 1\\right) \\cos{\\left(x^{2} - x + 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x \\cos{\\left(\\left(x^{2} - x\\right) + 1 \\right)}) = - x \\sin{\\left(x^{2} - x + 1 \\right)}$.\n3) So $f'(x) = - 6 x^{4} \\sin{\\left(\\left(x^{2} - x\\right) + 1 \\right)} + 9 x^{3} \\sin{\\left(\\left(x^{2} - x\\right) + 1 \\right)} - 3 x^{2} \\sin{\\left(\\left(x^{2} - x\\right) + 1 \\right)} + 9 x^{2} \\cos{\\left(\\left(x^{2} - x\\right) + 1 \\right)} + x e^{x} - 2 x \\sin{\\left(\\left(x^{2} - x\\right) + 1 \\right)} - x \\sin{\\left(x^{2} - x + 1 \\right)} - 6 x \\cos{\\left(\\left(x^{2} - x\\right) + 1 \\right)} + 2 e^{x} + \\sin{\\left(\\left(x^{2} - x\\right) + 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x \\cos{\\left(\\left(x^{2} - x\\right) + 1 \\right)}) = - x \\left(2 x - 1\\right) \\sin{\\left(\\left(x^{2} - x\\right) + 1 \\right)} + \\cos{\\left(\\left(x^{2} - x\\right) + 1 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 1\\right) e^{x} - \\left(2 x - 1\\right) \\left(x \\left(3 x \\left(x - 1\\right) + 1\\right) + 1\\right) \\sin{\\left(\\left(x^{2} - x\\right) + 1 \\right)} + \\left(9 x^{2} - 6 x + 1\\right) \\cos{\\left(\\left(x^{2} - x\\right) + 1 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 1\\right) e^{x} - \\left(2 x - 1\\right) \\left(x \\left(3 x \\left(x - 1\\right) + 1\\right) + 1\\right) \\sin{\\left(\\left(x^{2} - x\\right) + 1 \\right)} + \\left(9 x^{2} - 6 x + 1\\right) \\cos{\\left(\\left(x^{2} - x\\right) + 1 \\right)} + e^{x}$", "final_answer": "\\left(x + 1\\right) e^{x} - \\left(2 x - 1\\right) \\left(x \\left(3 x \\left(x - 1\\right) + 1\\right) + 1\\right) \\sin{\\left(\\left(x^{2} - x\\right) + 1 \\right)} + \\left(9 x^{2} - 6 x + 1\\right) \\cos{\\left(\\left(x^{2} - x\\right) + 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5e2aaaffe5b9b8c9", "problem_id": "321181df711be1a9", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + x - 2$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$1$, c=$-2$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $9$, so √Δ = $3$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $1$.\nx2 = $-2$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{1, -2\\}$", "final_answer": "x \\in \\{1, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "731b3ea00034054f", "problem_id": "321181df711be1a9", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + x - 2$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$1$, c=$-2$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $9$, so √Δ = $3$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $1$.\n7) x2 = $-2$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{1, -2\\}$", "final_answer": "x \\in \\{1, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ea4826981af1cc1", "problem_id": "d2e16742ae55e12c", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A cyclist rides at 71 miles per hour for 1 hours, then at 65 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 71 × 1 = 71 miles.\n2) Second leg distance = 65 × 2 = 130 miles.\n3) Total distance = 71 + 130 = 201 miles.\nFinal answer: $201$", "final_answer": "201", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7502677b3bb51e7a", "problem_id": "d2e16742ae55e12c", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "A cyclist rides at 71 miles per hour for 1 hours, then at 65 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 71 × 1 = 71 miles.\nSecond leg distance = 65 × 2 = 130 miles.\nTotal distance = 71 + 130 = 201 miles.\nFinal answer: $201$", "final_answer": "201", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "37bdc90a70838ae9", "problem_id": "ccefcff6d4b64b05", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 7 x + 5$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$-7$, c=$5$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $9$, so √Δ = $3$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{5}{2}$.\nx2 = $1$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{5}{2}, 1\\}$", "final_answer": "x \\in \\{\\frac{5}{2}, 1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "65da9c0ec845fd6e", "problem_id": "ccefcff6d4b64b05", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 7 x + 5$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$-7$, c=$5$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $9$, so √Δ = $3$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{5}{2}$.\n7) x2 = $1$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{5}{2}, 1\\}$", "final_answer": "x \\in \\{\\frac{5}{2}, 1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0fb91b8440e92ea6", "problem_id": "b2ea521ab8090dcc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(2 x^{3} + x^{2} + 3 x - 3\\right) \\cos{\\left(2 x + 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x e^{x}) = 2 x e^{x}$.\n3) So $f'(x) = - 4 x^{3} \\sin{\\left(2 x + 2 \\right)} - 2 x^{2} \\sin{\\left(2 x + 2 \\right)} + 6 x^{2} \\cos{\\left(2 x + 2 \\right)} + 2 x e^{x} - 6 x \\sin{\\left(2 x + 2 \\right)} + 2 x \\cos{\\left(2 x + 2 \\right)} - 6 e^{x} + 6 \\sin{\\left(2 x + 2 \\right)} + 3 \\cos{\\left(2 x + 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x e^{x}) = 2 x e^{x} + 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x - 3\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x + 1\\right) + 3\\right) - 3\\right) \\sin{\\left(2 x + 2 \\right)} + \\left(6 x^{2} + 2 x + 3\\right) \\cos{\\left(2 x + 2 \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(x - 3\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x + 1\\right) + 3\\right) - 3\\right) \\sin{\\left(2 x + 2 \\right)} + \\left(6 x^{2} + 2 x + 3\\right) \\cos{\\left(2 x + 2 \\right)} + 2 e^{x}$", "final_answer": "2 \\left(x - 3\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x + 1\\right) + 3\\right) - 3\\right) \\sin{\\left(2 x + 2 \\right)} + \\left(6 x^{2} + 2 x + 3\\right) \\cos{\\left(2 x + 2 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "de7ac01c14c9969c", "problem_id": "ed11695a5f0aff60", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$- 3 x - 4 y$ = $-11$\n$3 x - 2 y$ = $35$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $3$: $- 9 x - 12 y$ = $-33$.\n3) Multiply the second equation by $-3$: $- 9 x + 6 y$ = $-105$.\n4) Subtract to eliminate x: $- 18 y$ = $72$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $9$.\nFinal answer: $(x,y)=\\left(9, -4\\right)$", "final_answer": "(x,y)=\\left(9, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fd5db918105659cc", "problem_id": "4664990e3143d624", "style": "answer_only", "instruction": "Final answer only.", "input": "Solve the congruences: x≡2 (mod 4), x≡2 (mod 13). Give the solution modulo 52.", "output": "$x\\equiv 2\\pmod{52}$", "final_answer": "x\\equiv 2\\pmod{52}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5ee86aab5c994fe7", "problem_id": "4664990e3143d624", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 4), x≡2 (mod 13). Give the solution modulo 52.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 52.\nCompute the CRT solution to get x ≡ 2 (mod 52).\nCheck: 2 mod 4=2 and mod 13=2.\nFinal answer: $x\\equiv 2\\pmod{52}$", "final_answer": "x\\equiv 2\\pmod{52}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bbb5d75c57f4447a", "problem_id": "25f24e850f171704", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$6 x - 7 y$ = $17$\n$x + 5 y$ = $9$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $6 x - 7 y$ = $17$.\nMultiply the second equation by $6$: $6 x + 30 y$ = $54$.\nSubtract to eliminate x: $- 37 y$ = $-37$.\nSolve for y: y = $1$.\nSubstitute back to find x: x = $4$.\nCheck: plug (x,y)=(4,1) into both equations.\nFinal answer: $(x,y)=\\left(4, 1\\right)$", "final_answer": "(x,y)=\\left(4, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7fdab7e82a657708", "problem_id": "25f24e850f171704", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$6 x - 7 y$ = $17$\n$x + 5 y$ = $9$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $6 x - 7 y$ = $17$.\n3) Multiply the second equation by $6$: $6 x + 30 y$ = $54$.\n4) Subtract to eliminate x: $- 37 y$ = $-37$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, 1\\right)$", "final_answer": "(x,y)=\\left(4, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4f6beabf4dcbce6a", "problem_id": "3619b6eff74d5878", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} x^{4} + 4 x^{3} + 3 x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $x^{4} + 4 x^{3} + 3 x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(0).\n3) Subtract: F(2 \\pi) − F(0) = $\\frac{4 \\pi \\left(- 10 \\pi - 5 + 10 \\pi^{2} + 20 \\pi^{3} + 8 \\pi^{4}\\right)}{5}$.\n4) Simplify the result.\nFinal answer: $\\frac{4\\cdot \\pi\\cdot \\left(- 10\\cdot \\pi - 5 + 10\\cdot \\pi^{2} + 20\\cdot \\pi^{3} + 8\\cdot \\pi^{4}\\right)}{5}$", "final_answer": "\\frac{4\\cdot \\pi\\cdot \\left(- 10\\cdot \\pi - 5 + 10\\cdot \\pi^{2} + 20\\cdot \\pi^{3} + 8\\cdot \\pi^{4}\\right)}{5}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5e0568890795e8e6", "problem_id": "3619b6eff74d5878", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} x^{4} + 4 x^{3} + 3 x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $x^{4} + 4 x^{3} + 3 x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2$.\nEvaluate at the bounds: F(2 \\pi) and F(0).\nSubtract: F(2 \\pi) − F(0) = $\\frac{4 \\pi \\left(- 10 \\pi - 5 + 10 \\pi^{2} + 20 \\pi^{3} + 8 \\pi^{4}\\right)}{5}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{4\\cdot \\pi\\cdot \\left(- 10\\cdot \\pi - 5 + 10\\cdot \\pi^{2} + 20\\cdot \\pi^{3} + 8\\cdot \\pi^{4}\\right)}{5}$", "final_answer": "\\frac{4\\cdot \\pi\\cdot \\left(- 10\\cdot \\pi - 5 + 10\\cdot \\pi^{2} + 20\\cdot \\pi^{3} + 8\\cdot \\pi^{4}\\right)}{5}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d29d9dda1a0ef2bb", "problem_id": "1ec5be26d86d7a17", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Two friends split $1410 in the ratio 7:3. How much does each receive?", "output": "Total ratio parts = 7+3 = 10.\nEach part = 1410 ÷ 10 = 141.\nFirst share = 7 × 141 = 987.\nSecond share = 3 × 141 = 423.\nCheck: 987+423=1410.\nFinal answer: $423$", "final_answer": "423", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "be51a7c8b5e4d84a", "problem_id": "1ec5be26d86d7a17", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Two friends split $1410 in the ratio 7:3. How much does each receive?", "output": "1) Total ratio parts = 7+3 = 10.\n2) Each part = 1410 ÷ 10 = 141.\n3) First share = 7 × 141 = 987.\n4) Second share = 3 × 141 = 423.\nFinal answer: $423$", "final_answer": "423", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "06e80e6a32fbc5da", "problem_id": "baecc643cf18cbf8", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the linear system:\n$4 x$ = $-28$\n$- 8 x + y$ = $53$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-8$: $- 32 x$ = $224$.\nMultiply the second equation by $4$: $- 32 x + 4 y$ = $212$.\nFinal answer: $(x,y)=\\left(-7, -3\\right)$", "final_answer": "(x,y)=\\left(-7, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3fdb299ac6ea5990", "problem_id": "baecc643cf18cbf8", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$4 x$ = $-28$\n$- 8 x + y$ = $53$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-8$: $- 32 x$ = $224$.\n3) Multiply the second equation by $4$: $- 32 x + 4 y$ = $212$.\n4) Subtract to eliminate x: $- 4 y$ = $12$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $-7$.\nFinal answer: $(x,y)=\\left(-7, -3\\right)$", "final_answer": "(x,y)=\\left(-7, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8f4117fa4713de35", "problem_id": "498c414e718751f0", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A cyclist rides at 55 miles per hour for 2 hours, then at 57 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 55 × 2 = 110 miles.\n2) Second leg distance = 57 × 5/2 = 285/2 miles.\n3) Total distance = 110 + 285/2 = 505/2 miles.\nFinal answer: $\\frac{505}{2}$", "final_answer": "\\frac{505}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1a707c7c0b048aac", "problem_id": "498c414e718751f0", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A cyclist rides at 55 miles per hour for 2 hours, then at 57 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 55 × 2 = 110 miles.\nSecond leg distance = 57 × 5/2 = 285/2 miles.\nTotal distance = 110 + 285/2 = 505/2 miles.\nFinal answer: $\\frac{505}{2}$", "final_answer": "\\frac{505}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d214c029acc1fe6f", "problem_id": "8d432d8c049fde0e", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$3 x - 8 y$ = $30$\n$- 4 x - y$ = $30$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $- 12 x + 32 y$ = $-120$.\n3) Multiply the second equation by $3$: $- 12 x - 3 y$ = $90$.\n4) Subtract to eliminate x: $35 y$ = $-210$.\n5) Solve for y: y = $-6$.\n6) Substitute back to find x: x = $-6$.\nFinal answer: $(x,y)=\\left(-6, -6\\right)$", "final_answer": "(x,y)=\\left(-6, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "267f438489a659cd", "problem_id": "8d432d8c049fde0e", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$3 x - 8 y$ = $30$\n$- 4 x - y$ = $30$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $- 12 x + 32 y$ = $-120$.\nMultiply the second equation by $3$: $- 12 x - 3 y$ = $90$.\nSubtract to eliminate x: $35 y$ = $-210$.\nSolve for y: y = $-6$.\nSubstitute back to find x: x = $-6$.\nCheck: plug (x,y)=(-6,-6) into both equations.\nFinal answer: $(x,y)=\\left(-6, -6\\right)$", "final_answer": "(x,y)=\\left(-6, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bb9a4ee951861ef9", "problem_id": "5f4b0485eba0d08c", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the multiplicative inverse of 12 modulo 23.", "output": "1) We want x such that 12x ≡ 1 (mod 23).\n2) Use extended Euclid to find s,t with 12s + 23t = 1.\n3) One Bezout identity is: 12·(2) + 23·(-1) = 1.\n4) Reduce s modulo 23: x ≡ 2 (mod 23).\nFinal answer: $x\\equiv 2\\pmod{23}$", "final_answer": "x\\equiv 2\\pmod{23}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1345b556109c7a8b", "problem_id": "5f4b0485eba0d08c", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Find the multiplicative inverse of 12 modulo 23.", "output": "We want x such that 12x ≡ 1 (mod 23).\nUse extended Euclid to find s,t with 12s + 23t = 1.\nOne Bezout identity is: 12·(2) + 23·(-1) = 1.\nFinal answer: $x\\equiv 2\\pmod{23}$", "final_answer": "x\\equiv 2\\pmod{23}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a2bddd1636dee4c", "problem_id": "0a1be025f503c38a", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{2} + x + 3$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{3} - 3 x^{2} + x + 2$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $6 x^{2} - 6 x + 1$.\nEvaluate at a: p'(2) = $13$.\nCheck: the identity gives $13$.\nFinal answer: $13$", "final_answer": "13", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a2ac5a9cb2aaf20", "problem_id": "d54f72c07c949c2c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x^{2} - 3 x - 9$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{2} - 4 x - 9.\n3) Then the limit is -25.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -23.\nFinal answer: $-23$", "final_answer": "-23", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0e3c423529adad56", "problem_id": "a7df38b982a395d9", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A cyclist rides at 30 miles per hour for 1/2 hours, then at 60 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 30 × 1/2 = 15 miles.\nSecond leg distance = 60 × 2 = 120 miles.\nTotal distance = 15 + 120 = 135 miles.\nFinal answer: $135$", "final_answer": "135", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c5eb06c08bcbacf1", "problem_id": "a7df38b982a395d9", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A cyclist rides at 30 miles per hour for 1/2 hours, then at 60 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 30 × 1/2 = 15 miles.\n2) Second leg distance = 60 × 2 = 120 miles.\n3) Total distance = 15 + 120 = 135 miles.\nFinal answer: $135$", "final_answer": "135", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c4f05ddbbb96a7d7", "problem_id": "af012ffd7dec7f05", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 19 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=19, k=5, p=$\\frac{1}{6}$ and simplify.\n4) Here C(19,5) = 11628.\nFinal answer: $\\frac{1971435546875}{16926659444736}$", "final_answer": "\\frac{1971435546875}{16926659444736}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a012cd0f96b63c33", "problem_id": "af012ffd7dec7f05", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 19 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=19, k=5, p=$\\frac{1}{6}$ and simplify.\nHere C(19,5) = 11628.\nCheck: expression simplifies to $\\frac{1971435546875}{16926659444736}$.\nFinal answer: $\\frac{1971435546875}{16926659444736}$", "final_answer": "\\frac{1971435546875}{16926659444736}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72f32b0a2bf02dee", "problem_id": "be0609e87ac3b150", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Two friends split $470 in the ratio 8:2. How much does each receive?", "output": "Total ratio parts = 8+2 = 10.\nEach part = 470 ÷ 10 = 47.\nFirst share = 8 × 47 = 376.\nSecond share = 2 × 47 = 94.\nCheck: 376+94=470.\nFinal answer: $94$", "final_answer": "94", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f30ba1ea0a431a16", "problem_id": "be0609e87ac3b150", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Two friends split $470 in the ratio 8:2. How much does each receive?", "output": "1) Total ratio parts = 8+2 = 10.\n2) Each part = 470 ÷ 10 = 47.\n3) First share = 8 × 47 = 376.\n4) Second share = 2 × 47 = 94.\nFinal answer: $94$", "final_answer": "94", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d67355e2d3602eb1", "problem_id": "feb17313a85e305d", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{x}{3} - \\frac{35}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$- \\frac{1}{3}$, c=$- \\frac{35}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{841}{9}$, so √Δ = $\\frac{29}{3}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{5}{2}$.\n7) x2 = $- \\frac{7}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{5}{2}, - \\frac{7}{3}\\}$", "final_answer": "x \\in \\{\\frac{5}{2}, - \\frac{7}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "86aaadb481bc0716", "problem_id": "feb17313a85e305d", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{x}{3} - \\frac{35}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$- \\frac{1}{3}$, c=$- \\frac{35}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{841}{9}$, so √Δ = $\\frac{29}{3}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{5}{2}$.\nx2 = $- \\frac{7}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{5}{2}, - \\frac{7}{3}\\}$", "final_answer": "x \\in \\{\\frac{5}{2}, - \\frac{7}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7f25a9380ac7521d", "problem_id": "d8d35b6d9871079c", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- \\frac{3 x^{2}}{2} - x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1}{x - 4}$.", "output": "1) Let f(x) = $\\frac{- \\frac{3 x^{2}}{2} - x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1}{x - 4}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{3 x^{2} + 2 x + \\left(x - 4\\right) \\left(- 6 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{2 \\left(x - 4\\right)^{2}}$.\nFinal answer: $\\frac{3\\cdot x^{2} + 2\\cdot x + \\left(x - 4\\right)\\cdot \\left(- 6\\cdot x + \\left(x - 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x - 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right) - \\left(2\\cdot x - 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 2}{2\\cdot \\left(x - 4\\right)^{2}}$", "final_answer": "\\frac{3\\cdot x^{2} + 2\\cdot x + \\left(x - 4\\right)\\cdot \\left(- 6\\cdot x + \\left(x - 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x - 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right) - \\left(2\\cdot x - 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 2}{2\\cdot \\left(x - 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8798901fa8a5fc92", "problem_id": "f0edbf272a0e2609", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} + x^{2} - 2 x + 4$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x^{3}) = - 6 x^{3}$.\n3) So $f'(x) = - 6 x^{3} + 2 x - 2$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x^{3}) = - 6 x^{2}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- 3 x^{2} + x - 1\\right)$.\nFinal answer: $2 \\left(- 3 x^{2} + x - 1\\right)$", "final_answer": "2 \\left(- 3 x^{2} + x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "cf286f6b2c080c16", "problem_id": "434fb9ff87e5e1ff", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{3} - 3 x^{2} + 2 x + \\left(x^{2} + 3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{x^{2} + 3 x + 7}$.", "output": "1) Let f(x) = $\\frac{- x^{3} - 3 x^{2} + 2 x + \\left(x^{2} + 3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{x^{2} + 3 x + 7}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{2 \\left(- 2 x - 3\\right) \\left(- x^{3} - 3 x^{2} + 2 x + \\left(x^{2} + 3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2\\right) + \\left(x^{2} + 3 x + 7\\right) \\left(- 6 x^{2} - 12 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} + 3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 3 x + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x^{2} + 3 x + 7\\right)^{2}}$.\nFinal answer: $\\frac{\\left(- 4\\cdot x - 6\\right)\\cdot \\left(- x^{3} - 3\\cdot x^{2} + 2\\cdot x + \\left(x^{2} + 3\\cdot x + 7\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 2\\right) + \\left(x^{2} + 3\\cdot x + 7\\right)\\cdot \\left(- 6\\cdot x^{2} - 12\\cdot x + \\left(4\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(x^{2} + 3\\cdot x + 7\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 6\\cdot x + 14\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\right)}{2\\cdot \\left(x^{2} + 3\\cdot x + 7\\right)^{2}}$", "final_answer": "\\frac{\\left(- 4\\cdot x - 6\\right)\\cdot \\left(- x^{3} - 3\\cdot x^{2} + 2\\cdot x + \\left(x^{2} + 3\\cdot x + 7\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 2\\right) + \\left(x^{2} + 3\\cdot x + 7\\right)\\cdot \\left(- 6\\cdot x^{2} - 12\\cdot x + \\left(4\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(x^{2} + 3\\cdot x + 7\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 6\\cdot x + 14\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\right)}{2\\cdot \\left(x^{2} + 3\\cdot x + 7\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b8f16c8565fbed1e", "problem_id": "90ce192073de0a89", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{4} - 2 x^{3} + x^{2} - 2 x$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(2 x^{4}) = 8 x^{4}$.\n3) So $f'(x) = 8 x^{4} - 6 x^{2} + 2 x - 2$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(2 x^{4}) = 8 x^{3}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(4 x^{3} - 3 x^{2} + x - 1\\right)$.\nFinal answer: $2 \\left(4 x^{3} - 3 x^{2} + x - 1\\right)$", "final_answer": "2 \\left(4 x^{3} - 3 x^{2} + x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ef453ffcb5fc1fd7", "problem_id": "f8fd563c0bb7ee55", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} - 2 x^{3} - 2 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{3} - 2 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\n3) Subtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{3 \\pi^{2}}{8} - \\frac{\\pi^{4}}{32} - \\frac{\\pi^{3}}{12} + 2$.\n4) Simplify the result.\nFinal answer: $- \\frac{3\\cdot \\pi^{2}}{8} - \\frac{\\pi^{4}}{32} - \\frac{\\pi^{3}}{12} + 2$", "final_answer": "- \\frac{3\\cdot \\pi^{2}}{8} - \\frac{\\pi^{4}}{32} - \\frac{\\pi^{3}}{12} + 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dfca1da08c427ff1", "problem_id": "f8fd563c0bb7ee55", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} - 2 x^{3} - 2 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{3} - 2 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\nSubtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{3 \\pi^{2}}{8} - \\frac{\\pi^{4}}{32} - \\frac{\\pi^{3}}{12} + 2$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{3\\cdot \\pi^{2}}{8} - \\frac{\\pi^{4}}{32} - \\frac{\\pi^{3}}{12} + 2$", "final_answer": "- \\frac{3\\cdot \\pi^{2}}{8} - \\frac{\\pi^{4}}{32} - \\frac{\\pi^{3}}{12} + 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3d6f3df947fb64d9", "problem_id": "6142cadaa8beebcc", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Worker A finishes a job in 17 hours and Worker B finishes it in 19 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/17 job/hour. B's rate = 1/19 job/hour.\n2) Combined rate = 1/17 + 1/19 = 36/323 job/hour.\n3) Time = 1 ÷ (combined rate) = 323/36 hours.\nFinal answer: $\\frac{323}{36}$", "final_answer": "\\frac{323}{36}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b805b47364e959d1", "problem_id": "6142cadaa8beebcc", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Worker A finishes a job in 17 hours and Worker B finishes it in 19 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/17 job/hour. B's rate = 1/19 job/hour.\nCombined rate = 1/17 + 1/19 = 36/323 job/hour.\nTime = 1 ÷ (combined rate) = 323/36 hours.\nCheck: (combined rate)×(time) = 36/323×323/36 = 1.\nFinal answer: $\\frac{323}{36}$", "final_answer": "\\frac{323}{36}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e912e0c857ad0d3a", "problem_id": "afbbba79142d40f8", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡0 (mod 4), x≡8 (mod 13). Give the solution modulo 52.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 52.\n3) Compute the CRT solution to get x ≡ 8 (mod 52).\nFinal answer: $x\\equiv 8\\pmod{52}$", "final_answer": "x\\equiv 8\\pmod{52}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e664ab2cfd63dc05", "problem_id": "afbbba79142d40f8", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡0 (mod 4), x≡8 (mod 13). Give the solution modulo 52.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 52.\nCompute the CRT solution to get x ≡ 8 (mod 52).\nCheck: 8 mod 4=0 and mod 13=8.\nFinal answer: $x\\equiv 8\\pmod{52}$", "final_answer": "x\\equiv 8\\pmod{52}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "20c4d8fd33ed3904", "problem_id": "e08e509797875464", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$- 8 x - 8 y$ = $48$\n$- 6 x - 2 y$ = $8$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-6$: $48 x + 48 y$ = $-288$.\n3) Multiply the second equation by $-8$: $48 x + 16 y$ = $-64$.\n4) Subtract to eliminate x: $32 y$ = $-224$.\n5) Solve for y: y = $-7$.\n6) Substitute back to find x: x = $1$.\nFinal answer: $(x,y)=\\left(1, -7\\right)$", "final_answer": "(x,y)=\\left(1, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "de2a7c7cb9deec06", "problem_id": "e08e509797875464", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$- 8 x - 8 y$ = $48$\n$- 6 x - 2 y$ = $8$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-6$: $48 x + 48 y$ = $-288$.\nMultiply the second equation by $-8$: $48 x + 16 y$ = $-64$.\nSubtract to eliminate x: $32 y$ = $-224$.\nSolve for y: y = $-7$.\nSubstitute back to find x: x = $1$.\nCheck: plug (x,y)=(1,-7) into both equations.\nFinal answer: $(x,y)=\\left(1, -7\\right)$", "final_answer": "(x,y)=\\left(1, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d6b8c10e0e9b038c", "problem_id": "3b686d1a3403215a", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{9 \\pi^{2}}{8} - 2 + \\frac{3 \\pi^{3}}{8} + \\frac{9 \\pi}{2}$.\n4) Simplify the result.\nFinal answer: $- \\frac{9\\cdot \\pi^{2}}{8} - 2 + \\frac{3\\cdot \\pi^{3}}{8} + \\frac{9\\cdot \\pi}{2}$", "final_answer": "- \\frac{9\\cdot \\pi^{2}}{8} - 2 + \\frac{3\\cdot \\pi^{3}}{8} + \\frac{9\\cdot \\pi}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "79897f89c0f076d2", "problem_id": "3b686d1a3403215a", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{9 \\pi^{2}}{8} - 2 + \\frac{3 \\pi^{3}}{8} + \\frac{9 \\pi}{2}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{9\\cdot \\pi^{2}}{8} - 2 + \\frac{3\\cdot \\pi^{3}}{8} + \\frac{9\\cdot \\pi}{2}$", "final_answer": "- \\frac{9\\cdot \\pi^{2}}{8} - 2 + \\frac{3\\cdot \\pi^{3}}{8} + \\frac{9\\cdot \\pi}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d45ea8d4e0dc776d", "problem_id": "5e9445dd9905de61", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$5 x - 6 y$ = $36$\n$5 y$ = $-30$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $5$: $25 y$ = $-150$.\n4) Subtract to eliminate x: $- 25 y$ = $150$.\n5) Solve for y: y = $-6$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, -6\\right)$", "final_answer": "(x,y)=\\left(0, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e8f9f68ae9800e8a", "problem_id": "5e9445dd9905de61", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the linear system:\n$5 x - 6 y$ = $36$\n$5 y$ = $-30$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $5$: $25 y$ = $-150$.\nFinal answer: $(x,y)=\\left(0, -6\\right)$", "final_answer": "(x,y)=\\left(0, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8cbbc7c79fd89aba", "problem_id": "ea6bc4716b35e45b", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A disease has prevalence $\\frac{1}{100}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{117}{2000}$.\n3) Compute P(D|+) = $\\frac{2}{13}$.\nFinal answer: $\\frac{2}{13}$", "final_answer": "\\frac{2}{13}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bcf642af397e470b", "problem_id": "ea6bc4716b35e45b", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A disease has prevalence $\\frac{1}{100}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{117}{2000}$.\nCompute P(D|+) = $\\frac{2}{13}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{2}{13}$", "final_answer": "\\frac{2}{13}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "52eec0dafaf76fdb", "problem_id": "220da9974e5a4ac9", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Worker A finishes a job in 14 hours and Worker B finishes it in 7 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/14 job/hour. B's rate = 1/7 job/hour.\n2) Combined rate = 1/14 + 1/7 = 3/14 job/hour.\n3) Time = 1 ÷ (combined rate) = 14/3 hours.\nFinal answer: $\\frac{14}{3}$", "final_answer": "\\frac{14}{3}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d78a792afab02b63", "problem_id": "220da9974e5a4ac9", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Worker A finishes a job in 14 hours and Worker B finishes it in 7 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/14 job/hour. B's rate = 1/7 job/hour.\nCombined rate = 1/14 + 1/7 = 3/14 job/hour.\nTime = 1 ÷ (combined rate) = 14/3 hours.\nCheck: (combined rate)×(time) = 3/14×14/3 = 1.\nFinal answer: $\\frac{14}{3}$", "final_answer": "\\frac{14}{3}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "97482bd43e98ffec", "problem_id": "7fb6f6de4181faea", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A cyclist rides at 35 miles per hour for 2 hours, then at 32 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 35 × 2 = 70 miles.\n2) Second leg distance = 32 × 5/2 = 80 miles.\n3) Total distance = 70 + 80 = 150 miles.\nFinal answer: $150$", "final_answer": "150", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ba00fb85864be394", "problem_id": "7fb6f6de4181faea", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 35 miles per hour for 2 hours, then at 32 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 35 × 2 = 70 miles.\nSecond leg distance = 32 × 5/2 = 80 miles.\nTotal distance = 70 + 80 = 150 miles.\nCheck: adding both legs gives 150.\nFinal answer: $150$", "final_answer": "150", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c31b0f325496a356", "problem_id": "146b0a6432aa0ef0", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "You want 55 liters of a 45% salt solution. You have a 20% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 55−x liters is 50% solution.\nSalt equation: 20x + 50(55−x) = 45·55.\n(20-50)x = (45-50)·55 ⇒ x = 55/6.\nFinal answer: $55/6 L of 20% and 275/6 L of 50%$", "final_answer": "55/6 L of 20% and 275/6 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0a517dbc671c49dd", "problem_id": "146b0a6432aa0ef0", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "You want 55 liters of a 45% salt solution. You have a 20% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 55−x liters is 50% solution.\n2) Salt equation: 20x + 50(55−x) = 45·55.\n3) (20-50)x = (45-50)·55 ⇒ x = 55/6.\n4) Then 55−x = 275/6.\nFinal answer: $55/6 L of 20% and 275/6 L of 50%$", "final_answer": "55/6 L of 20% and 275/6 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9f9c5765e5e595f0", "problem_id": "e2cf496453b09b14", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 15 red balls and 10 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(25,8).\n3) Favorable samples: choose k reds and n−k blues: C(15,2)·C(10,6).\n4) Compute totals: C(25,8) = 1081575, and favorable = 22050.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{98}{4807}$", "final_answer": "\\frac{98}{4807}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d6f5320b04b25d60", "problem_id": "e2cf496453b09b14", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 15 red balls and 10 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(25,8).\nFavorable samples: choose k reds and n−k blues: C(15,2)·C(10,6).\nCompute totals: C(25,8) = 1081575, and favorable = 22050.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{98}{4807}$.\nFinal answer: $\\frac{98}{4807}$", "final_answer": "\\frac{98}{4807}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8e76f5768f74528d", "problem_id": "a937a9e4c68fb6f5", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the equation of the tangent line to $y=4 x^{3} + 3 x + 4$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $12 x^{2} + 3$.\n2) Slope at x=-2: m = f'(-2) = $51$.\n3) Point on curve: (-2, f(-2)) = (-2, $-34$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -34 = 51(x − -2).\n6) Optionally expand: y = $51 x + 68$.\nFinal answer: $y = 51\\cdot x + 68$", "final_answer": "y = 51\\cdot x + 68", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d5c0355f5966a9d", "problem_id": "1973eb501f26a2f2", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} + 2 x + \\left(x^{2} - x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{x^{2} - x + 7}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7}) = \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 7}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{4 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 15 x^{2} - 14 x + 49} + \\frac{15 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{30 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{2 x^{2}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 15 x^{2} - 14 x + 49} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 2 x + 14} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 7} - \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7} - \\frac{14 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{28 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{40 x}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7} + \\frac{49 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{98 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{34}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7}) = \\frac{x^{2} \\left(1 - 2 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(x^{2} - x\\right) + 7\\right)^{2}} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(x^{2} - x\\right) + 7\\right)} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7} + \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(x - 1\\right) + 7\\right) \\left(- 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x - 1\\right) + 7\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(x - 1\\right) + 7\\right) \\left(- 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x - 1\\right) + 7\\right)^{2}}$", "final_answer": "\\frac{2 \\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(x - 1\\right) + 7\\right) \\left(- 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x - 1\\right) + 7\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7434219d07fb35be", "problem_id": "e9aa77d0bde413f9", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{6} x + \\frac{23 x}{10} - \\frac{1}{2} + \\frac{\\sqrt{6}}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{23}{10} - \\sqrt{6}$, c=$- \\frac{1}{2} + \\frac{\\sqrt{6}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{1329}{100} - \\frac{27 \\sqrt{6}}{5}$, so √Δ = $\\sqrt{\\frac{1329}{100} - \\frac{27 \\sqrt{6}}{5}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{23}{20} + \\frac{\\sqrt{1329 - 540 \\sqrt{6}}}{20} + \\frac{\\sqrt{6}}{2}$.\n7) x2 = $- \\frac{23}{20} - \\frac{\\sqrt{1329 - 540 \\sqrt{6}}}{20} + \\frac{\\sqrt{6}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{23}{20} + \\frac{\\sqrt{1329 - 540\\cdot \\sqrt{6}}}{20} + \\frac{\\sqrt{6}}{2}, - \\frac{23}{20} - \\frac{\\sqrt{1329 - 540\\cdot \\sqrt{6}}}{20} + \\frac{\\sqrt{6}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{23}{20} + \\frac{\\sqrt{1329 - 540\\cdot \\sqrt{6}}}{20} + \\frac{\\sqrt{6}}{2}, - \\frac{23}{20} - \\frac{\\sqrt{1329 - 540\\cdot \\sqrt{6}}}{20} + \\frac{\\sqrt{6}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d86962e5379c6280", "problem_id": "e9aa77d0bde413f9", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{6} x + \\frac{23 x}{10} - \\frac{1}{2} + \\frac{\\sqrt{6}}{5}$ = 0.", "output": "$x \\in \\{- \\frac{23}{20} + \\frac{\\sqrt{1329 - 540\\cdot \\sqrt{6}}}{20} + \\frac{\\sqrt{6}}{2}, - \\frac{23}{20} - \\frac{\\sqrt{1329 - 540\\cdot \\sqrt{6}}}{20} + \\frac{\\sqrt{6}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{23}{20} + \\frac{\\sqrt{1329 - 540\\cdot \\sqrt{6}}}{20} + \\frac{\\sqrt{6}}{2}, - \\frac{23}{20} - \\frac{\\sqrt{1329 - 540\\cdot \\sqrt{6}}}{20} + \\frac{\\sqrt{6}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1dacac4b0b81a3a0", "problem_id": "6f2facd11a3ea6fb", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡1 (mod 11), x≡7 (mod 10). Give the solution modulo 110.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 110.\nCompute the CRT solution to get x ≡ 67 (mod 110).\nCheck: 67 mod 11=1 and mod 10=7.\nFinal answer: $x\\equiv 67\\pmod{110}$", "final_answer": "x\\equiv 67\\pmod{110}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "708732fce0f39938", "problem_id": "6f2facd11a3ea6fb", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡1 (mod 11), x≡7 (mod 10). Give the solution modulo 110.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 110.\n3) Compute the CRT solution to get x ≡ 67 (mod 110).\nFinal answer: $x\\equiv 67\\pmod{110}$", "final_answer": "x\\equiv 67\\pmod{110}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a3eb5327c0fe0902", "problem_id": "adf3b2d15c594cf1", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=24$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=24 and k=10.\n4) Compute C(33,9) = 38567100.\nFinal answer: $38567100$", "final_answer": "38567100", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8d24bdb282308c7a", "problem_id": "adf3b2d15c594cf1", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=24$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=24 and k=10.\nCompute C(33,9) = 38567100.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $38567100$", "final_answer": "38567100", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c86536975cce47c4", "problem_id": "111b9ac673257213", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 6 x - 24$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} - 6 x - 24$ = $3 \\left(x - 4\\right) \\left(x + 2\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $4$.\nFrom the second factor: x = $-2$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{4, -2\\}$", "final_answer": "x \\in \\{4, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "47772f219f810685", "problem_id": "111b9ac673257213", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 6 x - 24$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} - 6 x - 24$ = $3 \\left(x - 4\\right) \\left(x + 2\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $4$.\n5) From the second factor: x = $-2$.\nFinal answer: $x \\in \\{4, -2\\}$", "final_answer": "x \\in \\{4, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ad7f2d5d61c76abb", "problem_id": "a8d54a88bb4e2537", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Two friends split $104 in the ratio 1:7. How much does each receive?", "output": "Total ratio parts = 1+7 = 8.\nEach part = 104 ÷ 8 = 13.\nFirst share = 1 × 13 = 13.\nFinal answer: $91$", "final_answer": "91", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dc1fea2d6a168e5f", "problem_id": "a8d54a88bb4e2537", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Two friends split $104 in the ratio 1:7. How much does each receive?", "output": "1) Total ratio parts = 1+7 = 8.\n2) Each part = 104 ÷ 8 = 13.\n3) First share = 1 × 13 = 13.\n4) Second share = 7 × 13 = 91.\nFinal answer: $91$", "final_answer": "91", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "79dab4ab90ee43d1", "problem_id": "2b88eb96e7476d3d", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{3} \\sin{\\left(x \\right)} - x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $2 x^{3} \\sin{\\left(x \\right)} - x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - x^{2}\\cdot \\cos{\\left(x \\right)} - 2\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}$", "final_answer": "2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - x^{2}\\cdot \\cos{\\left(x \\right)} - 2\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d8c017fc4deff730", "problem_id": "2b88eb96e7476d3d", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{3} \\sin{\\left(x \\right)} - x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $2 x^{3} \\sin{\\left(x \\right)} - x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - x^{2}\\cdot \\cos{\\left(x \\right)} - 2\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}$", "final_answer": "2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - x^{2}\\cdot \\cos{\\left(x \\right)} - 2\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "248c2bb8f647742f", "problem_id": "f4f54f1b57e3d6a2", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A disease has prevalence $\\frac{2}{25}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{4}{25}$.\n3) Compute P(D|+) = $\\frac{17}{40}$.\nFinal answer: $\\frac{17}{40}$", "final_answer": "\\frac{17}{40}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a371ca9ffa4462b4", "problem_id": "fd2bcac6548591c0", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\cos{\\left(2 x \\right)}) = - \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\cos{\\left(2 x \\right)}) = - 2 \\sin{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$", "final_answer": "x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ae548198178a3bfb", "problem_id": "6a1845fa22bbe199", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the congruences: x≡0 (mod 5), x≡10 (mod 11). Give the solution modulo 55.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 55.\n3) Compute the CRT solution to get x ≡ 10 (mod 55).\nFinal answer: $x\\equiv 10\\pmod{55}$", "final_answer": "x\\equiv 10\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a18e4c7bdb3e2a36", "problem_id": "6a1845fa22bbe199", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡0 (mod 5), x≡10 (mod 11). Give the solution modulo 55.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 55.\nCompute the CRT solution to get x ≡ 10 (mod 55).\nCheck: 10 mod 5=0 and mod 11=10.\nFinal answer: $x\\equiv 10\\pmod{55}$", "final_answer": "x\\equiv 10\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bcb8292f1e77e0a9", "problem_id": "3152143d55272e21", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - x^{2} - x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)}$ at $x=2$.", "output": "Differentiate: f'(x) = $- 9 x^{2} - 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 1$.\nSlope at x=2: m = f'(2) = $-41 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\nPoint on curve: (2, f(2)) = (2, $-30 + \\cos{\\left(4 \\right)} + e$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -30 + \\cos{\\left(4 \\right)} + e = -41 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\nOptionally expand: y = $- 41 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 52$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 41\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 52$", "final_answer": "y = - 41\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 52", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "83eb6c59e420c5cc", "problem_id": "3152143d55272e21", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - x^{2} - x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)}$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 9 x^{2} - 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 1$.\n2) Slope at x=2: m = f'(2) = $-41 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\n3) Point on curve: (2, f(2)) = (2, $-30 + \\cos{\\left(4 \\right)} + e$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -30 + \\cos{\\left(4 \\right)} + e = -41 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\n6) Optionally expand: y = $- 41 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 52$.\nFinal answer: $y = - 41\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 52$", "final_answer": "y = - 41\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 52", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c1eef9479de3f021", "problem_id": "a51df341928c6f2c", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the multiplicative inverse of 8 modulo 19.", "output": "1) We want x such that 8x ≡ 1 (mod 19).\n2) Use extended Euclid to find s,t with 8s + 19t = 1.\n3) One Bezout identity is: 8·(-7) + 19·(3) = 1.\n4) Reduce s modulo 19: x ≡ 12 (mod 19).\nFinal answer: $x\\equiv 12\\pmod{19}$", "final_answer": "x\\equiv 12\\pmod{19}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f96e4c8b1863f949", "problem_id": "a51df341928c6f2c", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 8 modulo 19.", "output": "We want x such that 8x ≡ 1 (mod 19).\nUse extended Euclid to find s,t with 8s + 19t = 1.\nOne Bezout identity is: 8·(-7) + 19·(3) = 1.\nReduce s modulo 19: x ≡ 12 (mod 19).\nCheck: 8·12 mod 19 = 1.\nFinal answer: $x\\equiv 12\\pmod{19}$", "final_answer": "x\\equiv 12\\pmod{19}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b4b3849941e08adc", "problem_id": "4a41ce2a613a58bd", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the linear system:\n$2 x + y$ = $4$\n$- 4 x + y$ = $-20$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $- 8 x - 4 y$ = $-16$.\n3) Multiply the second equation by $2$: $- 8 x + 2 y$ = $-40$.\n4) Subtract to eliminate x: $- 6 y$ = $24$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, -4\\right)$", "final_answer": "(x,y)=\\left(4, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "90a12fd8df92f0f6", "problem_id": "30d6cf9a97495d00", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{x^{2} + 3}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 3}) = \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} - \\frac{2 x^{4}}{2 x^{4} + 12 x^{2} + 18} + \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 6 x^{2} + 9} + \\frac{6 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} + \\frac{12 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} - \\frac{12 x^{2}}{2 x^{4} + 12 x^{2} + 18} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 6} - \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3} + \\frac{36 x}{2 x^{4} + 12 x^{2} + 18} - \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 3} + \\frac{9 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} + \\frac{18 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} - \\frac{18}{2 x^{4} + 12 x^{2} + 18}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 3}) = - \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3\\right)^{2}} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x^{2} + 3\\right)} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3} + \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 3}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 4 x \\left(- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} + 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x^{2} + 3\\right)^{2}}$.\nFinal answer: $\\frac{- 4 x \\left(- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} + 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x^{2} + 3\\right)^{2}}$", "final_answer": "\\frac{- 4 x \\left(- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} + 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x^{2} + 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "994e744219769a5c", "problem_id": "b3b96579d7b196e2", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the equation of the tangent line to $y=x^{3} - 2 x^{2} - x - 2$ at $x=-1$.", "output": "Differentiate: f'(x) = $3 x^{2} - 4 x - 1$.\nSlope at x=-1: m = f'(-1) = $6$.\nPoint on curve: (-1, f(-1)) = (-1, $-4$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -4 = 6(x − -1).\nOptionally expand: y = $6 x + 2$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 6\\cdot x + 2$", "final_answer": "y = 6\\cdot x + 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8cc6361018bde532", "problem_id": "02f5ba068679179f", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(- x^{3} + 2 x^{2} - 2 x - 2\\right) \\cos{\\left(2 x - 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x^{2} \\cos{\\left(2 x - 2 \\right)}) = - 2 x^{2} \\sin{\\left(2 x - 2 \\right)}$.\n3) So $f'(x) = 2 x^{3} \\sin{\\left(2 x - 2 \\right)} - 2 x^{2} \\sin{\\left(2 x - 2 \\right)} - 3 x^{2} \\cos{\\left(2 x - 2 \\right)} + x e^{x} + 4 x \\sin{\\left(2 x - 2 \\right)} - 2 e^{x} + 4 \\sin{\\left(2 x - 2 \\right)} - 2 \\cos{\\left(2 x - 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x^{2} \\cos{\\left(2 x - 2 \\right)}) = - 4 x^{2} \\sin{\\left(2 x - 2 \\right)} + 4 x \\cos{\\left(2 x - 2 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x - 3\\right) e^{x} - 2 \\left(x \\left(x \\left(2 - x\\right) - 2\\right) - 2\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(- 3 x^{2} + 4 x - 2\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}$.\nFinal answer: $\\left(x - 3\\right) e^{x} - 2 \\left(x \\left(x \\left(2 - x\\right) - 2\\right) - 2\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(- 3 x^{2} + 4 x - 2\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}$", "final_answer": "\\left(x - 3\\right) e^{x} - 2 \\left(x \\left(x \\left(2 - x\\right) - 2\\right) - 2\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(- 3 x^{2} + 4 x - 2\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c143da0c21435fdd", "problem_id": "35a7a6013cb5cb70", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A cyclist rides at 52 miles per hour for 3/2 hours, then at 88 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 52 × 3/2 = 78 miles.\n2) Second leg distance = 88 × 1 = 88 miles.\n3) Total distance = 78 + 88 = 166 miles.\nFinal answer: $166$", "final_answer": "166", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a40a7edd46af62e", "problem_id": "35a7a6013cb5cb70", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A cyclist rides at 52 miles per hour for 3/2 hours, then at 88 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 52 × 3/2 = 78 miles.\nSecond leg distance = 88 × 1 = 88 miles.\nTotal distance = 78 + 88 = 166 miles.\nCheck: adding both legs gives 166.\nFinal answer: $166$", "final_answer": "166", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "96863dd7650ce8c9", "problem_id": "1f8f9df2523c1315", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Two friends split $960 in the ratio 3:5. How much does each receive?", "output": "Total ratio parts = 3+5 = 8.\nEach part = 960 ÷ 8 = 120.\nFirst share = 3 × 120 = 360.\nFinal answer: $600$", "final_answer": "600", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "491bf5ff889080ad", "problem_id": "1f8f9df2523c1315", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Two friends split $960 in the ratio 3:5. How much does each receive?", "output": "Total ratio parts = 3+5 = 8.\nEach part = 960 ÷ 8 = 120.\nFirst share = 3 × 120 = 360.\nSecond share = 5 × 120 = 600.\nCheck: 360+600=960.\nFinal answer: $600$", "final_answer": "600", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "166e9c02503d9259", "problem_id": "614271b3aeb2dc06", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} - 3 x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\n2) Evaluate at the bounds: F(2 \\pi) and F(0).\n3) Subtract: F(2 \\pi) − F(0) = $\\frac{2 \\pi \\left(- 48 \\pi^{4} - 20 \\pi^{3} - 10 \\pi + 5 + 20 \\pi^{2}\\right)}{5}$.\n4) Simplify the result.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(- 48\\cdot \\pi^{4} - 20\\cdot \\pi^{3} - 10\\cdot \\pi + 5 + 20\\cdot \\pi^{2}\\right)}{5}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(- 48\\cdot \\pi^{4} - 20\\cdot \\pi^{3} - 10\\cdot \\pi + 5 + 20\\cdot \\pi^{2}\\right)}{5}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "23936b6b7815ab29", "problem_id": "614271b3aeb2dc06", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} - 3 x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\nEvaluate at the bounds: F(2 \\pi) and F(0).\nSubtract: F(2 \\pi) − F(0) = $\\frac{2 \\pi \\left(- 48 \\pi^{4} - 20 \\pi^{3} - 10 \\pi + 5 + 20 \\pi^{2}\\right)}{5}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(- 48\\cdot \\pi^{4} - 20\\cdot \\pi^{3} - 10\\cdot \\pi + 5 + 20\\cdot \\pi^{2}\\right)}{5}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(- 48\\cdot \\pi^{4} - 20\\cdot \\pi^{3} - 10\\cdot \\pi + 5 + 20\\cdot \\pi^{2}\\right)}{5}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2c5c33fca9a863da", "problem_id": "b7cb9ea75a26e522", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 25 red balls and 10 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(35,9).\nFavorable samples: choose k reds and n−k blues: C(25,4)·C(10,5).\nCompute totals: C(35,9) = 70607460, and favorable = 3187800.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{690}{15283}$.\nFinal answer: $\\frac{690}{15283}$", "final_answer": "\\frac{690}{15283}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c189a8fbb82429e9", "problem_id": "b7cb9ea75a26e522", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "An urn has 25 red balls and 10 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(35,9).\n3) Favorable samples: choose k reds and n−k blues: C(25,4)·C(10,5).\n4) Compute totals: C(35,9) = 70607460, and favorable = 3187800.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{690}{15283}$", "final_answer": "\\frac{690}{15283}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e81635f770396345", "problem_id": "318c4a3da672021a", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$- x + 3 y$ = $4$\n$- 3 x + 3 y$ = $-6$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $3 x - 9 y$ = $-12$.\n3) Multiply the second equation by $-1$: $3 x - 3 y$ = $6$.\n4) Subtract to eliminate x: $- 6 y$ = $-18$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, 3\\right)$", "final_answer": "(x,y)=\\left(5, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec9e1fc5be2cd707", "problem_id": "f175f8884f0e5adc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} x^{3} - 2 x^{2} + 6 x - 24$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to x^{3} - 2 x^{2} + 5 x - 24.\n3) Then the limit is -140.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: -144.\nFinal answer: $-144$", "final_answer": "-144", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "eba5bd07481c47bc", "problem_id": "8583802c0bda0482", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{11 x}{3} - \\frac{\\sqrt{11} x}{2} + \\frac{5 \\sqrt{11}}{6} + \\frac{10}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{11}{3} - \\frac{\\sqrt{11}}{2}$, c=$\\frac{5 \\sqrt{11}}{6} + \\frac{10}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{\\sqrt{11}}{3} + \\frac{103}{36}$, so √Δ = $\\sqrt{\\frac{\\sqrt{11}}{3} + \\frac{103}{36}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{11}}{4} + \\frac{\\sqrt{12 \\sqrt{11} + 103}}{12} + \\frac{11}{6}$.\n7) x2 = $- \\frac{\\sqrt{12 \\sqrt{11} + 103}}{12} + \\frac{\\sqrt{11}}{4} + \\frac{11}{6}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{11}}{4} + \\frac{\\sqrt{12\\cdot \\sqrt{11} + 103}}{12} + \\frac{11}{6}, - \\frac{\\sqrt{12\\cdot \\sqrt{11} + 103}}{12} + \\frac{\\sqrt{11}}{4} + \\frac{11}{6}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{11}}{4} + \\frac{\\sqrt{12\\cdot \\sqrt{11} + 103}}{12} + \\frac{11}{6}, - \\frac{\\sqrt{12\\cdot \\sqrt{11} + 103}}{12} + \\frac{\\sqrt{11}}{4} + \\frac{11}{6}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4a4a821a2227958d", "problem_id": "8583802c0bda0482", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{11 x}{3} - \\frac{\\sqrt{11} x}{2} + \\frac{5 \\sqrt{11}}{6} + \\frac{10}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{11}{3} - \\frac{\\sqrt{11}}{2}$, c=$\\frac{5 \\sqrt{11}}{6} + \\frac{10}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{\\sqrt{11}}{3} + \\frac{103}{36}$, so √Δ = $\\sqrt{\\frac{\\sqrt{11}}{3} + \\frac{103}{36}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{11}}{4} + \\frac{\\sqrt{12 \\sqrt{11} + 103}}{12} + \\frac{11}{6}$.\nx2 = $- \\frac{\\sqrt{12 \\sqrt{11} + 103}}{12} + \\frac{\\sqrt{11}}{4} + \\frac{11}{6}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{11}}{4} + \\frac{\\sqrt{12\\cdot \\sqrt{11} + 103}}{12} + \\frac{11}{6}, - \\frac{\\sqrt{12\\cdot \\sqrt{11} + 103}}{12} + \\frac{\\sqrt{11}}{4} + \\frac{11}{6}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{11}}{4} + \\frac{\\sqrt{12\\cdot \\sqrt{11} + 103}}{12} + \\frac{11}{6}, - \\frac{\\sqrt{12\\cdot \\sqrt{11} + 103}}{12} + \\frac{\\sqrt{11}}{4} + \\frac{11}{6}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ad9ec51dcd7798f5", "problem_id": "e404dd7172894a04", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 7 red balls and 11 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(18,4).\nFavorable samples: choose k reds and n−k blues: C(7,1)·C(11,3).\nCompute totals: C(18,4) = 3060, and favorable = 1155.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{77}{204}$.\nFinal answer: $\\frac{77}{204}$", "final_answer": "\\frac{77}{204}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f5dbf4d335fa228b", "problem_id": "e404dd7172894a04", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "An urn has 7 red balls and 11 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(18,4).\n3) Favorable samples: choose k reds and n−k blues: C(7,1)·C(11,3).\n4) Compute totals: C(18,4) = 3060, and favorable = 1155.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{77}{204}$", "final_answer": "\\frac{77}{204}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9539d4a3bcca4d97", "problem_id": "44410728f2ef1171", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "An urn has 17 red balls and 15 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(32,7).\n3) Favorable samples: choose k reds and n−k blues: C(17,1)·C(15,6).\n4) Compute totals: C(32,7) = 3365856, and favorable = 85085.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{6545}{258912}$", "final_answer": "\\frac{6545}{258912}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "05c668ea4eaa43b1", "problem_id": "44410728f2ef1171", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "An urn has 17 red balls and 15 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(32,7).\nFavorable samples: choose k reds and n−k blues: C(17,1)·C(15,6).\nFinal answer: $\\frac{6545}{258912}$", "final_answer": "\\frac{6545}{258912}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f8740f95ac1aa619", "problem_id": "97cbb6047e4471c5", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 3 x^{2} + 9 x + 18$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{2} + 8 x + 18.\n3) Then the limit is 46.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 48.\nFinal answer: $48$", "final_answer": "48", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "03fa7d39e4b335eb", "problem_id": "68130d6277ed8575", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 15 times, what is the probability of exactly 8 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=15, k=8, p=$\\frac{3}{7}$ and simplify.\n4) Here C(15,8) = 6435.\nFinal answer: $\\frac{691733053440}{4747561509943}$", "final_answer": "\\frac{691733053440}{4747561509943}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "36676bb3a3b5977c", "problem_id": "b4ac2e49c555db6c", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 11 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=6, p=$\\frac{2}{5}$ and simplify.\n4) Here C(11,6) = 462.\nFinal answer: $\\frac{7185024}{48828125}$", "final_answer": "\\frac{7185024}{48828125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d5720a04a119af5c", "problem_id": "37c1fa38fb04f96b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 2 x \\sin{\\left(x \\right)}) = - 2 x \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 2 x \\sin{\\left(x \\right)}) = - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- x^{3} \\cos{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 3 \\sin{\\left(2 x \\right)}\\right)$.\nFinal answer: $2 \\left(- x^{3} \\cos{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 3 \\sin{\\left(2 x \\right)}\\right)$", "final_answer": "2 \\left(- x^{3} \\cos{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 3 \\sin{\\left(2 x \\right)}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "07e6589925e43749", "problem_id": "2b8ba01075122a2d", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=24$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=24 and k=5.\n4) Compute C(28,4) = 20475.\nFinal answer: $20475$", "final_answer": "20475", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c013d8ec57296163", "problem_id": "2b8ba01075122a2d", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=24$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=24 and k=5.\nFinal answer: $20475$", "final_answer": "20475", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "51482605dcc9e160", "problem_id": "89d6d184019b9d06", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(3 x^{2} + 1\\right) \\cos{\\left(3 x + 2 \\right)}$.", "output": "$6\\cdot x\\cdot \\cos{\\left(3\\cdot x + 2 \\right)} + \\left(x - 3\\right)\\cdot e^{x} - \\left(9\\cdot x^{2} + 3\\right)\\cdot \\sin{\\left(3\\cdot x + 2 \\right)} + e^{x}$", "final_answer": "6\\cdot x\\cdot \\cos{\\left(3\\cdot x + 2 \\right)} + \\left(x - 3\\right)\\cdot e^{x} - \\left(9\\cdot x^{2} + 3\\right)\\cdot \\sin{\\left(3\\cdot x + 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d476d5411e23cb3", "problem_id": "89d6d184019b9d06", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(3 x^{2} + 1\\right) \\cos{\\left(3 x + 2 \\right)}$.", "output": "1) Let f(x) = $\\left(x - 3\\right) e^{x} + \\left(3 x^{2} + 1\\right) \\cos{\\left(3 x + 2 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $6 x \\cos{\\left(3 x + 2 \\right)} + \\left(x - 3\\right) e^{x} - 3 \\left(3 x^{2} + 1\\right) \\sin{\\left(3 x + 2 \\right)} + e^{x}$.\nFinal answer: $6\\cdot x\\cdot \\cos{\\left(3\\cdot x + 2 \\right)} + \\left(x - 3\\right)\\cdot e^{x} - \\left(9\\cdot x^{2} + 3\\right)\\cdot \\sin{\\left(3\\cdot x + 2 \\right)} + e^{x}$", "final_answer": "6\\cdot x\\cdot \\cos{\\left(3\\cdot x + 2 \\right)} + \\left(x - 3\\right)\\cdot e^{x} - \\left(9\\cdot x^{2} + 3\\right)\\cdot \\sin{\\left(3\\cdot x + 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2106e9c7af08efde", "problem_id": "427f4ac468aefa28", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A store sells a backpack for $70. It is discounted by 10%. You buy 10 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 70 × (1 − 10/100) = 70 × 90/100 = 63.\n2) Subtotal = 10 × 63 = 630.\n3) Total with tax = 630 × 108/100 = 3402/5.\nFinal answer: $\\frac{3402}{5}$", "final_answer": "\\frac{3402}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d87fb79c08a36eb1", "problem_id": "427f4ac468aefa28", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A store sells a backpack for $70. It is discounted by 10%. You buy 10 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 70 × (1 − 10/100) = 70 × 90/100 = 63.\nSubtotal = 10 × 63 = 630.\nTotal with tax = 630 × 108/100 = 3402/5.\nFinal answer: $\\frac{3402}{5}$", "final_answer": "\\frac{3402}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "24455fc256121a35", "problem_id": "7657725ec4aa3516", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Worker A finishes a job in 9 hours and Worker B finishes it in 10 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/9 job/hour. B's rate = 1/10 job/hour.\n2) Combined rate = 1/9 + 1/10 = 19/90 job/hour.\n3) Time = 1 ÷ (combined rate) = 90/19 hours.\nFinal answer: $\\frac{90}{19}$", "final_answer": "\\frac{90}{19}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "07d954ff530efee5", "problem_id": "7657725ec4aa3516", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Worker A finishes a job in 9 hours and Worker B finishes it in 10 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/9 job/hour. B's rate = 1/10 job/hour.\nCombined rate = 1/9 + 1/10 = 19/90 job/hour.\nTime = 1 ÷ (combined rate) = 90/19 hours.\nCheck: (combined rate)×(time) = 19/90×90/19 = 1.\nFinal answer: $\\frac{90}{19}$", "final_answer": "\\frac{90}{19}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7e6485333de96829", "problem_id": "740fc6c695589186", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡10 (mod 13). Give the solution modulo 39.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 39.\n3) Compute the CRT solution to get x ≡ 10 (mod 39).\nFinal answer: $x\\equiv 10\\pmod{39}$", "final_answer": "x\\equiv 10\\pmod{39}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a86cf9bbf0af417", "problem_id": "740fc6c695589186", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡10 (mod 13). Give the solution modulo 39.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 39.\nCompute the CRT solution to get x ≡ 10 (mod 39).\nCheck: 10 mod 3=1 and mod 13=10.\nFinal answer: $x\\equiv 10\\pmod{39}$", "final_answer": "x\\equiv 10\\pmod{39}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fae990ae976090c2", "problem_id": "1973eb501f26a2f2", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} + 2 x + \\left(x^{2} - x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{x^{2} - x + 7}$.", "output": "1) Let f(x) = $\\frac{- x^{2} + 2 x + \\left(x^{2} - x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{x^{2} - x + 7}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{2 \\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + \\left(x^{2} - x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x^{2} - x + 7\\right) \\left(- 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} - x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - x + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x^{2} - x + 7\\right)^{2}}$.\nFinal answer: $\\frac{\\left(2 - 4\\cdot x\\right)\\cdot \\left(- x^{2} + 2\\cdot x + \\left(x^{2} - x + 7\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 3\\right) + \\left(x^{2} - x + 7\\right)\\cdot \\left(- 4\\cdot x + \\left(4\\cdot x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(x^{2} - x + 7\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} - 2\\cdot x + 14\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\right)}{2\\cdot \\left(x^{2} - x + 7\\right)^{2}}$", "final_answer": "\\frac{\\left(2 - 4\\cdot x\\right)\\cdot \\left(- x^{2} + 2\\cdot x + \\left(x^{2} - x + 7\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 3\\right) + \\left(x^{2} - x + 7\\right)\\cdot \\left(- 4\\cdot x + \\left(4\\cdot x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(x^{2} - x + 7\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} - 2\\cdot x + 14\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\right)}{2\\cdot \\left(x^{2} - x + 7\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "902c5ebbe1e24f83", "problem_id": "f988366da0f25484", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{3} - x^{2} - 3 x + \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{3 x - 2}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 2}) = \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 2}$.\n3) So $f'(x) = - \\frac{36 x^{3}}{18 x^{2} - 24 x + 8} + \\frac{9 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} - 24 x + 8} + \\frac{18 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} - 24 x + 8} + \\frac{30 x^{2}}{18 x^{2} - 24 x + 8} - \\frac{12 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} - 24 x + 8} - \\frac{24 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} - 24 x + 8} + \\frac{8 x}{18 x^{2} - 24 x + 8} + \\frac{9 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{2} - 12 x + 4} - \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x - 4} + \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 2} - \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 2} + \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} - 24 x + 8} + \\frac{8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} - 24 x + 8} + \\frac{24}{18 x^{2} - 24 x + 8} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 2}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 2}) = \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(3 x - 2\\right)} + \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 2} - \\frac{9 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(3 x - 2\\right)^{2}} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 2}$.\nTherefore the correct derivative is $f'(x) = \\frac{18 x^{3} + 6 x^{2} + 18 x + \\left(3 x - 2\\right) \\left(- 18 x^{2} - 4 x + \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6\\right) - 6 \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 12}{2 \\left(3 x - 2\\right)^{2}}$.\nFinal answer: $\\frac{18 x^{3} + 6 x^{2} + 18 x + \\left(3 x - 2\\right) \\left(- 18 x^{2} - 4 x + \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6\\right) - 6 \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 12}{2 \\left(3 x - 2\\right)^{2}}$", "final_answer": "\\frac{18 x^{3} + 6 x^{2} + 18 x + \\left(3 x - 2\\right) \\left(- 18 x^{2} - 4 x + \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6\\right) - 6 \\left(3 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 12}{2 \\left(3 x - 2\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "38c64e7efbbc7703", "problem_id": "b6247d562fb05d69", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} - 4 x + 3$.", "output": "1) Differentiate term-by-term:\n2) d/dx(3) = 0.\n3) d/dx(- 4 x) = -4.\n4) d/dx(2 x^{2}) = 4 x.\n5) Combine: f'(x) = 4 \\left(x - 1\\right).\nFinal answer: $4\\cdot x - 4$", "final_answer": "4\\cdot x - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c0c7789ca25f7eb3", "problem_id": "b6247d562fb05d69", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} - 4 x + 3$.", "output": "Differentiate term-by-term:\nd/dx(3) = 0.\nd/dx(- 4 x) = -4.\nFinal answer: $4\\cdot x - 4$", "final_answer": "4\\cdot x - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "334c3a6d2a7bdd11", "problem_id": "12717b20a9259407", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $2 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4be92c2d2337cf3e", "problem_id": "12717b20a9259407", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $2 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f21500f31753c556", "problem_id": "22f3bab636f40a94", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the multiplicative inverse of 8 modulo 31.", "output": "1) We want x such that 8x ≡ 1 (mod 31).\n2) Use extended Euclid to find s,t with 8s + 31t = 1.\n3) One Bezout identity is: 8·(4) + 31·(-1) = 1.\n4) Reduce s modulo 31: x ≡ 4 (mod 31).\nFinal answer: $x\\equiv 4\\pmod{31}$", "final_answer": "x\\equiv 4\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c374151cc07ef320", "problem_id": "bda48a2c50db7cae", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} \\sin{\\left(x \\right)} - 3 x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 \\cos{\\left(2 x \\right)}) = - 3 \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 3 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 \\cos{\\left(2 x \\right)}) = - 6 \\sin{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.\nFinal answer: $x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$", "final_answer": "x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "440697e65dbd6a5e", "problem_id": "491d2091f114a1b1", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$- 3 x - 3 y$ = $-27$\n$- 5 x + 8 y$ = $46$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-5$: $15 x + 15 y$ = $135$.\nMultiply the second equation by $-3$: $15 x - 24 y$ = $-138$.\nSubtract to eliminate x: $39 y$ = $273$.\nSolve for y: y = $7$.\nSubstitute back to find x: x = $2$.\nCheck: plug (x,y)=(2,7) into both equations.\nFinal answer: $(x,y)=\\left(2, 7\\right)$", "final_answer": "(x,y)=\\left(2, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e8041add68f90044", "problem_id": "491d2091f114a1b1", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$- 3 x - 3 y$ = $-27$\n$- 5 x + 8 y$ = $46$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-5$: $15 x + 15 y$ = $135$.\n3) Multiply the second equation by $-3$: $15 x - 24 y$ = $-138$.\n4) Subtract to eliminate x: $39 y$ = $273$.\n5) Solve for y: y = $7$.\n6) Substitute back to find x: x = $2$.\nFinal answer: $(x,y)=\\left(2, 7\\right)$", "final_answer": "(x,y)=\\left(2, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1d31980896e5fbe1", "problem_id": "0dfb54c352e18c5c", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=23$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=23 and k=10.\n4) Compute C(32,9) = 28048800.\nFinal answer: $28048800$", "final_answer": "28048800", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "40896a492fcf3a6b", "problem_id": "0dfb54c352e18c5c", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=23$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=23 and k=10.\nCompute C(32,9) = 28048800.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $28048800$", "final_answer": "28048800", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d2ab5ad69c20b7a5", "problem_id": "dfd78a42f9e17578", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Find the equation of the tangent line to $y=x^{2} + 2 \\sin{\\left(x \\right)} + 3$ at $x=0$.", "output": "Differentiate: f'(x) = $2 x + 2 \\cos{\\left(x \\right)}$.\nSlope at x=0: m = f'(0) = $2$.\nPoint on curve: (0, f(0)) = (0, $3$).\nFinal answer: $y = 2\\cdot x + 3$", "final_answer": "y = 2\\cdot x + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e6dc005d43711a9", "problem_id": "dfd78a42f9e17578", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the equation of the tangent line to $y=x^{2} + 2 \\sin{\\left(x \\right)} + 3$ at $x=0$.", "output": "1) Differentiate: f'(x) = $2 x + 2 \\cos{\\left(x \\right)}$.\n2) Slope at x=0: m = f'(0) = $2$.\n3) Point on curve: (0, f(0)) = (0, $3$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 3 = 2(x − 0).\n6) Optionally expand: y = $2 x + 3$.\nFinal answer: $y = 2\\cdot x + 3$", "final_answer": "y = 2\\cdot x + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b297bde725e53ed9", "problem_id": "881a09c5d047fe81", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the linear system:\n$- 10 x + 2 y$ = $-6$\n$7 x - 5 y$ = $-3$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $7$: $- 70 x + 14 y$ = $-42$.\nMultiply the second equation by $-10$: $- 70 x + 50 y$ = $30$.\nFinal answer: $(x,y)=\\left(1, 2\\right)$", "final_answer": "(x,y)=\\left(1, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c233b28481c1150f", "problem_id": "881a09c5d047fe81", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$- 10 x + 2 y$ = $-6$\n$7 x - 5 y$ = $-3$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $7$: $- 70 x + 14 y$ = $-42$.\n3) Multiply the second equation by $-10$: $- 70 x + 50 y$ = $30$.\n4) Subtract to eliminate x: $- 36 y$ = $-72$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $1$.\nFinal answer: $(x,y)=\\left(1, 2\\right)$", "final_answer": "(x,y)=\\left(1, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ee2082f59dfb115d", "problem_id": "d82db2e445c4625c", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Worker A finishes a job in 22 hours and Worker B finishes it in 11 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/22 job/hour. B's rate = 1/11 job/hour.\n2) Combined rate = 1/22 + 1/11 = 3/22 job/hour.\n3) Time = 1 ÷ (combined rate) = 22/3 hours.\nFinal answer: $\\frac{22}{3}$", "final_answer": "\\frac{22}{3}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "01e474938c7e2491", "problem_id": "d82db2e445c4625c", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Worker A finishes a job in 22 hours and Worker B finishes it in 11 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/22 job/hour. B's rate = 1/11 job/hour.\nCombined rate = 1/22 + 1/11 = 3/22 job/hour.\nTime = 1 ÷ (combined rate) = 22/3 hours.\nCheck: (combined rate)×(time) = 3/22×22/3 = 1.\nFinal answer: $\\frac{22}{3}$", "final_answer": "\\frac{22}{3}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8ac1e174ac636209", "problem_id": "2ec4757f8d1d140e", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Two friends split $924 in the ratio 1:5. How much does each receive?", "output": "1) Total ratio parts = 1+5 = 6.\n2) Each part = 924 ÷ 6 = 154.\n3) First share = 1 × 154 = 154.\n4) Second share = 5 × 154 = 770.\nFinal answer: $770$", "final_answer": "770", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a56a83a077ba1d28", "problem_id": "2ec4757f8d1d140e", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $924 in the ratio 1:5. How much does each receive?", "output": "Total ratio parts = 1+5 = 6.\nEach part = 924 ÷ 6 = 154.\nFirst share = 1 × 154 = 154.\nSecond share = 5 × 154 = 770.\nCheck: 154+770=924.\nFinal answer: $770$", "final_answer": "770", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eaf2a0f876cd56b2", "problem_id": "dfac23609a570bd4", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 20 red balls and 30 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(50,8).\n3) Favorable samples: choose k reds and n−k blues: C(20,2)·C(30,6).\n4) Compute totals: C(50,8) = 536878650, and favorable = 112817250.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{107445}{511313}$", "final_answer": "\\frac{107445}{511313}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3c728dc212e3112a", "problem_id": "dfac23609a570bd4", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "An urn has 20 red balls and 30 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(50,8).\nFavorable samples: choose k reds and n−k blues: C(20,2)·C(30,6).\nFinal answer: $\\frac{107445}{511313}$", "final_answer": "\\frac{107445}{511313}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c4370f31ef86bc7f", "problem_id": "e19a7cb0d139eedd", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Worker A finishes a job in 16 hours and Worker B finishes it in 14 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/16 job/hour. B's rate = 1/14 job/hour.\n2) Combined rate = 1/16 + 1/14 = 15/112 job/hour.\n3) Time = 1 ÷ (combined rate) = 112/15 hours.\nFinal answer: $\\frac{112}{15}$", "final_answer": "\\frac{112}{15}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "33c175a82c33182c", "problem_id": "e19a7cb0d139eedd", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Worker A finishes a job in 16 hours and Worker B finishes it in 14 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/16 job/hour. B's rate = 1/14 job/hour.\nCombined rate = 1/16 + 1/14 = 15/112 job/hour.\nTime = 1 ÷ (combined rate) = 112/15 hours.\nCheck: (combined rate)×(time) = 15/112×112/15 = 1.\nFinal answer: $\\frac{112}{15}$", "final_answer": "\\frac{112}{15}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3cebcfc8a192980a", "problem_id": "8becc4ad099fc928", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 3 x^{2} + 12 x - 35$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} + 11 x - 35.\n3) Then the limit is -95.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -98.\nFinal answer: $-98$", "final_answer": "-98", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d13f7eff2726eee5", "problem_id": "21d05eaad5516f9d", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{7 x}{5} - \\frac{2 \\sqrt{2} x}{3} - \\frac{6}{5} - \\frac{2 \\sqrt{2}}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{7}{5} - \\frac{2 \\sqrt{2}}{3}$, c=$- \\frac{6}{5} - \\frac{2 \\sqrt{2}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{52 \\sqrt{2}}{15} + \\frac{1721}{225}$, so √Δ = $\\sqrt{\\frac{52 \\sqrt{2}}{15} + \\frac{1721}{225}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{2}}{3} + \\frac{7}{10} + \\frac{\\sqrt{780 \\sqrt{2} + 1721}}{30}$.\n7) x2 = $- \\frac{\\sqrt{780 \\sqrt{2} + 1721}}{30} + \\frac{\\sqrt{2}}{3} + \\frac{7}{10}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{2}}{3} + \\frac{7}{10} + \\frac{\\sqrt{780\\cdot \\sqrt{2} + 1721}}{30}, - \\frac{\\sqrt{780\\cdot \\sqrt{2} + 1721}}{30} + \\frac{\\sqrt{2}}{3} + \\frac{7}{10}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{2}}{3} + \\frac{7}{10} + \\frac{\\sqrt{780\\cdot \\sqrt{2} + 1721}}{30}, - \\frac{\\sqrt{780\\cdot \\sqrt{2} + 1721}}{30} + \\frac{\\sqrt{2}}{3} + \\frac{7}{10}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "76445789b5e83408", "problem_id": "b37b20769ddd11d9", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Worker A finishes a job in 19 hours and Worker B finishes it in 10 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/19 job/hour. B's rate = 1/10 job/hour.\n2) Combined rate = 1/19 + 1/10 = 29/190 job/hour.\n3) Time = 1 ÷ (combined rate) = 190/29 hours.\nFinal answer: $\\frac{190}{29}$", "final_answer": "\\frac{190}{29}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "40da4d96700c3138", "problem_id": "b37b20769ddd11d9", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Worker A finishes a job in 19 hours and Worker B finishes it in 10 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/19 job/hour. B's rate = 1/10 job/hour.\nCombined rate = 1/19 + 1/10 = 29/190 job/hour.\nTime = 1 ÷ (combined rate) = 190/29 hours.\nCheck: (combined rate)×(time) = 29/190×190/29 = 1.\nFinal answer: $\\frac{190}{29}$", "final_answer": "\\frac{190}{29}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f34965612b9041a9", "problem_id": "3861645a96784d24", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - 2 x^{3} - 7 x^{2} - 19 x - 57$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 2 x^{4} - x^{3} + 2 x^{2} - 1$ and a = 3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 8 x^{3} - 3 x^{2} + 4 x$.\n5) Evaluate at a: p'(3) = $-231$.\nFinal answer: $-231$", "final_answer": "-231", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1d31df8bfe4e2a28", "problem_id": "3861645a96784d24", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - 2 x^{3} - 7 x^{2} - 19 x - 57$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 2 x^{4} - x^{3} + 2 x^{2} - 1$ and a = 3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 8 x^{3} - 3 x^{2} + 4 x$.\nEvaluate at a: p'(3) = $-231$.\nCheck: the identity gives $-231$.\nFinal answer: $-231$", "final_answer": "-231", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a85eb63a2ba33a3f", "problem_id": "94828040a2d82298", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A store sells a backpack for $370. It is discounted by 20%. You buy 13 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 370 × (1 − 20/100) = 370 × 80/100 = 296.\n2) Subtotal = 13 × 296 = 3848.\n3) Total with tax = 3848 × 105/100 = 20202/5.\nFinal answer: $\\frac{20202}{5}$", "final_answer": "\\frac{20202}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ffcf5119bad66220", "problem_id": "94828040a2d82298", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A store sells a backpack for $370. It is discounted by 20%. You buy 13 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 370 × (1 − 20/100) = 370 × 80/100 = 296.\nSubtotal = 13 × 296 = 3848.\nTotal with tax = 3848 × 105/100 = 20202/5.\nCheck: recomputing the formula yields 20202/5.\nFinal answer: $\\frac{20202}{5}$", "final_answer": "\\frac{20202}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e89a2b826ca5c79a", "problem_id": "3861645a96784d24", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - 2 x^{3} - 7 x^{2} - 19 x - 57$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{3} - 7 x^{2} - 20 x - 57.\n3) Then the limit is -234.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: -231.\nFinal answer: $-231$", "final_answer": "-231", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ceb3127ff10088eb", "problem_id": "48a712a72aecc8de", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(- x^{3} + 2 x^{2} + x + 2\\right) \\cos{\\left(3 x - 1 \\right)}$.", "output": "1) Let f(x) = $\\left(x - 3\\right) e^{x} + \\left(- x^{3} + 2 x^{2} + x + 2\\right) \\cos{\\left(3 x - 1 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\left(x - 3\\right) e^{x} + \\left(- 3 x^{2} + 4 x + 1\\right) \\cos{\\left(3 x - 1 \\right)} - 3 \\left(- x^{3} + 2 x^{2} + x + 2\\right) \\sin{\\left(3 x - 1 \\right)} + e^{x}$.\nFinal answer: $\\left(x - 3\\right)\\cdot e^{x} + \\left(- 3\\cdot x^{2} + 4\\cdot x + 1\\right)\\cdot \\cos{\\left(3\\cdot x - 1 \\right)} - \\left(- 3\\cdot x^{3} + 6\\cdot x^{2} + 3\\cdot x + 6\\right)\\cdot \\sin{\\left(3\\cdot x - 1 \\right)} + e^{x}$", "final_answer": "\\left(x - 3\\right)\\cdot e^{x} + \\left(- 3\\cdot x^{2} + 4\\cdot x + 1\\right)\\cdot \\cos{\\left(3\\cdot x - 1 \\right)} - \\left(- 3\\cdot x^{3} + 6\\cdot x^{2} + 3\\cdot x + 6\\right)\\cdot \\sin{\\left(3\\cdot x - 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "518a99d9e2221c6b", "problem_id": "aa0789ad8a21e477", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - 2 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{65 \\pi^{3}}{12} - \\frac{15 \\pi^{2}}{8} - 5 \\pi - 2$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{65\\cdot \\pi^{3}}{12} - \\frac{15\\cdot \\pi^{2}}{8} - 5\\cdot \\pi - 2$", "final_answer": "- \\frac{65\\cdot \\pi^{3}}{12} - \\frac{15\\cdot \\pi^{2}}{8} - 5\\cdot \\pi - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6e0ac956db8bb9d4", "problem_id": "aa0789ad8a21e477", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - 2 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{65 \\pi^{3}}{12} - \\frac{15 \\pi^{2}}{8} - 5 \\pi - 2$.\n4) Simplify the result.\nFinal answer: $- \\frac{65\\cdot \\pi^{3}}{12} - \\frac{15\\cdot \\pi^{2}}{8} - 5\\cdot \\pi - 2$", "final_answer": "- \\frac{65\\cdot \\pi^{3}}{12} - \\frac{15\\cdot \\pi^{2}}{8} - 5\\cdot \\pi - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5934feecdb172c3d", "problem_id": "276e32001b053792", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - 2 x^{2} - 11 x - 41$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 2 x^{3} - 3 x^{2} + 3 x + 3$ and a = 4.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 6 x^{2} - 6 x + 3$.\nEvaluate at a: p'(4) = $-117$.\nCheck: the identity gives $-117$.\nFinal answer: $-117$", "final_answer": "-117", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8493ca0d285f9f38", "problem_id": "276e32001b053792", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - 2 x^{2} - 11 x - 41$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 2 x^{3} - 3 x^{2} + 3 x + 3$ and a = 4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 6 x^{2} - 6 x + 3$.\n5) Evaluate at a: p'(4) = $-117$.\nFinal answer: $-117$", "final_answer": "-117", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "70b5bf4c5870f894", "problem_id": "ffa6768218331c19", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 10 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=5, p=$\\frac{1}{2}$ and simplify.\n4) Here C(10,5) = 252.\nFinal answer: $\\frac{63}{256}$", "final_answer": "\\frac{63}{256}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "480052b41b428793", "problem_id": "ffa6768218331c19", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 10 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=5, p=$\\frac{1}{2}$ and simplify.\nHere C(10,5) = 252.\nCheck: expression simplifies to $\\frac{63}{256}$.\nFinal answer: $\\frac{63}{256}$", "final_answer": "\\frac{63}{256}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8494e7d28ccfd9d7", "problem_id": "2d81ab2fdfc7ca34", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡1 (mod 4), x≡3 (mod 11). Give the solution modulo 44.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 44.\n3) Compute the CRT solution to get x ≡ 25 (mod 44).\nFinal answer: $x\\equiv 25\\pmod{44}$", "final_answer": "x\\equiv 25\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "49df79df24b353d2", "problem_id": "2d81ab2fdfc7ca34", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡1 (mod 4), x≡3 (mod 11). Give the solution modulo 44.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 44.\nCompute the CRT solution to get x ≡ 25 (mod 44).\nCheck: 25 mod 4=1 and mod 11=3.\nFinal answer: $x\\equiv 25\\pmod{44}$", "final_answer": "x\\equiv 25\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4109ce86476aad8a", "problem_id": "049f5ad9ccf6fdaa", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 4 x^{3} + 2 x^{2} + x - 3$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-3) = 0.\n3) d/dx(x) = 1.\n4) d/dx(- 4 x^{3}) = - 12 x^{2}.\n5) d/dx(2 x^{2}) = 4 x.\n6) Combine: f'(x) = - 12 x^{2} + 4 x + 1.\nFinal answer: $- 12\\cdot x^{2} + 4\\cdot x + 1$", "final_answer": "- 12\\cdot x^{2} + 4\\cdot x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4dab0cfb02ccbb6d", "problem_id": "86ad5e3e1f9c98d9", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - x^{2} + 5 x - 10$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} + 4 x - 10.\n3) Then the limit is -22.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -24.\nFinal answer: $-24$", "final_answer": "-24", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c13e6ccf8fbe92bc", "problem_id": "7d7cfff9f44d025f", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$- x + 12 y$ = $78$\n$7 x - 2 y$ = $28$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $7$: $- 7 x + 84 y$ = $546$.\nMultiply the second equation by $-1$: $- 7 x + 2 y$ = $-28$.\nSubtract to eliminate x: $82 y$ = $574$.\nSolve for y: y = $7$.\nSubstitute back to find x: x = $6$.\nCheck: plug (x,y)=(6,7) into both equations.\nFinal answer: $(x,y)=\\left(6, 7\\right)$", "final_answer": "(x,y)=\\left(6, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7bc337c7208f7eea", "problem_id": "7d7cfff9f44d025f", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$- x + 12 y$ = $78$\n$7 x - 2 y$ = $28$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $7$: $- 7 x + 84 y$ = $546$.\n3) Multiply the second equation by $-1$: $- 7 x + 2 y$ = $-28$.\n4) Subtract to eliminate x: $82 y$ = $574$.\n5) Solve for y: y = $7$.\n6) Substitute back to find x: x = $6$.\nFinal answer: $(x,y)=\\left(6, 7\\right)$", "final_answer": "(x,y)=\\left(6, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aa164c87cedb4ef1", "problem_id": "e929f7a079880e5e", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 18 x + 24$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} + 18 x + 24$ = $3 \\left(x + 2\\right) \\left(x + 4\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFinal answer: $x \\in \\{-2, -4\\}$", "final_answer": "x \\in \\{-2, -4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "edb17f7520a056a0", "problem_id": "e929f7a079880e5e", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 18 x + 24$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} + 18 x + 24$ = $3 \\left(x + 2\\right) \\left(x + 4\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-2$.\n5) From the second factor: x = $-4$.\nFinal answer: $x \\in \\{-2, -4\\}$", "final_answer": "x \\in \\{-2, -4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "32d0791173c0f356", "problem_id": "20f0283ad6ab131b", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the multiplicative inverse of 23 modulo 31.", "output": "1) We want x such that 23x ≡ 1 (mod 31).\n2) Use extended Euclid to find s,t with 23s + 31t = 1.\n3) One Bezout identity is: 23·(-4) + 31·(3) = 1.\n4) Reduce s modulo 31: x ≡ 27 (mod 31).\nFinal answer: $x\\equiv 27\\pmod{31}$", "final_answer": "x\\equiv 27\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "390d289db1abfa9d", "problem_id": "20f0283ad6ab131b", "style": "answer_only", "instruction": "Final answer only.", "input": "Find the multiplicative inverse of 23 modulo 31.", "output": "$x\\equiv 27\\pmod{31}$", "final_answer": "x\\equiv 27\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b66cb16ff820f6c3", "problem_id": "f57fbe6bf2d1281f", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 24 red balls and 17 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(41,7).\nFavorable samples: choose k reds and n−k blues: C(24,1)·C(17,6).\nCompute totals: C(41,7) = 22481940, and favorable = 297024.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1904}{144115}$.\nFinal answer: $\\frac{1904}{144115}$", "final_answer": "\\frac{1904}{144115}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1471f0d75785841e", "problem_id": "f57fbe6bf2d1281f", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "An urn has 24 red balls and 17 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(41,7).\n3) Favorable samples: choose k reds and n−k blues: C(24,1)·C(17,6).\n4) Compute totals: C(41,7) = 22481940, and favorable = 297024.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1904}{144115}$", "final_answer": "\\frac{1904}{144115}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff667e5035fd09bd", "problem_id": "35f56044230f1a2f", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A disease has prevalence $\\frac{1}{200}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{341}{10000}$.\n3) Compute P(D|+) = $\\frac{85}{682}$.\nFinal answer: $\\frac{85}{682}$", "final_answer": "\\frac{85}{682}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3a621c5b65c17743", "problem_id": "6f207e9759e1c11d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{3} - \\frac{3 x}{2} + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}) = - \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} - \\frac{2 x^{4}}{2 x^{4} - 12 x^{2} + 18} - \\frac{6 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} - \\frac{12 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} + \\frac{21 x^{2}}{2 x^{4} - 12 x^{2} + 18} - \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 6 x^{2} + 9} + \\frac{9 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} + \\frac{18 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} + \\frac{9}{2 x^{4} - 12 x^{2} + 18} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 6} + \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3} - \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}) = \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - 3\\right)^{2}} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x^{2} - 3\\right)} - \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 2 x \\left(- 2 x^{3} - 3 x + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 3\\right)}{2 \\left(x^{2} - 3\\right)^{2}}$.\nFinal answer: $\\frac{- 2 x \\left(- 2 x^{3} - 3 x + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 3\\right)}{2 \\left(x^{2} - 3\\right)^{2}}$", "final_answer": "\\frac{- 2 x \\left(- 2 x^{3} - 3 x + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 3\\right)}{2 \\left(x^{2} - 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c2f57830ef15b7c9", "problem_id": "694ac8f6f8da0488", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} - x^{2} + 7 x - 26$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{3} + 3 x^{2} + 2 x + 2$ and a = -4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 3 x^{2} + 6 x + 2$.\n5) Evaluate at a: p'(-4) = $-70$.\nFinal answer: $-70$", "final_answer": "-70", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b28c5af6db5bc006", "problem_id": "694ac8f6f8da0488", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} - x^{2} + 7 x - 26$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{3} + 3 x^{2} + 2 x + 2$ and a = -4.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 3 x^{2} + 6 x + 2$.\nEvaluate at a: p'(-4) = $-70$.\nCheck: the identity gives $-70$.\nFinal answer: $-70$", "final_answer": "-70", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1651e4169994b493", "problem_id": "7804733167ce3a44", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the congruences: x≡3 (mod 5), x≡6 (mod 11). Give the solution modulo 55.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 55.\n3) Compute the CRT solution to get x ≡ 28 (mod 55).\nFinal answer: $x\\equiv 28\\pmod{55}$", "final_answer": "x\\equiv 28\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7270488b08d080d0", "problem_id": "7804733167ce3a44", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡3 (mod 5), x≡6 (mod 11). Give the solution modulo 55.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 55.\nCompute the CRT solution to get x ≡ 28 (mod 55).\nCheck: 28 mod 5=3 and mod 11=6.\nFinal answer: $x\\equiv 28\\pmod{55}$", "final_answer": "x\\equiv 28\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "908052129deed425", "problem_id": "120f2ff03d0345c1", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $625. It is discounted by 5%. You buy 4 backpacks. What is the total cost?", "output": "Discounted price = 625 × (1 − 5/100) = 625 × 95/100 = 2375/4.\nSubtotal = 4 × 2375/4 = 2375.\nCheck: recomputing the formula yields 2375.\nFinal answer: $2375$", "final_answer": "2375", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4413d206aa7e823c", "problem_id": "120f2ff03d0345c1", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A store sells a backpack for $625. It is discounted by 5%. You buy 4 backpacks. What is the total cost?", "output": "1) Discounted price = 625 × (1 − 5/100) = 625 × 95/100 = 2375/4.\n2) Subtotal = 4 × 2375/4 = 2375.\nFinal answer: $2375$", "final_answer": "2375", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6ca0a1fa7180f75f", "problem_id": "899f3de1a7c47ba9", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A store sells a backpack for $860. It is discounted by 40%. You buy 9 backpacks. What is the total cost?", "output": "1) Discounted price = 860 × (1 − 40/100) = 860 × 60/100 = 516.\n2) Subtotal = 9 × 516 = 4644.\nFinal answer: $4644$", "final_answer": "4644", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b7b3acd4a95addf8", "problem_id": "899f3de1a7c47ba9", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $860. It is discounted by 40%. You buy 9 backpacks. What is the total cost?", "output": "Discounted price = 860 × (1 − 40/100) = 860 × 60/100 = 516.\nSubtotal = 9 × 516 = 4644.\nCheck: recomputing the formula yields 4644.\nFinal answer: $4644$", "final_answer": "4644", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "994766377e963a09", "problem_id": "a4ede402d7c489a1", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the linear system:\n$4 y$ = $-24$\n$7 x - y$ = $-57$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $7$: $28 y$ = $-168$.\n3) Multiply the second equation by $0$: $0$ = $0$.\n4) Subtract to eliminate x: $28 y$ = $-168$.\n5) Solve for y: y = $-6$.\n6) Substitute back to find x: x = $-9$.\nFinal answer: $(x,y)=\\left(-9, -6\\right)$", "final_answer": "(x,y)=\\left(-9, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "94a706fa99d24ddf", "problem_id": "a4ede402d7c489a1", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$4 y$ = $-24$\n$7 x - y$ = $-57$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $7$: $28 y$ = $-168$.\nMultiply the second equation by $0$: $0$ = $0$.\nSubtract to eliminate x: $28 y$ = $-168$.\nSolve for y: y = $-6$.\nSubstitute back to find x: x = $-9$.\nCheck: plug (x,y)=(-9,-6) into both equations.\nFinal answer: $(x,y)=\\left(-9, -6\\right)$", "final_answer": "(x,y)=\\left(-9, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b1ed6269fc3b2f51", "problem_id": "2def7120b8bc2987", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡4 (mod 5), x≡0 (mod 11). Give the solution modulo 55.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 55.\n3) Compute the CRT solution to get x ≡ 44 (mod 55).\nFinal answer: $x\\equiv 44\\pmod{55}$", "final_answer": "x\\equiv 44\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "37be484e7ee77f99", "problem_id": "c70b1fa0bce0719e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} 2 - x$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to 2 - 2 x.\n3) Then the limit is 10.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: 6.\nFinal answer: $6$", "final_answer": "6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "76d1b2a001082fe5", "problem_id": "f566dc96e05e4897", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$- 6 x - 10 y$ = $-68$\n$10 y$ = $80$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $-6$: $- 60 y$ = $-480$.\nSubtract to eliminate x: $60 y$ = $480$.\nSolve for y: y = $8$.\nSubstitute back to find x: x = $-2$.\nCheck: plug (x,y)=(-2,8) into both equations.\nFinal answer: $(x,y)=\\left(-2, 8\\right)$", "final_answer": "(x,y)=\\left(-2, 8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d5de8275b0200aef", "problem_id": "f566dc96e05e4897", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$- 6 x - 10 y$ = $-68$\n$10 y$ = $80$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $-6$: $- 60 y$ = $-480$.\n4) Subtract to eliminate x: $60 y$ = $480$.\n5) Solve for y: y = $8$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, 8\\right)$", "final_answer": "(x,y)=\\left(-2, 8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d37d9665997f03ac", "problem_id": "a685f8fb07378a34", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A store sells a backpack for $142. It is discounted by 40%. You buy 5 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 142 × (1 − 40/100) = 142 × 60/100 = 426/5.\n2) Subtotal = 5 × 426/5 = 426.\n3) Total with tax = 426 × 110/100 = 2343/5.\nFinal answer: $\\frac{2343}{5}$", "final_answer": "\\frac{2343}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ccf4a3f16e1567ea", "problem_id": "83a3b0f783dfd312", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 4 x^{3} + 3 x^{2} - 4 x - 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(3 x^{2}) = 6 x^{2}$.\n3) So $f'(x) = - 6 x^{2} - 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(3 x^{2}) = 6 x$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- 6 x^{2} + 3 x - 2\\right)$.\nFinal answer: $2 \\left(- 6 x^{2} + 3 x - 2\\right)$", "final_answer": "2 \\left(- 6 x^{2} + 3 x - 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "11939a77e9dc6fc8", "problem_id": "d357cd92977cc6a3", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} + x^{2} + 2 x - 1$ at $x=1$.", "output": "Differentiate: f'(x) = $9 x^{2} + 2 x + 2$.\nSlope at x=1: m = f'(1) = $13$.\nPoint on curve: (1, f(1)) = (1, $5$).\nFinal answer: $y = 13\\cdot x - 8$", "final_answer": "y = 13\\cdot x - 8", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1ec3d31f31a6a8ac", "problem_id": "d357cd92977cc6a3", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Find the equation of the tangent line to $y=3 x^{3} + x^{2} + 2 x - 1$ at $x=1$.", "output": "1) Differentiate: f'(x) = $9 x^{2} + 2 x + 2$.\n2) Slope at x=1: m = f'(1) = $13$.\n3) Point on curve: (1, f(1)) = (1, $5$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 5 = 13(x − 1).\n6) Optionally expand: y = $13 x - 8$.\nFinal answer: $y = 13\\cdot x - 8$", "final_answer": "y = 13\\cdot x - 8", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ba45e7cb106941f5", "problem_id": "cfdf4f6436c9f9fc", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Two friends split $138 in the ratio 1:2. How much does each receive?", "output": "1) Total ratio parts = 1+2 = 3.\n2) Each part = 138 ÷ 3 = 46.\n3) First share = 1 × 46 = 46.\n4) Second share = 2 × 46 = 92.\nFinal answer: $92$", "final_answer": "92", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "03df7a5b57a2a2c7", "problem_id": "cfdf4f6436c9f9fc", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Two friends split $138 in the ratio 1:2. How much does each receive?", "output": "Total ratio parts = 1+2 = 3.\nEach part = 138 ÷ 3 = 46.\nFirst share = 1 × 46 = 46.\nSecond share = 2 × 46 = 92.\nCheck: 46+92=138.\nFinal answer: $92$", "final_answer": "92", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "085b98f70268300d", "problem_id": "61f85626b8382b88", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$3 x - 2 y$ = $4$\n$- x + 4 y$ = $-8$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $- 3 x + 2 y$ = $-4$.\n3) Multiply the second equation by $3$: $- 3 x + 12 y$ = $-24$.\n4) Subtract to eliminate x: $- 10 y$ = $20$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, -2\\right)$", "final_answer": "(x,y)=\\left(0, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e0e0fcdca489668c", "problem_id": "61f85626b8382b88", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$3 x - 2 y$ = $4$\n$- x + 4 y$ = $-8$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $- 3 x + 2 y$ = $-4$.\nMultiply the second equation by $3$: $- 3 x + 12 y$ = $-24$.\nSubtract to eliminate x: $- 10 y$ = $20$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $0$.\nCheck: plug (x,y)=(0,-2) into both equations.\nFinal answer: $(x,y)=\\left(0, -2\\right)$", "final_answer": "(x,y)=\\left(0, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ee772aa9a9380ccb", "problem_id": "c35f3ad016c56f9f", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A cyclist rides at 69 miles per hour for 1/2 hours, then at 67 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 69 × 1/2 = 69/2 miles.\n2) Second leg distance = 67 × 2 = 134 miles.\n3) Total distance = 69/2 + 134 = 337/2 miles.\nFinal answer: $\\frac{337}{2}$", "final_answer": "\\frac{337}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "86d78e1f93bf4893", "problem_id": "c35f3ad016c56f9f", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A cyclist rides at 69 miles per hour for 1/2 hours, then at 67 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 69 × 1/2 = 69/2 miles.\nSecond leg distance = 67 × 2 = 134 miles.\nTotal distance = 69/2 + 134 = 337/2 miles.\nFinal answer: $\\frac{337}{2}$", "final_answer": "\\frac{337}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "936bec3f41fea0f7", "problem_id": "5c3aeed26667a874", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Two friends split $231 in the ratio 2:5. How much does each receive?", "output": "Total ratio parts = 2+5 = 7.\nEach part = 231 ÷ 7 = 33.\nFirst share = 2 × 33 = 66.\nSecond share = 5 × 33 = 165.\nCheck: 66+165=231.\nFinal answer: $165$", "final_answer": "165", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3d670889fefe864c", "problem_id": "5c3aeed26667a874", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Two friends split $231 in the ratio 2:5. How much does each receive?", "output": "1) Total ratio parts = 2+5 = 7.\n2) Each part = 231 ÷ 7 = 33.\n3) First share = 2 × 33 = 66.\n4) Second share = 5 × 33 = 165.\nFinal answer: $165$", "final_answer": "165", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aa989c2d0be1d2ae", "problem_id": "73ead386aba4855a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Worker A finishes a job in 20 hours and Worker B finishes it in 7 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/20 job/hour. B's rate = 1/7 job/hour.\nCombined rate = 1/20 + 1/7 = 27/140 job/hour.\nTime = 1 ÷ (combined rate) = 140/27 hours.\nCheck: (combined rate)×(time) = 27/140×140/27 = 1.\nFinal answer: $\\frac{140}{27}$", "final_answer": "\\frac{140}{27}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "00e100afaeafbfbd", "problem_id": "73ead386aba4855a", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Worker A finishes a job in 20 hours and Worker B finishes it in 7 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/20 job/hour. B's rate = 1/7 job/hour.\nCombined rate = 1/20 + 1/7 = 27/140 job/hour.\nTime = 1 ÷ (combined rate) = 140/27 hours.\nFinal answer: $\\frac{140}{27}$", "final_answer": "\\frac{140}{27}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b7fc866aa2ee7e3", "problem_id": "ad00cbb0d8082aa5", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the equation of the tangent line to $y=x^{2} + 3 x + 2 \\sin{\\left(x \\right)} + 1$ at $x=1$.", "output": "Differentiate: f'(x) = $2 x + 2 \\cos{\\left(x \\right)} + 3$.\nSlope at x=1: m = f'(1) = $2 \\cos{\\left(1 \\right)} + 5$.\nPoint on curve: (1, f(1)) = (1, $2 \\sin{\\left(1 \\right)} + 5$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 2 \\sin{\\left(1 \\right)} + 5 = 2 \\cos{\\left(1 \\right)} + 5(x − 1).\nOptionally expand: y = $2 x \\cos{\\left(1 \\right)} + 5 x - 2 \\cos{\\left(1 \\right)} + 2 \\sin{\\left(1 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 5\\cdot x - 2\\cdot \\cos{\\left(1 \\right)} + 2\\cdot \\sin{\\left(1 \\right)}$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 5\\cdot x - 2\\cdot \\cos{\\left(1 \\right)} + 2\\cdot \\sin{\\left(1 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "abe5a66fe38cc80d", "problem_id": "ad00cbb0d8082aa5", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the equation of the tangent line to $y=x^{2} + 3 x + 2 \\sin{\\left(x \\right)} + 1$ at $x=1$.", "output": "1) Differentiate: f'(x) = $2 x + 2 \\cos{\\left(x \\right)} + 3$.\n2) Slope at x=1: m = f'(1) = $2 \\cos{\\left(1 \\right)} + 5$.\n3) Point on curve: (1, f(1)) = (1, $2 \\sin{\\left(1 \\right)} + 5$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 2 \\sin{\\left(1 \\right)} + 5 = 2 \\cos{\\left(1 \\right)} + 5(x − 1).\n6) Optionally expand: y = $2 x \\cos{\\left(1 \\right)} + 5 x - 2 \\cos{\\left(1 \\right)} + 2 \\sin{\\left(1 \\right)}$.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 5\\cdot x - 2\\cdot \\cos{\\left(1 \\right)} + 2\\cdot \\sin{\\left(1 \\right)}$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 5\\cdot x - 2\\cdot \\cos{\\left(1 \\right)} + 2\\cdot \\sin{\\left(1 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "20ab993a2e7e4f0b", "problem_id": "48a712a72aecc8de", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(- x^{3} + 2 x^{2} + x + 2\\right) \\cos{\\left(3 x - 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 \\cos{\\left(3 x - 1 \\right)}) = - 2 \\sin{\\left(3 x - 1 \\right)}$.\n3) So $f'(x) = 3 x^{3} \\sin{\\left(3 x - 1 \\right)} - 6 x^{2} \\sin{\\left(3 x - 1 \\right)} - 3 x^{2} \\cos{\\left(3 x - 1 \\right)} + x e^{x} - 3 x \\sin{\\left(3 x - 1 \\right)} + 4 x \\cos{\\left(3 x - 1 \\right)} - 2 e^{x} - 2 \\sin{\\left(3 x - 1 \\right)} + \\cos{\\left(3 x - 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 \\cos{\\left(3 x - 1 \\right)}) = - 6 \\sin{\\left(3 x - 1 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x - 3\\right) e^{x} - 3 \\left(x \\left(x \\left(2 - x\\right) + 1\\right) + 2\\right) \\sin{\\left(3 x - 1 \\right)} + \\left(- 3 x^{2} + 4 x + 1\\right) \\cos{\\left(3 x - 1 \\right)} + e^{x}$.\nFinal answer: $\\left(x - 3\\right) e^{x} - 3 \\left(x \\left(x \\left(2 - x\\right) + 1\\right) + 2\\right) \\sin{\\left(3 x - 1 \\right)} + \\left(- 3 x^{2} + 4 x + 1\\right) \\cos{\\left(3 x - 1 \\right)} + e^{x}$", "final_answer": "\\left(x - 3\\right) e^{x} - 3 \\left(x \\left(x \\left(2 - x\\right) + 1\\right) + 2\\right) \\sin{\\left(3 x - 1 \\right)} + \\left(- 3 x^{2} + 4 x + 1\\right) \\cos{\\left(3 x - 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5f546b24b7ff3695", "problem_id": "f4ac4a51985af0c7", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 3 x^{3} - 4 x^{2} - 9 x - 17$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 3 x^{4} + 2 x^{3} - x^{2} + x + 2$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 12 x^{3} + 6 x^{2} - 2 x + 1$.\nEvaluate at a: p'(2) = $-75$.\nCheck: the identity gives $-75$.\nFinal answer: $-75$", "final_answer": "-75", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "296eff59e60e4524", "problem_id": "7722f0ada4aa6358", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A store sells a backpack for $170. It is discounted by 30%. You buy 3 backpacks. What is the total cost?", "output": "1) Discounted price = 170 × (1 − 30/100) = 170 × 70/100 = 119.\n2) Subtotal = 3 × 119 = 357.\nFinal answer: $357$", "final_answer": "357", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c88a3b87dde3a811", "problem_id": "7722f0ada4aa6358", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A store sells a backpack for $170. It is discounted by 30%. You buy 3 backpacks. What is the total cost?", "output": "Discounted price = 170 × (1 − 30/100) = 170 × 70/100 = 119.\nSubtotal = 3 × 119 = 357.\nCheck: recomputing the formula yields 357.\nFinal answer: $357$", "final_answer": "357", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8ce70c40f01479c6", "problem_id": "9bdae11998e35e9d", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$3 x + 4 y$ = $20$\n$4 x + 4 y$ = $24$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $12 x + 16 y$ = $80$.\n3) Multiply the second equation by $3$: $12 x + 12 y$ = $72$.\n4) Subtract to eliminate x: $4 y$ = $8$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, 2\\right)$", "final_answer": "(x,y)=\\left(4, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "acea269486873259", "problem_id": "9bdae11998e35e9d", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$3 x + 4 y$ = $20$\n$4 x + 4 y$ = $24$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $12 x + 16 y$ = $80$.\nMultiply the second equation by $3$: $12 x + 12 y$ = $72$.\nSubtract to eliminate x: $4 y$ = $8$.\nSolve for y: y = $2$.\nSubstitute back to find x: x = $4$.\nCheck: plug (x,y)=(4,2) into both equations.\nFinal answer: $(x,y)=\\left(4, 2\\right)$", "final_answer": "(x,y)=\\left(4, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "48ac6d764c912d46", "problem_id": "5d74551ae3071453", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "How many binary strings of length 26 contain exactly 4 ones with no two ones adjacent?", "output": "1) Reserve 3 mandatory zeros between the 4 ones.\n2) This leaves n−k+1 = 23 valid slots for the ones.\n3) Choose the slots: C(23,4) = 8855.\nFinal answer: $8855$", "final_answer": "8855", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "11a596395cf5e4ee", "problem_id": "5d74551ae3071453", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many binary strings of length 26 contain exactly 4 ones with no two ones adjacent?", "output": "Reserve 3 mandatory zeros between the 4 ones.\nThis leaves n−k+1 = 23 valid slots for the ones.\nChoose the slots: C(23,4) = 8855.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $8855$", "final_answer": "8855", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "36d4367aeacbc963", "problem_id": "c4aa75c753ae9dfe", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x^{2} - 5 x - 13$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{2} - 6 x - 13.\n3) Then the limit is -33.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -31.\nFinal answer: $-31$", "final_answer": "-31", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "508ef591c09519f7", "problem_id": "77fa3c5735e2c03b", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the multiplicative inverse of 27 modulo 29.", "output": "We want x such that 27x ≡ 1 (mod 29).\nUse extended Euclid to find s,t with 27s + 29t = 1.\nOne Bezout identity is: 27·(14) + 29·(-13) = 1.\nReduce s modulo 29: x ≡ 14 (mod 29).\nCheck: 27·14 mod 29 = 1.\nFinal answer: $x\\equiv 14\\pmod{29}$", "final_answer": "x\\equiv 14\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "36db5e07e0f4adb2", "problem_id": "95c3b6af1665f977", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A cyclist rides at 39 miles per hour for 3/2 hours, then at 33 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 39 × 3/2 = 117/2 miles.\n2) Second leg distance = 33 × 1 = 33 miles.\n3) Total distance = 117/2 + 33 = 183/2 miles.\nFinal answer: $\\frac{183}{2}$", "final_answer": "\\frac{183}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f6432e456856a9f3", "problem_id": "95c3b6af1665f977", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A cyclist rides at 39 miles per hour for 3/2 hours, then at 33 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 39 × 3/2 = 117/2 miles.\nSecond leg distance = 33 × 1 = 33 miles.\nTotal distance = 117/2 + 33 = 183/2 miles.\nCheck: adding both legs gives 183/2.\nFinal answer: $\\frac{183}{2}$", "final_answer": "\\frac{183}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9b120ea1dad21232", "problem_id": "078e8c05ec56c6a0", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "An urn has 26 red balls and 30 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(56,8).\nFavorable samples: choose k reds and n−k blues: C(26,4)·C(30,4).\nFinal answer: $\\frac{20010}{69377}$", "final_answer": "\\frac{20010}{69377}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b1d8162f21741f5a", "problem_id": "078e8c05ec56c6a0", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "An urn has 26 red balls and 30 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(56,8).\n3) Favorable samples: choose k reds and n−k blues: C(26,4)·C(30,4).\n4) Compute totals: C(56,8) = 1420494075, and favorable = 409704750.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{20010}{69377}$", "final_answer": "\\frac{20010}{69377}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "742487fa395a3b97", "problem_id": "12c25dd7e7d903fc", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 15 times, what is the probability of exactly 7 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=15, k=7, p=$\\frac{3}{7}$ and simplify.\n4) Here C(15,7) = 6435.\nFinal answer: $\\frac{922310737920}{4747561509943}$", "final_answer": "\\frac{922310737920}{4747561509943}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c274513dab40274c", "problem_id": "48dbd6030f3b25ea", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 x^{3} \\sin{\\left(x \\right)}) = 3 x^{3} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = 3 x^{3} \\cos{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 x^{3} \\sin{\\left(x \\right)}) = 3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)}$.\nFinal answer: $3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)}$", "final_answer": "3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "da668125a06c6b47", "problem_id": "aea62d0bd3e023ac", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "You want 26 liters of a 45% salt solution. You have a 30% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 26−x liters is 60% solution.\n2) Salt equation: 30x + 60(26−x) = 45·26.\n3) (30-60)x = (45-60)·26 ⇒ x = 13.\n4) Then 26−x = 13.\nFinal answer: $13 L of 30% and 13 L of 60%$", "final_answer": "13 L of 30% and 13 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "54b6a75982d667d0", "problem_id": "aea62d0bd3e023ac", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 26 liters of a 45% salt solution. You have a 30% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 26−x liters is 60% solution.\nSalt equation: 30x + 60(26−x) = 45·26.\n(30-60)x = (45-60)·26 ⇒ x = 13.\nThen 26−x = 13.\nCheck: 30·13 + 60·13 = 45·26.\nFinal answer: $13 L of 30% and 13 L of 60%$", "final_answer": "13 L of 30% and 13 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d75fe8033fd6c097", "problem_id": "82177c44a16bcd88", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$- x + 2 y$ = $5$\n$- 5 x - 12 y$ = $-41$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-5$: $5 x - 10 y$ = $-25$.\n3) Multiply the second equation by $-1$: $5 x + 12 y$ = $41$.\n4) Subtract to eliminate x: $- 22 y$ = $-66$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $1$.\nFinal answer: $(x,y)=\\left(1, 3\\right)$", "final_answer": "(x,y)=\\left(1, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "34d786ccd79d59c4", "problem_id": "82177c44a16bcd88", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$- x + 2 y$ = $5$\n$- 5 x - 12 y$ = $-41$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-5$: $5 x - 10 y$ = $-25$.\nMultiply the second equation by $-1$: $5 x + 12 y$ = $41$.\nSubtract to eliminate x: $- 22 y$ = $-66$.\nSolve for y: y = $3$.\nSubstitute back to find x: x = $1$.\nCheck: plug (x,y)=(1,3) into both equations.\nFinal answer: $(x,y)=\\left(1, 3\\right)$", "final_answer": "(x,y)=\\left(1, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "24a1c0e8c023c726", "problem_id": "00d7544c4e2b4629", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "You want 26 liters of a 25% salt solution. You have a 20% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 26−x liters is 40% solution.\n2) Salt equation: 20x + 40(26−x) = 25·26.\n3) (20-40)x = (25-40)·26 ⇒ x = 39/2.\n4) Then 26−x = 13/2.\nFinal answer: $39/2 L of 20% and 13/2 L of 40%$", "final_answer": "39/2 L of 20% and 13/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f8d5d048140366e3", "problem_id": "00d7544c4e2b4629", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "You want 26 liters of a 25% salt solution. You have a 20% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 26−x liters is 40% solution.\nSalt equation: 20x + 40(26−x) = 25·26.\n(20-40)x = (25-40)·26 ⇒ x = 39/2.\nThen 26−x = 13/2.\nCheck: 20·39/2 + 40·13/2 = 25·26.\nFinal answer: $39/2 L of 20% and 13/2 L of 40%$", "final_answer": "39/2 L of 20% and 13/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "844c246bbec2075d", "problem_id": "05f49dcda8be9ed5", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡2 (mod 5), x≡2 (mod 8). Give the solution modulo 40.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 40.\n3) Compute the CRT solution to get x ≡ 2 (mod 40).\nFinal answer: $x\\equiv 2\\pmod{40}$", "final_answer": "x\\equiv 2\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3a621c5b65c17743", "problem_id": "6f207e9759e1c11d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{3} - \\frac{3 x}{2} + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}) = - \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} - \\frac{2 x^{4}}{2 x^{4} - 12 x^{2} + 18} - \\frac{6 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} - \\frac{12 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} + \\frac{21 x^{2}}{2 x^{4} - 12 x^{2} + 18} - \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 6 x^{2} + 9} + \\frac{9 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} + \\frac{18 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} + \\frac{9}{2 x^{4} - 12 x^{2} + 18} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 6} + \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3} - \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}) = \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - 3\\right)^{2}} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x^{2} - 3\\right)} - \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 2 x \\left(- 2 x^{3} - 3 x + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 3\\right)}{2 \\left(x^{2} - 3\\right)^{2}}$.\nFinal answer: $\\frac{- 2 x \\left(- 2 x^{3} - 3 x + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 3\\right)}{2 \\left(x^{2} - 3\\right)^{2}}$", "final_answer": "\\frac{- 2 x \\left(- 2 x^{3} - 3 x + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 3\\right)}{2 \\left(x^{2} - 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f582075eb8fc2a41", "problem_id": "b6d24d26387d09c8", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$- 2 x$ = $0$\n$4 x - y$ = $-1$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $- 8 x$ = $0$.\n3) Multiply the second equation by $-2$: $- 8 x + 2 y$ = $2$.\n4) Subtract to eliminate x: $- 2 y$ = $-2$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, 1\\right)$", "final_answer": "(x,y)=\\left(0, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "656647d2ed6ef063", "problem_id": "ca1db2df9672f21b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x \\sin{\\left(x \\right)}) = x \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x \\sin{\\left(x \\right)}) = x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$.\nFinal answer: $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$", "final_answer": "- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "df469203dfcd716c", "problem_id": "75f8e9253b1e8bd0", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the linear system:\n$- 2 x - y$ = $4$\n$- 2 x + 2 y$ = $10$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $4 x + 2 y$ = $-8$.\nMultiply the second equation by $-2$: $4 x - 4 y$ = $-20$.\nFinal answer: $(x,y)=\\left(-3, 2\\right)$", "final_answer": "(x,y)=\\left(-3, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "40c22e43834f3446", "problem_id": "75f8e9253b1e8bd0", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$- 2 x - y$ = $4$\n$- 2 x + 2 y$ = $10$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $4 x + 2 y$ = $-8$.\n3) Multiply the second equation by $-2$: $4 x - 4 y$ = $-20$.\n4) Subtract to eliminate x: $6 y$ = $12$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, 2\\right)$", "final_answer": "(x,y)=\\left(-3, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d9c098d3af6f234", "problem_id": "d9be432c3b43a0bc", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - \\frac{24 x}{5} + \\frac{9}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$- \\frac{24}{5}$, c=$\\frac{9}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{36}{25}$, so √Δ = $\\frac{6}{5}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $1$.\n7) x2 = $\\frac{3}{5}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{1, \\frac{3}{5}\\}$", "final_answer": "x \\in \\{1, \\frac{3}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff3f101aa9c0b575", "problem_id": "d14866f0ac52d8c3", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(3 x^{2} + x - 2\\right) \\cos{\\left(3 x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 2 \\cos{\\left(3 x + 3 \\right)}) = 2 \\sin{\\left(3 x + 3 \\right)}$.\n3) So $f'(x) = - 9 x^{2} \\sin{\\left(3 x + 3 \\right)} + 2 x e^{x} - 3 x \\sin{\\left(3 x + 3 \\right)} + 6 x \\cos{\\left(3 x + 3 \\right)} + 4 e^{x} + 2 \\sin{\\left(3 x + 3 \\right)} + \\cos{\\left(3 x + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 2 \\cos{\\left(3 x + 3 \\right)}) = 6 \\sin{\\left(3 x + 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x + 1\\right) e^{x} + \\left(6 x + 1\\right) \\cos{\\left(3 x + 3 \\right)} - 3 \\left(x \\left(3 x + 1\\right) - 2\\right) \\sin{\\left(3 x + 3 \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(x + 1\\right) e^{x} + \\left(6 x + 1\\right) \\cos{\\left(3 x + 3 \\right)} - 3 \\left(x \\left(3 x + 1\\right) - 2\\right) \\sin{\\left(3 x + 3 \\right)} + 2 e^{x}$", "final_answer": "2 \\left(x + 1\\right) e^{x} + \\left(6 x + 1\\right) \\cos{\\left(3 x + 3 \\right)} - 3 \\left(x \\left(3 x + 1\\right) - 2\\right) \\sin{\\left(3 x + 3 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "821da45ad8a04982", "problem_id": "25645746f223cd9f", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{6} x - \\frac{x}{6} - \\frac{35}{6} - \\frac{7 \\sqrt{6}}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\sqrt{6} - \\frac{1}{6}$, c=$- \\frac{35}{6} - \\frac{7 \\sqrt{6}}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{29 \\sqrt{6}}{3} + \\frac{1057}{36}$, so √Δ = $\\sqrt{\\frac{29 \\sqrt{6}}{3} + \\frac{1057}{36}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{1}{12} + \\frac{\\sqrt{6}}{2} + \\frac{\\sqrt{348 \\sqrt{6} + 1057}}{12}$.\n7) x2 = $- \\frac{\\sqrt{348 \\sqrt{6} + 1057}}{12} + \\frac{1}{12} + \\frac{\\sqrt{6}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{1}{12} + \\frac{\\sqrt{6}}{2} + \\frac{\\sqrt{348\\cdot \\sqrt{6} + 1057}}{12}, - \\frac{\\sqrt{348\\cdot \\sqrt{6} + 1057}}{12} + \\frac{1}{12} + \\frac{\\sqrt{6}}{2}\\}$", "final_answer": "x \\in \\{\\frac{1}{12} + \\frac{\\sqrt{6}}{2} + \\frac{\\sqrt{348\\cdot \\sqrt{6} + 1057}}{12}, - \\frac{\\sqrt{348\\cdot \\sqrt{6} + 1057}}{12} + \\frac{1}{12} + \\frac{\\sqrt{6}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4e3d104c8d6df5e6", "problem_id": "25645746f223cd9f", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{6} x - \\frac{x}{6} - \\frac{35}{6} - \\frac{7 \\sqrt{6}}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\sqrt{6} - \\frac{1}{6}$, c=$- \\frac{35}{6} - \\frac{7 \\sqrt{6}}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{29 \\sqrt{6}}{3} + \\frac{1057}{36}$, so √Δ = $\\sqrt{\\frac{29 \\sqrt{6}}{3} + \\frac{1057}{36}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{1}{12} + \\frac{\\sqrt{6}}{2} + \\frac{\\sqrt{348 \\sqrt{6} + 1057}}{12}$.\nx2 = $- \\frac{\\sqrt{348 \\sqrt{6} + 1057}}{12} + \\frac{1}{12} + \\frac{\\sqrt{6}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{1}{12} + \\frac{\\sqrt{6}}{2} + \\frac{\\sqrt{348\\cdot \\sqrt{6} + 1057}}{12}, - \\frac{\\sqrt{348\\cdot \\sqrt{6} + 1057}}{12} + \\frac{1}{12} + \\frac{\\sqrt{6}}{2}\\}$", "final_answer": "x \\in \\{\\frac{1}{12} + \\frac{\\sqrt{6}}{2} + \\frac{\\sqrt{348\\cdot \\sqrt{6} + 1057}}{12}, - \\frac{\\sqrt{348\\cdot \\sqrt{6} + 1057}}{12} + \\frac{1}{12} + \\frac{\\sqrt{6}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9405ca72bad2e529", "problem_id": "9cdc337fcbc4c4f6", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$- 3 x + y$ = $-6$\n$- 3 x - 3 y$ = $-18$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $9 x - 3 y$ = $18$.\n3) Multiply the second equation by $-3$: $9 x + 9 y$ = $54$.\n4) Subtract to eliminate x: $- 12 y$ = $-36$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, 3\\right)$", "final_answer": "(x,y)=\\left(3, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1e1444f45274b111", "problem_id": "9cdc337fcbc4c4f6", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$- 3 x + y$ = $-6$\n$- 3 x - 3 y$ = $-18$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $9 x - 3 y$ = $18$.\nMultiply the second equation by $-3$: $9 x + 9 y$ = $54$.\nSubtract to eliminate x: $- 12 y$ = $-36$.\nSolve for y: y = $3$.\nSubstitute back to find x: x = $3$.\nCheck: plug (x,y)=(3,3) into both equations.\nFinal answer: $(x,y)=\\left(3, 3\\right)$", "final_answer": "(x,y)=\\left(3, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "86a9fd4cb028bef0", "problem_id": "4616d3914728c9f7", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 5 x + 2$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$-5$, c=$2$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $9$, so √Δ = $3$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $2$.\n7) x2 = $\\frac{1}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{2, \\frac{1}{2}\\}$", "final_answer": "x \\in \\{2, \\frac{1}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f5ba10d06c09047", "problem_id": "80cd2d9d70c1bc48", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{3} \\sin{\\left(x \\right)} - x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x \\sin{\\left(x \\right)}) = 2 x \\cos{\\left(x \\right)}$.\n3) So $f'(x) = 2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x \\sin{\\left(x \\right)}) = 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)}$.\nFinal answer: $2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)}$", "final_answer": "2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ae927d9b7cc4718c", "problem_id": "5271d56e18843f66", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 4 x^{2} - 4 x - 2$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-2) = 0.\n3) d/dx(- 4 x) = -4.\n4) d/dx(- 4 x^{2}) = - 8 x.\n5) Combine: f'(x) = 4 \\left(- 2 x - 1\\right).\nFinal answer: $- 8\\cdot x - 4$", "final_answer": "- 8\\cdot x - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2afb4160cca5be5f", "problem_id": "5271d56e18843f66", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 4 x^{2} - 4 x - 2$.", "output": "Differentiate term-by-term:\nd/dx(-2) = 0.\nd/dx(- 4 x) = -4.\nd/dx(- 4 x^{2}) = - 8 x.\nCombine: f'(x) = 4 \\left(- 2 x - 1\\right).\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 8\\cdot x - 4$", "final_answer": "- 8\\cdot x - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1ee84b1dadfe80c3", "problem_id": "31a5bae293ebc08a", "style": "answer_only", "instruction": "Final answer only.", "input": "Worker A finishes a job in 18 hours and Worker B finishes it in 20 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "$\\frac{180}{19}$", "final_answer": "\\frac{180}{19}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "711ce61b06dc0d30", "problem_id": "31a5bae293ebc08a", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Worker A finishes a job in 18 hours and Worker B finishes it in 20 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/18 job/hour. B's rate = 1/20 job/hour.\n2) Combined rate = 1/18 + 1/20 = 19/180 job/hour.\n3) Time = 1 ÷ (combined rate) = 180/19 hours.\nFinal answer: $\\frac{180}{19}$", "final_answer": "\\frac{180}{19}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "316a1219b4d8116b", "problem_id": "22a6b89a050d0312", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$- x + 4 y$ = $-3$\n$2 y$ = $-4$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $-1$: $- 2 y$ = $4$.\n4) Subtract to eliminate x: $2 y$ = $-4$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, -2\\right)$", "final_answer": "(x,y)=\\left(-5, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "572696fefd81be48", "problem_id": "22a6b89a050d0312", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$- x + 4 y$ = $-3$\n$2 y$ = $-4$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $-1$: $- 2 y$ = $4$.\nSubtract to eliminate x: $2 y$ = $-4$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $-5$.\nCheck: plug (x,y)=(-5,-2) into both equations.\nFinal answer: $(x,y)=\\left(-5, -2\\right)$", "final_answer": "(x,y)=\\left(-5, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4fbbc2ce429cf357", "problem_id": "0d425e0cc01df70b", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 23 red balls and 11 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(34,11).\n3) Favorable samples: choose k reds and n−k blues: C(23,2)·C(11,9).\n4) Compute totals: C(34,11) = 286097760, and favorable = 13915.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{2783}{57219552}$", "final_answer": "\\frac{2783}{57219552}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8c47276f7719435c", "problem_id": "0d425e0cc01df70b", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 23 red balls and 11 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(34,11).\nFavorable samples: choose k reds and n−k blues: C(23,2)·C(11,9).\nCompute totals: C(34,11) = 286097760, and favorable = 13915.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{2783}{57219552}$.\nFinal answer: $\\frac{2783}{57219552}$", "final_answer": "\\frac{2783}{57219552}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8cee9e8934ef8638", "problem_id": "1f230a1d61b09558", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$4 x + y$ = $-27$\n$- x - 3 y$ = $-7$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $- 4 x - y$ = $27$.\n3) Multiply the second equation by $4$: $- 4 x - 12 y$ = $-28$.\n4) Subtract to eliminate x: $11 y$ = $55$.\n5) Solve for y: y = $5$.\n6) Substitute back to find x: x = $-8$.\nFinal answer: $(x,y)=\\left(-8, 5\\right)$", "final_answer": "(x,y)=\\left(-8, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "492b01ce2bf3a2f5", "problem_id": "1f230a1d61b09558", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$4 x + y$ = $-27$\n$- x - 3 y$ = $-7$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $- 4 x - y$ = $27$.\nMultiply the second equation by $4$: $- 4 x - 12 y$ = $-28$.\nSubtract to eliminate x: $11 y$ = $55$.\nSolve for y: y = $5$.\nSubstitute back to find x: x = $-8$.\nCheck: plug (x,y)=(-8,5) into both equations.\nFinal answer: $(x,y)=\\left(-8, 5\\right)$", "final_answer": "(x,y)=\\left(-8, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a11204338e0f1a70", "problem_id": "5bf1c679d39ecb2e", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + x + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{3 x + 4}$.", "output": "Let f(x) = $\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + x + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{3 x + 4}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{- 18 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6 x + \\left(3 x + 4\\right) \\left(3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 10 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right) - 24 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 12}{2 \\left(3 x + 4\\right)^{2}}$.\nFinal answer: $\\frac{- 18\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 6\\cdot x + \\left(3\\cdot x + 4\\right)\\cdot \\left(3\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 10\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 8\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\right) - 24\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 12}{2\\cdot \\left(3\\cdot x + 4\\right)^{2}}$", "final_answer": "\\frac{- 18\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 6\\cdot x + \\left(3\\cdot x + 4\\right)\\cdot \\left(3\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 10\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 8\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\right) - 24\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 12}{2\\cdot \\left(3\\cdot x + 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0fed00cff841593f", "problem_id": "5bf1c679d39ecb2e", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + x + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{3 x + 4}$.", "output": "1) Let f(x) = $\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + x + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{3 x + 4}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{- 18 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6 x + \\left(3 x + 4\\right) \\left(3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 10 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right) - 24 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 12}{2 \\left(3 x + 4\\right)^{2}}$.\nFinal answer: $\\frac{- 18\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 6\\cdot x + \\left(3\\cdot x + 4\\right)\\cdot \\left(3\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 10\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 8\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\right) - 24\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 12}{2\\cdot \\left(3\\cdot x + 4\\right)^{2}}$", "final_answer": "\\frac{- 18\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 6\\cdot x + \\left(3\\cdot x + 4\\right)\\cdot \\left(3\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 10\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 8\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\right) - 24\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 12}{2\\cdot \\left(3\\cdot x + 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "13f79035b5787d5d", "problem_id": "b6b7f1022cf272c9", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A cyclist rides at 54 miles per hour for 2 hours, then at 83 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 54 × 2 = 108 miles.\n2) Second leg distance = 83 × 1 = 83 miles.\n3) Total distance = 108 + 83 = 191 miles.\nFinal answer: $191$", "final_answer": "191", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4be0336597053de9", "problem_id": "29f1f9e398c80276", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 3 x^{3} + 9 x^{2} + 17 x + 31$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} + 9 x^{2} + 16 x + 31.\n3) Then the limit is 123.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 125.\nFinal answer: $125$", "final_answer": "125", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "4f02a54d3a6106c9", "problem_id": "852da1c4da75a7a7", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{3} x + \\frac{38 x}{5} - \\frac{24 \\sqrt{3}}{5} + \\frac{48}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{38}{5} - 3 \\sqrt{3}$, c=$\\frac{48}{5} - \\frac{24 \\sqrt{3}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{1159}{25} - \\frac{132 \\sqrt{3}}{5}$, so √Δ = $\\sqrt{\\frac{1159}{25} - \\frac{132 \\sqrt{3}}{5}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{19}{5} + \\frac{\\sqrt{1159 - 660 \\sqrt{3}}}{10} + \\frac{3 \\sqrt{3}}{2}$.\n7) x2 = $- \\frac{19}{5} - \\frac{\\sqrt{1159 - 660 \\sqrt{3}}}{10} + \\frac{3 \\sqrt{3}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{19}{5} + \\frac{\\sqrt{1159 - 660\\cdot \\sqrt{3}}}{10} + \\frac{3\\cdot \\sqrt{3}}{2}, - \\frac{19}{5} - \\frac{\\sqrt{1159 - 660\\cdot \\sqrt{3}}}{10} + \\frac{3\\cdot \\sqrt{3}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{19}{5} + \\frac{\\sqrt{1159 - 660\\cdot \\sqrt{3}}}{10} + \\frac{3\\cdot \\sqrt{3}}{2}, - \\frac{19}{5} - \\frac{\\sqrt{1159 - 660\\cdot \\sqrt{3}}}{10} + \\frac{3\\cdot \\sqrt{3}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6b50fdbd776dba43", "problem_id": "852da1c4da75a7a7", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{3} x + \\frac{38 x}{5} - \\frac{24 \\sqrt{3}}{5} + \\frac{48}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{38}{5} - 3 \\sqrt{3}$, c=$\\frac{48}{5} - \\frac{24 \\sqrt{3}}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{1159}{25} - \\frac{132 \\sqrt{3}}{5}$, so √Δ = $\\sqrt{\\frac{1159}{25} - \\frac{132 \\sqrt{3}}{5}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{19}{5} + \\frac{\\sqrt{1159 - 660 \\sqrt{3}}}{10} + \\frac{3 \\sqrt{3}}{2}$.\nx2 = $- \\frac{19}{5} - \\frac{\\sqrt{1159 - 660 \\sqrt{3}}}{10} + \\frac{3 \\sqrt{3}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{19}{5} + \\frac{\\sqrt{1159 - 660\\cdot \\sqrt{3}}}{10} + \\frac{3\\cdot \\sqrt{3}}{2}, - \\frac{19}{5} - \\frac{\\sqrt{1159 - 660\\cdot \\sqrt{3}}}{10} + \\frac{3\\cdot \\sqrt{3}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{19}{5} + \\frac{\\sqrt{1159 - 660\\cdot \\sqrt{3}}}{10} + \\frac{3\\cdot \\sqrt{3}}{2}, - \\frac{19}{5} - \\frac{\\sqrt{1159 - 660\\cdot \\sqrt{3}}}{10} + \\frac{3\\cdot \\sqrt{3}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fbe8e8abd86ac9a5", "problem_id": "130dadf694997c19", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- 2 x^{2} + 2 x + 1\\right) \\cos{\\left(x - 1 \\right)}$.", "output": "1) Let f(x) = $2 \\left(x + 1\\right) e^{x} + \\left(- 2 x^{2} + 2 x + 1\\right) \\cos{\\left(x - 1 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $2 \\left(1 - 2 x\\right) \\cos{\\left(x - 1 \\right)} + 2 \\left(x + 1\\right) e^{x} - \\left(- 2 x^{2} + 2 x + 1\\right) \\sin{\\left(x - 1 \\right)} + 2 e^{x}$.\nFinal answer: $\\left(2 - 4\\cdot x\\right)\\cdot \\cos{\\left(x - 1 \\right)} + \\left(2\\cdot x + 2\\right)\\cdot e^{x} - \\left(- 2\\cdot x^{2} + 2\\cdot x + 1\\right)\\cdot \\sin{\\left(x - 1 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2 - 4\\cdot x\\right)\\cdot \\cos{\\left(x - 1 \\right)} + \\left(2\\cdot x + 2\\right)\\cdot e^{x} - \\left(- 2\\cdot x^{2} + 2\\cdot x + 1\\right)\\cdot \\sin{\\left(x - 1 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9e6e1940b9ffa7d4", "problem_id": "130dadf694997c19", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- 2 x^{2} + 2 x + 1\\right) \\cos{\\left(x - 1 \\right)}$.", "output": "Let f(x) = $2 \\left(x + 1\\right) e^{x} + \\left(- 2 x^{2} + 2 x + 1\\right) \\cos{\\left(x - 1 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $2 \\left(1 - 2 x\\right) \\cos{\\left(x - 1 \\right)} + 2 \\left(x + 1\\right) e^{x} - \\left(- 2 x^{2} + 2 x + 1\\right) \\sin{\\left(x - 1 \\right)} + 2 e^{x}$.\nFinal answer: $\\left(2 - 4\\cdot x\\right)\\cdot \\cos{\\left(x - 1 \\right)} + \\left(2\\cdot x + 2\\right)\\cdot e^{x} - \\left(- 2\\cdot x^{2} + 2\\cdot x + 1\\right)\\cdot \\sin{\\left(x - 1 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2 - 4\\cdot x\\right)\\cdot \\cos{\\left(x - 1 \\right)} + \\left(2\\cdot x + 2\\right)\\cdot e^{x} - \\left(- 2\\cdot x^{2} + 2\\cdot x + 1\\right)\\cdot \\sin{\\left(x - 1 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bc3196b29ccf9025", "problem_id": "e94b9007718cfc98", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} x - 5$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{2} - 2 x - 3$ and a = -3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $2 x - 2$.\n5) Evaluate at a: p'(-3) = $-8$.\nFinal answer: $-8$", "final_answer": "-8", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d036f1822b4d8d5f", "problem_id": "e94b9007718cfc98", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} x - 5$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $x^{2} - 2 x - 3$ and a = -3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-8$", "final_answer": "-8", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d2647d0bd6a443a6", "problem_id": "a790467a9e4bbfbb", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡3 (mod 11), x≡9 (mod 10). Give the solution modulo 110.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 110.\n3) Compute the CRT solution to get x ≡ 69 (mod 110).\nFinal answer: $x\\equiv 69\\pmod{110}$", "final_answer": "x\\equiv 69\\pmod{110}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2f45910d921d9486", "problem_id": "a790467a9e4bbfbb", "style": "answer_only", "instruction": "Final answer only.", "input": "Solve the congruences: x≡3 (mod 11), x≡9 (mod 10). Give the solution modulo 110.", "output": "$x\\equiv 69\\pmod{110}$", "final_answer": "x\\equiv 69\\pmod{110}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5f69787039a34e33", "problem_id": "8b75e195dc686de2", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 12 red balls and 26 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(38,11).\n3) Favorable samples: choose k reds and n−k blues: C(12,3)·C(26,8).\n4) Compute totals: C(38,11) = 1203322288, and favorable = 343700500.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{4522375}{15833188}$", "final_answer": "\\frac{4522375}{15833188}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4a98ad886dd7b81a", "problem_id": "8b75e195dc686de2", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 12 red balls and 26 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(38,11).\nFavorable samples: choose k reds and n−k blues: C(12,3)·C(26,8).\nCompute totals: C(38,11) = 1203322288, and favorable = 343700500.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{4522375}{15833188}$.\nFinal answer: $\\frac{4522375}{15833188}$", "final_answer": "\\frac{4522375}{15833188}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1d1fb1e433ab705a", "problem_id": "4c8979990304532a", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{2} x + \\frac{7 x}{2} - 3 \\sqrt{2} + \\frac{3}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{7}{2} - \\sqrt{2}$, c=$\\frac{3}{2} - 3 \\sqrt{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $5 \\sqrt{2} + \\frac{33}{4}$, so √Δ = $\\sqrt{5 \\sqrt{2} + \\frac{33}{4}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{7}{4} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{20 \\sqrt{2} + 33}}{4}$.\n7) x2 = $- \\frac{\\sqrt{20 \\sqrt{2} + 33}}{4} - \\frac{7}{4} + \\frac{\\sqrt{2}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{7}{4} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{20\\cdot \\sqrt{2} + 33}}{4}, - \\frac{\\sqrt{20\\cdot \\sqrt{2} + 33}}{4} - \\frac{7}{4} + \\frac{\\sqrt{2}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{7}{4} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{20\\cdot \\sqrt{2} + 33}}{4}, - \\frac{\\sqrt{20\\cdot \\sqrt{2} + 33}}{4} - \\frac{7}{4} + \\frac{\\sqrt{2}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "83609efb7c85ddad", "problem_id": "4c8979990304532a", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{2} x + \\frac{7 x}{2} - 3 \\sqrt{2} + \\frac{3}{2}$ = 0.", "output": "$x \\in \\{- \\frac{7}{4} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{20\\cdot \\sqrt{2} + 33}}{4}, - \\frac{\\sqrt{20\\cdot \\sqrt{2} + 33}}{4} - \\frac{7}{4} + \\frac{\\sqrt{2}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{7}{4} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{20\\cdot \\sqrt{2} + 33}}{4}, - \\frac{\\sqrt{20\\cdot \\sqrt{2} + 33}}{4} - \\frac{7}{4} + \\frac{\\sqrt{2}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1c0ed919e8ff7e0d", "problem_id": "20e6e34ac6bb1843", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x - 1$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{2} + x + 3$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $1 - 2 x$.\n5) Evaluate at a: p'(2) = $-3$.\nFinal answer: $-3$", "final_answer": "-3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "26528de5ad158479", "problem_id": "20e6e34ac6bb1843", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x - 1$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{2} + x + 3$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $1 - 2 x$.\nEvaluate at a: p'(2) = $-3$.\nCheck: the identity gives $-3$.\nFinal answer: $-3$", "final_answer": "-3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bb7cfc64e8ea724a", "problem_id": "f3d25eef9ce4a93d", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many binary strings of length 23 contain exactly 10 ones with no two ones adjacent?", "output": "Reserve 9 mandatory zeros between the 10 ones.\nThis leaves n−k+1 = 14 valid slots for the ones.\nChoose the slots: C(14,10) = 1001.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $1001$", "final_answer": "1001", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b23e9628b40fbd5", "problem_id": "f3d25eef9ce4a93d", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many binary strings of length 23 contain exactly 10 ones with no two ones adjacent?", "output": "1) Reserve 9 mandatory zeros between the 10 ones.\n2) This leaves n−k+1 = 14 valid slots for the ones.\n3) Choose the slots: C(14,10) = 1001.\nFinal answer: $1001$", "final_answer": "1001", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a86630f3b6037b2", "problem_id": "d1207321943d30de", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡1 (mod 5), x≡2 (mod 8). Give the solution modulo 40.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 40.\nCompute the CRT solution to get x ≡ 26 (mod 40).\nCheck: 26 mod 5=1 and mod 8=2.\nFinal answer: $x\\equiv 26\\pmod{40}$", "final_answer": "x\\equiv 26\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fe8b5da5aeecc217", "problem_id": "d1207321943d30de", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡1 (mod 5), x≡2 (mod 8). Give the solution modulo 40.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 40.\n3) Compute the CRT solution to get x ≡ 26 (mod 40).\nFinal answer: $x\\equiv 26\\pmod{40}$", "final_answer": "x\\equiv 26\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6bee09139bb9d5fd", "problem_id": "4381a2517f440a7a", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$3 x + 2 y$ = $-37$\n$4 x - 2 y$ = $-12$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $12 x + 8 y$ = $-148$.\n3) Multiply the second equation by $3$: $12 x - 6 y$ = $-36$.\n4) Subtract to eliminate x: $14 y$ = $-112$.\n5) Solve for y: y = $-8$.\n6) Substitute back to find x: x = $-7$.\nFinal answer: $(x,y)=\\left(-7, -8\\right)$", "final_answer": "(x,y)=\\left(-7, -8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b53e1930f6781e2", "problem_id": "4381a2517f440a7a", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$3 x + 2 y$ = $-37$\n$4 x - 2 y$ = $-12$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $12 x + 8 y$ = $-148$.\nMultiply the second equation by $3$: $12 x - 6 y$ = $-36$.\nSubtract to eliminate x: $14 y$ = $-112$.\nSolve for y: y = $-8$.\nSubstitute back to find x: x = $-7$.\nCheck: plug (x,y)=(-7,-8) into both equations.\nFinal answer: $(x,y)=\\left(-7, -8\\right)$", "final_answer": "(x,y)=\\left(-7, -8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3552b6fe68971f2f", "problem_id": "6ffbb966a04e4bba", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x + 3$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{2} + x + 2$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $2 x + 1$.\n5) Evaluate at a: p'(2) = $5$.\nFinal answer: $5$", "final_answer": "5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5e795c22faacf1ad", "problem_id": "6ffbb966a04e4bba", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x + 3$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $x^{2} + x + 2$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $2 x + 1$.\nEvaluate at a: p'(2) = $5$.\nCheck: the identity gives $5$.\nFinal answer: $5$", "final_answer": "5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ea88a7ee3d5843f5", "problem_id": "65041f1baeb1c468", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 2 \\sqrt{10} x - \\frac{x}{2} - \\frac{3}{2} + 3 \\sqrt{10}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- 2 \\sqrt{10} - \\frac{1}{2}$, c=$- \\frac{3}{2} + 3 \\sqrt{10}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{185}{4} - 10 \\sqrt{10}$, so √Δ = $\\sqrt{\\frac{185}{4} - 10 \\sqrt{10}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{1}{4} + \\frac{\\sqrt{185 - 40 \\sqrt{10}}}{4} + \\sqrt{10}$.\n7) x2 = $- \\frac{\\sqrt{185 - 40 \\sqrt{10}}}{4} + \\frac{1}{4} + \\sqrt{10}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{1}{4} + \\frac{\\sqrt{185 - 40\\cdot \\sqrt{10}}}{4} + \\sqrt{10}, - \\frac{\\sqrt{185 - 40\\cdot \\sqrt{10}}}{4} + \\frac{1}{4} + \\sqrt{10}\\}$", "final_answer": "x \\in \\{\\frac{1}{4} + \\frac{\\sqrt{185 - 40\\cdot \\sqrt{10}}}{4} + \\sqrt{10}, - \\frac{\\sqrt{185 - 40\\cdot \\sqrt{10}}}{4} + \\frac{1}{4} + \\sqrt{10}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "87eb03607694947b", "problem_id": "65041f1baeb1c468", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 2 \\sqrt{10} x - \\frac{x}{2} - \\frac{3}{2} + 3 \\sqrt{10}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- 2 \\sqrt{10} - \\frac{1}{2}$, c=$- \\frac{3}{2} + 3 \\sqrt{10}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{185}{4} - 10 \\sqrt{10}$, so √Δ = $\\sqrt{\\frac{185}{4} - 10 \\sqrt{10}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{1}{4} + \\frac{\\sqrt{185 - 40 \\sqrt{10}}}{4} + \\sqrt{10}$.\nx2 = $- \\frac{\\sqrt{185 - 40 \\sqrt{10}}}{4} + \\frac{1}{4} + \\sqrt{10}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{1}{4} + \\frac{\\sqrt{185 - 40\\cdot \\sqrt{10}}}{4} + \\sqrt{10}, - \\frac{\\sqrt{185 - 40\\cdot \\sqrt{10}}}{4} + \\frac{1}{4} + \\sqrt{10}\\}$", "final_answer": "x \\in \\{\\frac{1}{4} + \\frac{\\sqrt{185 - 40\\cdot \\sqrt{10}}}{4} + \\sqrt{10}, - \\frac{\\sqrt{185 - 40\\cdot \\sqrt{10}}}{4} + \\frac{1}{4} + \\sqrt{10}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d6ce316bd0c19c6e", "problem_id": "6f74cc5326d881b1", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} 2 x^{3} - 11 x^{2} + 47 x - 191$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{4} - 3 x^{3} + 3 x^{2} - 3 x - 1$ and a = -4.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $8 x^{3} - 9 x^{2} + 6 x - 3$.\nEvaluate at a: p'(-4) = $-683$.\nCheck: the identity gives $-683$.\nFinal answer: $-683$", "final_answer": "-683", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ed8ebf8e005d20b9", "problem_id": "6f74cc5326d881b1", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} 2 x^{3} - 11 x^{2} + 47 x - 191$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $2 x^{4} - 3 x^{3} + 3 x^{2} - 3 x - 1$ and a = -4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $8 x^{3} - 9 x^{2} + 6 x - 3$.\n5) Evaluate at a: p'(-4) = $-683$.\nFinal answer: $-683$", "final_answer": "-683", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3a4149a40b626f31", "problem_id": "41ad95bf45fa2778", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "How many binary strings of length 11 contain exactly 4 ones with no two ones adjacent?", "output": "1) Reserve 3 mandatory zeros between the 4 ones.\n2) This leaves n−k+1 = 8 valid slots for the ones.\n3) Choose the slots: C(8,4) = 70.\nFinal answer: $70$", "final_answer": "70", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e2046666820a7da1", "problem_id": "41ad95bf45fa2778", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many binary strings of length 11 contain exactly 4 ones with no two ones adjacent?", "output": "Reserve 3 mandatory zeros between the 4 ones.\nThis leaves n−k+1 = 8 valid slots for the ones.\nChoose the slots: C(8,4) = 70.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $70$", "final_answer": "70", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "12a453f8b0f66ea2", "problem_id": "5ce9f5fa5ecb2d45", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + \\frac{7 x}{6} - \\frac{5}{6}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{7}{6}$, c=$- \\frac{5}{6}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{169}{36}$, so √Δ = $\\frac{13}{6}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{1}{2}$.\n7) x2 = $- \\frac{5}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{1}{2}, - \\frac{5}{3}\\}$", "final_answer": "x \\in \\{\\frac{1}{2}, - \\frac{5}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d9d16b6cb2ee0f71", "problem_id": "5ce9f5fa5ecb2d45", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + \\frac{7 x}{6} - \\frac{5}{6}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{7}{6}$, c=$- \\frac{5}{6}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{169}{36}$, so √Δ = $\\frac{13}{6}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{1}{2}$.\nx2 = $- \\frac{5}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{1}{2}, - \\frac{5}{3}\\}$", "final_answer": "x \\in \\{\\frac{1}{2}, - \\frac{5}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ded8c607275dfb87", "problem_id": "27f79e2eda8e0e46", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "An urn has 10 red balls and 17 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(27,7).\n3) Favorable samples: choose k reds and n−k blues: C(10,1)·C(17,6).\n4) Compute totals: C(27,7) = 888030, and favorable = 123760.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{952}{6831}$", "final_answer": "\\frac{952}{6831}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d5c0355f5966a9d", "problem_id": "1973eb501f26a2f2", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} + 2 x + \\left(x^{2} - x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{x^{2} - x + 7}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7}) = \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 7}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{4 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 15 x^{2} - 14 x + 49} + \\frac{15 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{30 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{2 x^{2}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 15 x^{2} - 14 x + 49} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 2 x + 14} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 7} - \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7} - \\frac{14 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{28 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{40 x}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7} + \\frac{49 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{98 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{34}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7}) = \\frac{x^{2} \\left(1 - 2 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(x^{2} - x\\right) + 7\\right)^{2}} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(x^{2} - x\\right) + 7\\right)} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7} + \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(x - 1\\right) + 7\\right) \\left(- 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x - 1\\right) + 7\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(x - 1\\right) + 7\\right) \\left(- 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x - 1\\right) + 7\\right)^{2}}$", "final_answer": "\\frac{2 \\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(x - 1\\right) + 7\\right) \\left(- 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x - 1\\right) + 7\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7f6ca59e8da95305", "problem_id": "76564298ab312c2c", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 4 x^{4} + 2 x^{3} + x - 3\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $4 x^{4} + 2 x^{3} + x - 3$.\n3) Compute an antiderivative: F(x) = $\\frac{4 x^{5}}{5} + \\frac{x^{4}}{2} + \\frac{x^{2}}{2} - 3 x$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{4\\cdot x^{5}}{5} + \\frac{x^{4}}{2} + \\frac{x^{2}}{2} - 3\\cdot x$", "final_answer": "C + \\frac{4\\cdot x^{5}}{5} + \\frac{x^{4}}{2} + \\frac{x^{2}}{2} - 3\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a26b4bd38633f81b", "problem_id": "3f28178ada29b9c7", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 \\sin{\\left(x \\right)}) = 3 \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 \\sin{\\left(x \\right)}) = 3 \\cos{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$", "final_answer": "- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d9d2c709cd7b4395", "problem_id": "c908878b35798985", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Find the equation of the tangent line to $y=4 x^{2} + 1$ at $x=2$.", "output": "Differentiate: f'(x) = $8 x$.\nSlope at x=2: m = f'(2) = $16$.\nPoint on curve: (2, f(2)) = (2, $17$).\nFinal answer: $y = 16\\cdot x - 15$", "final_answer": "y = 16\\cdot x - 15", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d0165fb286b0aab6", "problem_id": "c908878b35798985", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the equation of the tangent line to $y=4 x^{2} + 1$ at $x=2$.", "output": "1) Differentiate: f'(x) = $8 x$.\n2) Slope at x=2: m = f'(2) = $16$.\n3) Point on curve: (2, f(2)) = (2, $17$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 17 = 16(x − 2).\n6) Optionally expand: y = $16 x - 15$.\nFinal answer: $y = 16\\cdot x - 15$", "final_answer": "y = 16\\cdot x - 15", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5dbbe5ecd024adcb", "problem_id": "43d8442665bda382", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 13 red balls and 27 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(40,11).\n3) Favorable samples: choose k reds and n−k blues: C(13,4)·C(27,7).\n4) Compute totals: C(40,11) = 2311801440, and favorable = 634941450.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1628055}{5927696}$", "final_answer": "\\frac{1628055}{5927696}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "50866e186c385446", "problem_id": "43d8442665bda382", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 13 red balls and 27 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(40,11).\nFavorable samples: choose k reds and n−k blues: C(13,4)·C(27,7).\nCompute totals: C(40,11) = 2311801440, and favorable = 634941450.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1628055}{5927696}$.\nFinal answer: $\\frac{1628055}{5927696}$", "final_answer": "\\frac{1628055}{5927696}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ecd7a2d1eb3979f2", "problem_id": "69a9cca92b22c7fa", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{2} + \\left(2 x^{2} + 3 x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{2 x^{2} + 3 x - 6}$.", "output": "Let f(x) = $\\frac{- 3 x^{2} + \\left(2 x^{2} + 3 x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{2 x^{2} + 3 x - 6}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{2 \\left(- 4 x - 3\\right) \\left(- 3 x^{2} + \\left(2 x^{2} + 3 x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(2 x^{2} + 3 x - 6\\right) \\left(- 12 x + 2 \\left(4 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(2 x^{2} + 3 x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x^{2} + 3 x - 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)}\\right)}{2 \\left(2 x^{2} + 3 x - 6\\right)^{2}}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\frac{\\left(- 8\\cdot x - 6\\right)\\cdot \\left(- 3\\cdot x^{2} + \\left(2\\cdot x^{2} + 3\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 3\\right) + \\left(2\\cdot x^{2} + 3\\cdot x - 6\\right)\\cdot \\left(- 12\\cdot x + \\left(8\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 3\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x^{2} + 6\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)}\\right)}{2\\cdot \\left(2\\cdot x^{2} + 3\\cdot x - 6\\right)^{2}}$", "final_answer": "\\frac{\\left(- 8\\cdot x - 6\\right)\\cdot \\left(- 3\\cdot x^{2} + \\left(2\\cdot x^{2} + 3\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 3\\right) + \\left(2\\cdot x^{2} + 3\\cdot x - 6\\right)\\cdot \\left(- 12\\cdot x + \\left(8\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 3\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x^{2} + 6\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)}\\right)}{2\\cdot \\left(2\\cdot x^{2} + 3\\cdot x - 6\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "01129d7c44528b1f", "problem_id": "69a9cca92b22c7fa", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{2} + \\left(2 x^{2} + 3 x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{2 x^{2} + 3 x - 6}$.", "output": "1) Let f(x) = $\\frac{- 3 x^{2} + \\left(2 x^{2} + 3 x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{2 x^{2} + 3 x - 6}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{2 \\left(- 4 x - 3\\right) \\left(- 3 x^{2} + \\left(2 x^{2} + 3 x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(2 x^{2} + 3 x - 6\\right) \\left(- 12 x + 2 \\left(4 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(2 x^{2} + 3 x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x^{2} + 3 x - 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)}\\right)}{2 \\left(2 x^{2} + 3 x - 6\\right)^{2}}$.\nFinal answer: $\\frac{\\left(- 8\\cdot x - 6\\right)\\cdot \\left(- 3\\cdot x^{2} + \\left(2\\cdot x^{2} + 3\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 3\\right) + \\left(2\\cdot x^{2} + 3\\cdot x - 6\\right)\\cdot \\left(- 12\\cdot x + \\left(8\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 3\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x^{2} + 6\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)}\\right)}{2\\cdot \\left(2\\cdot x^{2} + 3\\cdot x - 6\\right)^{2}}$", "final_answer": "\\frac{\\left(- 8\\cdot x - 6\\right)\\cdot \\left(- 3\\cdot x^{2} + \\left(2\\cdot x^{2} + 3\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 3\\right) + \\left(2\\cdot x^{2} + 3\\cdot x - 6\\right)\\cdot \\left(- 12\\cdot x + \\left(8\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 3\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x^{2} + 6\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)}\\right)}{2\\cdot \\left(2\\cdot x^{2} + 3\\cdot x - 6\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a9e27080c745f3f2", "problem_id": "c7a839d2fc8d32f7", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "An urn has 22 red balls and 17 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(39,8).\nFavorable samples: choose k reds and n−k blues: C(22,3)·C(17,5).\nFinal answer: $\\frac{980}{6327}$", "final_answer": "\\frac{980}{6327}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6fb8380a846ea29d", "problem_id": "c7a839d2fc8d32f7", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 22 red balls and 17 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(39,8).\n3) Favorable samples: choose k reds and n−k blues: C(22,3)·C(17,5).\n4) Compute totals: C(39,8) = 61523748, and favorable = 9529520.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{980}{6327}$", "final_answer": "\\frac{980}{6327}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c09c691d369d1f5c", "problem_id": "e6a37a26e894287e", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 2$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 3 x^{2} + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\n2) Slope at x=2: m = f'(2) = $-14 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\n3) Point on curve: (2, f(2)) = (2, $-10 + \\cos{\\left(4 \\right)} + e$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -10 + \\cos{\\left(4 \\right)} + e = -14 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\n6) Optionally expand: y = $- 14 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 18$.\nFinal answer: $y = - 14\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 18$", "final_answer": "y = - 14\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 18", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "401df3e7bb1bf2b2", "problem_id": "f3d3b4e037485f8c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(x^{2} - 3 x - 2\\right) \\cos{\\left(- 3 x^{2} + 3 x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 2 \\cos{\\left(\\left(- 3 x^{2} + 3 x\\right) + 3 \\right)}) = 2 \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)}$.\n3) So $f'(x) = 6 x^{3} \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} - 21 x^{2} \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} + x e^{x} + 9 x \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} + 2 x \\cos{\\left(\\left(- 3 x^{2} + 3 x\\right) + 3 \\right)} + 3 e^{x} + 2 \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} - 3 \\cos{\\left(\\left(- 3 x^{2} + 3 x\\right) + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 2 \\cos{\\left(\\left(- 3 x^{2} + 3 x\\right) + 3 \\right)}) = 2 \\left(3 - 6 x\\right) \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = - 3 \\left(1 - 2 x\\right) \\left(x \\left(x - 3\\right) - 2\\right) \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} + \\left(x + 2\\right) e^{x} + \\left(2 x - 3\\right) \\cos{\\left(\\left(- 3 x^{2} + 3 x\\right) + 3 \\right)} + e^{x}$.\nFinal answer: $- 3 \\left(1 - 2 x\\right) \\left(x \\left(x - 3\\right) - 2\\right) \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} + \\left(x + 2\\right) e^{x} + \\left(2 x - 3\\right) \\cos{\\left(\\left(- 3 x^{2} + 3 x\\right) + 3 \\right)} + e^{x}$", "final_answer": "- 3 \\left(1 - 2 x\\right) \\left(x \\left(x - 3\\right) - 2\\right) \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} + \\left(x + 2\\right) e^{x} + \\left(2 x - 3\\right) \\cos{\\left(\\left(- 3 x^{2} + 3 x\\right) + 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "1763276c3ed1b35f", "problem_id": "6c85ab2b764c17a9", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} x^{3} + 4 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $x^{3} + 4 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1$.\n2) Evaluate at the bounds: F(2 \\pi) and F(0).\n3) Subtract: F(2 \\pi) − F(0) = $- 2 \\pi + \\frac{32 \\pi^{3}}{3} + 4 \\pi^{4}$.\n4) Simplify the result.\nFinal answer: $- 2\\cdot \\pi + \\frac{32\\cdot \\pi^{3}}{3} + 4\\cdot \\pi^{4}$", "final_answer": "- 2\\cdot \\pi + \\frac{32\\cdot \\pi^{3}}{3} + 4\\cdot \\pi^{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5cfb832e4207c872", "problem_id": "6c85ab2b764c17a9", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} x^{3} + 4 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $x^{3} + 4 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1$.\nEvaluate at the bounds: F(2 \\pi) and F(0).\nSubtract: F(2 \\pi) − F(0) = $- 2 \\pi + \\frac{32 \\pi^{3}}{3} + 4 \\pi^{4}$.\nFinal answer: $- 2\\cdot \\pi + \\frac{32\\cdot \\pi^{3}}{3} + 4\\cdot \\pi^{4}$", "final_answer": "- 2\\cdot \\pi + \\frac{32\\cdot \\pi^{3}}{3} + 4\\cdot \\pi^{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "787d8c0002f679a8", "problem_id": "2ac0a7034cf072b8", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Worker A finishes a job in 13 hours and Worker B finishes it in 9 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/13 job/hour. B's rate = 1/9 job/hour.\nCombined rate = 1/13 + 1/9 = 22/117 job/hour.\nTime = 1 ÷ (combined rate) = 117/22 hours.\nFinal answer: $\\frac{117}{22}$", "final_answer": "\\frac{117}{22}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "760c6066629641a5", "problem_id": "2ac0a7034cf072b8", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Worker A finishes a job in 13 hours and Worker B finishes it in 9 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/13 job/hour. B's rate = 1/9 job/hour.\n2) Combined rate = 1/13 + 1/9 = 22/117 job/hour.\n3) Time = 1 ÷ (combined rate) = 117/22 hours.\nFinal answer: $\\frac{117}{22}$", "final_answer": "\\frac{117}{22}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a189c7fb6bbfe890", "problem_id": "641573428b827539", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "You want 38 liters of a 35% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 38−x liters is 70% solution.\n2) Salt equation: 25x + 70(38−x) = 35·38.\n3) (25-70)x = (35-70)·38 ⇒ x = 266/9.\n4) Then 38−x = 76/9.\nFinal answer: $266/9 L of 25% and 76/9 L of 70%$", "final_answer": "266/9 L of 25% and 76/9 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1fc7a6131cc0ed59", "problem_id": "4303602b72d4f841", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} 3 x^{3} - 11 x^{2} + 47 x - 191$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} - 11 x^{2} + 46 x - 191.\n3) Then the limit is -743.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: -747.\nFinal answer: $-747$", "final_answer": "-747", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "584f5b7d83054d07", "problem_id": "f5efe803ad178b5d", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the linear system:\n$8 x + 5 y$ = $34$\n$4 x - 12 y$ = $-12$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $32 x + 20 y$ = $136$.\n3) Multiply the second equation by $8$: $32 x - 96 y$ = $-96$.\n4) Subtract to eliminate x: $116 y$ = $232$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, 2\\right)$", "final_answer": "(x,y)=\\left(3, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ebf5604c10bd9642", "problem_id": "f5efe803ad178b5d", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the linear system:\n$8 x + 5 y$ = $34$\n$4 x - 12 y$ = $-12$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $32 x + 20 y$ = $136$.\nMultiply the second equation by $8$: $32 x - 96 y$ = $-96$.\nFinal answer: $(x,y)=\\left(3, 2\\right)$", "final_answer": "(x,y)=\\left(3, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e2ae5eeeb2ec6c74", "problem_id": "5679110f9b355af6", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡4 (mod 5), x≡4 (mod 9). Give the solution modulo 45.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 45.\nCompute the CRT solution to get x ≡ 4 (mod 45).\nCheck: 4 mod 5=4 and mod 9=4.\nFinal answer: $x\\equiv 4\\pmod{45}$", "final_answer": "x\\equiv 4\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7c6af7d950f65e06", "problem_id": "5679110f9b355af6", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the congruences: x≡4 (mod 5), x≡4 (mod 9). Give the solution modulo 45.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 45.\n3) Compute the CRT solution to get x ≡ 4 (mod 45).\nFinal answer: $x\\equiv 4\\pmod{45}$", "final_answer": "x\\equiv 4\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "726cd2b1a06f4a57", "problem_id": "7c240afa2dedd616", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$3 x + y$ = $-8$\n$x - 4 y$ = $19$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $3 x + y$ = $-8$.\n3) Multiply the second equation by $3$: $3 x - 12 y$ = $57$.\n4) Subtract to eliminate x: $13 y$ = $-65$.\n5) Solve for y: y = $-5$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, -5\\right)$", "final_answer": "(x,y)=\\left(-1, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "829b3f8e2716da15", "problem_id": "7c240afa2dedd616", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$3 x + y$ = $-8$\n$x - 4 y$ = $19$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $3 x + y$ = $-8$.\nMultiply the second equation by $3$: $3 x - 12 y$ = $57$.\nSubtract to eliminate x: $13 y$ = $-65$.\nSolve for y: y = $-5$.\nSubstitute back to find x: x = $-1$.\nCheck: plug (x,y)=(-1,-5) into both equations.\nFinal answer: $(x,y)=\\left(-1, -5\\right)$", "final_answer": "(x,y)=\\left(-1, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "12571bb6712c783e", "problem_id": "65c0f015255bc211", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$2 x - 7 y$ = $30$\n$2 x + 6 y$ = $-48$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $4 x - 14 y$ = $60$.\n3) Multiply the second equation by $2$: $4 x + 12 y$ = $-96$.\n4) Subtract to eliminate x: $- 26 y$ = $156$.\n5) Solve for y: y = $-6$.\n6) Substitute back to find x: x = $-6$.\nFinal answer: $(x,y)=\\left(-6, -6\\right)$", "final_answer": "(x,y)=\\left(-6, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2162b404fefd1844", "problem_id": "65c0f015255bc211", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the linear system:\n$2 x - 7 y$ = $30$\n$2 x + 6 y$ = $-48$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $2$: $4 x - 14 y$ = $60$.\nMultiply the second equation by $2$: $4 x + 12 y$ = $-96$.\nFinal answer: $(x,y)=\\left(-6, -6\\right)$", "final_answer": "(x,y)=\\left(-6, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7e112fde0310db4c", "problem_id": "2176bb3f0159c6b8", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Worker A finishes a job in 16 hours and Worker B finishes it in 5 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/16 job/hour. B's rate = 1/5 job/hour.\n2) Combined rate = 1/16 + 1/5 = 21/80 job/hour.\n3) Time = 1 ÷ (combined rate) = 80/21 hours.\nFinal answer: $\\frac{80}{21}$", "final_answer": "\\frac{80}{21}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "24b24e86017dd2cf", "problem_id": "2176bb3f0159c6b8", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Worker A finishes a job in 16 hours and Worker B finishes it in 5 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/16 job/hour. B's rate = 1/5 job/hour.\nCombined rate = 1/16 + 1/5 = 21/80 job/hour.\nTime = 1 ÷ (combined rate) = 80/21 hours.\nFinal answer: $\\frac{80}{21}$", "final_answer": "\\frac{80}{21}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2e358c34f7e9f935", "problem_id": "682304ae76510be2", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $360 in the ratio 7:3. How much does each receive?", "output": "1) Total ratio parts = 7+3 = 10.\n2) Each part = 360 ÷ 10 = 36.\n3) First share = 7 × 36 = 252.\n4) Second share = 3 × 36 = 108.\nFinal answer: $108$", "final_answer": "108", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4b6214079af98a74", "problem_id": "08dad0ffebbc8c3d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x - 4$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 2 x - 4.\n3) Then the limit is -8.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -6.\nFinal answer: $-6$", "final_answer": "-6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "51266dd5417881b0", "problem_id": "630e9af161b24ceb", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "You want 17 liters of a 25% salt solution. You have a 15% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 17−x liters is 50% solution.\n2) Salt equation: 15x + 50(17−x) = 25·17.\n3) (15-50)x = (25-50)·17 ⇒ x = 85/7.\n4) Then 17−x = 34/7.\nFinal answer: $85/7 L of 15% and 34/7 L of 50%$", "final_answer": "85/7 L of 15% and 34/7 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db655db7eabf6968", "problem_id": "f7d4f6af5c069b4a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{2} + 3 x - 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- x^{2}) = - 2 x^{2}$.\n3) So $f'(x) = 3 - 2 x^{2}$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- x^{2}) = - 2 x$.\nTherefore the correct derivative is $f'(x) = 3 - 2 x$.\nFinal answer: $3 - 2 x$", "final_answer": "3 - 2 x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2810fd6b6aa3ec4e", "problem_id": "087801e9febc0a3d", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡9 (mod 11), x≡1 (mod 7). Give the solution modulo 77.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 77.\n3) Compute the CRT solution to get x ≡ 64 (mod 77).\nFinal answer: $x\\equiv 64\\pmod{77}$", "final_answer": "x\\equiv 64\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "279c83af3d555b43", "problem_id": "087801e9febc0a3d", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡9 (mod 11), x≡1 (mod 7). Give the solution modulo 77.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 77.\nCompute the CRT solution to get x ≡ 64 (mod 77).\nCheck: 64 mod 11=9 and mod 7=1.\nFinal answer: $x\\equiv 64\\pmod{77}$", "final_answer": "x\\equiv 64\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d9770c33c5441f9e", "problem_id": "24896f284808cc3e", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{10} x - \\frac{25}{9} + \\frac{5 \\sqrt{10}}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\sqrt{10}$, c=$- \\frac{25}{9} + \\frac{5 \\sqrt{10}}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{\\sqrt{10}\\cdot \\left(\\sqrt{19 - 6\\cdot \\sqrt{10}} + 3\\right)}{6}, \\frac{\\sqrt{10}\\cdot \\left(3 - \\sqrt{19 - 6\\cdot \\sqrt{10}}\\right)}{6}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{10}\\cdot \\left(\\sqrt{19 - 6\\cdot \\sqrt{10}} + 3\\right)}{6}, \\frac{\\sqrt{10}\\cdot \\left(3 - \\sqrt{19 - 6\\cdot \\sqrt{10}}\\right)}{6}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ec61be6a0ea2014", "problem_id": "24896f284808cc3e", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{10} x - \\frac{25}{9} + \\frac{5 \\sqrt{10}}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\sqrt{10}$, c=$- \\frac{25}{9} + \\frac{5 \\sqrt{10}}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{190}{9} - \\frac{20 \\sqrt{10}}{3}$, so √Δ = $\\sqrt{\\frac{190}{9} - \\frac{20 \\sqrt{10}}{3}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{10} \\left(\\sqrt{19 - 6 \\sqrt{10}} + 3\\right)}{6}$.\nx2 = $\\frac{\\sqrt{10} \\left(3 - \\sqrt{19 - 6 \\sqrt{10}}\\right)}{6}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{10}\\cdot \\left(\\sqrt{19 - 6\\cdot \\sqrt{10}} + 3\\right)}{6}, \\frac{\\sqrt{10}\\cdot \\left(3 - \\sqrt{19 - 6\\cdot \\sqrt{10}}\\right)}{6}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{10}\\cdot \\left(\\sqrt{19 - 6\\cdot \\sqrt{10}} + 3\\right)}{6}, \\frac{\\sqrt{10}\\cdot \\left(3 - \\sqrt{19 - 6\\cdot \\sqrt{10}}\\right)}{6}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "06d671b111dd6a2e", "problem_id": "dec60a28e7c0f77d", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the multiplicative inverse of 20 modulo 23.", "output": "1) We want x such that 20x ≡ 1 (mod 23).\n2) Use extended Euclid to find s,t with 20s + 23t = 1.\n3) One Bezout identity is: 20·(-8) + 23·(7) = 1.\n4) Reduce s modulo 23: x ≡ 15 (mod 23).\nFinal answer: $x\\equiv 15\\pmod{23}$", "final_answer": "x\\equiv 15\\pmod{23}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "36b49d4c2e6cd64f", "problem_id": "dec60a28e7c0f77d", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the multiplicative inverse of 20 modulo 23.", "output": "We want x such that 20x ≡ 1 (mod 23).\nUse extended Euclid to find s,t with 20s + 23t = 1.\nOne Bezout identity is: 20·(-8) + 23·(7) = 1.\nReduce s modulo 23: x ≡ 15 (mod 23).\nCheck: 20·15 mod 23 = 1.\nFinal answer: $x\\equiv 15\\pmod{23}$", "final_answer": "x\\equiv 15\\pmod{23}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b3e5598bd0dc7d39", "problem_id": "145328d3f21fe968", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(- 3 x^{2} - 2 x - 2\\right) \\cos{\\left(2 x^{2} - x + 1 \\right)}$.", "output": "Let f(x) = $\\left(x + 1\\right) e^{x} + \\left(- 3 x^{2} - 2 x - 2\\right) \\cos{\\left(2 x^{2} - x + 1 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $2 \\left(- 3 x - 1\\right) \\cos{\\left(2 x^{2} - x + 1 \\right)} + \\left(x + 1\\right) e^{x} - \\left(4 x - 1\\right) \\left(- 3 x^{2} - 2 x - 2\\right) \\sin{\\left(2 x^{2} - x + 1 \\right)} + e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\left(- 6\\cdot x - 2\\right)\\cdot \\cos{\\left(2\\cdot x^{2} - x + 1 \\right)} + \\left(x + 1\\right)\\cdot e^{x} - \\left(4\\cdot x - 1\\right)\\cdot \\left(- 3\\cdot x^{2} - 2\\cdot x - 2\\right)\\cdot \\sin{\\left(2\\cdot x^{2} - x + 1 \\right)} + e^{x}$", "final_answer": "\\left(- 6\\cdot x - 2\\right)\\cdot \\cos{\\left(2\\cdot x^{2} - x + 1 \\right)} + \\left(x + 1\\right)\\cdot e^{x} - \\left(4\\cdot x - 1\\right)\\cdot \\left(- 3\\cdot x^{2} - 2\\cdot x - 2\\right)\\cdot \\sin{\\left(2\\cdot x^{2} - x + 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "439686413363b7c4", "problem_id": "145328d3f21fe968", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(- 3 x^{2} - 2 x - 2\\right) \\cos{\\left(2 x^{2} - x + 1 \\right)}$.", "output": "1) Let f(x) = $\\left(x + 1\\right) e^{x} + \\left(- 3 x^{2} - 2 x - 2\\right) \\cos{\\left(2 x^{2} - x + 1 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $2 \\left(- 3 x - 1\\right) \\cos{\\left(2 x^{2} - x + 1 \\right)} + \\left(x + 1\\right) e^{x} - \\left(4 x - 1\\right) \\left(- 3 x^{2} - 2 x - 2\\right) \\sin{\\left(2 x^{2} - x + 1 \\right)} + e^{x}$.\nFinal answer: $\\left(- 6\\cdot x - 2\\right)\\cdot \\cos{\\left(2\\cdot x^{2} - x + 1 \\right)} + \\left(x + 1\\right)\\cdot e^{x} - \\left(4\\cdot x - 1\\right)\\cdot \\left(- 3\\cdot x^{2} - 2\\cdot x - 2\\right)\\cdot \\sin{\\left(2\\cdot x^{2} - x + 1 \\right)} + e^{x}$", "final_answer": "\\left(- 6\\cdot x - 2\\right)\\cdot \\cos{\\left(2\\cdot x^{2} - x + 1 \\right)} + \\left(x + 1\\right)\\cdot e^{x} - \\left(4\\cdot x - 1\\right)\\cdot \\left(- 3\\cdot x^{2} - 2\\cdot x - 2\\right)\\cdot \\sin{\\left(2\\cdot x^{2} - x + 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a8b655701f473cce", "problem_id": "71cbbc2f6cb59ded", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 4 x^{2} + 4 x + 3\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $4 x^{2} + 4 x + 3$.\n3) Compute an antiderivative: F(x) = $\\frac{4 x^{3}}{3} + 2 x^{2} + 3 x$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{4\\cdot x^{3}}{3} + 2\\cdot x^{2} + 3\\cdot x$", "final_answer": "C + \\frac{4\\cdot x^{3}}{3} + 2\\cdot x^{2} + 3\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "23fa4291ced85e97", "problem_id": "71cbbc2f6cb59ded", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 4 x^{2} + 4 x + 3\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $4 x^{2} + 4 x + 3$.\nCompute an antiderivative: F(x) = $\\frac{4 x^{3}}{3} + 2 x^{2} + 3 x$.\nFinal answer: $C + \\frac{4\\cdot x^{3}}{3} + 2\\cdot x^{2} + 3\\cdot x$", "final_answer": "C + \\frac{4\\cdot x^{3}}{3} + 2\\cdot x^{2} + 3\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7a568a19f42f2733", "problem_id": "3e496c4edb618a4d", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 53 liters of a 30% salt solution. You have a 10% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 53−x liters is 40% solution.\nSalt equation: 10x + 40(53−x) = 30·53.\n(10-40)x = (30-40)·53 ⇒ x = 53/3.\nThen 53−x = 106/3.\nCheck: 10·53/3 + 40·106/3 = 30·53.\nFinal answer: $53/3 L of 10% and 106/3 L of 40%$", "final_answer": "53/3 L of 10% and 106/3 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "76fe35584a40a063", "problem_id": "3e496c4edb618a4d", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "You want 53 liters of a 30% salt solution. You have a 10% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 53−x liters is 40% solution.\n2) Salt equation: 10x + 40(53−x) = 30·53.\n3) (10-40)x = (30-40)·53 ⇒ x = 53/3.\n4) Then 53−x = 106/3.\nFinal answer: $53/3 L of 10% and 106/3 L of 40%$", "final_answer": "53/3 L of 10% and 106/3 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3780a3583e880723", "problem_id": "8467c87a2c2e5df1", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Two friends split $1246 in the ratio 4:3. How much does each receive?", "output": "1) Total ratio parts = 4+3 = 7.\n2) Each part = 1246 ÷ 7 = 178.\n3) First share = 4 × 178 = 712.\n4) Second share = 3 × 178 = 534.\nFinal answer: $534$", "final_answer": "534", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b5474a5288c8fc4", "problem_id": "8467c87a2c2e5df1", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Two friends split $1246 in the ratio 4:3. How much does each receive?", "output": "Total ratio parts = 4+3 = 7.\nEach part = 1246 ÷ 7 = 178.\nFirst share = 4 × 178 = 712.\nFinal answer: $534$", "final_answer": "534", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "660432cdd4d0a1d2", "problem_id": "bace532deb08738f", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=43$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=43 and k=10.\n4) Compute C(52,9) = 3679075400.\nFinal answer: $3679075400$", "final_answer": "3679075400", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2dd534f19b59c931", "problem_id": "bace532deb08738f", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=43$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=43 and k=10.\nCompute C(52,9) = 3679075400.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $3679075400$", "final_answer": "3679075400", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4746de7965cbf404", "problem_id": "c9caa646a1f664d6", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} + 4 x - 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(2 x^{2}) = 4 x^{2}$.\n3) So $f'(x) = 4 x^{2} + 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(2 x^{2}) = 4 x$.\nTherefore the correct derivative is $f'(x) = 4 \\left(x + 1\\right)$.\nFinal answer: $4 \\left(x + 1\\right)$", "final_answer": "4 \\left(x + 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c6d25b54d82182ee", "problem_id": "5405a4e305bc4f5a", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_3=26$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=26 and k=3.\nCompute C(28,2) = 378.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $378$", "final_answer": "378", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ba07cc1a4f575240", "problem_id": "848b171121e4e6cd", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{13 x}{3} - \\frac{8}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$\\frac{13}{3}$, c=$- \\frac{8}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{361}{9}$, so √Δ = $\\frac{19}{3}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{1}{2}$.\n7) x2 = $- \\frac{8}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{1}{2}, - \\frac{8}{3}\\}$", "final_answer": "x \\in \\{\\frac{1}{2}, - \\frac{8}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8310f4e48b42432c", "problem_id": "848b171121e4e6cd", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{13 x}{3} - \\frac{8}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$\\frac{13}{3}$, c=$- \\frac{8}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{361}{9}$, so √Δ = $\\frac{19}{3}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{1}{2}$.\nx2 = $- \\frac{8}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{1}{2}, - \\frac{8}{3}\\}$", "final_answer": "x \\in \\{\\frac{1}{2}, - \\frac{8}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "53e60e6b71f39f90", "problem_id": "949f29576266dd70", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 14 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=14, k=5, p=$\\frac{1}{2}$ and simplify.\n4) Here C(14,5) = 2002.\nFinal answer: $\\frac{1001}{8192}$", "final_answer": "\\frac{1001}{8192}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a2b31292da1d693", "problem_id": "949f29576266dd70", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 14 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=14, k=5, p=$\\frac{1}{2}$ and simplify.\nHere C(14,5) = 2002.\nCheck: expression simplifies to $\\frac{1001}{8192}$.\nFinal answer: $\\frac{1001}{8192}$", "final_answer": "\\frac{1001}{8192}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b99d6858ac6ccd8", "problem_id": "69a58c012843560c", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$7 x - 3 y$ = $26$\n$- 3 x - 6 y$ = $18$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $- 21 x + 9 y$ = $-78$.\n3) Multiply the second equation by $7$: $- 21 x - 42 y$ = $126$.\n4) Subtract to eliminate x: $51 y$ = $-204$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $2$.\nFinal answer: $(x,y)=\\left(2, -4\\right)$", "final_answer": "(x,y)=\\left(2, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "13c1cf09bb3331f1", "problem_id": "69a58c012843560c", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$7 x - 3 y$ = $26$\n$- 3 x - 6 y$ = $18$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $- 21 x + 9 y$ = $-78$.\nMultiply the second equation by $7$: $- 21 x - 42 y$ = $126$.\nSubtract to eliminate x: $51 y$ = $-204$.\nSolve for y: y = $-4$.\nSubstitute back to find x: x = $2$.\nCheck: plug (x,y)=(2,-4) into both equations.\nFinal answer: $(x,y)=\\left(2, -4\\right)$", "final_answer": "(x,y)=\\left(2, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6d14d4e065097af2", "problem_id": "5e4af5b0c6eaba18", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the linear system:\n$- 8 x + 8 y$ = $-72$\n$3 x + 6 y$ = $18$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $- 24 x + 24 y$ = $-216$.\nMultiply the second equation by $-8$: $- 24 x - 48 y$ = $-144$.\nFinal answer: $(x,y)=\\left(8, -1\\right)$", "final_answer": "(x,y)=\\left(8, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9fdd45cdd691a781", "problem_id": "5e4af5b0c6eaba18", "style": "answer_only", "instruction": "Final answer only.", "input": "Solve the linear system:\n$- 8 x + 8 y$ = $-72$\n$3 x + 6 y$ = $18$", "output": "$(x,y)=\\left(8, -1\\right)$", "final_answer": "(x,y)=\\left(8, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3bbaee0cab7bfcfe", "problem_id": "1e10cce3995b9efe", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 4 - 3 x^{2}\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $4 - 3 x^{2}$.\n3) Compute an antiderivative: F(x) = $- x^{3} + 4 x$.\n4) Include the constant of integration C.\nFinal answer: $C - x^{3} + 4\\cdot x$", "final_answer": "C - x^{3} + 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a763fa9d9605b1fe", "problem_id": "1e10cce3995b9efe", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 4 - 3 x^{2}\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $4 - 3 x^{2}$.\nCompute an antiderivative: F(x) = $- x^{3} + 4 x$.\nFinal answer: $C - x^{3} + 4\\cdot x$", "final_answer": "C - x^{3} + 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d009ca3196073603", "problem_id": "6caad702bff11203", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A store sells a backpack for $820. It is discounted by 15%. You buy 13 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 820 × (1 − 15/100) = 820 × 85/100 = 697.\n2) Subtotal = 13 × 697 = 9061.\n3) Total with tax = 9061 × 107/100 = 969527/100.\nFinal answer: $\\frac{969527}{100}$", "final_answer": "\\frac{969527}{100}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9696133798df5ad6", "problem_id": "6caad702bff11203", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A store sells a backpack for $820. It is discounted by 15%. You buy 13 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 820 × (1 − 15/100) = 820 × 85/100 = 697.\nSubtotal = 13 × 697 = 9061.\nTotal with tax = 9061 × 107/100 = 969527/100.\nCheck: recomputing the formula yields 969527/100.\nFinal answer: $\\frac{969527}{100}$", "final_answer": "\\frac{969527}{100}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9f76f0fbde3ee838", "problem_id": "bc22b95c5715b950", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 3 \\sqrt{10} x + 2 x - 2 \\sqrt{10}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$2 - 3 \\sqrt{10}$, c=$- 2 \\sqrt{10}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $12 \\sqrt{10} + 94$, so √Δ = $\\sqrt{12 \\sqrt{10} + 94}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{1}{3} + \\frac{\\sqrt{10}}{2} + \\frac{\\sqrt{12 \\sqrt{10} + 94}}{6}$.\n7) x2 = $- \\frac{\\sqrt{12 \\sqrt{10} + 94}}{6} - \\frac{1}{3} + \\frac{\\sqrt{10}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{1}{3} + \\frac{\\sqrt{10}}{2} + \\frac{\\sqrt{12\\cdot \\sqrt{10} + 94}}{6}, - \\frac{\\sqrt{12\\cdot \\sqrt{10} + 94}}{6} - \\frac{1}{3} + \\frac{\\sqrt{10}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{1}{3} + \\frac{\\sqrt{10}}{2} + \\frac{\\sqrt{12\\cdot \\sqrt{10} + 94}}{6}, - \\frac{\\sqrt{12\\cdot \\sqrt{10} + 94}}{6} - \\frac{1}{3} + \\frac{\\sqrt{10}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a16507cc3f3c7f6c", "problem_id": "bc22b95c5715b950", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 3 \\sqrt{10} x + 2 x - 2 \\sqrt{10}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$2 - 3 \\sqrt{10}$, c=$- 2 \\sqrt{10}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{- \\frac{1}{3} + \\frac{\\sqrt{10}}{2} + \\frac{\\sqrt{12\\cdot \\sqrt{10} + 94}}{6}, - \\frac{\\sqrt{12\\cdot \\sqrt{10} + 94}}{6} - \\frac{1}{3} + \\frac{\\sqrt{10}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{1}{3} + \\frac{\\sqrt{10}}{2} + \\frac{\\sqrt{12\\cdot \\sqrt{10} + 94}}{6}, - \\frac{\\sqrt{12\\cdot \\sqrt{10} + 94}}{6} - \\frac{1}{3} + \\frac{\\sqrt{10}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c181db3f2fd1b323", "problem_id": "82933cdd7235b0df", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡2 (mod 4), x≡2 (mod 7). Give the solution modulo 28.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 28.\nCompute the CRT solution to get x ≡ 2 (mod 28).\nCheck: 2 mod 4=2 and mod 7=2.\nFinal answer: $x\\equiv 2\\pmod{28}$", "final_answer": "x\\equiv 2\\pmod{28}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "08735042904424b3", "problem_id": "82933cdd7235b0df", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡2 (mod 4), x≡2 (mod 7). Give the solution modulo 28.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 28.\n3) Compute the CRT solution to get x ≡ 2 (mod 28).\nFinal answer: $x\\equiv 2\\pmod{28}$", "final_answer": "x\\equiv 2\\pmod{28}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "501f1a6a713a24a2", "problem_id": "9558efb153f9dcef", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Worker A finishes a job in 12 hours and Worker B finishes it in 20 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/12 job/hour. B's rate = 1/20 job/hour.\n2) Combined rate = 1/12 + 1/20 = 2/15 job/hour.\n3) Time = 1 ÷ (combined rate) = 15/2 hours.\nFinal answer: $\\frac{15}{2}$", "final_answer": "\\frac{15}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e38e1f236e21461d", "problem_id": "9558efb153f9dcef", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Worker A finishes a job in 12 hours and Worker B finishes it in 20 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/12 job/hour. B's rate = 1/20 job/hour.\nCombined rate = 1/12 + 1/20 = 2/15 job/hour.\nTime = 1 ÷ (combined rate) = 15/2 hours.\nFinal answer: $\\frac{15}{2}$", "final_answer": "\\frac{15}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c96570f998087586", "problem_id": "0954e66808e05141", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(2 x^{3} - x^{2}\\right) \\cos{\\left(3 x^{2} + 2 x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(4 e^{x}) = 4 e^{x}$.\n3) So $f'(x) = - 12 x^{4} \\sin{\\left(\\left(3 x^{2} + 2 x\\right) + 3 \\right)} + 2 x^{3} \\sin{\\left(\\left(3 x^{2} + 2 x\\right) + 3 \\right)} + 2 x^{2} \\sin{\\left(\\left(3 x^{2} + 2 x\\right) + 3 \\right)} + 6 x^{2} \\cos{\\left(\\left(3 x^{2} + 2 x\\right) + 3 \\right)} + 2 x e^{x} - 2 x \\cos{\\left(\\left(3 x^{2} + 2 x\\right) + 3 \\right)} + 6 e^{x}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(4 e^{x}) = 4 e^{x}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- x^{2} \\left(2 x - 1\\right) \\left(3 x + 1\\right) \\sin{\\left(\\left(3 x^{2} + 2 x\\right) + 3 \\right)} + x \\left(3 x - 1\\right) \\cos{\\left(\\left(3 x^{2} + 2 x\\right) + 3 \\right)} + \\left(x + 2\\right) e^{x} + e^{x}\\right)$.\nFinal answer: $2 \\left(- x^{2} \\left(2 x - 1\\right) \\left(3 x + 1\\right) \\sin{\\left(\\left(3 x^{2} + 2 x\\right) + 3 \\right)} + x \\left(3 x - 1\\right) \\cos{\\left(\\left(3 x^{2} + 2 x\\right) + 3 \\right)} + \\left(x + 2\\right) e^{x} + e^{x}\\right)$", "final_answer": "2 \\left(- x^{2} \\left(2 x - 1\\right) \\left(3 x + 1\\right) \\sin{\\left(\\left(3 x^{2} + 2 x\\right) + 3 \\right)} + x \\left(3 x - 1\\right) \\cos{\\left(\\left(3 x^{2} + 2 x\\right) + 3 \\right)} + \\left(x + 2\\right) e^{x} + e^{x}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "bfaf846290ecc059", "problem_id": "c3d29a5296faec1b", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "How many binary strings of length 25 contain exactly 4 ones with no two ones adjacent?", "output": "1) Reserve 3 mandatory zeros between the 4 ones.\n2) This leaves n−k+1 = 22 valid slots for the ones.\n3) Choose the slots: C(22,4) = 7315.\nFinal answer: $7315$", "final_answer": "7315", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "498841a21aa1c0ba", "problem_id": "a3386a4e5f9c4b10", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A store sells a backpack for $615. It is discounted by 15%. You buy 5 backpacks. What is the total cost?", "output": "1) Discounted price = 615 × (1 − 15/100) = 615 × 85/100 = 2091/4.\n2) Subtotal = 5 × 2091/4 = 10455/4.\nFinal answer: $\\frac{10455}{4}$", "final_answer": "\\frac{10455}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a00f3993ed797ce7", "problem_id": "a3386a4e5f9c4b10", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $615. It is discounted by 15%. You buy 5 backpacks. What is the total cost?", "output": "Discounted price = 615 × (1 − 15/100) = 615 × 85/100 = 2091/4.\nSubtotal = 5 × 2091/4 = 10455/4.\nCheck: recomputing the formula yields 10455/4.\nFinal answer: $\\frac{10455}{4}$", "final_answer": "\\frac{10455}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3b875aa441817020", "problem_id": "ed573eed7b8392df", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 4 x^{2} - x - 2\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 4 x^{2} - x - 2$.\nCompute an antiderivative: F(x) = $- \\frac{4 x^{3}}{3} - \\frac{x^{2}}{2} - 2 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - \\frac{4\\cdot x^{3}}{3} - \\frac{x^{2}}{2} - 2\\cdot x$", "final_answer": "C - \\frac{4\\cdot x^{3}}{3} - \\frac{x^{2}}{2} - 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bcc8517dbbbae4c7", "problem_id": "ed573eed7b8392df", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 4 x^{2} - x - 2\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 4 x^{2} - x - 2$.\n3) Compute an antiderivative: F(x) = $- \\frac{4 x^{3}}{3} - \\frac{x^{2}}{2} - 2 x$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{4\\cdot x^{3}}{3} - \\frac{x^{2}}{2} - 2\\cdot x$", "final_answer": "C - \\frac{4\\cdot x^{3}}{3} - \\frac{x^{2}}{2} - 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ae836eb54ca01503", "problem_id": "5d65a7b3a1c5fed4", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_6=38$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_6=26.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=26 and k=6.\nCompute C(31,5) = 169911.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $169911$", "final_answer": "169911", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e2ad38e681b87e99", "problem_id": "5d65a7b3a1c5fed4", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_6=38$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_6=26.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=26 and k=6.\n4) Compute C(31,5) = 169911.\nFinal answer: $169911$", "final_answer": "169911", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "59651e2966c3659f", "problem_id": "b542dbdd43abada7", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "How many binary strings of length 8 contain exactly 4 ones with no two ones adjacent?", "output": "1) Reserve 3 mandatory zeros between the 4 ones.\n2) This leaves n−k+1 = 5 valid slots for the ones.\n3) Choose the slots: C(5,4) = 5.\nFinal answer: $5$", "final_answer": "5", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "00045d6058b65f68", "problem_id": "b542dbdd43abada7", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many binary strings of length 8 contain exactly 4 ones with no two ones adjacent?", "output": "Reserve 3 mandatory zeros between the 4 ones.\nThis leaves n−k+1 = 5 valid slots for the ones.\nChoose the slots: C(5,4) = 5.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $5$", "final_answer": "5", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b29908f1b94bd06f", "problem_id": "f15df6072b3a1877", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A store sells a backpack for $284. It is discounted by 20%. You buy 13 backpacks. What is the total cost?", "output": "1) Discounted price = 284 × (1 − 20/100) = 284 × 80/100 = 1136/5.\n2) Subtotal = 13 × 1136/5 = 14768/5.\nFinal answer: $\\frac{14768}{5}$", "final_answer": "\\frac{14768}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "86301520ccd0149e", "problem_id": "f15df6072b3a1877", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "A store sells a backpack for $284. It is discounted by 20%. You buy 13 backpacks. What is the total cost?", "output": "Discounted price = 284 × (1 − 20/100) = 284 × 80/100 = 1136/5.\nSubtotal = 13 × 1136/5 = 14768/5.\nFinal answer: $\\frac{14768}{5}$", "final_answer": "\\frac{14768}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "44660acf5206d9cc", "problem_id": "7ad3a2f26a730e42", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{3}$. If it is flipped 18 times, what is the probability of exactly 10 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=18, k=10, p=$\\frac{1}{3}$ and simplify.\n4) Here C(18,10) = 43758.\nFinal answer: $\\frac{1244672}{43046721}$", "final_answer": "\\frac{1244672}{43046721}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dfb07a6679291506", "problem_id": "7ad3a2f26a730e42", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{3}$. If it is flipped 18 times, what is the probability of exactly 10 heads?", "output": "$\\frac{1244672}{43046721}$", "final_answer": "\\frac{1244672}{43046721}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8991a450e626434f", "problem_id": "dccfc2f73f19ae4e", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A cyclist rides at 75 miles per hour for 3/2 hours, then at 58 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 75 × 3/2 = 225/2 miles.\n2) Second leg distance = 58 × 3/2 = 87 miles.\n3) Total distance = 225/2 + 87 = 399/2 miles.\nFinal answer: $\\frac{399}{2}$", "final_answer": "\\frac{399}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a3dc5f55075b4776", "problem_id": "dccfc2f73f19ae4e", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 75 miles per hour for 3/2 hours, then at 58 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 75 × 3/2 = 225/2 miles.\nSecond leg distance = 58 × 3/2 = 87 miles.\nTotal distance = 225/2 + 87 = 399/2 miles.\nCheck: adding both legs gives 399/2.\nFinal answer: $\\frac{399}{2}$", "final_answer": "\\frac{399}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ac9bdc4008281da", "problem_id": "991b343f8ba9a15c", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 23 red balls and 16 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(39,7).\nFavorable samples: choose k reds and n−k blues: C(23,2)·C(16,5).\nCompute totals: C(39,7) = 15380937, and favorable = 1105104.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{2576}{35853}$.\nFinal answer: $\\frac{2576}{35853}$", "final_answer": "\\frac{2576}{35853}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "988d5e66d135b888", "problem_id": "991b343f8ba9a15c", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 23 red balls and 16 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(39,7).\n3) Favorable samples: choose k reds and n−k blues: C(23,2)·C(16,5).\n4) Compute totals: C(39,7) = 15380937, and favorable = 1105104.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{2576}{35853}$", "final_answer": "\\frac{2576}{35853}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aa1fa479e2165354", "problem_id": "f7d8aa5901e81d57", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 22 red balls and 24 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(46,12).\n3) Favorable samples: choose k reds and n−k blues: C(22,1)·C(24,11).\n4) Compute totals: C(46,12) = 38910617655, and favorable = 54915168.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{5984}{4240015}$", "final_answer": "\\frac{5984}{4240015}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "932f449e483e6888", "problem_id": "f7d8aa5901e81d57", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 22 red balls and 24 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(46,12).\nFavorable samples: choose k reds and n−k blues: C(22,1)·C(24,11).\nCompute totals: C(46,12) = 38910617655, and favorable = 54915168.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{5984}{4240015}$.\nFinal answer: $\\frac{5984}{4240015}$", "final_answer": "\\frac{5984}{4240015}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "da05f92507114b7d", "problem_id": "77cc431c232c394e", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A cyclist rides at 63 miles per hour for 3/2 hours, then at 85 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 63 × 3/2 = 189/2 miles.\n2) Second leg distance = 85 × 2 = 170 miles.\n3) Total distance = 189/2 + 170 = 529/2 miles.\nFinal answer: $\\frac{529}{2}$", "final_answer": "\\frac{529}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7df705bd1a07a26c", "problem_id": "77cc431c232c394e", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A cyclist rides at 63 miles per hour for 3/2 hours, then at 85 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 63 × 3/2 = 189/2 miles.\nSecond leg distance = 85 × 2 = 170 miles.\nTotal distance = 189/2 + 170 = 529/2 miles.\nCheck: adding both legs gives 529/2.\nFinal answer: $\\frac{529}{2}$", "final_answer": "\\frac{529}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cfcd2d9d3c0733b9", "problem_id": "9b231151634be5ce", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A store sells a backpack for $136. It is discounted by 30%. You buy 11 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 136 × (1 − 30/100) = 136 × 70/100 = 476/5.\n2) Subtotal = 11 × 476/5 = 5236/5.\n3) Total with tax = 5236/5 × 105/100 = 27489/25.\nFinal answer: $\\frac{27489}{25}$", "final_answer": "\\frac{27489}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c66a8c0cfa7e62c1", "problem_id": "9b231151634be5ce", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $136. It is discounted by 30%. You buy 11 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 136 × (1 − 30/100) = 136 × 70/100 = 476/5.\nSubtotal = 11 × 476/5 = 5236/5.\nTotal with tax = 5236/5 × 105/100 = 27489/25.\nCheck: recomputing the formula yields 27489/25.\nFinal answer: $\\frac{27489}{25}$", "final_answer": "\\frac{27489}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "82a9438c7ee2aebf", "problem_id": "7929e79695c9415b", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡2 (mod 5), x≡1 (mod 11). Give the solution modulo 55.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 55.\n3) Compute the CRT solution to get x ≡ 12 (mod 55).\nFinal answer: $x\\equiv 12\\pmod{55}$", "final_answer": "x\\equiv 12\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "01a551435d7f7b89", "problem_id": "7929e79695c9415b", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 5), x≡1 (mod 11). Give the solution modulo 55.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 55.\nCompute the CRT solution to get x ≡ 12 (mod 55).\nFinal answer: $x\\equiv 12\\pmod{55}$", "final_answer": "x\\equiv 12\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a02250d5cd88581", "problem_id": "4790e6acdd25f72c", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Two friends split $1440 in the ratio 1:7. How much does each receive?", "output": "1) Total ratio parts = 1+7 = 8.\n2) Each part = 1440 ÷ 8 = 180.\n3) First share = 1 × 180 = 180.\n4) Second share = 7 × 180 = 1260.\nFinal answer: $1260$", "final_answer": "1260", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a2464c629f08821d", "problem_id": "4790e6acdd25f72c", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $1440 in the ratio 1:7. How much does each receive?", "output": "Total ratio parts = 1+7 = 8.\nEach part = 1440 ÷ 8 = 180.\nFirst share = 1 × 180 = 180.\nSecond share = 7 × 180 = 1260.\nCheck: 180+1260=1440.\nFinal answer: $1260$", "final_answer": "1260", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "914a6fa1bfdd1ba2", "problem_id": "caa83b32882466d9", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$2 x - 8 y$ = $-2$\n$- 11 x - 5 y$ = $11$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-11$: $- 22 x + 88 y$ = $22$.\n3) Multiply the second equation by $2$: $- 22 x - 10 y$ = $22$.\n4) Subtract to eliminate x: $98 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, 0\\right)$", "final_answer": "(x,y)=\\left(-1, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "889c0dce012e50fa", "problem_id": "caa83b32882466d9", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$2 x - 8 y$ = $-2$\n$- 11 x - 5 y$ = $11$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-11$: $- 22 x + 88 y$ = $22$.\nMultiply the second equation by $2$: $- 22 x - 10 y$ = $22$.\nSubtract to eliminate x: $98 y$ = $0$.\nSolve for y: y = $0$.\nSubstitute back to find x: x = $-1$.\nCheck: plug (x,y)=(-1,0) into both equations.\nFinal answer: $(x,y)=\\left(-1, 0\\right)$", "final_answer": "(x,y)=\\left(-1, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7753c5f23acfd1f9", "problem_id": "5e394626e5324ede", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "You want 70 liters of a 35% salt solution. You have a 30% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 70−x liters is 40% solution.\n2) Salt equation: 30x + 40(70−x) = 35·70.\n3) (30-40)x = (35-40)·70 ⇒ x = 35.\n4) Then 70−x = 35.\nFinal answer: $35 L of 30% and 35 L of 40%$", "final_answer": "35 L of 30% and 35 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a15088c2c1728d46", "problem_id": "5e394626e5324ede", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "You want 70 liters of a 35% salt solution. You have a 30% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 70−x liters is 40% solution.\nSalt equation: 30x + 40(70−x) = 35·70.\n(30-40)x = (35-40)·70 ⇒ x = 35.\nFinal answer: $35 L of 30% and 35 L of 40%$", "final_answer": "35 L of 30% and 35 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4dd93129290421ff", "problem_id": "b126ef5dcee8eb4f", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Two friends split $2158 in the ratio 8:5. How much does each receive?", "output": "1) Total ratio parts = 8+5 = 13.\n2) Each part = 2158 ÷ 13 = 166.\n3) First share = 8 × 166 = 1328.\n4) Second share = 5 × 166 = 830.\nFinal answer: $830$", "final_answer": "830", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "30f5a41450ae2a8e", "problem_id": "b126ef5dcee8eb4f", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $2158 in the ratio 8:5. How much does each receive?", "output": "Total ratio parts = 8+5 = 13.\nEach part = 2158 ÷ 13 = 166.\nFirst share = 8 × 166 = 1328.\nSecond share = 5 × 166 = 830.\nCheck: 1328+830=2158.\nFinal answer: $830$", "final_answer": "830", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "59ce6d18a807ab32", "problem_id": "fc33bfdb8b69c430", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=17$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=17 and k=7.\n4) Compute C(23,6) = 100947.\nFinal answer: $100947$", "final_answer": "100947", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8352827cb13b1464", "problem_id": "fc33bfdb8b69c430", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=17$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=17 and k=7.\nFinal answer: $100947$", "final_answer": "100947", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1598df9f5f427ff2", "problem_id": "31a4230d41352928", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the linear system:\n$- x - 5 y$ = $23$\n$- 2 x + 5 y$ = $-29$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $2 x + 10 y$ = $-46$.\nMultiply the second equation by $-1$: $2 x - 5 y$ = $29$.\nFinal answer: $(x,y)=\\left(2, -5\\right)$", "final_answer": "(x,y)=\\left(2, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ceea773c77289a55", "problem_id": "31a4230d41352928", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$- x - 5 y$ = $23$\n$- 2 x + 5 y$ = $-29$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $2 x + 10 y$ = $-46$.\n3) Multiply the second equation by $-1$: $2 x - 5 y$ = $29$.\n4) Subtract to eliminate x: $15 y$ = $-75$.\n5) Solve for y: y = $-5$.\n6) Substitute back to find x: x = $2$.\nFinal answer: $(x,y)=\\left(2, -5\\right)$", "final_answer": "(x,y)=\\left(2, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8440c7b49d4bc057", "problem_id": "b95dcc8bfda8043f", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Worker A finishes a job in 15 hours and Worker B finishes it in 6 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/15 job/hour. B's rate = 1/6 job/hour.\nCombined rate = 1/15 + 1/6 = 7/30 job/hour.\nTime = 1 ÷ (combined rate) = 30/7 hours.\nCheck: (combined rate)×(time) = 7/30×30/7 = 1.\nFinal answer: $\\frac{30}{7}$", "final_answer": "\\frac{30}{7}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "233f2dacc70308f9", "problem_id": "b95dcc8bfda8043f", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Worker A finishes a job in 15 hours and Worker B finishes it in 6 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/15 job/hour. B's rate = 1/6 job/hour.\n2) Combined rate = 1/15 + 1/6 = 7/30 job/hour.\n3) Time = 1 ÷ (combined rate) = 30/7 hours.\nFinal answer: $\\frac{30}{7}$", "final_answer": "\\frac{30}{7}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d2a69608f3dae421", "problem_id": "6ffbb966a04e4bba", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x + 3$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 3.\n3) Then the limit is 3.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 5.\nFinal answer: $5$", "final_answer": "5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0684e694ecd52446", "problem_id": "090fe2c831109cd9", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$- 5 x + y$ = $29$\n$- x - 5 y$ = $-15$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $5 x - y$ = $-29$.\n3) Multiply the second equation by $-5$: $5 x + 25 y$ = $75$.\n4) Subtract to eliminate x: $- 26 y$ = $-104$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, 4\\right)$", "final_answer": "(x,y)=\\left(-5, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a605d3d2186af77b", "problem_id": "34b6c2ce859d11de", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A cyclist rides at 55 miles per hour for 1 hours, then at 45 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 55 × 1 = 55 miles.\n2) Second leg distance = 45 × 3 = 135 miles.\n3) Total distance = 55 + 135 = 190 miles.\nFinal answer: $190$", "final_answer": "190", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c9207fa00425cae0", "problem_id": "34b6c2ce859d11de", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A cyclist rides at 55 miles per hour for 1 hours, then at 45 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 55 × 1 = 55 miles.\nSecond leg distance = 45 × 3 = 135 miles.\nTotal distance = 55 + 135 = 190 miles.\nCheck: adding both legs gives 190.\nFinal answer: $190$", "final_answer": "190", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3e93e7cf9e548eb4", "problem_id": "8ffe8a1022f7e1d1", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 2 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 2 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 3$.\n3) Compute an antiderivative: F(x) = $- \\frac{2 x^{3}}{3} - \\frac{3 x^{2}}{2} + 3 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{2\\cdot x^{3}}{3} - \\frac{3\\cdot x^{2}}{2} + 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{2\\cdot x^{3}}{3} - \\frac{3\\cdot x^{2}}{2} + 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "328e769527a2add7", "problem_id": "9324b99eb1983050", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(3 x^{2} - 2 x - 3\\right) \\cos{\\left(2 x^{2} + x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\n3) So $f'(x) = - 12 x^{3} \\sin{\\left(2 x^{2} + x \\right)} + 5 x^{2} \\sin{\\left(2 x^{2} + x \\right)} + x e^{x} + 14 x \\sin{\\left(2 x^{2} + x \\right)} + 6 x \\cos{\\left(2 x^{2} + x \\right)} + 3 e^{x} + 3 \\sin{\\left(2 x^{2} + x \\right)} - 2 \\cos{\\left(2 x^{2} + x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 2\\right) e^{x} + 2 \\left(3 x - 1\\right) \\cos{\\left(2 x^{2} + x \\right)} - \\left(4 x + 1\\right) \\left(x \\left(3 x - 2\\right) - 3\\right) \\sin{\\left(2 x^{2} + x \\right)} + e^{x}$.\nFinal answer: $\\left(x + 2\\right) e^{x} + 2 \\left(3 x - 1\\right) \\cos{\\left(2 x^{2} + x \\right)} - \\left(4 x + 1\\right) \\left(x \\left(3 x - 2\\right) - 3\\right) \\sin{\\left(2 x^{2} + x \\right)} + e^{x}$", "final_answer": "\\left(x + 2\\right) e^{x} + 2 \\left(3 x - 1\\right) \\cos{\\left(2 x^{2} + x \\right)} - \\left(4 x + 1\\right) \\left(x \\left(3 x - 2\\right) - 3\\right) \\sin{\\left(2 x^{2} + x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "16e6bd46744f1e45", "problem_id": "bc7a25d41745f97b", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Two friends split $702 in the ratio 9:9. How much does each receive?", "output": "Total ratio parts = 9+9 = 18.\nEach part = 702 ÷ 18 = 39.\nFirst share = 9 × 39 = 351.\nSecond share = 9 × 39 = 351.\nCheck: 351+351=702.\nFinal answer: $351$", "final_answer": "351", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4ff41804a5b352d6", "problem_id": "bc7a25d41745f97b", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Two friends split $702 in the ratio 9:9. How much does each receive?", "output": "1) Total ratio parts = 9+9 = 18.\n2) Each part = 702 ÷ 18 = 39.\n3) First share = 9 × 39 = 351.\n4) Second share = 9 × 39 = 351.\nFinal answer: $351$", "final_answer": "351", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "21aa923bff394ba0", "problem_id": "717837817d9354f8", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 4 x - 30$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} - 4 x - 30$ = $2 \\left(x - 5\\right) \\left(x + 3\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $5$.\nFrom the second factor: x = $-3$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{5, -3\\}$", "final_answer": "x \\in \\{5, -3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "01f35208fadfa7e8", "problem_id": "717837817d9354f8", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 4 x - 30$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 4 x - 30$ = $2 \\left(x - 5\\right) \\left(x + 3\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $5$.\n5) From the second factor: x = $-3$.\nFinal answer: $x \\in \\{5, -3\\}$", "final_answer": "x \\in \\{5, -3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "48f58e0136e6b9d6", "problem_id": "bc07347abcdf219c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 5 - x$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 5 - 2 x.\n3) Then the limit is 9.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 7.\nFinal answer: $7$", "final_answer": "7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "71eb5236313d0aa4", "problem_id": "f3de6b1b5b09bd33", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 7 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=7, k=2, p=$\\frac{1}{6}$ and simplify.\n4) Here C(7,2) = 21.\nFinal answer: $\\frac{21875}{93312}$", "final_answer": "\\frac{21875}{93312}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "873ae1f0a05496ca", "problem_id": "57256686c6aea78e", "style": "answer_only", "instruction": "Final answer only.", "input": "How many binary strings of length 20 contain exactly 3 ones with no two ones adjacent?", "output": "$816$", "final_answer": "816", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "233cf4d6f61e4fbc", "problem_id": "57256686c6aea78e", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "How many binary strings of length 20 contain exactly 3 ones with no two ones adjacent?", "output": "1) Reserve 2 mandatory zeros between the 3 ones.\n2) This leaves n−k+1 = 18 valid slots for the ones.\n3) Choose the slots: C(18,3) = 816.\nFinal answer: $816$", "final_answer": "816", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "add161d774c67826", "problem_id": "c0cd4b2c614ac0ec", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A store sells a backpack for $128. It is discounted by 20%. You buy 4 backpacks. What is the total cost?", "output": "1) Discounted price = 128 × (1 − 20/100) = 128 × 80/100 = 512/5.\n2) Subtotal = 4 × 512/5 = 2048/5.\nFinal answer: $\\frac{2048}{5}$", "final_answer": "\\frac{2048}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f1eed1270915ff6d", "problem_id": "c0cd4b2c614ac0ec", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A store sells a backpack for $128. It is discounted by 20%. You buy 4 backpacks. What is the total cost?", "output": "Discounted price = 128 × (1 − 20/100) = 128 × 80/100 = 512/5.\nSubtotal = 4 × 512/5 = 2048/5.\nCheck: recomputing the formula yields 2048/5.\nFinal answer: $\\frac{2048}{5}$", "final_answer": "\\frac{2048}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "436a5bfaf154ce7d", "problem_id": "e87817263420430d", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(x^{3} + 2 x^{2} + 2 x + 3\\right) \\cos{\\left(3 x - 2 \\right)}$.", "output": "Let f(x) = $\\left(x + 1\\right) e^{x} + \\left(x^{3} + 2 x^{2} + 2 x + 3\\right) \\cos{\\left(3 x - 2 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\left(x + 1\\right) e^{x} + \\left(3 x^{2} + 4 x + 2\\right) \\cos{\\left(3 x - 2 \\right)} - 3 \\left(x^{3} + 2 x^{2} + 2 x + 3\\right) \\sin{\\left(3 x - 2 \\right)} + e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\left(x + 1\\right)\\cdot e^{x} + \\left(3\\cdot x^{2} + 4\\cdot x + 2\\right)\\cdot \\cos{\\left(3\\cdot x - 2 \\right)} - \\left(3\\cdot x^{3} + 6\\cdot x^{2} + 6\\cdot x + 9\\right)\\cdot \\sin{\\left(3\\cdot x - 2 \\right)} + e^{x}$", "final_answer": "\\left(x + 1\\right)\\cdot e^{x} + \\left(3\\cdot x^{2} + 4\\cdot x + 2\\right)\\cdot \\cos{\\left(3\\cdot x - 2 \\right)} - \\left(3\\cdot x^{3} + 6\\cdot x^{2} + 6\\cdot x + 9\\right)\\cdot \\sin{\\left(3\\cdot x - 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b471cf351e786430", "problem_id": "4166a418f2c3a4dd", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x \\sin{\\left(x \\right)}) = - x \\cos{\\left(x \\right)}$.\n3) So $f'(x) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x \\sin{\\left(x \\right)}) = - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$", "final_answer": "x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8e708f379b08e0ca", "problem_id": "bc0d606f2a4ad813", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Worker A finishes a job in 5 hours and Worker B finishes it in 16 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/5 job/hour. B's rate = 1/16 job/hour.\n2) Combined rate = 1/5 + 1/16 = 21/80 job/hour.\n3) Time = 1 ÷ (combined rate) = 80/21 hours.\nFinal answer: $\\frac{80}{21}$", "final_answer": "\\frac{80}{21}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "76a642fb43e3c9aa", "problem_id": "bc0d606f2a4ad813", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Worker A finishes a job in 5 hours and Worker B finishes it in 16 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/5 job/hour. B's rate = 1/16 job/hour.\nCombined rate = 1/5 + 1/16 = 21/80 job/hour.\nTime = 1 ÷ (combined rate) = 80/21 hours.\nCheck: (combined rate)×(time) = 21/80×80/21 = 1.\nFinal answer: $\\frac{80}{21}$", "final_answer": "\\frac{80}{21}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8353a5ff253e89a1", "problem_id": "c881272959fc3038", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the linear system:\n$x + y$ = $0$\n$4 x + 2 y$ = $-4$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $4 x + 4 y$ = $0$.\n3) Multiply the second equation by $1$: $4 x + 2 y$ = $-4$.\n4) Subtract to eliminate x: $2 y$ = $4$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, 2\\right)$", "final_answer": "(x,y)=\\left(-2, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0d1be347395ae702", "problem_id": "c881272959fc3038", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$x + y$ = $0$\n$4 x + 2 y$ = $-4$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $4 x + 4 y$ = $0$.\nMultiply the second equation by $1$: $4 x + 2 y$ = $-4$.\nSubtract to eliminate x: $2 y$ = $4$.\nSolve for y: y = $2$.\nSubstitute back to find x: x = $-2$.\nCheck: plug (x,y)=(-2,2) into both equations.\nFinal answer: $(x,y)=\\left(-2, 2\\right)$", "final_answer": "(x,y)=\\left(-2, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "75f545be53d993d5", "problem_id": "cbfc2742c4c3ebdf", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- \\frac{x^{2}}{2} + x + \\left(x^{2} - x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\frac{1}{2}}{x^{2} - x + 1}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 1}) = \\frac{e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 1}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} - \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} - \\frac{4 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} + \\frac{3 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} + \\frac{6 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} - \\frac{x^{2}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} - \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} - \\frac{4 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} - \\frac{4 x}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} + \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 1} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} + \\frac{2 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} + \\frac{3}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 1} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 2 x + 2} + \\frac{e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 1} - \\frac{e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 1}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 1}) = \\frac{\\left(1 - 2 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(x^{2} - x\\right) + 1\\right)^{2}} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(x^{2} - x\\right) + 1\\right)} + \\frac{e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 1}$.\nTherefore the correct derivative is $f'(x) = \\frac{\\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + 2 \\left(x \\left(x - 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(x \\left(x - 1\\right) + 1\\right) \\left(- 2 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{2 \\left(x \\left(x - 1\\right) + 1\\right)^{2}}$.\nFinal answer: $\\frac{\\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + 2 \\left(x \\left(x - 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(x \\left(x - 1\\right) + 1\\right) \\left(- 2 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{2 \\left(x \\left(x - 1\\right) + 1\\right)^{2}}$", "final_answer": "\\frac{\\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + 2 \\left(x \\left(x - 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(x \\left(x - 1\\right) + 1\\right) \\left(- 2 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{2 \\left(x \\left(x - 1\\right) + 1\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5e01f47f22718689", "problem_id": "c767e3686f25fffb", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the congruences: x≡0 (mod 7), x≡0 (mod 11). Give the solution modulo 77.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 77.\n3) Compute the CRT solution to get x ≡ 0 (mod 77).\nFinal answer: $x\\equiv 0\\pmod{77}$", "final_answer": "x\\equiv 0\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "925ad04e49264485", "problem_id": "7fd543964e73311d", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A cyclist rides at 67 miles per hour for 1 hours, then at 87 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 67 × 1 = 67 miles.\n2) Second leg distance = 87 × 1 = 87 miles.\n3) Total distance = 67 + 87 = 154 miles.\nFinal answer: $154$", "final_answer": "154", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2f681244da02677a", "problem_id": "7fd543964e73311d", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "A cyclist rides at 67 miles per hour for 1 hours, then at 87 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 67 × 1 = 67 miles.\nSecond leg distance = 87 × 1 = 87 miles.\nTotal distance = 67 + 87 = 154 miles.\nFinal answer: $154$", "final_answer": "154", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72adb2933f889d34", "problem_id": "406bbc8dbed9d972", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Two friends split $1360 in the ratio 3:7. How much does each receive?", "output": "1) Total ratio parts = 3+7 = 10.\n2) Each part = 1360 ÷ 10 = 136.\n3) First share = 3 × 136 = 408.\n4) Second share = 7 × 136 = 952.\nFinal answer: $952$", "final_answer": "952", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "00747cb6a1c8171c", "problem_id": "406bbc8dbed9d972", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Two friends split $1360 in the ratio 3:7. How much does each receive?", "output": "Total ratio parts = 3+7 = 10.\nEach part = 1360 ÷ 10 = 136.\nFirst share = 3 × 136 = 408.\nSecond share = 7 × 136 = 952.\nCheck: 408+952=1360.\nFinal answer: $952$", "final_answer": "952", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "25e166df41027fa4", "problem_id": "a1a0bddbda13f46d", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{2} - 2 x + 4\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $2 x^{2} - 2 x + 4$.\n3) Compute an antiderivative: F(x) = $\\frac{2 x^{3}}{3} - x^{2} + 4 x$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{2\\cdot x^{3}}{3} - x^{2} + 4\\cdot x$", "final_answer": "C + \\frac{2\\cdot x^{3}}{3} - x^{2} + 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "172b3ed1c86d5dab", "problem_id": "a1a0bddbda13f46d", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{2} - 2 x + 4\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $2 x^{2} - 2 x + 4$.\nCompute an antiderivative: F(x) = $\\frac{2 x^{3}}{3} - x^{2} + 4 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{2\\cdot x^{3}}{3} - x^{2} + 4\\cdot x$", "final_answer": "C + \\frac{2\\cdot x^{3}}{3} - x^{2} + 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "58758b6047092148", "problem_id": "ac7af901556d5195", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Two friends split $60 in the ratio 7:3. How much does each receive?", "output": "1) Total ratio parts = 7+3 = 10.\n2) Each part = 60 ÷ 10 = 6.\n3) First share = 7 × 6 = 42.\n4) Second share = 3 × 6 = 18.\nFinal answer: $18$", "final_answer": "18", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ffcf93fa03d32e48", "problem_id": "ac7af901556d5195", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Two friends split $60 in the ratio 7:3. How much does each receive?", "output": "Total ratio parts = 7+3 = 10.\nEach part = 60 ÷ 10 = 6.\nFirst share = 7 × 6 = 42.\nFinal answer: $18$", "final_answer": "18", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8bdbd4c2f49132f8", "problem_id": "2a1b6cf70f7774c2", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} - 2 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} + 1$ at $x=-1$.", "output": "Differentiate: f'(x) = $- 3 x^{2} - 4 x + 2 \\cos{\\left(x \\right)} - 2$.\nSlope at x=-1: m = f'(-1) = $-1 + 2 \\cos{\\left(1 \\right)}$.\nPoint on curve: (-1, f(-1)) = (-1, $2 - 2 \\sin{\\left(1 \\right)}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 2 - 2 \\sin{\\left(1 \\right)} = -1 + 2 \\cos{\\left(1 \\right)}(x − -1).\nOptionally expand: y = $- x + 2 x \\cos{\\left(1 \\right)} - 2 \\sin{\\left(1 \\right)} + 1 + 2 \\cos{\\left(1 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - x + 2\\cdot x\\cdot \\cos{\\left(1 \\right)} - 2\\cdot \\sin{\\left(1 \\right)} + 1 + 2\\cdot \\cos{\\left(1 \\right)}$", "final_answer": "y = - x + 2\\cdot x\\cdot \\cos{\\left(1 \\right)} - 2\\cdot \\sin{\\left(1 \\right)} + 1 + 2\\cdot \\cos{\\left(1 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "98d7b7fabafb4a8b", "problem_id": "2a1b6cf70f7774c2", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} - 2 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} + 1$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $- 3 x^{2} - 4 x + 2 \\cos{\\left(x \\right)} - 2$.\n2) Slope at x=-1: m = f'(-1) = $-1 + 2 \\cos{\\left(1 \\right)}$.\n3) Point on curve: (-1, f(-1)) = (-1, $2 - 2 \\sin{\\left(1 \\right)}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 2 - 2 \\sin{\\left(1 \\right)} = -1 + 2 \\cos{\\left(1 \\right)}(x − -1).\n6) Optionally expand: y = $- x + 2 x \\cos{\\left(1 \\right)} - 2 \\sin{\\left(1 \\right)} + 1 + 2 \\cos{\\left(1 \\right)}$.\nFinal answer: $y = - x + 2\\cdot x\\cdot \\cos{\\left(1 \\right)} - 2\\cdot \\sin{\\left(1 \\right)} + 1 + 2\\cdot \\cos{\\left(1 \\right)}$", "final_answer": "y = - x + 2\\cdot x\\cdot \\cos{\\left(1 \\right)} - 2\\cdot \\sin{\\left(1 \\right)} + 1 + 2\\cdot \\cos{\\left(1 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "da1120174273cecf", "problem_id": "5a679705a6a7e278", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=20$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=20 and k=6.\n4) Compute C(25,5) = 53130.\nFinal answer: $53130$", "final_answer": "53130", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4a5aeaa1060c8806", "problem_id": "5a679705a6a7e278", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=20$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=20 and k=6.\nFinal answer: $53130$", "final_answer": "53130", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f4b4f6c5987222cd", "problem_id": "42dabb638da3d9c2", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "You want 63 liters of a 50% salt solution. You have a 30% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 63−x liters is 70% solution.\nSalt equation: 30x + 70(63−x) = 50·63.\n(30-70)x = (50-70)·63 ⇒ x = 63/2.\nThen 63−x = 63/2.\nCheck: 30·63/2 + 70·63/2 = 50·63.\nFinal answer: $63/2 L of 30% and 63/2 L of 70%$", "final_answer": "63/2 L of 30% and 63/2 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b9e3780188d09d95", "problem_id": "42dabb638da3d9c2", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "You want 63 liters of a 50% salt solution. You have a 30% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 63−x liters is 70% solution.\n2) Salt equation: 30x + 70(63−x) = 50·63.\n3) (30-70)x = (50-70)·63 ⇒ x = 63/2.\n4) Then 63−x = 63/2.\nFinal answer: $63/2 L of 30% and 63/2 L of 70%$", "final_answer": "63/2 L of 30% and 63/2 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fad9373c40ee5f51", "problem_id": "b43f27f0210897ef", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 24 red balls and 14 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(38,9).\n3) Favorable samples: choose k reds and n−k blues: C(24,4)·C(14,5).\n4) Compute totals: C(38,9) = 163011640, and favorable = 21273252.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{483483}{3704810}$", "final_answer": "\\frac{483483}{3704810}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "31a1d71c358460f1", "problem_id": "b43f27f0210897ef", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 24 red balls and 14 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(38,9).\nFavorable samples: choose k reds and n−k blues: C(24,4)·C(14,5).\nCompute totals: C(38,9) = 163011640, and favorable = 21273252.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{483483}{3704810}$.\nFinal answer: $\\frac{483483}{3704810}$", "final_answer": "\\frac{483483}{3704810}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4acec3ada1ca5eba", "problem_id": "4ef76f011bd2b562", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Worker A finishes a job in 6 hours and Worker B finishes it in 11 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/6 job/hour. B's rate = 1/11 job/hour.\n2) Combined rate = 1/6 + 1/11 = 17/66 job/hour.\n3) Time = 1 ÷ (combined rate) = 66/17 hours.\nFinal answer: $\\frac{66}{17}$", "final_answer": "\\frac{66}{17}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bde1a1c1844fec11", "problem_id": "4ef76f011bd2b562", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Worker A finishes a job in 6 hours and Worker B finishes it in 11 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/6 job/hour. B's rate = 1/11 job/hour.\nCombined rate = 1/6 + 1/11 = 17/66 job/hour.\nTime = 1 ÷ (combined rate) = 66/17 hours.\nCheck: (combined rate)×(time) = 17/66×66/17 = 1.\nFinal answer: $\\frac{66}{17}$", "final_answer": "\\frac{66}{17}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "954041d5265bd364", "problem_id": "e6d79d5e4af4b0bb", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "How many binary strings of length 20 contain exactly 4 ones with no two ones adjacent?", "output": "1) Reserve 3 mandatory zeros between the 4 ones.\n2) This leaves n−k+1 = 17 valid slots for the ones.\n3) Choose the slots: C(17,4) = 2380.\nFinal answer: $2380$", "final_answer": "2380", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "484d40e7daaf4bb8", "problem_id": "e6d79d5e4af4b0bb", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many binary strings of length 20 contain exactly 4 ones with no two ones adjacent?", "output": "Reserve 3 mandatory zeros between the 4 ones.\nThis leaves n−k+1 = 17 valid slots for the ones.\nChoose the slots: C(17,4) = 2380.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $2380$", "final_answer": "2380", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a805b3ec3f8c5ff5", "problem_id": "f04bea27458a0686", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 6 times, what is the probability of exactly 2 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=6, k=2, p=$\\frac{1}{8}$ and simplify.\nHere C(6,2) = 15.\nCheck: expression simplifies to $\\frac{36015}{262144}$.\nFinal answer: $\\frac{36015}{262144}$", "final_answer": "\\frac{36015}{262144}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4aa7726a6e320434", "problem_id": "f04bea27458a0686", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 6 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=6, k=2, p=$\\frac{1}{8}$ and simplify.\n4) Here C(6,2) = 15.\nFinal answer: $\\frac{36015}{262144}$", "final_answer": "\\frac{36015}{262144}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d8eeb4dbc462cffc", "problem_id": "7b28e84eb1f88d32", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 17 times, what is the probability of exactly 8 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=17, k=8, p=$\\frac{2}{3}$ and simplify.\n4) Here C(17,8) = 24310.\nFinal answer: $\\frac{6223360}{129140163}$", "final_answer": "\\frac{6223360}{129140163}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "435d36289e1865af", "problem_id": "7b28e84eb1f88d32", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 17 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=17, k=8, p=$\\frac{2}{3}$ and simplify.\nFinal answer: $\\frac{6223360}{129140163}$", "final_answer": "\\frac{6223360}{129140163}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9aa5be96960f4c2a", "problem_id": "87133f1ccbc533c0", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A disease has prevalence $\\frac{3}{100}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{151}{2000}$.\n3) Compute P(D|+) = $\\frac{54}{151}$.\nFinal answer: $\\frac{54}{151}$", "final_answer": "\\frac{54}{151}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5bb0a03eb586dc6b", "problem_id": "87133f1ccbc533c0", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A disease has prevalence $\\frac{3}{100}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{151}{2000}$.\nCompute P(D|+) = $\\frac{54}{151}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{54}{151}$", "final_answer": "\\frac{54}{151}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f56d502a3e59ae36", "problem_id": "bb3e1182e1fb7a95", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "You want 36 liters of a 45% salt solution. You have a 30% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 36−x liters is 60% solution.\n2) Salt equation: 30x + 60(36−x) = 45·36.\n3) (30-60)x = (45-60)·36 ⇒ x = 18.\n4) Then 36−x = 18.\nFinal answer: $18 L of 30% and 18 L of 60%$", "final_answer": "18 L of 30% and 18 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a6dc968ae8e51b1f", "problem_id": "e3905de450773855", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} - x - 4$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to - 2 x - 4.\n3) Then the limit is -6.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: -5.\nFinal answer: $-5$", "final_answer": "-5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "cb0cd047dbf777fb", "problem_id": "adcd20268f12c9e2", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{17 x}{15} - \\frac{\\sqrt{10} x}{3} - \\frac{6}{5} + \\frac{3 \\sqrt{10}}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{17}{15} - \\frac{\\sqrt{10}}{3}$, c=$- \\frac{6}{5} + \\frac{3 \\sqrt{10}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{1619}{225} - \\frac{74 \\sqrt{10}}{45}$, so √Δ = $\\sqrt{\\frac{1619}{225} - \\frac{74 \\sqrt{10}}{45}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{10}}{6} + \\frac{17}{30} + \\frac{\\sqrt{1619 - 370 \\sqrt{10}}}{30}$.\n7) x2 = $- \\frac{\\sqrt{1619 - 370 \\sqrt{10}}}{30} + \\frac{\\sqrt{10}}{6} + \\frac{17}{30}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{10}}{6} + \\frac{17}{30} + \\frac{\\sqrt{1619 - 370\\cdot \\sqrt{10}}}{30}, - \\frac{\\sqrt{1619 - 370\\cdot \\sqrt{10}}}{30} + \\frac{\\sqrt{10}}{6} + \\frac{17}{30}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{10}}{6} + \\frac{17}{30} + \\frac{\\sqrt{1619 - 370\\cdot \\sqrt{10}}}{30}, - \\frac{\\sqrt{1619 - 370\\cdot \\sqrt{10}}}{30} + \\frac{\\sqrt{10}}{6} + \\frac{17}{30}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "700ac08de9216ea8", "problem_id": "adcd20268f12c9e2", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{17 x}{15} - \\frac{\\sqrt{10} x}{3} - \\frac{6}{5} + \\frac{3 \\sqrt{10}}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{17}{15} - \\frac{\\sqrt{10}}{3}$, c=$- \\frac{6}{5} + \\frac{3 \\sqrt{10}}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{1619}{225} - \\frac{74 \\sqrt{10}}{45}$, so √Δ = $\\sqrt{\\frac{1619}{225} - \\frac{74 \\sqrt{10}}{45}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{10}}{6} + \\frac{17}{30} + \\frac{\\sqrt{1619 - 370 \\sqrt{10}}}{30}$.\nx2 = $- \\frac{\\sqrt{1619 - 370 \\sqrt{10}}}{30} + \\frac{\\sqrt{10}}{6} + \\frac{17}{30}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{10}}{6} + \\frac{17}{30} + \\frac{\\sqrt{1619 - 370\\cdot \\sqrt{10}}}{30}, - \\frac{\\sqrt{1619 - 370\\cdot \\sqrt{10}}}{30} + \\frac{\\sqrt{10}}{6} + \\frac{17}{30}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{10}}{6} + \\frac{17}{30} + \\frac{\\sqrt{1619 - 370\\cdot \\sqrt{10}}}{30}, - \\frac{\\sqrt{1619 - 370\\cdot \\sqrt{10}}}{30} + \\frac{\\sqrt{10}}{6} + \\frac{17}{30}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "11dc777f3c03fb91", "problem_id": "efd797e912d16f69", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$7 x + 3 y$ = $70$\n$- 3 x - 5 y$ = $-56$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $- 21 x - 9 y$ = $-210$.\n3) Multiply the second equation by $7$: $- 21 x - 35 y$ = $-392$.\n4) Subtract to eliminate x: $26 y$ = $182$.\n5) Solve for y: y = $7$.\n6) Substitute back to find x: x = $7$.\nFinal answer: $(x,y)=\\left(7, 7\\right)$", "final_answer": "(x,y)=\\left(7, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9528c651f15326c2", "problem_id": "efd797e912d16f69", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$7 x + 3 y$ = $70$\n$- 3 x - 5 y$ = $-56$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $- 21 x - 9 y$ = $-210$.\nMultiply the second equation by $7$: $- 21 x - 35 y$ = $-392$.\nSubtract to eliminate x: $26 y$ = $182$.\nSolve for y: y = $7$.\nSubstitute back to find x: x = $7$.\nCheck: plug (x,y)=(7,7) into both equations.\nFinal answer: $(x,y)=\\left(7, 7\\right)$", "final_answer": "(x,y)=\\left(7, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b4c38ce4f34eee12", "problem_id": "024917e8a978a81f", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=40$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=40 and k=7.\nFinal answer: $9366819$", "final_answer": "9366819", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d84e67cbe95b8046", "problem_id": "024917e8a978a81f", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=40$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=40 and k=7.\n4) Compute C(46,6) = 9366819.\nFinal answer: $9366819$", "final_answer": "9366819", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e66c7244964a1007", "problem_id": "7058a2f3b9fb8bad", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Two friends split $1092 in the ratio 6:1. How much does each receive?", "output": "Total ratio parts = 6+1 = 7.\nEach part = 1092 ÷ 7 = 156.\nFirst share = 6 × 156 = 936.\nSecond share = 1 × 156 = 156.\nCheck: 936+156=1092.\nFinal answer: $156$", "final_answer": "156", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "59953ae4b6114696", "problem_id": "11024044ec7ec250", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{3} + 4 x^{2} + 9 x + 20$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} + 4 x^{2} + 8 x + 20.\n3) Then the limit is 68.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 70.\nFinal answer: $70$", "final_answer": "70", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "689ef61f8091cc6b", "problem_id": "66c20e08eccfdaa4", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + \\frac{42 x}{5} + \\frac{24}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$\\frac{42}{5}$, c=$\\frac{24}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{324}{25}$, so √Δ = $\\frac{18}{5}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{4}{5}$.\n7) x2 = $-2$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{4}{5}, -2\\}$", "final_answer": "x \\in \\{- \\frac{4}{5}, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "da4578e1ad3cb192", "problem_id": "66c20e08eccfdaa4", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + \\frac{42 x}{5} + \\frac{24}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$\\frac{42}{5}$, c=$\\frac{24}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{324}{25}$, so √Δ = $\\frac{18}{5}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{4}{5}$.\nx2 = $-2$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{4}{5}, -2\\}$", "final_answer": "x \\in \\{- \\frac{4}{5}, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6386b1392a1ca6c7", "problem_id": "ebcf7336d9c2489a", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A store sells a backpack for $173. It is discounted by 40%. You buy 14 backpacks. What is the total cost?", "output": "1) Discounted price = 173 × (1 − 40/100) = 173 × 60/100 = 519/5.\n2) Subtotal = 14 × 519/5 = 7266/5.\nFinal answer: $\\frac{7266}{5}$", "final_answer": "\\frac{7266}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "81389176c2b44e08", "problem_id": "ebcf7336d9c2489a", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "A store sells a backpack for $173. It is discounted by 40%. You buy 14 backpacks. What is the total cost?", "output": "Discounted price = 173 × (1 − 40/100) = 173 × 60/100 = 519/5.\nSubtotal = 14 × 519/5 = 7266/5.\nFinal answer: $\\frac{7266}{5}$", "final_answer": "\\frac{7266}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec1b7ed97aa5304a", "problem_id": "3e15c4b54b0085e7", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 14 x + 24$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 14 x + 24$ = $2 \\left(x - 4\\right) \\left(x - 3\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $4$.\n5) From the second factor: x = $3$.\nFinal answer: $x \\in \\{4, 3\\}$", "final_answer": "x \\in \\{4, 3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "26cf5ed9988e0e31", "problem_id": "3e15c4b54b0085e7", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 14 x + 24$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} - 14 x + 24$ = $2 \\left(x - 4\\right) \\left(x - 3\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $4$.\nFrom the second factor: x = $3$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{4, 3\\}$", "final_answer": "x \\in \\{4, 3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6d18c33464141d72", "problem_id": "2d2af815c2c3f46c", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Worker A finishes a job in 6 hours and Worker B finishes it in 14 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/6 job/hour. B's rate = 1/14 job/hour.\n2) Combined rate = 1/6 + 1/14 = 5/21 job/hour.\n3) Time = 1 ÷ (combined rate) = 21/5 hours.\nFinal answer: $\\frac{21}{5}$", "final_answer": "\\frac{21}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff991d5d3c3614bf", "problem_id": "2d2af815c2c3f46c", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Worker A finishes a job in 6 hours and Worker B finishes it in 14 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "$\\frac{21}{5}$", "final_answer": "\\frac{21}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bee78b5457c9bd9f", "problem_id": "194d9bb3a855018c", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡2 (mod 13). Give the solution modulo 39.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 39.\nCompute the CRT solution to get x ≡ 28 (mod 39).\nCheck: 28 mod 3=1 and mod 13=2.\nFinal answer: $x\\equiv 28\\pmod{39}$", "final_answer": "x\\equiv 28\\pmod{39}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5cdd5015d6ebe2b3", "problem_id": "194d9bb3a855018c", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡2 (mod 13). Give the solution modulo 39.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 39.\n3) Compute the CRT solution to get x ≡ 28 (mod 39).\nFinal answer: $x\\equiv 28\\pmod{39}$", "final_answer": "x\\equiv 28\\pmod{39}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f844d668d6b7ae20", "problem_id": "0fcce856a8f0352c", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "You want 60 liters of a 40% salt solution. You have a 15% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 60−x liters is 50% solution.\n2) Salt equation: 15x + 50(60−x) = 40·60.\n3) (15-50)x = (40-50)·60 ⇒ x = 120/7.\n4) Then 60−x = 300/7.\nFinal answer: $120/7 L of 15% and 300/7 L of 50%$", "final_answer": "120/7 L of 15% and 300/7 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e67dd90e79620e34", "problem_id": "2aff119ccb94f44f", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 2 x^{2} \\sin{\\left(x \\right)}) = - 2 x^{2} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 2 x^{2} \\sin{\\left(x \\right)}) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$", "final_answer": "x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e52ba605497942ad", "problem_id": "89296b6225dcd545", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "An urn has 11 red balls and 6 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(17,7).\n3) Favorable samples: choose k reds and n−k blues: C(11,3)·C(6,4).\n4) Compute totals: C(17,7) = 19448, and favorable = 2475.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{225}{1768}$", "final_answer": "\\frac{225}{1768}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3c2333a2533ee030", "problem_id": "89296b6225dcd545", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 11 red balls and 6 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(17,7).\nFavorable samples: choose k reds and n−k blues: C(11,3)·C(6,4).\nCompute totals: C(17,7) = 19448, and favorable = 2475.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{225}{1768}$.\nFinal answer: $\\frac{225}{1768}$", "final_answer": "\\frac{225}{1768}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9cc34c6654aa45a7", "problem_id": "a2c3a8b7fb694a52", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Worker A finishes a job in 4 hours and Worker B finishes it in 7 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/4 job/hour. B's rate = 1/7 job/hour.\n2) Combined rate = 1/4 + 1/7 = 11/28 job/hour.\n3) Time = 1 ÷ (combined rate) = 28/11 hours.\nFinal answer: $\\frac{28}{11}$", "final_answer": "\\frac{28}{11}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "583429959354af03", "problem_id": "5271d56e18843f66", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 4 x^{2} - 4 x - 2$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 4 x) = - 4 x$.\n3) So $f'(x) = - 12 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 4 x) = -4$.\nTherefore the correct derivative is $f'(x) = 4 \\left(- 2 x - 1\\right)$.\nFinal answer: $4 \\left(- 2 x - 1\\right)$", "final_answer": "4 \\left(- 2 x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6909c383e8096a3a", "problem_id": "e81f69fbac60da18", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} + 2 x - 4$.", "output": "Differentiate term-by-term:\nd/dx(-4) = 0.\nd/dx(x^{2}) = 2 x.\nFinal answer: $2\\cdot x + 2$", "final_answer": "2\\cdot x + 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a2aa7ddb39c71f59", "problem_id": "e81f69fbac60da18", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} + 2 x - 4$.", "output": "Differentiate term-by-term:\nd/dx(-4) = 0.\nd/dx(x^{2}) = 2 x.\nd/dx(2 x) = 2.\nCombine: f'(x) = 2 \\left(x + 1\\right).\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $2\\cdot x + 2$", "final_answer": "2\\cdot x + 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9630c3598ed73b42", "problem_id": "4394e832e7c0b94c", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$3 x + 6 y$ = $-30$\n$- 6 x - 3 y$ = $6$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-6$: $- 18 x - 36 y$ = $180$.\nMultiply the second equation by $3$: $- 18 x - 9 y$ = $18$.\nSubtract to eliminate x: $- 27 y$ = $162$.\nSolve for y: y = $-6$.\nSubstitute back to find x: x = $2$.\nCheck: plug (x,y)=(2,-6) into both equations.\nFinal answer: $(x,y)=\\left(2, -6\\right)$", "final_answer": "(x,y)=\\left(2, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1b742716e02630be", "problem_id": "4394e832e7c0b94c", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$3 x + 6 y$ = $-30$\n$- 6 x - 3 y$ = $6$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-6$: $- 18 x - 36 y$ = $180$.\n3) Multiply the second equation by $3$: $- 18 x - 9 y$ = $18$.\n4) Subtract to eliminate x: $- 27 y$ = $162$.\n5) Solve for y: y = $-6$.\n6) Substitute back to find x: x = $2$.\nFinal answer: $(x,y)=\\left(2, -6\\right)$", "final_answer": "(x,y)=\\left(2, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "33568354aa144be3", "problem_id": "6b48787f749e62c7", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=36$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=36 and k=6.\n4) Compute C(41,5) = 749398.\nFinal answer: $749398$", "final_answer": "749398", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "99dabc6df45b3442", "problem_id": "6b48787f749e62c7", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=36$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=36 and k=6.\nCompute C(41,5) = 749398.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $749398$", "final_answer": "749398", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d8abc890bee05a0a", "problem_id": "f73ac11c8ffc93ef", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{4 x}{3} - \\frac{70}{9}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$- \\frac{4}{3}$, c=$- \\frac{70}{9}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $64$, so √Δ = $8$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{7}{3}$.\n7) x2 = $- \\frac{5}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{7}{3}, - \\frac{5}{3}\\}$", "final_answer": "x \\in \\{\\frac{7}{3}, - \\frac{5}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "069f46de751362c9", "problem_id": "f73ac11c8ffc93ef", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{4 x}{3} - \\frac{70}{9}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$- \\frac{4}{3}$, c=$- \\frac{70}{9}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{7}{3}, - \\frac{5}{3}\\}$", "final_answer": "x \\in \\{\\frac{7}{3}, - \\frac{5}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "09626b7ab61b641c", "problem_id": "469c632b2a2d3132", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 41 modulo 53.", "output": "We want x such that 41x ≡ 1 (mod 53).\nUse extended Euclid to find s,t with 41s + 53t = 1.\nOne Bezout identity is: 41·(22) + 53·(-17) = 1.\nReduce s modulo 53: x ≡ 22 (mod 53).\nCheck: 41·22 mod 53 = 1.\nFinal answer: $x\\equiv 22\\pmod{53}$", "final_answer": "x\\equiv 22\\pmod{53}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c39fae38363e1158", "problem_id": "0d1ed7df721c833d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(3 x^{2} + x + 1\\right) \\cos{\\left(x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x \\cos{\\left(x + 3 \\right)}) = - x \\sin{\\left(x + 3 \\right)}$.\n3) So $f'(x) = - 3 x^{2} \\sin{\\left(x + 3 \\right)} + 2 x e^{x} - x \\sin{\\left(x + 3 \\right)} + 6 x \\cos{\\left(x + 3 \\right)} - 4 e^{x} - \\sin{\\left(x + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x \\cos{\\left(x + 3 \\right)}) = - x \\sin{\\left(x + 3 \\right)} + \\cos{\\left(x + 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x - 3\\right) e^{x} + \\left(6 x + 1\\right) \\cos{\\left(x + 3 \\right)} - \\left(x \\left(3 x + 1\\right) + 1\\right) \\sin{\\left(x + 3 \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(x - 3\\right) e^{x} + \\left(6 x + 1\\right) \\cos{\\left(x + 3 \\right)} - \\left(x \\left(3 x + 1\\right) + 1\\right) \\sin{\\left(x + 3 \\right)} + 2 e^{x}$", "final_answer": "2 \\left(x - 3\\right) e^{x} + \\left(6 x + 1\\right) \\cos{\\left(x + 3 \\right)} - \\left(x \\left(3 x + 1\\right) + 1\\right) \\sin{\\left(x + 3 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ce8b9f348b96be7f", "problem_id": "799ba01f5b1e3201", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 3$ at $x=2$.", "output": "1) Differentiate: f'(x) = $9 x^{2} + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\n2) Slope at x=2: m = f'(2) = $\\frac{e}{2} - 2 \\sin{\\left(4 \\right)} + 34$.\n3) Point on curve: (2, f(2)) = (2, $\\cos{\\left(4 \\right)} + e + 17$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(4 \\right)} + e + 17 = \\frac{e}{2} - 2 \\sin{\\left(4 \\right)} + 34(x − 2).\n6) Optionally expand: y = $\\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 34 x - 51 + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$.\nFinal answer: $y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 34\\cdot x - 51 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$", "final_answer": "y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 34\\cdot x - 51 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1b63eb0da7474919", "problem_id": "799ba01f5b1e3201", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 3$ at $x=2$.", "output": "Differentiate: f'(x) = $9 x^{2} + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\nSlope at x=2: m = f'(2) = $\\frac{e}{2} - 2 \\sin{\\left(4 \\right)} + 34$.\nPoint on curve: (2, f(2)) = (2, $\\cos{\\left(4 \\right)} + e + 17$).\nFinal answer: $y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 34\\cdot x - 51 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$", "final_answer": "y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 34\\cdot x - 51 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "923941e662b70d7a", "problem_id": "a59a6976d7da736b", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the congruences: x≡1 (mod 7), x≡3 (mod 5). Give the solution modulo 35.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 35.\n3) Compute the CRT solution to get x ≡ 8 (mod 35).\nFinal answer: $x\\equiv 8\\pmod{35}$", "final_answer": "x\\equiv 8\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3407799a0a7febf5", "problem_id": "a59a6976d7da736b", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡1 (mod 7), x≡3 (mod 5). Give the solution modulo 35.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 35.\nCompute the CRT solution to get x ≡ 8 (mod 35).\nCheck: 8 mod 7=1 and mod 5=3.\nFinal answer: $x\\equiv 8\\pmod{35}$", "final_answer": "x\\equiv 8\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2b595ecc8579b201", "problem_id": "556d2b3f8051a180", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Two friends split $684 in the ratio 9:3. How much does each receive?", "output": "Total ratio parts = 9+3 = 12.\nEach part = 684 ÷ 12 = 57.\nFirst share = 9 × 57 = 513.\nFinal answer: $171$", "final_answer": "171", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e2b686360d8f1fcb", "problem_id": "556d2b3f8051a180", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Two friends split $684 in the ratio 9:3. How much does each receive?", "output": "1) Total ratio parts = 9+3 = 12.\n2) Each part = 684 ÷ 12 = 57.\n3) First share = 9 × 57 = 513.\n4) Second share = 3 × 57 = 171.\nFinal answer: $171$", "final_answer": "171", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "52ee543fd4fe12fa", "problem_id": "7259d64bd33b2019", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 4 x^{3} - x^{2} - 2 x - 1\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $4 x^{3} - x^{2} - 2 x - 1$.\n3) Compute an antiderivative: F(x) = $x^{4} - \\frac{x^{3}}{3} - x^{2} - x$.\n4) Include the constant of integration C.\nFinal answer: $C + x^{4} - \\frac{x^{3}}{3} - x^{2} - x$", "final_answer": "C + x^{4} - \\frac{x^{3}}{3} - x^{2} - x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a236e5427c808215", "problem_id": "80f53ffa2a56b3ea", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} - 3 x - 4$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to - 4 x - 4.\n3) Then the limit is -8.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: -7.\nFinal answer: $-7$", "final_answer": "-7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c23a70eab59bba89", "problem_id": "c739be763fd3a448", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "How many binary strings of length 19 contain exactly 3 ones with no two ones adjacent?", "output": "Reserve 2 mandatory zeros between the 3 ones.\nThis leaves n−k+1 = 17 valid slots for the ones.\nChoose the slots: C(17,3) = 680.\nFinal answer: $680$", "final_answer": "680", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "786fec604d47cce9", "problem_id": "c739be763fd3a448", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many binary strings of length 19 contain exactly 3 ones with no two ones adjacent?", "output": "Reserve 2 mandatory zeros between the 3 ones.\nThis leaves n−k+1 = 17 valid slots for the ones.\nChoose the slots: C(17,3) = 680.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $680$", "final_answer": "680", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9703bc3e339bfa4d", "problem_id": "3aa6f76cc6768715", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 7 red balls and 14 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(21,7).\n3) Favorable samples: choose k reds and n−k blues: C(7,3)·C(14,4).\n4) Compute totals: C(21,7) = 116280, and favorable = 35035.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{7007}{23256}$", "final_answer": "\\frac{7007}{23256}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "705ebcb9dbbc0b67", "problem_id": "09726800bf81c01d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} + 4 x^{2} + 3 x$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(3 x) = 3 x$.\n3) So $f'(x) = 3 x^{2} + 11 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(3 x) = 3$.\nTherefore the correct derivative is $f'(x) = 3 x^{2} + 8 x + 3$.\nFinal answer: $3 x^{2} + 8 x + 3$", "final_answer": "3 x^{2} + 8 x + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "eee13d83b5446d97", "problem_id": "575a85a033bbab08", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{3} + x^{2} - 2 x - 2\\,dx$.", "output": "$C + \\frac{x^{4}}{2} + \\frac{x^{3}}{3} - x^{2} - 2\\cdot x$", "final_answer": "C + \\frac{x^{4}}{2} + \\frac{x^{3}}{3} - x^{2} - 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "06ea04fae2ddd381", "problem_id": "575a85a033bbab08", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{3} + x^{2} - 2 x - 2\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $2 x^{3} + x^{2} - 2 x - 2$.\n3) Compute an antiderivative: F(x) = $\\frac{x^{4}}{2} + \\frac{x^{3}}{3} - x^{2} - 2 x$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{x^{4}}{2} + \\frac{x^{3}}{3} - x^{2} - 2\\cdot x$", "final_answer": "C + \\frac{x^{4}}{2} + \\frac{x^{3}}{3} - x^{2} - 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "47d0101a6b88f621", "problem_id": "81439d83112399fd", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 4 x - 48$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} + 4 x - 48$ = $2 \\left(x - 4\\right) \\left(x + 6\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $4$.\n5) From the second factor: x = $-6$.\nFinal answer: $x \\in \\{4, -6\\}$", "final_answer": "x \\in \\{4, -6\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9abbda168b0d359b", "problem_id": "936e13033f401138", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 3 x^{2} + 9 x + 19$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{2} + 8 x + 19.\n3) Then the limit is 47.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 49.\nFinal answer: $49$", "final_answer": "49", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "56583a70e3c92b94", "problem_id": "ec402b48406a19af", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - x^{3} - 6 x^{2} - 15 x - 45$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{4} - 3 x^{3} + 3 x^{2} + 3$ and a = 3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 4 x^{3} - 9 x^{2} + 6 x$.\n5) Evaluate at a: p'(3) = $-171$.\nFinal answer: $-171$", "final_answer": "-171", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6c249bf4548d4098", "problem_id": "0639cd931b08127e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x - 1$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 3 x - 1.\n3) Then the limit is -7.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -5.\nFinal answer: $-5$", "final_answer": "-5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0ab74a75c18e82f7", "problem_id": "8f8dd3066bfe3b75", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{x}{2} - \\frac{3}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$- \\frac{1}{2}$, c=$- \\frac{3}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{49}{4}$, so √Δ = $\\frac{7}{2}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $1$.\n7) x2 = $- \\frac{3}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{1, - \\frac{3}{4}\\}$", "final_answer": "x \\in \\{1, - \\frac{3}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "13e2ea967d212512", "problem_id": "8f8dd3066bfe3b75", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{x}{2} - \\frac{3}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$- \\frac{1}{2}$, c=$- \\frac{3}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{1, - \\frac{3}{4}\\}$", "final_answer": "x \\in \\{1, - \\frac{3}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "334bb99829609b6a", "problem_id": "9d3d72eaf058d2a7", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} 3 x^{3} + 6 x^{2} + 15 x + 48$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} + 6 x^{2} + 14 x + 48.\n3) Then the limit is 225.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: 228.\nFinal answer: $228$", "final_answer": "228", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3707c615a4d3a562", "problem_id": "f2cb09bae9900703", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "An urn has 20 red balls and 15 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(35,11).\n3) Favorable samples: choose k reds and n−k blues: C(20,3)·C(15,8).\n4) Compute totals: C(35,11) = 417225900, and favorable = 7335900.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1881}{106981}$", "final_answer": "\\frac{1881}{106981}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e22d15960df3c381", "problem_id": "965274e30a071e8a", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Worker A finishes a job in 14 hours and Worker B finishes it in 12 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/14 job/hour. B's rate = 1/12 job/hour.\n2) Combined rate = 1/14 + 1/12 = 13/84 job/hour.\n3) Time = 1 ÷ (combined rate) = 84/13 hours.\nFinal answer: $\\frac{84}{13}$", "final_answer": "\\frac{84}{13}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b90721d86ebdff62", "problem_id": "27b3ab511b08e064", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} - 2 x - 5$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to - 3 x - 5.\n3) Then the limit is -8.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: -7.\nFinal answer: $-7$", "final_answer": "-7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7f8117933257718d", "problem_id": "bd3f19c2064f892b", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 2 x - 60$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 2 x - 60$ = $2 \\left(x - 6\\right) \\left(x + 5\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $6$.\n5) From the second factor: x = $-5$.\nFinal answer: $x \\in \\{6, -5\\}$", "final_answer": "x \\in \\{6, -5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b2b3369beb61f5b6", "problem_id": "bd3f19c2064f892b", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 2 x - 60$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} - 2 x - 60$ = $2 \\left(x - 6\\right) \\left(x + 5\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $6$.\nFrom the second factor: x = $-5$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{6, -5\\}$", "final_answer": "x \\in \\{6, -5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ce3278c536fd6347", "problem_id": "dfe45db6d235951c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 2 x^{2} \\sin{\\left(x \\right)}) = - 2 x^{2} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 2 x^{2} \\sin{\\left(x \\right)}) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$", "final_answer": "- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ceb538049c205448", "problem_id": "4445437d930236b5", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 x - 4$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} - 3 x - 4$ = $\\left(x - 4\\right) \\left(x + 1\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $4$.\n5) From the second factor: x = $-1$.\nFinal answer: $x \\in \\{4, -1\\}$", "final_answer": "x \\in \\{4, -1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "450d5d41c9492660", "problem_id": "30f0a13b5f4c0c0d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- 2 x^{2} + 2 x + 3\\right) \\cos{\\left(x^{2} - 3 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\n3) So $f'(x) = 4 x^{3} \\sin{\\left(x^{2} - 3 x \\right)} - 10 x^{2} \\sin{\\left(x^{2} - 3 x \\right)} + 2 x e^{x} - 4 x \\cos{\\left(x^{2} - 3 x \\right)} + 4 e^{x} + 9 \\sin{\\left(x^{2} - 3 x \\right)} + 2 \\cos{\\left(x^{2} - 3 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(1 - 2 x\\right) \\cos{\\left(x^{2} - 3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - \\left(2 x - 3\\right) \\left(2 x \\left(1 - x\\right) + 3\\right) \\sin{\\left(x^{2} - 3 x \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(1 - 2 x\\right) \\cos{\\left(x^{2} - 3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - \\left(2 x - 3\\right) \\left(2 x \\left(1 - x\\right) + 3\\right) \\sin{\\left(x^{2} - 3 x \\right)} + 2 e^{x}$", "final_answer": "2 \\left(1 - 2 x\\right) \\cos{\\left(x^{2} - 3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - \\left(2 x - 3\\right) \\left(2 x \\left(1 - x\\right) + 3\\right) \\sin{\\left(x^{2} - 3 x \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "9a7f8c08f44e12f2", "problem_id": "7e720df1b2ccb53d", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "You want 47 liters of a 40% salt solution. You have a 10% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 47−x liters is 50% solution.\nSalt equation: 10x + 50(47−x) = 40·47.\n(10-50)x = (40-50)·47 ⇒ x = 47/4.\nThen 47−x = 141/4.\nCheck: 10·47/4 + 50·141/4 = 40·47.\nFinal answer: $47/4 L of 10% and 141/4 L of 50%$", "final_answer": "47/4 L of 10% and 141/4 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6885e9387a5acbac", "problem_id": "b673a251fca107aa", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Worker A finishes a job in 4 hours and Worker B finishes it in 18 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/4 job/hour. B's rate = 1/18 job/hour.\n2) Combined rate = 1/4 + 1/18 = 11/36 job/hour.\n3) Time = 1 ÷ (combined rate) = 36/11 hours.\nFinal answer: $\\frac{36}{11}$", "final_answer": "\\frac{36}{11}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b5b5ede065489e11", "problem_id": "b673a251fca107aa", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Worker A finishes a job in 4 hours and Worker B finishes it in 18 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/4 job/hour. B's rate = 1/18 job/hour.\nCombined rate = 1/4 + 1/18 = 11/36 job/hour.\nTime = 1 ÷ (combined rate) = 36/11 hours.\nCheck: (combined rate)×(time) = 11/36×36/11 = 1.\nFinal answer: $\\frac{36}{11}$", "final_answer": "\\frac{36}{11}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bec13667c2026f49", "problem_id": "f5018be827ed1199", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A cyclist rides at 27 miles per hour for 1/2 hours, then at 65 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 27 × 1/2 = 27/2 miles.\n2) Second leg distance = 65 × 3/2 = 195/2 miles.\n3) Total distance = 27/2 + 195/2 = 111 miles.\nFinal answer: $111$", "final_answer": "111", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "556c2a20b6ad0575", "problem_id": "f5018be827ed1199", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A cyclist rides at 27 miles per hour for 1/2 hours, then at 65 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 27 × 1/2 = 27/2 miles.\nSecond leg distance = 65 × 3/2 = 195/2 miles.\nTotal distance = 27/2 + 195/2 = 111 miles.\nCheck: adding both legs gives 111.\nFinal answer: $111$", "final_answer": "111", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3327213f74df82bd", "problem_id": "d318f4de332dff5e", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{5 \\pi}{2} - 2 + \\frac{15 \\pi^{2}}{4} + \\frac{65 \\pi^{3}}{24}$.\n4) Simplify the result.\nFinal answer: $- \\frac{5\\cdot \\pi}{2} - 2 + \\frac{15\\cdot \\pi^{2}}{4} + \\frac{65\\cdot \\pi^{3}}{24}$", "final_answer": "- \\frac{5\\cdot \\pi}{2} - 2 + \\frac{15\\cdot \\pi^{2}}{4} + \\frac{65\\cdot \\pi^{3}}{24}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "68617aa323db0e50", "problem_id": "d318f4de332dff5e", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{5 \\pi}{2} - 2 + \\frac{15 \\pi^{2}}{4} + \\frac{65 \\pi^{3}}{24}$.\nFinal answer: $- \\frac{5\\cdot \\pi}{2} - 2 + \\frac{15\\cdot \\pi^{2}}{4} + \\frac{65\\cdot \\pi^{3}}{24}$", "final_answer": "- \\frac{5\\cdot \\pi}{2} - 2 + \\frac{15\\cdot \\pi^{2}}{4} + \\frac{65\\cdot \\pi^{3}}{24}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2b83fcbbb89a439a", "problem_id": "cedb01e9e98d1f49", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} - 2 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $- 6 \\pi^{3} - 4 + \\frac{3 \\pi^{2}}{2} + 9 \\pi$.\n4) Simplify the result.\nFinal answer: $- 6\\cdot \\pi^{3} - 4 + \\frac{3\\cdot \\pi^{2}}{2} + 9\\cdot \\pi$", "final_answer": "- 6\\cdot \\pi^{3} - 4 + \\frac{3\\cdot \\pi^{2}}{2} + 9\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e3de2cb880109fac", "problem_id": "cedb01e9e98d1f49", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} - 2 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $- 6 \\pi^{3} - 4 + \\frac{3 \\pi^{2}}{2} + 9 \\pi$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- 6\\cdot \\pi^{3} - 4 + \\frac{3\\cdot \\pi^{2}}{2} + 9\\cdot \\pi$", "final_answer": "- 6\\cdot \\pi^{3} - 4 + \\frac{3\\cdot \\pi^{2}}{2} + 9\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "841a82365e4c8d63", "problem_id": "a361a3c9da028b9d", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{39 x}{10} - \\frac{27}{10}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{39}{10}$, c=$- \\frac{27}{10}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{9}{2}, - \\frac{3}{5}\\}$", "final_answer": "x \\in \\{\\frac{9}{2}, - \\frac{3}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0bf9426daeeefc8e", "problem_id": "a361a3c9da028b9d", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{39 x}{10} - \\frac{27}{10}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{39}{10}$, c=$- \\frac{27}{10}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{2601}{100}$, so √Δ = $\\frac{51}{10}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{9}{2}$.\nx2 = $- \\frac{3}{5}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{9}{2}, - \\frac{3}{5}\\}$", "final_answer": "x \\in \\{\\frac{9}{2}, - \\frac{3}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "67e75c7decbdc73d", "problem_id": "a340a88c47730b76", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "An urn has 18 red balls and 22 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(40,8).\n3) Favorable samples: choose k reds and n−k blues: C(18,3)·C(22,5).\n4) Compute totals: C(40,8) = 76904685, and favorable = 21488544.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{672}{2405}$", "final_answer": "\\frac{672}{2405}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c68041f641122f56", "problem_id": "a340a88c47730b76", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 18 red balls and 22 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(40,8).\nFavorable samples: choose k reds and n−k blues: C(18,3)·C(22,5).\nCompute totals: C(40,8) = 76904685, and favorable = 21488544.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{672}{2405}$.\nFinal answer: $\\frac{672}{2405}$", "final_answer": "\\frac{672}{2405}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b42916884596f65", "problem_id": "20e6e34ac6bb1843", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x - 1$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 2 x - 1.\n3) Then the limit is -5.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -3.\nFinal answer: $-3$", "final_answer": "-3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2dcfffb68d68c52d", "problem_id": "38b84d4b95339037", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_4=23$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=23 and k=4.\n4) Compute C(26,3) = 2600.\nFinal answer: $2600$", "final_answer": "2600", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "906bf7b720f9be6a", "problem_id": "870dda6b62f53feb", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 22 times, what is the probability of exactly 10 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=22, k=10, p=$\\frac{1}{8}$ and simplify.\n4) Here C(22,10) = 646646.\nFinal answer: $\\frac{4475206501688923}{36893488147419103232}$", "final_answer": "\\frac{4475206501688923}{36893488147419103232}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "15cba08ca151597e", "problem_id": "870dda6b62f53feb", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 22 times, what is the probability of exactly 10 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=22, k=10, p=$\\frac{1}{8}$ and simplify.\nHere C(22,10) = 646646.\nCheck: expression simplifies to $\\frac{4475206501688923}{36893488147419103232}$.\nFinal answer: $\\frac{4475206501688923}{36893488147419103232}$", "final_answer": "\\frac{4475206501688923}{36893488147419103232}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8b06d0c8a8e06beb", "problem_id": "04990651917d000b", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_5=45$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_5=30.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=30 and k=5.\n4) Compute C(34,4) = 46376.\nFinal answer: $46376$", "final_answer": "46376", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec50007775283665", "problem_id": "04990651917d000b", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_5=45$?", "output": "Let y_i=x_i-3, then y_i≥0 and y_1+…+y_5=30.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=30 and k=5.\nCompute C(34,4) = 46376.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $46376$", "final_answer": "46376", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cfa8e2221d7f23ad", "problem_id": "d6bb2ee279f828cb", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} + x + 3$.", "output": "1) Differentiate term-by-term:\n2) d/dx(3) = 0.\n3) d/dx(x) = 1.\n4) d/dx(3 x^{2}) = 6 x.\n5) Combine: f'(x) = 6 x + 1.\nFinal answer: $6\\cdot x + 1$", "final_answer": "6\\cdot x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6ae7172aecf23a91", "problem_id": "d6bb2ee279f828cb", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} + x + 3$.", "output": "Differentiate term-by-term:\nd/dx(3) = 0.\nd/dx(x) = 1.\nd/dx(3 x^{2}) = 6 x.\nCombine: f'(x) = 6 x + 1.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $6\\cdot x + 1$", "final_answer": "6\\cdot x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0972d3e312a111f7", "problem_id": "15a878ae69b5201a", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(3 x^{3} - 3 x^{2} + 3 x\\right) \\cos{\\left(3 x + 2 \\right)}$.", "output": "1) Let f(x) = $2 \\left(x - 3\\right) e^{x} + \\left(3 x^{3} - 3 x^{2} + 3 x\\right) \\cos{\\left(3 x + 2 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- 9 x \\left(x^{2} - x + 1\\right) \\sin{\\left(3 x + 2 \\right)} + 2 \\left(x - 3\\right) e^{x} + 3 \\left(3 x^{2} - 2 x + 1\\right) \\cos{\\left(3 x + 2 \\right)} + 2 e^{x}$.\nFinal answer: $- 9\\cdot x\\cdot \\left(x^{2} - x + 1\\right)\\cdot \\sin{\\left(3\\cdot x + 2 \\right)} + \\left(2\\cdot x - 6\\right)\\cdot e^{x} + \\left(9\\cdot x^{2} - 6\\cdot x + 3\\right)\\cdot \\cos{\\left(3\\cdot x + 2 \\right)} + 2\\cdot e^{x}$", "final_answer": "- 9\\cdot x\\cdot \\left(x^{2} - x + 1\\right)\\cdot \\sin{\\left(3\\cdot x + 2 \\right)} + \\left(2\\cdot x - 6\\right)\\cdot e^{x} + \\left(9\\cdot x^{2} - 6\\cdot x + 3\\right)\\cdot \\cos{\\left(3\\cdot x + 2 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6dfa209b4bbf3f02", "problem_id": "d8d35b6d9871079c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- \\frac{3 x^{2}}{2} - x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1}{x - 4}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}) = \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4}$.\n3) So $f'(x) = \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{3 x^{2}}{2 x^{2} - 16 x + 32} - \\frac{8 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{16 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{24 x}{2 x^{2} - 16 x + 32} + \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 8 x + 16} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 8} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} - \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} + \\frac{16 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{32 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{10}{2 x^{2} - 16 x + 32} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}) = \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x - 4\\right)} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x - 4\\right)^{2}} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}$.\nTherefore the correct derivative is $f'(x) = \\frac{3 x^{2} + 2 x + \\left(x - 4\\right) \\left(- 6 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{2 \\left(x - 4\\right)^{2}}$.\nFinal answer: $\\frac{3 x^{2} + 2 x + \\left(x - 4\\right) \\left(- 6 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{2 \\left(x - 4\\right)^{2}}$", "final_answer": "\\frac{3 x^{2} + 2 x + \\left(x - 4\\right) \\left(- 6 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{2 \\left(x - 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a13ff01ac7d14313", "problem_id": "3bd33c59fe1be88f", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 2 x^{2} - 2 x + \\left(- 3 x^{2} + x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{- 3 x^{2} + x + 2}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- 3 x^{2} + x\\right) + 2}) = \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{- 3 x^{2} + x + 2}$.\n3) So $f'(x) = \\frac{9 x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 22 x^{2} + 8 x + 8} + \\frac{18 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 22 x^{2} + 8 x + 8} - \\frac{6 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 22 x^{2} + 8 x + 8} - \\frac{12 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 22 x^{2} + 8 x + 8} - \\frac{11 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 22 x^{2} + 8 x + 8} - \\frac{22 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 22 x^{2} + 8 x + 8} - \\frac{16 x^{2}}{18 x^{4} - 12 x^{3} - 22 x^{2} + 8 x + 8} - \\frac{6 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{4} - 6 x^{3} - 11 x^{2} + 4 x + 4} + \\frac{4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 22 x^{2} + 8 x + 8} + \\frac{8 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 22 x^{2} + 8 x + 8} - \\frac{52 x}{18 x^{4} - 12 x^{3} - 22 x^{2} + 8 x + 8} + \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{4} - 6 x^{3} - 11 x^{2} + 4 x + 4} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{- 3 x^{2} + x + 2} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{- 6 x^{2} + 2 x + 4} - \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(- 3 x^{2} + x\\right) + 2} + \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 22 x^{2} + 8 x + 8} + \\frac{8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 22 x^{2} + 8 x + 8} - \\frac{2}{18 x^{4} - 12 x^{3} - 22 x^{2} + 8 x + 8} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- 3 x^{2} + x\\right) + 2}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- 3 x^{2} + x\\right) + 2}) = \\frac{x \\left(6 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(- 3 x^{2} + x\\right) + 2\\right)^{2}} + \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(- 3 x^{2} + x\\right) + 2\\right)} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(- 3 x^{2} + x\\right) + 2} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- 3 x^{2} + x\\right) + 2}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(6 x - 1\\right) \\left(- 2 x^{2} - 2 x + \\left(x \\left(1 - 3 x\\right) + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x \\left(1 - 3 x\\right) + 2\\right) \\left(- 8 x + 2 \\left(1 - 6 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(1 - 3 x\\right) + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(1 - 3 x\\right) + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 4\\right)}{2 \\left(x \\left(1 - 3 x\\right) + 2\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(6 x - 1\\right) \\left(- 2 x^{2} - 2 x + \\left(x \\left(1 - 3 x\\right) + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x \\left(1 - 3 x\\right) + 2\\right) \\left(- 8 x + 2 \\left(1 - 6 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(1 - 3 x\\right) + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(1 - 3 x\\right) + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 4\\right)}{2 \\left(x \\left(1 - 3 x\\right) + 2\\right)^{2}}$", "final_answer": "\\frac{2 \\left(6 x - 1\\right) \\left(- 2 x^{2} - 2 x + \\left(x \\left(1 - 3 x\\right) + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x \\left(1 - 3 x\\right) + 2\\right) \\left(- 8 x + 2 \\left(1 - 6 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(1 - 3 x\\right) + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(1 - 3 x\\right) + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 4\\right)}{2 \\left(x \\left(1 - 3 x\\right) + 2\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ffc6dd88ad28fea8", "problem_id": "3dbf515312b9d240", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the multiplicative inverse of 15 modulo 31.", "output": "We want x such that 15x ≡ 1 (mod 31).\nUse extended Euclid to find s,t with 15s + 31t = 1.\nOne Bezout identity is: 15·(-2) + 31·(1) = 1.\nReduce s modulo 31: x ≡ 29 (mod 31).\nCheck: 15·29 mod 31 = 1.\nFinal answer: $x\\equiv 29\\pmod{31}$", "final_answer": "x\\equiv 29\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a216446f96aefe7b", "problem_id": "cbe014d638771f38", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A cyclist rides at 45 miles per hour for 1 hours, then at 32 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 45 × 1 = 45 miles.\nSecond leg distance = 32 × 3/2 = 48 miles.\nTotal distance = 45 + 48 = 93 miles.\nCheck: adding both legs gives 93.\nFinal answer: $93$", "final_answer": "93", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d6f3853a3f4148c", "problem_id": "cbe014d638771f38", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A cyclist rides at 45 miles per hour for 1 hours, then at 32 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 45 × 1 = 45 miles.\n2) Second leg distance = 32 × 3/2 = 48 miles.\n3) Total distance = 45 + 48 = 93 miles.\nFinal answer: $93$", "final_answer": "93", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ce500885d44e11b2", "problem_id": "53eab84fbd01f058", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A cyclist rides at 46 miles per hour for 1 hours, then at 57 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 46 × 1 = 46 miles.\n2) Second leg distance = 57 × 1 = 57 miles.\n3) Total distance = 46 + 57 = 103 miles.\nFinal answer: $103$", "final_answer": "103", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f78bb2d981e201a1", "problem_id": "53eab84fbd01f058", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A cyclist rides at 46 miles per hour for 1 hours, then at 57 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 46 × 1 = 46 miles.\nSecond leg distance = 57 × 1 = 57 miles.\nTotal distance = 46 + 57 = 103 miles.\nCheck: adding both legs gives 103.\nFinal answer: $103$", "final_answer": "103", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "97744e441e91e372", "problem_id": "48af45789f17eebf", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 10 x + 25$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} - 10 x + 25$ = $\\left(x - 5\\right)^{2}$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $5$.\n5) From the second factor: x = $5$.\nFinal answer: $x \\in \\{5, 5\\}$", "final_answer": "x \\in \\{5, 5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ba08a219d4e88ba3", "problem_id": "f4ac4a51985af0c7", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 3 x^{3} - 4 x^{2} - 9 x - 17$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{3} - 4 x^{2} - 10 x - 17.\n3) Then the limit is -77.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -75.\nFinal answer: $-75$", "final_answer": "-75", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b7bff861226e35c5", "problem_id": "f60f4d107407ebd7", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 20 red balls and 30 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(50,9).\n3) Favorable samples: choose k reds and n−k blues: C(20,2)·C(30,7).\n4) Compute totals: C(50,9) = 2505433700, and favorable = 386802000.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{3868020}{25054337}$", "final_answer": "\\frac{3868020}{25054337}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6d6765ea5a1b35cb", "problem_id": "6ba99aa955fab2e0", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} + 4 x$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x^{3}) = - 6 x^{3}$.\n3) So $f'(x) = 4 - 6 x^{3}$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x^{3}) = - 6 x^{2}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(2 - 3 x^{2}\\right)$.\nFinal answer: $2 \\left(2 - 3 x^{2}\\right)$", "final_answer": "2 \\left(2 - 3 x^{2}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "40a37abbba2f9447", "problem_id": "5f8ba781ab0c528e", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 14 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=14, k=3, p=$\\frac{1}{2}$ and simplify.\n4) Here C(14,3) = 364.\nFinal answer: $\\frac{91}{4096}$", "final_answer": "\\frac{91}{4096}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a583cbfc13e4e9a7", "problem_id": "ba579723ea38e816", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} x^{2} + 4 x + 14$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to x^{2} + 3 x + 14.\n3) Then the limit is 32.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: 35.\nFinal answer: $35$", "final_answer": "35", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d988c5ac24d54dc7", "problem_id": "aff686506b7b54d8", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{2 \\sqrt{7} x}{3} + \\frac{x}{6} - \\frac{5}{2} + \\sqrt{7}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{1}{6} - \\frac{2 \\sqrt{7}}{3}$, c=$- \\frac{5}{2} + \\sqrt{7}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{473}{36} - \\frac{38 \\sqrt{7}}{9}$, so √Δ = $\\sqrt{\\frac{473}{36} - \\frac{38 \\sqrt{7}}{9}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{1}{12} + \\frac{\\sqrt{473 - 152 \\sqrt{7}}}{12} + \\frac{\\sqrt{7}}{3}$.\n7) x2 = $- \\frac{\\sqrt{473 - 152 \\sqrt{7}}}{12} - \\frac{1}{12} + \\frac{\\sqrt{7}}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{1}{12} + \\frac{\\sqrt{473 - 152\\cdot \\sqrt{7}}}{12} + \\frac{\\sqrt{7}}{3}, - \\frac{\\sqrt{473 - 152\\cdot \\sqrt{7}}}{12} - \\frac{1}{12} + \\frac{\\sqrt{7}}{3}\\}$", "final_answer": "x \\in \\{- \\frac{1}{12} + \\frac{\\sqrt{473 - 152\\cdot \\sqrt{7}}}{12} + \\frac{\\sqrt{7}}{3}, - \\frac{\\sqrt{473 - 152\\cdot \\sqrt{7}}}{12} - \\frac{1}{12} + \\frac{\\sqrt{7}}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2259bdaf2cdac53c", "problem_id": "aff686506b7b54d8", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{2 \\sqrt{7} x}{3} + \\frac{x}{6} - \\frac{5}{2} + \\sqrt{7}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{1}{6} - \\frac{2 \\sqrt{7}}{3}$, c=$- \\frac{5}{2} + \\sqrt{7}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{473}{36} - \\frac{38 \\sqrt{7}}{9}$, so √Δ = $\\sqrt{\\frac{473}{36} - \\frac{38 \\sqrt{7}}{9}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{1}{12} + \\frac{\\sqrt{473 - 152 \\sqrt{7}}}{12} + \\frac{\\sqrt{7}}{3}$.\nx2 = $- \\frac{\\sqrt{473 - 152 \\sqrt{7}}}{12} - \\frac{1}{12} + \\frac{\\sqrt{7}}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{1}{12} + \\frac{\\sqrt{473 - 152\\cdot \\sqrt{7}}}{12} + \\frac{\\sqrt{7}}{3}, - \\frac{\\sqrt{473 - 152\\cdot \\sqrt{7}}}{12} - \\frac{1}{12} + \\frac{\\sqrt{7}}{3}\\}$", "final_answer": "x \\in \\{- \\frac{1}{12} + \\frac{\\sqrt{473 - 152\\cdot \\sqrt{7}}}{12} + \\frac{\\sqrt{7}}{3}, - \\frac{\\sqrt{473 - 152\\cdot \\sqrt{7}}}{12} - \\frac{1}{12} + \\frac{\\sqrt{7}}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "38fd5f0c1ee3464a", "problem_id": "fd96d3fac5550024", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} - x^{2} + 2 x + 4$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 6 x^{2} - 2 x + 2$.\n2) Slope at x=2: m = f'(2) = $-26$.\n3) Point on curve: (2, f(2)) = (2, $-12$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -12 = -26(x − 2).\n6) Optionally expand: y = $40 - 26 x$.\nFinal answer: $y = 40 - 26\\cdot x$", "final_answer": "y = 40 - 26\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "164f830848dd7852", "problem_id": "fd96d3fac5550024", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} - x^{2} + 2 x + 4$ at $x=2$.", "output": "Differentiate: f'(x) = $- 6 x^{2} - 2 x + 2$.\nSlope at x=2: m = f'(2) = $-26$.\nPoint on curve: (2, f(2)) = (2, $-12$).\nFinal answer: $y = 40 - 26\\cdot x$", "final_answer": "y = 40 - 26\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "20647c01050239d3", "problem_id": "3acc0169fa5474ac", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡3 (mod 4), x≡0 (mod 11). Give the solution modulo 44.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 44.\n3) Compute the CRT solution to get x ≡ 11 (mod 44).\nFinal answer: $x\\equiv 11\\pmod{44}$", "final_answer": "x\\equiv 11\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c11162be9db2a946", "problem_id": "3acc0169fa5474ac", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the congruences: x≡3 (mod 4), x≡0 (mod 11). Give the solution modulo 44.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 44.\nCompute the CRT solution to get x ≡ 11 (mod 44).\nFinal answer: $x\\equiv 11\\pmod{44}$", "final_answer": "x\\equiv 11\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "88613059af824bcb", "problem_id": "be26d1616e591c11", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Two friends split $70 in the ratio 8:6. How much does each receive?", "output": "Total ratio parts = 8+6 = 14.\nEach part = 70 ÷ 14 = 5.\nFirst share = 8 × 5 = 40.\nSecond share = 6 × 5 = 30.\nCheck: 40+30=70.\nFinal answer: $30$", "final_answer": "30", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eb6a9c8d1bce7a47", "problem_id": "be26d1616e591c11", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Two friends split $70 in the ratio 8:6. How much does each receive?", "output": "1) Total ratio parts = 8+6 = 14.\n2) Each part = 70 ÷ 14 = 5.\n3) First share = 8 × 5 = 40.\n4) Second share = 6 × 5 = 30.\nFinal answer: $30$", "final_answer": "30", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a8640418f0334d6a", "problem_id": "2f0c65bbc84c390b", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the equation of the tangent line to $y=x^{3} + x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 1$ at $x=2$.", "output": "Differentiate: f'(x) = $3 x^{2} + 2 x + 2 \\cos{\\left(x \\right)} - 2$.\nSlope at x=2: m = f'(2) = $2 \\cos{\\left(2 \\right)} + 14$.\nPoint on curve: (2, f(2)) = (2, $2 \\sin{\\left(2 \\right)} + 7$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 2 \\sin{\\left(2 \\right)} + 7 = 2 \\cos{\\left(2 \\right)} + 14(x − 2).\nOptionally expand: y = $2 x \\cos{\\left(2 \\right)} + 14 x - 21 - 4 \\cos{\\left(2 \\right)} + 2 \\sin{\\left(2 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 14\\cdot x - 21 - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 14\\cdot x - 21 - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dd6a092b9c105edb", "problem_id": "2f0c65bbc84c390b", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the equation of the tangent line to $y=x^{3} + x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 1$ at $x=2$.", "output": "1) Differentiate: f'(x) = $3 x^{2} + 2 x + 2 \\cos{\\left(x \\right)} - 2$.\n2) Slope at x=2: m = f'(2) = $2 \\cos{\\left(2 \\right)} + 14$.\n3) Point on curve: (2, f(2)) = (2, $2 \\sin{\\left(2 \\right)} + 7$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 2 \\sin{\\left(2 \\right)} + 7 = 2 \\cos{\\left(2 \\right)} + 14(x − 2).\n6) Optionally expand: y = $2 x \\cos{\\left(2 \\right)} + 14 x - 21 - 4 \\cos{\\left(2 \\right)} + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 14\\cdot x - 21 - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 14\\cdot x - 21 - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c3d0c0f4feb5641c", "problem_id": "377a66ba0aba23fc", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_7=25$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_7=11.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=11 and k=7.\n4) Compute C(17,6) = 12376.\nFinal answer: $12376$", "final_answer": "12376", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7e81956ac4c9a5c4", "problem_id": "377a66ba0aba23fc", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_7=25$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_7=11.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=11 and k=7.\nCompute C(17,6) = 12376.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $12376$", "final_answer": "12376", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9e3a982290fe65f1", "problem_id": "18c5e76a659c4a4a", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{2 \\sqrt{7} x}{3} + \\frac{x}{6} - \\frac{1}{3} + \\frac{\\sqrt{7}}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{1}{6} - \\frac{2 \\sqrt{7}}{3}$, c=$- \\frac{1}{3} + \\frac{\\sqrt{7}}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{161}{36} - \\frac{14 \\sqrt{7}}{9}$, so √Δ = $\\sqrt{\\frac{161}{36} - \\frac{14 \\sqrt{7}}{9}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{1}{12} + \\frac{\\sqrt{161 - 56 \\sqrt{7}}}{12} + \\frac{\\sqrt{7}}{3}$.\n7) x2 = $- \\frac{\\sqrt{161 - 56 \\sqrt{7}}}{12} - \\frac{1}{12} + \\frac{\\sqrt{7}}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{1}{12} + \\frac{\\sqrt{161 - 56\\cdot \\sqrt{7}}}{12} + \\frac{\\sqrt{7}}{3}, - \\frac{\\sqrt{161 - 56\\cdot \\sqrt{7}}}{12} - \\frac{1}{12} + \\frac{\\sqrt{7}}{3}\\}$", "final_answer": "x \\in \\{- \\frac{1}{12} + \\frac{\\sqrt{161 - 56\\cdot \\sqrt{7}}}{12} + \\frac{\\sqrt{7}}{3}, - \\frac{\\sqrt{161 - 56\\cdot \\sqrt{7}}}{12} - \\frac{1}{12} + \\frac{\\sqrt{7}}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6009964b99bcd385", "problem_id": "18c5e76a659c4a4a", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{2 \\sqrt{7} x}{3} + \\frac{x}{6} - \\frac{1}{3} + \\frac{\\sqrt{7}}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{1}{6} - \\frac{2 \\sqrt{7}}{3}$, c=$- \\frac{1}{3} + \\frac{\\sqrt{7}}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{161}{36} - \\frac{14 \\sqrt{7}}{9}$, so √Δ = $\\sqrt{\\frac{161}{36} - \\frac{14 \\sqrt{7}}{9}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{1}{12} + \\frac{\\sqrt{161 - 56 \\sqrt{7}}}{12} + \\frac{\\sqrt{7}}{3}$.\nx2 = $- \\frac{\\sqrt{161 - 56 \\sqrt{7}}}{12} - \\frac{1}{12} + \\frac{\\sqrt{7}}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{1}{12} + \\frac{\\sqrt{161 - 56\\cdot \\sqrt{7}}}{12} + \\frac{\\sqrt{7}}{3}, - \\frac{\\sqrt{161 - 56\\cdot \\sqrt{7}}}{12} - \\frac{1}{12} + \\frac{\\sqrt{7}}{3}\\}$", "final_answer": "x \\in \\{- \\frac{1}{12} + \\frac{\\sqrt{161 - 56\\cdot \\sqrt{7}}}{12} + \\frac{\\sqrt{7}}{3}, - \\frac{\\sqrt{161 - 56\\cdot \\sqrt{7}}}{12} - \\frac{1}{12} + \\frac{\\sqrt{7}}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "66c5522f7abdab5b", "problem_id": "8933a1f2e6b4c0bb", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "You want 55 liters of a 40% salt solution. You have a 30% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 55−x liters is 50% solution.\n2) Salt equation: 30x + 50(55−x) = 40·55.\n3) (30-50)x = (40-50)·55 ⇒ x = 55/2.\n4) Then 55−x = 55/2.\nFinal answer: $55/2 L of 30% and 55/2 L of 50%$", "final_answer": "55/2 L of 30% and 55/2 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f85fb0751caa8bdf", "problem_id": "8933a1f2e6b4c0bb", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 55 liters of a 40% salt solution. You have a 30% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 55−x liters is 50% solution.\nSalt equation: 30x + 50(55−x) = 40·55.\n(30-50)x = (40-50)·55 ⇒ x = 55/2.\nThen 55−x = 55/2.\nCheck: 30·55/2 + 50·55/2 = 40·55.\nFinal answer: $55/2 L of 30% and 55/2 L of 50%$", "final_answer": "55/2 L of 30% and 55/2 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "651d816abbb46ba6", "problem_id": "df21b0eaf4722c96", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A cyclist rides at 62 miles per hour for 3/2 hours, then at 28 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 62 × 3/2 = 93 miles.\nSecond leg distance = 28 × 3/2 = 42 miles.\nTotal distance = 93 + 42 = 135 miles.\nCheck: adding both legs gives 135.\nFinal answer: $135$", "final_answer": "135", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "60c22a753387782f", "problem_id": "df21b0eaf4722c96", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A cyclist rides at 62 miles per hour for 3/2 hours, then at 28 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 62 × 3/2 = 93 miles.\n2) Second leg distance = 28 × 3/2 = 42 miles.\n3) Total distance = 93 + 42 = 135 miles.\nFinal answer: $135$", "final_answer": "135", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a3fdd66d6ff0eedc", "problem_id": "73371f266a1cf3e6", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Two friends split $350 in the ratio 5:2. How much does each receive?", "output": "1) Total ratio parts = 5+2 = 7.\n2) Each part = 350 ÷ 7 = 50.\n3) First share = 5 × 50 = 250.\n4) Second share = 2 × 50 = 100.\nFinal answer: $100$", "final_answer": "100", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "771ca07c5a766426", "problem_id": "73371f266a1cf3e6", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Two friends split $350 in the ratio 5:2. How much does each receive?", "output": "Total ratio parts = 5+2 = 7.\nEach part = 350 ÷ 7 = 50.\nFirst share = 5 × 50 = 250.\nSecond share = 2 × 50 = 100.\nCheck: 250+100=350.\nFinal answer: $100$", "final_answer": "100", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8b2cea8df9c6f383", "problem_id": "a69f11ed3a78c6ab", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Two friends split $572 in the ratio 5:6. How much does each receive?", "output": "1) Total ratio parts = 5+6 = 11.\n2) Each part = 572 ÷ 11 = 52.\n3) First share = 5 × 52 = 260.\n4) Second share = 6 × 52 = 312.\nFinal answer: $312$", "final_answer": "312", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "213f03e3b21ecee3", "problem_id": "a69f11ed3a78c6ab", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Two friends split $572 in the ratio 5:6. How much does each receive?", "output": "Total ratio parts = 5+6 = 11.\nEach part = 572 ÷ 11 = 52.\nFirst share = 5 × 52 = 260.\nSecond share = 6 × 52 = 312.\nCheck: 260+312=572.\nFinal answer: $312$", "final_answer": "312", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4659e7a95b689a2a", "problem_id": "55af4600afa0ca2e", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} - x^{3} + x^{2} + x - 1$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{4} + 2 x^{3} - 2 x + 1$ and a = 1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 4 x^{3} + 6 x^{2} - 2$.\nEvaluate at a: p'(1) = $0$.\nCheck: the identity gives $0$.\nFinal answer: $0$", "final_answer": "0", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7e95a7bc369c383b", "problem_id": "55af4600afa0ca2e", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} - x^{3} + x^{2} + x - 1$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{4} + 2 x^{3} - 2 x + 1$ and a = 1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 4 x^{3} + 6 x^{2} - 2$.\n5) Evaluate at a: p'(1) = $0$.\nFinal answer: $0$", "final_answer": "0", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "38d131730d1176ef", "problem_id": "063f1f20774cbc77", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$- 4 x + 2 y$ = $-16$\n$- 2 x$ = $-10$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $8 x - 4 y$ = $32$.\n3) Multiply the second equation by $-4$: $8 x$ = $40$.\n4) Subtract to eliminate x: $- 4 y$ = $-8$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, 2\\right)$", "final_answer": "(x,y)=\\left(5, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "480823e8d01b71b4", "problem_id": "063f1f20774cbc77", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the linear system:\n$- 4 x + 2 y$ = $-16$\n$- 2 x$ = $-10$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $8 x - 4 y$ = $32$.\nMultiply the second equation by $-4$: $8 x$ = $40$.\nFinal answer: $(x,y)=\\left(5, 2\\right)$", "final_answer": "(x,y)=\\left(5, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dc0ae30faeb8918b", "problem_id": "e8327ac1d3351f7a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{13} x}{2} + \\frac{6 x}{5} - \\frac{8}{5} + \\frac{6 \\sqrt{13}}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{6}{5} - \\frac{3 \\sqrt{13}}{2}$, c=$- \\frac{8}{5} + \\frac{6 \\sqrt{13}}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{3709}{100} - \\frac{42 \\sqrt{13}}{5}$, so √Δ = $\\sqrt{\\frac{3709}{100} - \\frac{42 \\sqrt{13}}{5}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{3}{5} + \\frac{\\sqrt{3709 - 840 \\sqrt{13}}}{20} + \\frac{3 \\sqrt{13}}{4}$.\nx2 = $- \\frac{\\sqrt{3709 - 840 \\sqrt{13}}}{20} - \\frac{3}{5} + \\frac{3 \\sqrt{13}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{3}{5} + \\frac{\\sqrt{3709 - 840\\cdot \\sqrt{13}}}{20} + \\frac{3\\cdot \\sqrt{13}}{4}, - \\frac{\\sqrt{3709 - 840\\cdot \\sqrt{13}}}{20} - \\frac{3}{5} + \\frac{3\\cdot \\sqrt{13}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{3}{5} + \\frac{\\sqrt{3709 - 840\\cdot \\sqrt{13}}}{20} + \\frac{3\\cdot \\sqrt{13}}{4}, - \\frac{\\sqrt{3709 - 840\\cdot \\sqrt{13}}}{20} - \\frac{3}{5} + \\frac{3\\cdot \\sqrt{13}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8ae881cba5cbf6c8", "problem_id": "e8327ac1d3351f7a", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{13} x}{2} + \\frac{6 x}{5} - \\frac{8}{5} + \\frac{6 \\sqrt{13}}{5}$ = 0.", "output": "$x \\in \\{- \\frac{3}{5} + \\frac{\\sqrt{3709 - 840\\cdot \\sqrt{13}}}{20} + \\frac{3\\cdot \\sqrt{13}}{4}, - \\frac{\\sqrt{3709 - 840\\cdot \\sqrt{13}}}{20} - \\frac{3}{5} + \\frac{3\\cdot \\sqrt{13}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{3}{5} + \\frac{\\sqrt{3709 - 840\\cdot \\sqrt{13}}}{20} + \\frac{3\\cdot \\sqrt{13}}{4}, - \\frac{\\sqrt{3709 - 840\\cdot \\sqrt{13}}}{20} - \\frac{3}{5} + \\frac{3\\cdot \\sqrt{13}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "223a3e53c040f96b", "problem_id": "4457d2864ca7c3c3", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$- 3 x + 2 y$ = $-23$\n$- 4 x + 3 y$ = $-32$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $12 x - 8 y$ = $92$.\n3) Multiply the second equation by $-3$: $12 x - 9 y$ = $96$.\n4) Subtract to eliminate x: $y$ = $-4$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, -4\\right)$", "final_answer": "(x,y)=\\left(5, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "46ca39e646cff170", "problem_id": "4457d2864ca7c3c3", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- 3 x + 2 y$ = $-23$\n$- 4 x + 3 y$ = $-32$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $12 x - 8 y$ = $92$.\nMultiply the second equation by $-3$: $12 x - 9 y$ = $96$.\nSubtract to eliminate x: $y$ = $-4$.\nSolve for y: y = $-4$.\nSubstitute back to find x: x = $5$.\nCheck: plug (x,y)=(5,-4) into both equations.\nFinal answer: $(x,y)=\\left(5, -4\\right)$", "final_answer": "(x,y)=\\left(5, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "30ffecf0b1ae50c1", "problem_id": "6cae5ffa6130cd5f", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Two friends split $840 in the ratio 8:7. How much does each receive?", "output": "1) Total ratio parts = 8+7 = 15.\n2) Each part = 840 ÷ 15 = 56.\n3) First share = 8 × 56 = 448.\n4) Second share = 7 × 56 = 392.\nFinal answer: $392$", "final_answer": "392", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f8535656f438bd0f", "problem_id": "6cae5ffa6130cd5f", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $840 in the ratio 8:7. How much does each receive?", "output": "Total ratio parts = 8+7 = 15.\nEach part = 840 ÷ 15 = 56.\nFirst share = 8 × 56 = 448.\nSecond share = 7 × 56 = 392.\nCheck: 448+392=840.\nFinal answer: $392$", "final_answer": "392", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2f242991f8068fd7", "problem_id": "67a0b07ab9bf10f0", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$- 4 x + 4 y$ = $-36$\n$8 y$ = $-24$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $-4$: $- 32 y$ = $96$.\n4) Subtract to eliminate x: $32 y$ = $-96$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $6$.\nFinal answer: $(x,y)=\\left(6, -3\\right)$", "final_answer": "(x,y)=\\left(6, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0c0e186f7f4c7323", "problem_id": "67a0b07ab9bf10f0", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- 4 x + 4 y$ = $-36$\n$8 y$ = $-24$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $-4$: $- 32 y$ = $96$.\nSubtract to eliminate x: $32 y$ = $-96$.\nSolve for y: y = $-3$.\nSubstitute back to find x: x = $6$.\nCheck: plug (x,y)=(6,-3) into both equations.\nFinal answer: $(x,y)=\\left(6, -3\\right)$", "final_answer": "(x,y)=\\left(6, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "23f2b124f7d1a10e", "problem_id": "5b6e67763613245e", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 11), x≡6 (mod 8). Give the solution modulo 88.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 88.\nCompute the CRT solution to get x ≡ 46 (mod 88).\nCheck: 46 mod 11=2 and mod 8=6.\nFinal answer: $x\\equiv 46\\pmod{88}$", "final_answer": "x\\equiv 46\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "803ee782f3d31652", "problem_id": "5b6e67763613245e", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the congruences: x≡2 (mod 11), x≡6 (mod 8). Give the solution modulo 88.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 88.\nCompute the CRT solution to get x ≡ 46 (mod 88).\nFinal answer: $x\\equiv 46\\pmod{88}$", "final_answer": "x\\equiv 46\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d155b76654b3fe35", "problem_id": "73054d79e21bc3b9", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_3=23$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=23 and k=3.\n4) Compute C(25,2) = 300.\nFinal answer: $300$", "final_answer": "300", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6b4e99adb7c66afa", "problem_id": "73054d79e21bc3b9", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_3=23$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=23 and k=3.\nCompute C(25,2) = 300.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $300$", "final_answer": "300", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4f1ac4f888ff8377", "problem_id": "2651fa7eac0af2e4", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Two friends split $1360 in the ratio 5:5. How much does each receive?", "output": "1) Total ratio parts = 5+5 = 10.\n2) Each part = 1360 ÷ 10 = 136.\n3) First share = 5 × 136 = 680.\n4) Second share = 5 × 136 = 680.\nFinal answer: $680$", "final_answer": "680", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "53aff0a7dad16ee1", "problem_id": "2651fa7eac0af2e4", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $1360 in the ratio 5:5. How much does each receive?", "output": "Total ratio parts = 5+5 = 10.\nEach part = 1360 ÷ 10 = 136.\nFirst share = 5 × 136 = 680.\nSecond share = 5 × 136 = 680.\nCheck: 680+680=1360.\nFinal answer: $680$", "final_answer": "680", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "28ac8b8d551bdca5", "problem_id": "1b0146d968301fc6", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{3}$. If it is flipped 16 times, what is the probability of exactly 10 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=16, k=10, p=$\\frac{1}{3}$ and simplify.\n4) Here C(16,10) = 8008.\nFinal answer: $\\frac{512512}{43046721}$", "final_answer": "\\frac{512512}{43046721}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ce725a041c18eb8", "problem_id": "22a3089edc84880c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x - 3$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 2 x - 3.\n3) Then the limit is -7.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -5.\nFinal answer: $-5$", "final_answer": "-5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "38151e2fb546c3ff", "problem_id": "c89f708e23721452", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 26 red balls and 28 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(54,12).\n3) Favorable samples: choose k reds and n−k blues: C(26,4)·C(28,8).\n4) Compute totals: C(54,12) = 343006888770, and favorable = 46466169750.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{246675}{1820921}$", "final_answer": "\\frac{246675}{1820921}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "caffcf0a92e05ea0", "problem_id": "d37a8dd6d1d2c67d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{3} - 3 x^{2} - 2 x - 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 3 x^{2}) = - 6 x^{2}$.\n3) So $f'(x) = - 9 x^{2} - 2$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 3 x^{2}) = - 6 x$.\nTherefore the correct derivative is $f'(x) = - 3 x^{2} - 6 x - 2$.\nFinal answer: $- 3 x^{2} - 6 x - 2$", "final_answer": "- 3 x^{2} - 6 x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3f5ba10d06c09047", "problem_id": "80cd2d9d70c1bc48", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{3} \\sin{\\left(x \\right)} - x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x \\sin{\\left(x \\right)}) = 2 x \\cos{\\left(x \\right)}$.\n3) So $f'(x) = 2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x \\sin{\\left(x \\right)}) = 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)}$.\nFinal answer: $2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)}$", "final_answer": "2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5a05f2b5a4329ac9", "problem_id": "52f83643d177c4a4", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} - 4 x^{2} - 3 x + 1$.", "output": "1) Differentiate term-by-term:\n2) d/dx(1) = 0.\n3) d/dx(x^{3}) = 3 x^{2}.\n4) d/dx(- 4 x^{2}) = - 8 x.\n5) d/dx(- 3 x) = -3.\n6) Combine: f'(x) = 3 x^{2} - 8 x - 3.\nFinal answer: $3\\cdot x^{2} - 8\\cdot x - 3$", "final_answer": "3\\cdot x^{2} - 8\\cdot x - 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "927a031231ab0dc6", "problem_id": "ee29d0f0a8754305", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Two friends split $2244 in the ratio 8:9. How much does each receive?", "output": "1) Total ratio parts = 8+9 = 17.\n2) Each part = 2244 ÷ 17 = 132.\n3) First share = 8 × 132 = 1056.\n4) Second share = 9 × 132 = 1188.\nFinal answer: $1188$", "final_answer": "1188", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f43656860ae89990", "problem_id": "ee29d0f0a8754305", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $2244 in the ratio 8:9. How much does each receive?", "output": "Total ratio parts = 8+9 = 17.\nEach part = 2244 ÷ 17 = 132.\nFirst share = 8 × 132 = 1056.\nSecond share = 9 × 132 = 1188.\nCheck: 1056+1188=2244.\nFinal answer: $1188$", "final_answer": "1188", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2dbe8f54f223f59a", "problem_id": "ede8368177cd6391", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Two friends split $1320 in the ratio 6:9. How much does each receive?", "output": "1) Total ratio parts = 6+9 = 15.\n2) Each part = 1320 ÷ 15 = 88.\n3) First share = 6 × 88 = 528.\n4) Second share = 9 × 88 = 792.\nFinal answer: $792$", "final_answer": "792", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8bdc46d49e3b394a", "problem_id": "232140c6d24e0324", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} x^{2} - 6 x + 22$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to x^{2} - 7 x + 22.\n3) Then the limit is 66.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: 62.\nFinal answer: $62$", "final_answer": "62", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6fb513253dc8daa5", "problem_id": "60a1cda3c1d986c6", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A store sells a backpack for $352. It is discounted by 15%. You buy 6 backpacks. What is the total cost?", "output": "1) Discounted price = 352 × (1 − 15/100) = 352 × 85/100 = 1496/5.\n2) Subtotal = 6 × 1496/5 = 8976/5.\nFinal answer: $\\frac{8976}{5}$", "final_answer": "\\frac{8976}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3703d014d6541cba", "problem_id": "60a1cda3c1d986c6", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "A store sells a backpack for $352. It is discounted by 15%. You buy 6 backpacks. What is the total cost?", "output": "$\\frac{8976}{5}$", "final_answer": "\\frac{8976}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d7fd24cf3d5822e", "problem_id": "d4508b3a9beae98c", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{2} - 3 x + \\left(x^{2} - 3 x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3 x + 5}$.", "output": "1) Let f(x) = $\\frac{x^{2} - 3 x + \\left(x^{2} - 3 x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3 x + 5}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{2 \\left(3 - 2 x\\right) \\left(x^{2} - 3 x + \\left(x^{2} - 3 x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3 x + 5\\right) \\left(4 x + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} - 3 x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3 x + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x^{2} - 3 x + 5\\right)^{2}}$.\nFinal answer: $\\frac{\\left(6 - 4\\cdot x\\right)\\cdot \\left(x^{2} - 3\\cdot x + \\left(x^{2} - 3\\cdot x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\cdot x + 5\\right)\\cdot \\left(4\\cdot x + \\left(4\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(x^{2} - 3\\cdot x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} - 6\\cdot x + 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 6\\right)}{2\\cdot \\left(x^{2} - 3\\cdot x + 5\\right)^{2}}$", "final_answer": "\\frac{\\left(6 - 4\\cdot x\\right)\\cdot \\left(x^{2} - 3\\cdot x + \\left(x^{2} - 3\\cdot x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\cdot x + 5\\right)\\cdot \\left(4\\cdot x + \\left(4\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(x^{2} - 3\\cdot x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} - 6\\cdot x + 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 6\\right)}{2\\cdot \\left(x^{2} - 3\\cdot x + 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5e7c5f6f78fbed7f", "problem_id": "d4508b3a9beae98c", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{2} - 3 x + \\left(x^{2} - 3 x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3 x + 5}$.", "output": "Let f(x) = $\\frac{x^{2} - 3 x + \\left(x^{2} - 3 x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3 x + 5}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{2 \\left(3 - 2 x\\right) \\left(x^{2} - 3 x + \\left(x^{2} - 3 x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3 x + 5\\right) \\left(4 x + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} - 3 x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3 x + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x^{2} - 3 x + 5\\right)^{2}}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\frac{\\left(6 - 4\\cdot x\\right)\\cdot \\left(x^{2} - 3\\cdot x + \\left(x^{2} - 3\\cdot x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\cdot x + 5\\right)\\cdot \\left(4\\cdot x + \\left(4\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(x^{2} - 3\\cdot x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} - 6\\cdot x + 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 6\\right)}{2\\cdot \\left(x^{2} - 3\\cdot x + 5\\right)^{2}}$", "final_answer": "\\frac{\\left(6 - 4\\cdot x\\right)\\cdot \\left(x^{2} - 3\\cdot x + \\left(x^{2} - 3\\cdot x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\cdot x + 5\\right)\\cdot \\left(4\\cdot x + \\left(4\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(x^{2} - 3\\cdot x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} - 6\\cdot x + 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 6\\right)}{2\\cdot \\left(x^{2} - 3\\cdot x + 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "96032498a918016b", "problem_id": "ebb7976cad69e09e", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "An urn has 12 red balls and 25 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(37,7).\n3) Favorable samples: choose k reds and n−k blues: C(12,1)·C(25,6).\n4) Compute totals: C(37,7) = 10295472, and favorable = 2125200.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{4025}{19499}$", "final_answer": "\\frac{4025}{19499}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "16827dffec5665f1", "problem_id": "ebb7976cad69e09e", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 12 red balls and 25 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(37,7).\nFavorable samples: choose k reds and n−k blues: C(12,1)·C(25,6).\nCompute totals: C(37,7) = 10295472, and favorable = 2125200.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{4025}{19499}$.\nFinal answer: $\\frac{4025}{19499}$", "final_answer": "\\frac{4025}{19499}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ad0d0f6b9de6fb75", "problem_id": "b65d26ceeb9eeea5", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A cyclist rides at 44 miles per hour for 1 hours, then at 30 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 44 × 1 = 44 miles.\nSecond leg distance = 30 × 3 = 90 miles.\nTotal distance = 44 + 90 = 134 miles.\nCheck: adding both legs gives 134.\nFinal answer: $134$", "final_answer": "134", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "90796c2156853ad7", "problem_id": "b65d26ceeb9eeea5", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A cyclist rides at 44 miles per hour for 1 hours, then at 30 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 44 × 1 = 44 miles.\n2) Second leg distance = 30 × 3 = 90 miles.\n3) Total distance = 44 + 90 = 134 miles.\nFinal answer: $134$", "final_answer": "134", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "81274cab6e2529c1", "problem_id": "2b20de91d8350750", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{5} x}{2} + 5 x - 3 \\sqrt{5} + 6$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$5 - \\frac{3 \\sqrt{5}}{2}$, c=$6 - 3 \\sqrt{5}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{- \\frac{5}{2} + \\frac{\\sqrt{49 - 12\\cdot \\sqrt{5}}}{4} + \\frac{3\\cdot \\sqrt{5}}{4}, - \\frac{5}{2} - \\frac{\\sqrt{49 - 12\\cdot \\sqrt{5}}}{4} + \\frac{3\\cdot \\sqrt{5}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{5}{2} + \\frac{\\sqrt{49 - 12\\cdot \\sqrt{5}}}{4} + \\frac{3\\cdot \\sqrt{5}}{4}, - \\frac{5}{2} - \\frac{\\sqrt{49 - 12\\cdot \\sqrt{5}}}{4} + \\frac{3\\cdot \\sqrt{5}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "73b3fd1208881e08", "problem_id": "2b20de91d8350750", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{5} x}{2} + 5 x - 3 \\sqrt{5} + 6$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$5 - \\frac{3 \\sqrt{5}}{2}$, c=$6 - 3 \\sqrt{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{49}{4} - 3 \\sqrt{5}$, so √Δ = $\\sqrt{\\frac{49}{4} - 3 \\sqrt{5}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{5}{2} + \\frac{\\sqrt{49 - 12 \\sqrt{5}}}{4} + \\frac{3 \\sqrt{5}}{4}$.\n7) x2 = $- \\frac{5}{2} - \\frac{\\sqrt{49 - 12 \\sqrt{5}}}{4} + \\frac{3 \\sqrt{5}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{5}{2} + \\frac{\\sqrt{49 - 12\\cdot \\sqrt{5}}}{4} + \\frac{3\\cdot \\sqrt{5}}{4}, - \\frac{5}{2} - \\frac{\\sqrt{49 - 12\\cdot \\sqrt{5}}}{4} + \\frac{3\\cdot \\sqrt{5}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{5}{2} + \\frac{\\sqrt{49 - 12\\cdot \\sqrt{5}}}{4} + \\frac{3\\cdot \\sqrt{5}}{4}, - \\frac{5}{2} - \\frac{\\sqrt{49 - 12\\cdot \\sqrt{5}}}{4} + \\frac{3\\cdot \\sqrt{5}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "be200d8baf730202", "problem_id": "fa30c41e2f53e39c", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(- 2 x^{3} + 2 x^{2} - 2 x - 3\\right) \\cos{\\left(x - 3 \\right)}$.", "output": "1) Let f(x) = $2 \\left(x - 3\\right) e^{x} + \\left(- 2 x^{3} + 2 x^{2} - 2 x - 3\\right) \\cos{\\left(x - 3 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $2 \\left(x - 3\\right) e^{x} + 2 \\left(- 3 x^{2} + 2 x - 1\\right) \\cos{\\left(x - 3 \\right)} - \\left(- 2 x^{3} + 2 x^{2} - 2 x - 3\\right) \\sin{\\left(x - 3 \\right)} + 2 e^{x}$.\nFinal answer: $\\left(2\\cdot x - 6\\right)\\cdot e^{x} + \\left(- 6\\cdot x^{2} + 4\\cdot x - 2\\right)\\cdot \\cos{\\left(x - 3 \\right)} - \\left(- 2\\cdot x^{3} + 2\\cdot x^{2} - 2\\cdot x - 3\\right)\\cdot \\sin{\\left(x - 3 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x - 6\\right)\\cdot e^{x} + \\left(- 6\\cdot x^{2} + 4\\cdot x - 2\\right)\\cdot \\cos{\\left(x - 3 \\right)} - \\left(- 2\\cdot x^{3} + 2\\cdot x^{2} - 2\\cdot x - 3\\right)\\cdot \\sin{\\left(x - 3 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f1211eff87dfcbf2", "problem_id": "fa30c41e2f53e39c", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(- 2 x^{3} + 2 x^{2} - 2 x - 3\\right) \\cos{\\left(x - 3 \\right)}$.", "output": "Let f(x) = $2 \\left(x - 3\\right) e^{x} + \\left(- 2 x^{3} + 2 x^{2} - 2 x - 3\\right) \\cos{\\left(x - 3 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $2 \\left(x - 3\\right) e^{x} + 2 \\left(- 3 x^{2} + 2 x - 1\\right) \\cos{\\left(x - 3 \\right)} - \\left(- 2 x^{3} + 2 x^{2} - 2 x - 3\\right) \\sin{\\left(x - 3 \\right)} + 2 e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\left(2\\cdot x - 6\\right)\\cdot e^{x} + \\left(- 6\\cdot x^{2} + 4\\cdot x - 2\\right)\\cdot \\cos{\\left(x - 3 \\right)} - \\left(- 2\\cdot x^{3} + 2\\cdot x^{2} - 2\\cdot x - 3\\right)\\cdot \\sin{\\left(x - 3 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x - 6\\right)\\cdot e^{x} + \\left(- 6\\cdot x^{2} + 4\\cdot x - 2\\right)\\cdot \\cos{\\left(x - 3 \\right)} - \\left(- 2\\cdot x^{3} + 2\\cdot x^{2} - 2\\cdot x - 3\\right)\\cdot \\sin{\\left(x - 3 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2fae2edaea9bb970", "problem_id": "86a118c50610e51e", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $2208 in the ratio 8:8. How much does each receive?", "output": "1) Total ratio parts = 8+8 = 16.\n2) Each part = 2208 ÷ 16 = 138.\n3) First share = 8 × 138 = 1104.\n4) Second share = 8 × 138 = 1104.\nFinal answer: $1104$", "final_answer": "1104", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b27c7a97ac9299bf", "problem_id": "86a118c50610e51e", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Two friends split $2208 in the ratio 8:8. How much does each receive?", "output": "Total ratio parts = 8+8 = 16.\nEach part = 2208 ÷ 16 = 138.\nFirst share = 8 × 138 = 1104.\nFinal answer: $1104$", "final_answer": "1104", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6ec9c6ea4b319906", "problem_id": "0daa875cf48889be", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Two friends split $170 in the ratio 9:8. How much does each receive?", "output": "1) Total ratio parts = 9+8 = 17.\n2) Each part = 170 ÷ 17 = 10.\n3) First share = 9 × 10 = 90.\n4) Second share = 8 × 10 = 80.\nFinal answer: $80$", "final_answer": "80", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e3971c12697d6b79", "problem_id": "0daa875cf48889be", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $170 in the ratio 9:8. How much does each receive?", "output": "Total ratio parts = 9+8 = 17.\nEach part = 170 ÷ 17 = 10.\nFirst share = 9 × 10 = 90.\nSecond share = 8 × 10 = 80.\nCheck: 90+80=170.\nFinal answer: $80$", "final_answer": "80", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "307079c4cea425a2", "problem_id": "e167fc7024ef3dae", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=27$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=27 and k=10.\nCompute C(36,9) = 94143280.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $94143280$", "final_answer": "94143280", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9052ad84a5739c21", "problem_id": "e167fc7024ef3dae", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=27$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=27 and k=10.\n4) Compute C(36,9) = 94143280.\nFinal answer: $94143280$", "final_answer": "94143280", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "26782d080590025f", "problem_id": "9db794fc35f64fc0", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Two friends split $572 in the ratio 5:8. How much does each receive?", "output": "1) Total ratio parts = 5+8 = 13.\n2) Each part = 572 ÷ 13 = 44.\n3) First share = 5 × 44 = 220.\n4) Second share = 8 × 44 = 352.\nFinal answer: $352$", "final_answer": "352", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "098daf54b22c36b0", "problem_id": "e07c8f3c1d5f8a5b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - 3 x^{3} - 12 x^{2} - 38 x - 113$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{3} - 12 x^{2} - 39 x - 113.\n3) Then the limit is -419.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: -416.\nFinal answer: $-416$", "final_answer": "-416", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "af60de746dbe6a75", "problem_id": "fb32930df7fc5179", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_8=38$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_8=14.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=14 and k=8.\n4) Compute C(21,7) = 116280.\nFinal answer: $116280$", "final_answer": "116280", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b96d2fb6ee74274a", "problem_id": "fb32930df7fc5179", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_8=38$?", "output": "Let y_i=x_i-3, then y_i≥0 and y_1+…+y_8=14.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=14 and k=8.\nCompute C(21,7) = 116280.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $116280$", "final_answer": "116280", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "48b726b1f22ccac1", "problem_id": "d1d3f936bb748158", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_7=27$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_7=6.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=6 and k=7.\n4) Compute C(12,6) = 924.\nFinal answer: $924$", "final_answer": "924", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "520f11f54da0c375", "problem_id": "d1d3f936bb748158", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_7=27$?", "output": "Let y_i=x_i-3, then y_i≥0 and y_1+…+y_7=6.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=6 and k=7.\nFinal answer: $924$", "final_answer": "924", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7db4853410b56792", "problem_id": "2dea3d138e1f17a3", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 8 red balls and 14 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(22,8).\nFavorable samples: choose k reds and n−k blues: C(8,2)·C(14,6).\nCompute totals: C(22,8) = 319770, and favorable = 84084.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1274}{4845}$.\nFinal answer: $\\frac{1274}{4845}$", "final_answer": "\\frac{1274}{4845}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7469d2174ba462f6", "problem_id": "09726800bf81c01d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} + 4 x^{2} + 3 x$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x^{3}) = 3 x^{3}$.\n3) So $f'(x) = 3 x^{3} + 8 x + 3$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x^{3}) = 3 x^{2}$.\nTherefore the correct derivative is $f'(x) = 3 x^{2} + 8 x + 3$.\nFinal answer: $3 x^{2} + 8 x + 3$", "final_answer": "3 x^{2} + 8 x + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3eda8c3562504a8b", "problem_id": "17178a4d44e900ba", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{2} - 2 x$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $- 6 x - 2$.\n2) Slope at x=-2: m = f'(-2) = $10$.\n3) Point on curve: (-2, f(-2)) = (-2, $-8$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -8 = 10(x − -2).\n6) Optionally expand: y = $10 x + 12$.\nFinal answer: $y = 10\\cdot x + 12$", "final_answer": "y = 10\\cdot x + 12", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9130a94393447367", "problem_id": "17178a4d44e900ba", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{2} - 2 x$ at $x=-2$.", "output": "Differentiate: f'(x) = $- 6 x - 2$.\nSlope at x=-2: m = f'(-2) = $10$.\nPoint on curve: (-2, f(-2)) = (-2, $-8$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -8 = 10(x − -2).\nOptionally expand: y = $10 x + 12$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 10\\cdot x + 12$", "final_answer": "y = 10\\cdot x + 12", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "080a37cab04c6b8b", "problem_id": "748b16e9343f1a49", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A cyclist rides at 60 miles per hour for 1/2 hours, then at 32 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 60 × 1/2 = 30 miles.\n2) Second leg distance = 32 × 2 = 64 miles.\n3) Total distance = 30 + 64 = 94 miles.\nFinal answer: $94$", "final_answer": "94", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e7dd3d5efcb5c57f", "problem_id": "748b16e9343f1a49", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A cyclist rides at 60 miles per hour for 1/2 hours, then at 32 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 60 × 1/2 = 30 miles.\nSecond leg distance = 32 × 2 = 64 miles.\nTotal distance = 30 + 64 = 94 miles.\nCheck: adding both legs gives 94.\nFinal answer: $94$", "final_answer": "94", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fe08ce72221cc881", "problem_id": "3bebc22ff4bdd34d", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$- y$ = $-4$\n$3 x - 2 y$ = $-23$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $3$: $- 3 y$ = $-12$.\n3) Multiply the second equation by $0$: $0$ = $0$.\n4) Subtract to eliminate x: $- 3 y$ = $-12$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, 4\\right)$", "final_answer": "(x,y)=\\left(-5, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "561a2a5458947d7f", "problem_id": "3bebc22ff4bdd34d", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- y$ = $-4$\n$3 x - 2 y$ = $-23$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $- 3 y$ = $-12$.\nMultiply the second equation by $0$: $0$ = $0$.\nSubtract to eliminate x: $- 3 y$ = $-12$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $-5$.\nCheck: plug (x,y)=(-5,4) into both equations.\nFinal answer: $(x,y)=\\left(-5, 4\\right)$", "final_answer": "(x,y)=\\left(-5, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b30bdf027cec546c", "problem_id": "01a675386353afdf", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{3}$. If it is flipped 14 times, what is the probability of exactly 9 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=14, k=9, p=$\\frac{1}{3}$ and simplify.\nHere C(14,9) = 2002.\nCheck: expression simplifies to $\\frac{64064}{4782969}$.\nFinal answer: $\\frac{64064}{4782969}$", "final_answer": "\\frac{64064}{4782969}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e13f21b44e44796", "problem_id": "01a675386353afdf", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{3}$. If it is flipped 14 times, what is the probability of exactly 9 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=14, k=9, p=$\\frac{1}{3}$ and simplify.\n4) Here C(14,9) = 2002.\nFinal answer: $\\frac{64064}{4782969}$", "final_answer": "\\frac{64064}{4782969}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eefe78e4b7b13b79", "problem_id": "9f333624581004da", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{16 x}{15} - \\frac{\\sqrt{2} x}{3} - 1 - \\frac{\\sqrt{2}}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{16}{15} - \\frac{\\sqrt{2}}{3}$, c=$-1 - \\frac{\\sqrt{2}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{68 \\sqrt{2}}{45} + \\frac{134}{25}$, so √Δ = $\\sqrt{\\frac{68 \\sqrt{2}}{45} + \\frac{134}{25}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{2}}{6} + \\frac{8}{15} + \\frac{\\sqrt{340 \\sqrt{2} + 1206}}{30}$.\n7) x2 = $- \\frac{\\sqrt{340 \\sqrt{2} + 1206}}{30} + \\frac{\\sqrt{2}}{6} + \\frac{8}{15}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{2}}{6} + \\frac{8}{15} + \\frac{\\sqrt{340\\cdot \\sqrt{2} + 1206}}{30}, - \\frac{\\sqrt{340\\cdot \\sqrt{2} + 1206}}{30} + \\frac{\\sqrt{2}}{6} + \\frac{8}{15}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{2}}{6} + \\frac{8}{15} + \\frac{\\sqrt{340\\cdot \\sqrt{2} + 1206}}{30}, - \\frac{\\sqrt{340\\cdot \\sqrt{2} + 1206}}{30} + \\frac{\\sqrt{2}}{6} + \\frac{8}{15}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9cc4a761855c904f", "problem_id": "9f333624581004da", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{16 x}{15} - \\frac{\\sqrt{2} x}{3} - 1 - \\frac{\\sqrt{2}}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{16}{15} - \\frac{\\sqrt{2}}{3}$, c=$-1 - \\frac{\\sqrt{2}}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{68 \\sqrt{2}}{45} + \\frac{134}{25}$, so √Δ = $\\sqrt{\\frac{68 \\sqrt{2}}{45} + \\frac{134}{25}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{2}}{6} + \\frac{8}{15} + \\frac{\\sqrt{340 \\sqrt{2} + 1206}}{30}$.\nx2 = $- \\frac{\\sqrt{340 \\sqrt{2} + 1206}}{30} + \\frac{\\sqrt{2}}{6} + \\frac{8}{15}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{2}}{6} + \\frac{8}{15} + \\frac{\\sqrt{340\\cdot \\sqrt{2} + 1206}}{30}, - \\frac{\\sqrt{340\\cdot \\sqrt{2} + 1206}}{30} + \\frac{\\sqrt{2}}{6} + \\frac{8}{15}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{2}}{6} + \\frac{8}{15} + \\frac{\\sqrt{340\\cdot \\sqrt{2} + 1206}}{30}, - \\frac{\\sqrt{340\\cdot \\sqrt{2} + 1206}}{30} + \\frac{\\sqrt{2}}{6} + \\frac{8}{15}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e1d1806ead5a6cd3", "problem_id": "955f37ef1895c469", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- x^{3} + x^{2} - x + 1\\right) \\cos{\\left(x + 2 \\right)}$.", "output": "Let f(x) = $2 \\left(x + 1\\right) e^{x} + \\left(- x^{3} + x^{2} - x + 1\\right) \\cos{\\left(x + 2 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $2 \\left(x + 1\\right) e^{x} + \\left(- 3 x^{2} + 2 x - 1\\right) \\cos{\\left(x + 2 \\right)} - \\left(- x^{3} + x^{2} - x + 1\\right) \\sin{\\left(x + 2 \\right)} + 2 e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\left(2\\cdot x + 2\\right)\\cdot e^{x} + \\left(- 3\\cdot x^{2} + 2\\cdot x - 1\\right)\\cdot \\cos{\\left(x + 2 \\right)} - \\left(- x^{3} + x^{2} - x + 1\\right)\\cdot \\sin{\\left(x + 2 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x + 2\\right)\\cdot e^{x} + \\left(- 3\\cdot x^{2} + 2\\cdot x - 1\\right)\\cdot \\cos{\\left(x + 2 \\right)} - \\left(- x^{3} + x^{2} - x + 1\\right)\\cdot \\sin{\\left(x + 2 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c9158aaa9a1763ed", "problem_id": "955f37ef1895c469", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- x^{3} + x^{2} - x + 1\\right) \\cos{\\left(x + 2 \\right)}$.", "output": "1) Let f(x) = $2 \\left(x + 1\\right) e^{x} + \\left(- x^{3} + x^{2} - x + 1\\right) \\cos{\\left(x + 2 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $2 \\left(x + 1\\right) e^{x} + \\left(- 3 x^{2} + 2 x - 1\\right) \\cos{\\left(x + 2 \\right)} - \\left(- x^{3} + x^{2} - x + 1\\right) \\sin{\\left(x + 2 \\right)} + 2 e^{x}$.\nFinal answer: $\\left(2\\cdot x + 2\\right)\\cdot e^{x} + \\left(- 3\\cdot x^{2} + 2\\cdot x - 1\\right)\\cdot \\cos{\\left(x + 2 \\right)} - \\left(- x^{3} + x^{2} - x + 1\\right)\\cdot \\sin{\\left(x + 2 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x + 2\\right)\\cdot e^{x} + \\left(- 3\\cdot x^{2} + 2\\cdot x - 1\\right)\\cdot \\cos{\\left(x + 2 \\right)} - \\left(- x^{3} + x^{2} - x + 1\\right)\\cdot \\sin{\\left(x + 2 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e0c96deee50affa0", "problem_id": "64fc77bf138edd72", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_9=40$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_9=22.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=22 and k=9.\n4) Compute C(30,8) = 5852925.\nFinal answer: $5852925$", "final_answer": "5852925", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "492d544dbd1c8fdb", "problem_id": "0a9034d6bc05fb6a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{2} + 3 x + \\left(3 x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{3 x^{2} - x + 5}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(3 x^{2} - x\\right) + 5}) = \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x^{2} - x + 5}$.\n3) So $f'(x) = \\frac{9 x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 12 x^{3} + 62 x^{2} - 20 x + 50} + \\frac{18 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 12 x^{3} + 62 x^{2} - 20 x + 50} - \\frac{6 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 12 x^{3} + 62 x^{2} - 20 x + 50} - \\frac{12 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 12 x^{3} + 62 x^{2} - 20 x + 50} + \\frac{31 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 12 x^{3} + 62 x^{2} - 20 x + 50} + \\frac{62 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 12 x^{3} + 62 x^{2} - 20 x + 50} - \\frac{12 x^{2}}{18 x^{4} - 12 x^{3} + 62 x^{2} - 20 x + 50} - \\frac{10 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 12 x^{3} + 62 x^{2} - 20 x + 50} - \\frac{20 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 12 x^{3} + 62 x^{2} - 20 x + 50} - \\frac{72 x}{18 x^{4} - 12 x^{3} + 62 x^{2} - 20 x + 50} + \\frac{30 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{4} - 6 x^{3} + 31 x^{2} - 10 x + 25} + \\frac{25 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 12 x^{3} + 62 x^{2} - 20 x + 50} + \\frac{50 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 12 x^{3} + 62 x^{2} - 20 x + 50} + \\frac{32}{18 x^{4} - 12 x^{3} + 62 x^{2} - 20 x + 50} - \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{4} - 6 x^{3} + 31 x^{2} - 10 x + 25} - \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{2} - 2 x + 10} + \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x^{2} - x + 5} - \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(3 x^{2} - x\\right) + 5}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(3 x^{2} - x\\right) + 5}) = \\frac{5 \\left(1 - 6 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(3 x^{2} - x\\right) + 5\\right)^{2}} + \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(3 x^{2} - x\\right) + 5\\right)} + \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(3 x^{2} - x\\right) + 5}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(1 - 6 x\\right) \\left(- 3 x^{2} + 3 x + \\left(x \\left(3 x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(x \\left(3 x - 1\\right) + 5\\right) \\left(- 12 x + 2 \\left(6 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(3 x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(3 x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6\\right)}{2 \\left(x \\left(3 x - 1\\right) + 5\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(1 - 6 x\\right) \\left(- 3 x^{2} + 3 x + \\left(x \\left(3 x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(x \\left(3 x - 1\\right) + 5\\right) \\left(- 12 x + 2 \\left(6 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(3 x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(3 x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6\\right)}{2 \\left(x \\left(3 x - 1\\right) + 5\\right)^{2}}$", "final_answer": "\\frac{2 \\left(1 - 6 x\\right) \\left(- 3 x^{2} + 3 x + \\left(x \\left(3 x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(x \\left(3 x - 1\\right) + 5\\right) \\left(- 12 x + 2 \\left(6 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(3 x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(3 x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6\\right)}{2 \\left(x \\left(3 x - 1\\right) + 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c0152944852691ff", "problem_id": "7a1679055f58b946", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A store sells a backpack for $368. It is discounted by 40%. You buy 4 backpacks. What is the total cost?", "output": "1) Discounted price = 368 × (1 − 40/100) = 368 × 60/100 = 1104/5.\n2) Subtotal = 4 × 1104/5 = 4416/5.\nFinal answer: $\\frac{4416}{5}$", "final_answer": "\\frac{4416}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5f2cd077ad1ba399", "problem_id": "7a1679055f58b946", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A store sells a backpack for $368. It is discounted by 40%. You buy 4 backpacks. What is the total cost?", "output": "Discounted price = 368 × (1 − 40/100) = 368 × 60/100 = 1104/5.\nSubtotal = 4 × 1104/5 = 4416/5.\nCheck: recomputing the formula yields 4416/5.\nFinal answer: $\\frac{4416}{5}$", "final_answer": "\\frac{4416}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "837777350ca70232", "problem_id": "3044792d431e208a", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the equation of the tangent line to $y=- 4 x^{3} + x^{2} - 2 x + 3$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $- 12 x^{2} + 2 x - 2$.\n2) Slope at x=-2: m = f'(-2) = $-54$.\n3) Point on curve: (-2, f(-2)) = (-2, $43$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 43 = -54(x − -2).\n6) Optionally expand: y = $- 54 x - 65$.\nFinal answer: $y = - 54\\cdot x - 65$", "final_answer": "y = - 54\\cdot x - 65", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6f7a0407d7e9cd68", "problem_id": "3044792d431e208a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the equation of the tangent line to $y=- 4 x^{3} + x^{2} - 2 x + 3$ at $x=-2$.", "output": "Differentiate: f'(x) = $- 12 x^{2} + 2 x - 2$.\nSlope at x=-2: m = f'(-2) = $-54$.\nPoint on curve: (-2, f(-2)) = (-2, $43$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 43 = -54(x − -2).\nOptionally expand: y = $- 54 x - 65$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 54\\cdot x - 65$", "final_answer": "y = - 54\\cdot x - 65", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "99f775309f60813b", "problem_id": "fc90ae66a7fc4512", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 17 red balls and 13 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(30,8).\n3) Favorable samples: choose k reds and n−k blues: C(17,1)·C(13,7).\n4) Compute totals: C(30,8) = 5852925, and favorable = 29172.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{748}{150075}$", "final_answer": "\\frac{748}{150075}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b934208f2745e146", "problem_id": "b71bb7c7959cdfd8", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} - 2 x^{2} + 4 x - 4$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x^{3}) = - 6 x^{3}$.\n3) So $f'(x) = - 6 x^{3} - 4 x + 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x^{3}) = - 6 x^{2}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- 3 x^{2} - 2 x + 2\\right)$.\nFinal answer: $2 \\left(- 3 x^{2} - 2 x + 2\\right)$", "final_answer": "2 \\left(- 3 x^{2} - 2 x + 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a12768f4791d6d19", "problem_id": "6259a1bd8729cec8", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{65 \\pi^{3}}{24} - \\frac{15 \\pi^{2}}{8} - 2 + \\frac{15 \\pi}{2}$.\n4) Simplify the result.\nFinal answer: $- \\frac{65\\cdot \\pi^{3}}{24} - \\frac{15\\cdot \\pi^{2}}{8} - 2 + \\frac{15\\cdot \\pi}{2}$", "final_answer": "- \\frac{65\\cdot \\pi^{3}}{24} - \\frac{15\\cdot \\pi^{2}}{8} - 2 + \\frac{15\\cdot \\pi}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "396519b6f3e1782e", "problem_id": "6259a1bd8729cec8", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{65 \\pi^{3}}{24} - \\frac{15 \\pi^{2}}{8} - 2 + \\frac{15 \\pi}{2}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{65\\cdot \\pi^{3}}{24} - \\frac{15\\cdot \\pi^{2}}{8} - 2 + \\frac{15\\cdot \\pi}{2}$", "final_answer": "- \\frac{65\\cdot \\pi^{3}}{24} - \\frac{15\\cdot \\pi^{2}}{8} - 2 + \\frac{15\\cdot \\pi}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "faafdff69235add8", "problem_id": "29ebe8bb40065fcb", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} - 4 x$.", "output": "1) Differentiate term-by-term:\n2) d/dx(- 4 x) = -4.\n3) d/dx(3 x^{2}) = 6 x.\n4) Combine: f'(x) = 2 \\left(3 x - 2\\right).\nFinal answer: $6\\cdot x - 4$", "final_answer": "6\\cdot x - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "24655195081b8fee", "problem_id": "29ebe8bb40065fcb", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} - 4 x$.", "output": "$6\\cdot x - 4$", "final_answer": "6\\cdot x - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "637450044dc5fb7d", "problem_id": "bf21737ead3b37c3", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - x - 72$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} - x - 72$ = $\\left(x - 9\\right) \\left(x + 8\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $9$.\n5) From the second factor: x = $-8$.\nFinal answer: $x \\in \\{9, -8\\}$", "final_answer": "x \\in \\{9, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "845bda89282a9cc5", "problem_id": "bf21737ead3b37c3", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - x - 72$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} - x - 72$ = $\\left(x - 9\\right) \\left(x + 8\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $9$.\nFrom the second factor: x = $-8$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{9, -8\\}$", "final_answer": "x \\in \\{9, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "26e50231ef90f0f8", "problem_id": "39d6c4949a7e9622", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int x^{2} + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $x^{2} + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2$.\n3) Compute an antiderivative: F(x) = $\\frac{x^{3}}{3} + 2 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{x^{3}}{3} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{3}}{3} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "55d431ea236b4253", "problem_id": "39d6c4949a7e9622", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int x^{2} + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $x^{2} + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2$.\nCompute an antiderivative: F(x) = $\\frac{x^{3}}{3} + 2 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{x^{3}}{3} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{3}}{3} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8c274caf685c5b4d", "problem_id": "971bb1a8aa34b384", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - 3 x^{2} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $- 9 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\n2) Slope at x=-1: m = f'(-1) = $-5 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\n3) Point on curve: (-1, f(-1)) = (-1, $\\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 1$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 1 = -5 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}(x − -1).\n6) Optionally expand: y = $- 5 x + \\frac{x}{2 e^{\\frac{1}{2}}} + 2 x \\sin{\\left(2 \\right)} - 4 + \\cos{\\left(2 \\right)} + \\frac{3}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = - 5\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 4 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 5\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 4 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "31a2b50b1c4f0725", "problem_id": "971bb1a8aa34b384", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - 3 x^{2} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=-1$.", "output": "Differentiate: f'(x) = $- 9 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\nSlope at x=-1: m = f'(-1) = $-5 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nPoint on curve: (-1, f(-1)) = (-1, $\\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 1$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 1 = -5 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}(x − -1).\nOptionally expand: y = $- 5 x + \\frac{x}{2 e^{\\frac{1}{2}}} + 2 x \\sin{\\left(2 \\right)} - 4 + \\cos{\\left(2 \\right)} + \\frac{3}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 5\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 4 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 5\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 4 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "99c215ed21d1a96d", "problem_id": "05a569c346ba15c7", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=19$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=19 and k=7.\n4) Compute C(25,6) = 177100.\nFinal answer: $177100$", "final_answer": "177100", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d8c3c143714b979f", "problem_id": "05a569c346ba15c7", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=19$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=19 and k=7.\nFinal answer: $177100$", "final_answer": "177100", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8694969e4a38b0d0", "problem_id": "9e18fd5fbc2a57c0", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$- 8 x - 5 y$ = $-11$\n$5 x - 8 y$ = $18$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $5$: $- 40 x - 25 y$ = $-55$.\n3) Multiply the second equation by $-8$: $- 40 x + 64 y$ = $-144$.\n4) Subtract to eliminate x: $- 89 y$ = $89$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $2$.\nFinal answer: $(x,y)=\\left(2, -1\\right)$", "final_answer": "(x,y)=\\left(2, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c6438e46f6530793", "problem_id": "b834c3003c0c1aab", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - x^{2} + 4 x - 9$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} + 3 x - 9.\n3) Then the limit is -19.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -21.\nFinal answer: $-21$", "final_answer": "-21", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "846a8bfaea6d1972", "problem_id": "556ad55720259d27", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Find the multiplicative inverse of 40 modulo 47.", "output": "1) We want x such that 40x ≡ 1 (mod 47).\n2) Use extended Euclid to find s,t with 40s + 47t = 1.\n3) One Bezout identity is: 40·(20) + 47·(-17) = 1.\n4) Reduce s modulo 47: x ≡ 20 (mod 47).\nFinal answer: $x\\equiv 20\\pmod{47}$", "final_answer": "x\\equiv 20\\pmod{47}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "481c60fff925fa8f", "problem_id": "3f28178ada29b9c7", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 2 x^{2} \\sin{\\left(x \\right)}) = - 2 x^{2} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 2 x^{2} \\sin{\\left(x \\right)}) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$", "final_answer": "- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f22b4ae8c00db300", "problem_id": "474d447fe0e339c4", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "You want 37 liters of a 45% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 37−x liters is 70% solution.\n2) Salt equation: 25x + 70(37−x) = 45·37.\n3) (25-70)x = (45-70)·37 ⇒ x = 185/9.\n4) Then 37−x = 148/9.\nFinal answer: $185/9 L of 25% and 148/9 L of 70%$", "final_answer": "185/9 L of 25% and 148/9 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bbf2d9e924b89c83", "problem_id": "474d447fe0e339c4", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "You want 37 liters of a 45% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 37−x liters is 70% solution.\nSalt equation: 25x + 70(37−x) = 45·37.\n(25-70)x = (45-70)·37 ⇒ x = 185/9.\nThen 37−x = 148/9.\nCheck: 25·185/9 + 70·148/9 = 45·37.\nFinal answer: $185/9 L of 25% and 148/9 L of 70%$", "final_answer": "185/9 L of 25% and 148/9 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "70e4716d22e84f29", "problem_id": "ed153ad8d9f2ef36", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} - x^{4} - x^{3} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{4} - x^{3} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $- 2 \\pi - \\frac{\\pi^{5}}{80} + 4$.\nFinal answer: $- 2\\cdot \\pi - \\frac{\\pi^{5}}{80} + 4$", "final_answer": "- 2\\cdot \\pi - \\frac{\\pi^{5}}{80} + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "164a35102845c44d", "problem_id": "ed153ad8d9f2ef36", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} - x^{4} - x^{3} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{4} - x^{3} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $- 2 \\pi - \\frac{\\pi^{5}}{80} + 4$.\n4) Simplify the result.\nFinal answer: $- 2\\cdot \\pi - \\frac{\\pi^{5}}{80} + 4$", "final_answer": "- 2\\cdot \\pi - \\frac{\\pi^{5}}{80} + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "71cf4ee9ebca4c99", "problem_id": "4dd4cc37b96485d5", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} + x - 3$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-3) = 0.\n3) d/dx(x) = 1.\n4) d/dx(2 x^{2}) = 4 x.\n5) Combine: f'(x) = 4 x + 1.\nFinal answer: $4\\cdot x + 1$", "final_answer": "4\\cdot x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "67c535b8c14f8919", "problem_id": "4dd4cc37b96485d5", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} + x - 3$.", "output": "Differentiate term-by-term:\nd/dx(-3) = 0.\nd/dx(x) = 1.\nd/dx(2 x^{2}) = 4 x.\nCombine: f'(x) = 4 x + 1.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $4\\cdot x + 1$", "final_answer": "4\\cdot x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c663318a33d3af7e", "problem_id": "43807df5e2ea6c51", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{7 x}{3} - \\frac{49}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$\\frac{7}{3}$, c=$- \\frac{49}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{1225}{9}$, so √Δ = $\\frac{35}{3}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{7}{3}$.\nx2 = $- \\frac{7}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{7}{3}, - \\frac{7}{2}\\}$", "final_answer": "x \\in \\{\\frac{7}{3}, - \\frac{7}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1600626616ed4698", "problem_id": "43807df5e2ea6c51", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{7 x}{3} - \\frac{49}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$\\frac{7}{3}$, c=$- \\frac{49}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{1225}{9}$, so √Δ = $\\frac{35}{3}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{7}{3}$.\n7) x2 = $- \\frac{7}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{7}{3}, - \\frac{7}{2}\\}$", "final_answer": "x \\in \\{\\frac{7}{3}, - \\frac{7}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "112f28bb53204e92", "problem_id": "c2adfd708c8fce03", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Worker A finishes a job in 9 hours and Worker B finishes it in 15 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/9 job/hour. B's rate = 1/15 job/hour.\n2) Combined rate = 1/9 + 1/15 = 8/45 job/hour.\n3) Time = 1 ÷ (combined rate) = 45/8 hours.\nFinal answer: $\\frac{45}{8}$", "final_answer": "\\frac{45}{8}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "720f0e049b2b5adb", "problem_id": "c2adfd708c8fce03", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Worker A finishes a job in 9 hours and Worker B finishes it in 15 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/9 job/hour. B's rate = 1/15 job/hour.\nCombined rate = 1/9 + 1/15 = 8/45 job/hour.\nTime = 1 ÷ (combined rate) = 45/8 hours.\nCheck: (combined rate)×(time) = 8/45×45/8 = 1.\nFinal answer: $\\frac{45}{8}$", "final_answer": "\\frac{45}{8}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0b74f6d32d4c2d49", "problem_id": "ad3fd4dde05ca192", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Two friends split $990 in the ratio 9:1. How much does each receive?", "output": "1) Total ratio parts = 9+1 = 10.\n2) Each part = 990 ÷ 10 = 99.\n3) First share = 9 × 99 = 891.\n4) Second share = 1 × 99 = 99.\nFinal answer: $99$", "final_answer": "99", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e1ecf559ab0e8863", "problem_id": "ad3fd4dde05ca192", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $990 in the ratio 9:1. How much does each receive?", "output": "Total ratio parts = 9+1 = 10.\nEach part = 990 ÷ 10 = 99.\nFirst share = 9 × 99 = 891.\nSecond share = 1 × 99 = 99.\nCheck: 891+99=990.\nFinal answer: $99$", "final_answer": "99", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a363ae55d889982c", "problem_id": "2bd6b6daabf2ed36", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} - x^{2} + 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $- 6 x^{2} - 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 3$.\n2) Slope at x=-2: m = f'(-2) = $-17 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}$.\n3) Point on curve: (-2, f(-2)) = (-2, $\\cos{\\left(4 \\right)} + e^{-1} + 4$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(4 \\right)} + e^{-1} + 4 = -17 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}(x − -2).\n6) Optionally expand: y = $- 17 x + 2 x \\sin{\\left(4 \\right)} + \\frac{x}{2 e} - 30 + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$.\nFinal answer: $y = - 17\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 30 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$", "final_answer": "y = - 17\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 30 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e9fec1ca6843091f", "problem_id": "2bd6b6daabf2ed36", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} - x^{2} + 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=-2$.", "output": "$y = - 17\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 30 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$", "final_answer": "y = - 17\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 30 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8f5f69758817e770", "problem_id": "726d0b64c9bf2695", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$x + 7 y$ = $-10$\n$5 x + 7 y$ = $6$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $5$: $5 x + 35 y$ = $-50$.\n3) Multiply the second equation by $1$: $5 x + 7 y$ = $6$.\n4) Subtract to eliminate x: $28 y$ = $-56$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, -2\\right)$", "final_answer": "(x,y)=\\left(4, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "82972a2621eb089f", "problem_id": "726d0b64c9bf2695", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$x + 7 y$ = $-10$\n$5 x + 7 y$ = $6$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $5$: $5 x + 35 y$ = $-50$.\nMultiply the second equation by $1$: $5 x + 7 y$ = $6$.\nSubtract to eliminate x: $28 y$ = $-56$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $4$.\nCheck: plug (x,y)=(4,-2) into both equations.\nFinal answer: $(x,y)=\\left(4, -2\\right)$", "final_answer": "(x,y)=\\left(4, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d01272b354e4c7b2", "problem_id": "bfa05cd85d8d5b07", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 3 x^{3} - 8 x^{2} - 16 x - 30$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{3} - 8 x^{2} - 17 x - 30.\n3) Then the limit is -120.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -118.\nFinal answer: $-118$", "final_answer": "-118", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6d9b96628953aacb", "problem_id": "9b3d5552062f687d", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the multiplicative inverse of 8 modulo 13.", "output": "1) We want x such that 8x ≡ 1 (mod 13).\n2) Use extended Euclid to find s,t with 8s + 13t = 1.\n3) One Bezout identity is: 8·(5) + 13·(-3) = 1.\n4) Reduce s modulo 13: x ≡ 5 (mod 13).\nFinal answer: $x\\equiv 5\\pmod{13}$", "final_answer": "x\\equiv 5\\pmod{13}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "46190b96b800b767", "problem_id": "16f1f038bc1317df", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A disease has prevalence $\\frac{1}{50}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{121}{2500}$.\n3) Compute P(D|+) = $\\frac{95}{242}$.\nFinal answer: $\\frac{95}{242}$", "final_answer": "\\frac{95}{242}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "37be484e7ee77f99", "problem_id": "c70b1fa0bce0719e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} 2 - x$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to 2 - 2 x.\n3) Then the limit is 10.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: 6.\nFinal answer: $6$", "final_answer": "6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b817cb6968efbdac", "problem_id": "b0f406f7f811d903", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 9 red balls and 16 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(25,7).\n3) Favorable samples: choose k reds and n−k blues: C(9,3)·C(16,4).\n4) Compute totals: C(25,7) = 480700, and favorable = 152880.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{7644}{24035}$", "final_answer": "\\frac{7644}{24035}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dd4ab8b60a0b142a", "problem_id": "b0f406f7f811d903", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 9 red balls and 16 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(25,7).\nFavorable samples: choose k reds and n−k blues: C(9,3)·C(16,4).\nCompute totals: C(25,7) = 480700, and favorable = 152880.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{7644}{24035}$.\nFinal answer: $\\frac{7644}{24035}$", "final_answer": "\\frac{7644}{24035}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0c72000530d67204", "problem_id": "df542146d9adfae9", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Two friends split $2862 in the ratio 9:9. How much does each receive?", "output": "1) Total ratio parts = 9+9 = 18.\n2) Each part = 2862 ÷ 18 = 159.\n3) First share = 9 × 159 = 1431.\n4) Second share = 9 × 159 = 1431.\nFinal answer: $1431$", "final_answer": "1431", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c2a1a2a5f344b14a", "problem_id": "df542146d9adfae9", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Two friends split $2862 in the ratio 9:9. How much does each receive?", "output": "Total ratio parts = 9+9 = 18.\nEach part = 2862 ÷ 18 = 159.\nFirst share = 9 × 159 = 1431.\nSecond share = 9 × 159 = 1431.\nCheck: 1431+1431=2862.\nFinal answer: $1431$", "final_answer": "1431", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "663c1cd5146902ff", "problem_id": "2c3c11152bf84b2e", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Two friends split $320 in the ratio 6:2. How much does each receive?", "output": "1) Total ratio parts = 6+2 = 8.\n2) Each part = 320 ÷ 8 = 40.\n3) First share = 6 × 40 = 240.\n4) Second share = 2 × 40 = 80.\nFinal answer: $80$", "final_answer": "80", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e2ed5757c8f7273a", "problem_id": "1bea1c540a0aa966", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(- 2 x^{3} + x^{2} - x - 3\\right) \\cos{\\left(x + 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x e^{x}) = x e^{x}$.\n3) So $f'(x) = 2 x^{3} \\sin{\\left(x + 2 \\right)} - x^{2} \\sin{\\left(x + 2 \\right)} - 6 x^{2} \\cos{\\left(x + 2 \\right)} + x e^{x} + x \\sin{\\left(x + 2 \\right)} + 2 x \\cos{\\left(x + 2 \\right)} + e^{x} + 3 \\sin{\\left(x + 2 \\right)} - \\cos{\\left(x + 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x e^{x}) = x e^{x} + e^{x}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 1\\right) e^{x} - \\left(x \\left(x \\left(1 - 2 x\\right) - 1\\right) - 3\\right) \\sin{\\left(x + 2 \\right)} + \\left(- 6 x^{2} + 2 x - 1\\right) \\cos{\\left(x + 2 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 1\\right) e^{x} - \\left(x \\left(x \\left(1 - 2 x\\right) - 1\\right) - 3\\right) \\sin{\\left(x + 2 \\right)} + \\left(- 6 x^{2} + 2 x - 1\\right) \\cos{\\left(x + 2 \\right)} + e^{x}$", "final_answer": "\\left(x + 1\\right) e^{x} - \\left(x \\left(x \\left(1 - 2 x\\right) - 1\\right) - 3\\right) \\sin{\\left(x + 2 \\right)} + \\left(- 6 x^{2} + 2 x - 1\\right) \\cos{\\left(x + 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c4ce92ebf18560bf", "problem_id": "cdf9f801a0923420", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "You want 50 liters of a 32% salt solution. You have a 25% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 50−x liters is 40% solution.\n2) Salt equation: 25x + 40(50−x) = 32·50.\n3) (25-40)x = (32-40)·50 ⇒ x = 80/3.\n4) Then 50−x = 70/3.\nFinal answer: $80/3 L of 25% and 70/3 L of 40%$", "final_answer": "80/3 L of 25% and 70/3 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8b653dc3623906a9", "problem_id": "cdf9f801a0923420", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 50 liters of a 32% salt solution. You have a 25% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 50−x liters is 40% solution.\nSalt equation: 25x + 40(50−x) = 32·50.\n(25-40)x = (32-40)·50 ⇒ x = 80/3.\nThen 50−x = 70/3.\nCheck: 25·80/3 + 40·70/3 = 32·50.\nFinal answer: $80/3 L of 25% and 70/3 L of 40%$", "final_answer": "80/3 L of 25% and 70/3 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "70ba9a22b621ee74", "problem_id": "53e2d97f7f5d58dd", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{4} + x^{3} - 2 x^{2} + 4 x + 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x^{3}) = 3 x^{3}$.\n3) So $f'(x) = 15 x^{3} - 4 x + 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x^{3}) = 3 x^{2}$.\nTherefore the correct derivative is $f'(x) = 12 x^{3} + 3 x^{2} - 4 x + 4$.\nFinal answer: $12 x^{3} + 3 x^{2} - 4 x + 4$", "final_answer": "12 x^{3} + 3 x^{2} - 4 x + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "779029f246040c9a", "problem_id": "6f4a219767009a5a", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 30 miles per hour for 2 hours, then at 32 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 30 × 2 = 60 miles.\nSecond leg distance = 32 × 2 = 64 miles.\nTotal distance = 60 + 64 = 124 miles.\nCheck: adding both legs gives 124.\nFinal answer: $124$", "final_answer": "124", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6570383c743698f7", "problem_id": "00feadab028cc2a8", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=19$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=19 and k=10.\nCompute C(28,9) = 6906900.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $6906900$", "final_answer": "6906900", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b8d665994e3b1d89", "problem_id": "00feadab028cc2a8", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=19$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=19 and k=10.\n4) Compute C(28,9) = 6906900.\nFinal answer: $6906900$", "final_answer": "6906900", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8d0554bed51f2682", "problem_id": "3ec84c911d38e9d7", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡1 (mod 8), x≡5 (mod 7). Give the solution modulo 56.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 56.\n3) Compute the CRT solution to get x ≡ 33 (mod 56).\nFinal answer: $x\\equiv 33\\pmod{56}$", "final_answer": "x\\equiv 33\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d303e65d31d8452d", "problem_id": "3e52b4d472b1c18d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 \\cos{\\left(2 x \\right)}) = - 3 \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 3 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 \\cos{\\left(2 x \\right)}) = - 6 \\sin{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $- x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$", "final_answer": "- x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "fb9958ef947b0a9f", "problem_id": "7b99a4f5d0e51cd6", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$- 3 x - 2 y$ = $-3$\n$- 4 x - 4 y$ = $0$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $12 x + 8 y$ = $12$.\nMultiply the second equation by $-3$: $12 x + 12 y$ = $0$.\nSubtract to eliminate x: $- 4 y$ = $12$.\nSolve for y: y = $-3$.\nSubstitute back to find x: x = $3$.\nCheck: plug (x,y)=(3,-3) into both equations.\nFinal answer: $(x,y)=\\left(3, -3\\right)$", "final_answer": "(x,y)=\\left(3, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0123dbe6de919191", "problem_id": "7b99a4f5d0e51cd6", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$- 3 x - 2 y$ = $-3$\n$- 4 x - 4 y$ = $0$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $12 x + 8 y$ = $12$.\n3) Multiply the second equation by $-3$: $12 x + 12 y$ = $0$.\n4) Subtract to eliminate x: $- 4 y$ = $12$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, -3\\right)$", "final_answer": "(x,y)=\\left(3, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a9e51fd86655ba61", "problem_id": "b9c4519a6e9f14c8", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Worker A finishes a job in 19 hours and Worker B finishes it in 13 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/19 job/hour. B's rate = 1/13 job/hour.\n2) Combined rate = 1/19 + 1/13 = 32/247 job/hour.\n3) Time = 1 ÷ (combined rate) = 247/32 hours.\nFinal answer: $\\frac{247}{32}$", "final_answer": "\\frac{247}{32}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9e0df9f4a46b021c", "problem_id": "b9c4519a6e9f14c8", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Worker A finishes a job in 19 hours and Worker B finishes it in 13 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/19 job/hour. B's rate = 1/13 job/hour.\nCombined rate = 1/19 + 1/13 = 32/247 job/hour.\nTime = 1 ÷ (combined rate) = 247/32 hours.\nFinal answer: $\\frac{247}{32}$", "final_answer": "\\frac{247}{32}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8a653508261943a4", "problem_id": "e60d682246de5b16", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{17 x}{2} + \\frac{15}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$\\frac{17}{2}$, c=$\\frac{15}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{49}{4}$, so √Δ = $\\frac{7}{2}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{5}{4}$.\n7) x2 = $-3$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{5}{4}, -3\\}$", "final_answer": "x \\in \\{- \\frac{5}{4}, -3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3d156a89469058ce", "problem_id": "e60d682246de5b16", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{17 x}{2} + \\frac{15}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$\\frac{17}{2}$, c=$\\frac{15}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{49}{4}$, so √Δ = $\\frac{7}{2}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{5}{4}$.\nx2 = $-3$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{5}{4}, -3\\}$", "final_answer": "x \\in \\{- \\frac{5}{4}, -3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e31c7c96ffe047ad", "problem_id": "53b32fd47bd0580e", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 22 x + 48$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 22 x + 48$ = $2 \\left(x - 8\\right) \\left(x - 3\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $8$.\n5) From the second factor: x = $3$.\nFinal answer: $x \\in \\{8, 3\\}$", "final_answer": "x \\in \\{8, 3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b3b88ad68821254e", "problem_id": "53b32fd47bd0580e", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 22 x + 48$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} - 22 x + 48$ = $2 \\left(x - 8\\right) \\left(x - 3\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $8$.\nFrom the second factor: x = $3$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{8, 3\\}$", "final_answer": "x \\in \\{8, 3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b2b1195b72a5bfb", "problem_id": "aad8c5044a7d71cc", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "You want 50 liters of a 35% salt solution. You have a 20% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 50−x liters is 50% solution.\n2) Salt equation: 20x + 50(50−x) = 35·50.\n3) (20-50)x = (35-50)·50 ⇒ x = 25.\n4) Then 50−x = 25.\nFinal answer: $25 L of 20% and 25 L of 50%$", "final_answer": "25 L of 20% and 25 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c397fc5ff2aac23d", "problem_id": "aad8c5044a7d71cc", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 50 liters of a 35% salt solution. You have a 20% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 50−x liters is 50% solution.\nSalt equation: 20x + 50(50−x) = 35·50.\n(20-50)x = (35-50)·50 ⇒ x = 25.\nThen 50−x = 25.\nCheck: 20·25 + 50·25 = 35·50.\nFinal answer: $25 L of 20% and 25 L of 50%$", "final_answer": "25 L of 20% and 25 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f98cd50d5c1d25dd", "problem_id": "dfbda5979412d0af", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the congruences: x≡2 (mod 8), x≡6 (mod 11). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 50 (mod 88).\nFinal answer: $x\\equiv 50\\pmod{88}$", "final_answer": "x\\equiv 50\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "296d47511a5e0bcf", "problem_id": "dfbda5979412d0af", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 8), x≡6 (mod 11). Give the solution modulo 88.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 88.\nCompute the CRT solution to get x ≡ 50 (mod 88).\nCheck: 50 mod 8=2 and mod 11=6.\nFinal answer: $x\\equiv 50\\pmod{88}$", "final_answer": "x\\equiv 50\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eff3d072da3f1dae", "problem_id": "06439ed074353730", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} + x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 3$ at $x=0$.", "output": "1) Differentiate: f'(x) = $9 x^{2} + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 1$.\n2) Slope at x=0: m = f'(0) = $\\frac{3}{2}$.\n3) Point on curve: (0, f(0)) = (0, $-1$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -1 = \\frac{3}{2}(x − 0).\n6) Optionally expand: y = $\\frac{3 x}{2} - 1$.\nFinal answer: $y = \\frac{3\\cdot x}{2} - 1$", "final_answer": "y = \\frac{3\\cdot x}{2} - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bae50ca99e7b70c3", "problem_id": "06439ed074353730", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} + x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 3$ at $x=0$.", "output": "Differentiate: f'(x) = $9 x^{2} + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 1$.\nSlope at x=0: m = f'(0) = $\\frac{3}{2}$.\nPoint on curve: (0, f(0)) = (0, $-1$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -1 = \\frac{3}{2}(x − 0).\nOptionally expand: y = $\\frac{3 x}{2} - 1$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = \\frac{3\\cdot x}{2} - 1$", "final_answer": "y = \\frac{3\\cdot x}{2} - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b86432ac63c12987", "problem_id": "6745e709b7926c2e", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{2} + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 3 x^{2} + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}$.\nCompute an antiderivative: F(x) = $- x^{3} - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - x^{3} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - x^{3} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "40386f9774e56029", "problem_id": "6745e709b7926c2e", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{2} + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 3 x^{2} + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}$.\n3) Compute an antiderivative: F(x) = $- x^{3} - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C - x^{3} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - x^{3} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8ae4aad6a5850c1d", "problem_id": "42d9b81af7fabfc9", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} + 3$ at $x=-2$.", "output": "Differentiate: f'(x) = $- 6 x + 2 \\cos{\\left(x \\right)} + 3$.\nSlope at x=-2: m = f'(-2) = $2 \\cos{\\left(2 \\right)} + 15$.\nPoint on curve: (-2, f(-2)) = (-2, $-15 - 2 \\sin{\\left(2 \\right)}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -15 - 2 \\sin{\\left(2 \\right)} = 2 \\cos{\\left(2 \\right)} + 15(x − -2).\nOptionally expand: y = $2 x \\cos{\\left(2 \\right)} + 15 x - 2 \\sin{\\left(2 \\right)} + 4 \\cos{\\left(2 \\right)} + 15$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 15\\cdot x - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)} + 15$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 15\\cdot x - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)} + 15", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72784d869d6db218", "problem_id": "42d9b81af7fabfc9", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} + 3$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $- 6 x + 2 \\cos{\\left(x \\right)} + 3$.\n2) Slope at x=-2: m = f'(-2) = $2 \\cos{\\left(2 \\right)} + 15$.\n3) Point on curve: (-2, f(-2)) = (-2, $-15 - 2 \\sin{\\left(2 \\right)}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -15 - 2 \\sin{\\left(2 \\right)} = 2 \\cos{\\left(2 \\right)} + 15(x − -2).\n6) Optionally expand: y = $2 x \\cos{\\left(2 \\right)} + 15 x - 2 \\sin{\\left(2 \\right)} + 4 \\cos{\\left(2 \\right)} + 15$.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 15\\cdot x - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)} + 15$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 15\\cdot x - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)} + 15", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e6d977de51466ca8", "problem_id": "9674317a30440a64", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} - 3 x^{2} + 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $9 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 2$.\n2) Slope at x=-2: m = f'(-2) = $2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e} + 50$.\n3) Point on curve: (-2, f(-2)) = (-2, $-39 + \\cos{\\left(4 \\right)} + e^{-1}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -39 + \\cos{\\left(4 \\right)} + e^{-1} = 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e} + 50(x − -2).\n6) Optionally expand: y = $2 x \\sin{\\left(4 \\right)} + \\frac{x}{2 e} + 50 x + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 61$.\nFinal answer: $y = 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 50\\cdot x + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 61$", "final_answer": "y = 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 50\\cdot x + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 61", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "68c353ef223b8465", "problem_id": "9674317a30440a64", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} - 3 x^{2} + 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=-2$.", "output": "Differentiate: f'(x) = $9 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 2$.\nSlope at x=-2: m = f'(-2) = $2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e} + 50$.\nPoint on curve: (-2, f(-2)) = (-2, $-39 + \\cos{\\left(4 \\right)} + e^{-1}$).\nFinal answer: $y = 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 50\\cdot x + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 61$", "final_answer": "y = 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 50\\cdot x + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 61", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e0f5ebfc135ef487", "problem_id": "3a11b593ea2e9aa3", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} - 3 x^{3} + 2 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{3} + 2 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\n2) Evaluate at the bounds: F(\\pi) and F(0).\n3) Subtract: F(\\pi) − F(0) = $- \\frac{3 \\pi^{4}}{4} - 3 \\pi + 4 + \\frac{\\pi^{2}}{2} + \\frac{2 \\pi^{3}}{3}$.\n4) Simplify the result.\nFinal answer: $- \\frac{3\\cdot \\pi^{4}}{4} - 3\\cdot \\pi + 4 + \\frac{\\pi^{2}}{2} + \\frac{2\\cdot \\pi^{3}}{3}$", "final_answer": "- \\frac{3\\cdot \\pi^{4}}{4} - 3\\cdot \\pi + 4 + \\frac{\\pi^{2}}{2} + \\frac{2\\cdot \\pi^{3}}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f74d8184d1a401f", "problem_id": "3a11b593ea2e9aa3", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} - 3 x^{3} + 2 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{3} + 2 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\nEvaluate at the bounds: F(\\pi) and F(0).\nSubtract: F(\\pi) − F(0) = $- \\frac{3 \\pi^{4}}{4} - 3 \\pi + 4 + \\frac{\\pi^{2}}{2} + \\frac{2 \\pi^{3}}{3}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{3\\cdot \\pi^{4}}{4} - 3\\cdot \\pi + 4 + \\frac{\\pi^{2}}{2} + \\frac{2\\cdot \\pi^{3}}{3}$", "final_answer": "- \\frac{3\\cdot \\pi^{4}}{4} - 3\\cdot \\pi + 4 + \\frac{\\pi^{2}}{2} + \\frac{2\\cdot \\pi^{3}}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9df4dcc7bddcbc00", "problem_id": "fa07d8da2f3f1396", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the multiplicative inverse of 27 modulo 43.", "output": "1) We want x such that 27x ≡ 1 (mod 43).\n2) Use extended Euclid to find s,t with 27s + 43t = 1.\n3) One Bezout identity is: 27·(8) + 43·(-5) = 1.\n4) Reduce s modulo 43: x ≡ 8 (mod 43).\nFinal answer: $x\\equiv 8\\pmod{43}$", "final_answer": "x\\equiv 8\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9be0c1da549fb713", "problem_id": "fa07d8da2f3f1396", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the multiplicative inverse of 27 modulo 43.", "output": "We want x such that 27x ≡ 1 (mod 43).\nUse extended Euclid to find s,t with 27s + 43t = 1.\nOne Bezout identity is: 27·(8) + 43·(-5) = 1.\nReduce s modulo 43: x ≡ 8 (mod 43).\nCheck: 27·8 mod 43 = 1.\nFinal answer: $x\\equiv 8\\pmod{43}$", "final_answer": "x\\equiv 8\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c1f1dfa6b0b70f1d", "problem_id": "abf1af11eb0189e0", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "You want 52 liters of a 35% salt solution. You have a 10% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 52−x liters is 60% solution.\n2) Salt equation: 10x + 60(52−x) = 35·52.\n3) (10-60)x = (35-60)·52 ⇒ x = 26.\n4) Then 52−x = 26.\nFinal answer: $26 L of 10% and 26 L of 60%$", "final_answer": "26 L of 10% and 26 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bfc7209b3728f29d", "problem_id": "abf1af11eb0189e0", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "You want 52 liters of a 35% salt solution. You have a 10% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 52−x liters is 60% solution.\nSalt equation: 10x + 60(52−x) = 35·52.\n(10-60)x = (35-60)·52 ⇒ x = 26.\nFinal answer: $26 L of 10% and 26 L of 60%$", "final_answer": "26 L of 10% and 26 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e6120cb6ff8e5c5f", "problem_id": "6da6022af5bcbe72", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} 3 x^{3} - x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{3} - x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4$.\nEvaluate at the bounds: F(\\pi) and F(- \\pi).\nSubtract: F(\\pi) − F(- \\pi) = $\\frac{2 \\pi \\left(-12 - \\pi^{2}\\right)}{3}$.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(-12 - \\pi^{2}\\right)}{3}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(-12 - \\pi^{2}\\right)}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "69e2368c3e7b1d38", "problem_id": "6da6022af5bcbe72", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} 3 x^{3} - x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{3} - x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4$.\nEvaluate at the bounds: F(\\pi) and F(- \\pi).\nSubtract: F(\\pi) − F(- \\pi) = $\\frac{2 \\pi \\left(-12 - \\pi^{2}\\right)}{3}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(-12 - \\pi^{2}\\right)}{3}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(-12 - \\pi^{2}\\right)}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7f741cf3383c7bcd", "problem_id": "895a18a809d9dc53", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A cyclist rides at 77 miles per hour for 1/2 hours, then at 21 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 77 × 1/2 = 77/2 miles.\n2) Second leg distance = 21 × 3 = 63 miles.\n3) Total distance = 77/2 + 63 = 203/2 miles.\nFinal answer: $\\frac{203}{2}$", "final_answer": "\\frac{203}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "069ff4ca91d091f7", "problem_id": "895a18a809d9dc53", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A cyclist rides at 77 miles per hour for 1/2 hours, then at 21 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 77 × 1/2 = 77/2 miles.\nSecond leg distance = 21 × 3 = 63 miles.\nTotal distance = 77/2 + 63 = 203/2 miles.\nCheck: adding both legs gives 203/2.\nFinal answer: $\\frac{203}{2}$", "final_answer": "\\frac{203}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8126067d7aebf45b", "problem_id": "00063606ce0d8280", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 11 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=4, p=$\\frac{3}{8}$ and simplify.\n4) Here C(11,4) = 330.\nFinal answer: $\\frac{1044140625}{4294967296}$", "final_answer": "\\frac{1044140625}{4294967296}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a70568a4c086eba", "problem_id": "aae1fb5eaa1aecc6", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} - 3 x^{2} + 11 x - 41$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} + 10 x - 41.\n3) Then the limit is -129.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: -133.\nFinal answer: $-133$", "final_answer": "-133", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "899cf12bf3a45e07", "problem_id": "54c785f310737750", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡5 (mod 7), x≡3 (mod 5). Give the solution modulo 35.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 35.\n3) Compute the CRT solution to get x ≡ 33 (mod 35).\nFinal answer: $x\\equiv 33\\pmod{35}$", "final_answer": "x\\equiv 33\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d13834b5056fe116", "problem_id": "54c785f310737750", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡5 (mod 7), x≡3 (mod 5). Give the solution modulo 35.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 35.\nCompute the CRT solution to get x ≡ 33 (mod 35).\nCheck: 33 mod 7=5 and mod 5=3.\nFinal answer: $x\\equiv 33\\pmod{35}$", "final_answer": "x\\equiv 33\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "39b37decb739a48e", "problem_id": "d2a9e7285b29ec92", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A store sells a backpack for $262. It is discounted by 20%. You buy 8 backpacks. What is the total cost?", "output": "Discounted price = 262 × (1 − 20/100) = 262 × 80/100 = 1048/5.\nSubtotal = 8 × 1048/5 = 8384/5.\nCheck: recomputing the formula yields 8384/5.\nFinal answer: $\\frac{8384}{5}$", "final_answer": "\\frac{8384}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f7a90d804b463c74", "problem_id": "d2a9e7285b29ec92", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A store sells a backpack for $262. It is discounted by 20%. You buy 8 backpacks. What is the total cost?", "output": "1) Discounted price = 262 × (1 − 20/100) = 262 × 80/100 = 1048/5.\n2) Subtotal = 8 × 1048/5 = 8384/5.\nFinal answer: $\\frac{8384}{5}$", "final_answer": "\\frac{8384}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ce5f4f61293e094e", "problem_id": "437f052789f66e3d", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + \\frac{x}{3} - \\frac{4}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{1}{3}$, c=$- \\frac{4}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{1, - \\frac{4}{3}\\}$", "final_answer": "x \\in \\{1, - \\frac{4}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ad8ab6d0197cc52", "problem_id": "437f052789f66e3d", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + \\frac{x}{3} - \\frac{4}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{1}{3}$, c=$- \\frac{4}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{49}{9}$, so √Δ = $\\frac{7}{3}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $1$.\n7) x2 = $- \\frac{4}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{1, - \\frac{4}{3}\\}$", "final_answer": "x \\in \\{1, - \\frac{4}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3676a5c7ff72be54", "problem_id": "05d64cba78331910", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the multiplicative inverse of 32 modulo 37.", "output": "1) We want x such that 32x ≡ 1 (mod 37).\n2) Use extended Euclid to find s,t with 32s + 37t = 1.\n3) One Bezout identity is: 32·(-15) + 37·(13) = 1.\n4) Reduce s modulo 37: x ≡ 22 (mod 37).\nFinal answer: $x\\equiv 22\\pmod{37}$", "final_answer": "x\\equiv 22\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a61cafba83ff6582", "problem_id": "05d64cba78331910", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 32 modulo 37.", "output": "We want x such that 32x ≡ 1 (mod 37).\nUse extended Euclid to find s,t with 32s + 37t = 1.\nOne Bezout identity is: 32·(-15) + 37·(13) = 1.\nReduce s modulo 37: x ≡ 22 (mod 37).\nCheck: 32·22 mod 37 = 1.\nFinal answer: $x\\equiv 22\\pmod{37}$", "final_answer": "x\\equiv 22\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c6438e46f6530793", "problem_id": "b834c3003c0c1aab", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - x^{2} + 4 x - 9$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} + 3 x - 9.\n3) Then the limit is -19.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -21.\nFinal answer: $-21$", "final_answer": "-21", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "cbab430e47d7b100", "problem_id": "98bee1addb9fe092", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$x + 2 y$ = $-3$\n$3 x - 2 y$ = $-9$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $3$: $3 x + 6 y$ = $-9$.\n3) Multiply the second equation by $1$: $3 x - 2 y$ = $-9$.\n4) Subtract to eliminate x: $8 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, 0\\right)$", "final_answer": "(x,y)=\\left(-3, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b599904357e3a26c", "problem_id": "20fe04db216afe36", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 6 x - 20$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 6 x - 20$ = $2 \\left(x - 5\\right) \\left(x + 2\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $5$.\n5) From the second factor: x = $-2$.\nFinal answer: $x \\in \\{5, -2\\}$", "final_answer": "x \\in \\{5, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "309ea4e47cb20877", "problem_id": "20fe04db216afe36", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 6 x - 20$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} - 6 x - 20$ = $2 \\left(x - 5\\right) \\left(x + 2\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $5$.\nFrom the second factor: x = $-2$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{5, -2\\}$", "final_answer": "x \\in \\{5, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9fb9f7c31886beb6", "problem_id": "155b6a70ed236fbb", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} - x^{3} + 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{3} + 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $\\frac{\\pi \\left(-12 + \\pi^{2}\\right)}{4}$.\n4) Simplify the result.\nFinal answer: $\\frac{\\pi\\cdot \\left(-12 + \\pi^{2}\\right)}{4}$", "final_answer": "\\frac{\\pi\\cdot \\left(-12 + \\pi^{2}\\right)}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "10050f596bfba97e", "problem_id": "155b6a70ed236fbb", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} - x^{3} + 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{3} + 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $\\frac{\\pi \\left(-12 + \\pi^{2}\\right)}{4}$.\nFinal answer: $\\frac{\\pi\\cdot \\left(-12 + \\pi^{2}\\right)}{4}$", "final_answer": "\\frac{\\pi\\cdot \\left(-12 + \\pi^{2}\\right)}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "700ad70e8e12d2f9", "problem_id": "0df1d684f5a93113", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Worker A finishes a job in 21 hours and Worker B finishes it in 15 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/21 job/hour. B's rate = 1/15 job/hour.\nCombined rate = 1/21 + 1/15 = 4/35 job/hour.\nTime = 1 ÷ (combined rate) = 35/4 hours.\nFinal answer: $\\frac{35}{4}$", "final_answer": "\\frac{35}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e1a76603fe58ab47", "problem_id": "0df1d684f5a93113", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Worker A finishes a job in 21 hours and Worker B finishes it in 15 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/21 job/hour. B's rate = 1/15 job/hour.\n2) Combined rate = 1/21 + 1/15 = 4/35 job/hour.\n3) Time = 1 ÷ (combined rate) = 35/4 hours.\nFinal answer: $\\frac{35}{4}$", "final_answer": "\\frac{35}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "99ab31b0e056c0b8", "problem_id": "f0a07ba2d4b0ffe0", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡1 (mod 11), x≡1 (mod 8). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 1 (mod 88).\nFinal answer: $x\\equiv 1\\pmod{88}$", "final_answer": "x\\equiv 1\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "054ba9b11a39ed77", "problem_id": "f0a07ba2d4b0ffe0", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡1 (mod 11), x≡1 (mod 8). Give the solution modulo 88.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 88.\nCompute the CRT solution to get x ≡ 1 (mod 88).\nCheck: 1 mod 11=1 and mod 8=1.\nFinal answer: $x\\equiv 1\\pmod{88}$", "final_answer": "x\\equiv 1\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6401f8090a348dc9", "problem_id": "fe88d3ab437c78a9", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} - x^{2} - 3 x - 3$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{3} - 2 x^{2} - 2$ and a = 1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 3 x^{2} - 4 x$.\nEvaluate at a: p'(1) = $-7$.\nCheck: the identity gives $-7$.\nFinal answer: $-7$", "final_answer": "-7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4716fb1e3de49bc0", "problem_id": "b7addf236b39dd38", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} - x + 2$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(2 x^{2}) = 4 x^{2}$.\n3) So $f'(x) = 4 x^{2} - 1$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(2 x^{2}) = 4 x$.\nTherefore the correct derivative is $f'(x) = 4 x - 1$.\nFinal answer: $4 x - 1$", "final_answer": "4 x - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "181ec9d5bd8900af", "problem_id": "b45600416317fb03", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "You want 26 liters of a 35% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 26−x liters is 50% solution.\n2) Salt equation: 25x + 50(26−x) = 35·26.\n3) (25-50)x = (35-50)·26 ⇒ x = 78/5.\n4) Then 26−x = 52/5.\nFinal answer: $78/5 L of 25% and 52/5 L of 50%$", "final_answer": "78/5 L of 25% and 52/5 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "787dc9c49d802177", "problem_id": "b45600416317fb03", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 26 liters of a 35% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 26−x liters is 50% solution.\nSalt equation: 25x + 50(26−x) = 35·26.\n(25-50)x = (35-50)·26 ⇒ x = 78/5.\nThen 26−x = 52/5.\nCheck: 25·78/5 + 50·52/5 = 35·26.\nFinal answer: $78/5 L of 25% and 52/5 L of 50%$", "final_answer": "78/5 L of 25% and 52/5 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "73dfed43af93c11c", "problem_id": "eea40a22a7873d55", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_8=26$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_8=10.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=10 and k=8.\n4) Compute C(17,7) = 19448.\nFinal answer: $19448$", "final_answer": "19448", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d6b278945b14fa6a", "problem_id": "ec3295382b14f1e2", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{3} + 2 x^{2} + 3 x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{3 x + 4}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 4}) = \\frac{4 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 4}$.\n3) So $f'(x) = - \\frac{36 x^{3}}{18 x^{2} + 48 x + 32} + \\frac{9 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{18 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} + 48 x + 32} - \\frac{60 x^{2}}{18 x^{2} + 48 x + 32} + \\frac{24 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{48 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{32 x}{18 x^{2} + 48 x + 32} + \\frac{16 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{32 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{18}{18 x^{2} + 48 x + 32} + \\frac{12 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{2} + 24 x + 16} + \\frac{4 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 4} - \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 4} - \\frac{4 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 4}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 4}) = \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 4} + \\frac{4 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 4} - \\frac{12 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(3 x + 4\\right)^{2}}$.\nTherefore the correct derivative is $f'(x) = \\frac{18 x^{3} - 12 x^{2} - 18 x + \\left(3 x + 4\\right) \\left(- 18 x^{2} + 8 x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 6 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6}{2 \\left(3 x + 4\\right)^{2}}$.\nFinal answer: $\\frac{18 x^{3} - 12 x^{2} - 18 x + \\left(3 x + 4\\right) \\left(- 18 x^{2} + 8 x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 6 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6}{2 \\left(3 x + 4\\right)^{2}}$", "final_answer": "\\frac{18 x^{3} - 12 x^{2} - 18 x + \\left(3 x + 4\\right) \\left(- 18 x^{2} + 8 x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 6 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6}{2 \\left(3 x + 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "926c663da7304abb", "problem_id": "9543f6370bdca2d5", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A store sells a backpack for $81. It is discounted by 10%. You buy 5 backpacks. What is the total cost?", "output": "1) Discounted price = 81 × (1 − 10/100) = 81 × 90/100 = 729/10.\n2) Subtotal = 5 × 729/10 = 729/2.\nFinal answer: $\\frac{729}{2}$", "final_answer": "\\frac{729}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b8ffefda4769ebbf", "problem_id": "9543f6370bdca2d5", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A store sells a backpack for $81. It is discounted by 10%. You buy 5 backpacks. What is the total cost?", "output": "Discounted price = 81 × (1 − 10/100) = 81 × 90/100 = 729/10.\nSubtotal = 5 × 729/10 = 729/2.\nCheck: recomputing the formula yields 729/2.\nFinal answer: $\\frac{729}{2}$", "final_answer": "\\frac{729}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d41d37f5bdfeef34", "problem_id": "dfd3e4ed467f6f22", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 11 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=6, p=$\\frac{3}{8}$ and simplify.\n4) Here C(11,6) = 462.\nFinal answer: $\\frac{526246875}{4294967296}$", "final_answer": "\\frac{526246875}{4294967296}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6cda872eb9d5c8ef", "problem_id": "dfd3e4ed467f6f22", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 11 times, what is the probability of exactly 6 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=11, k=6, p=$\\frac{3}{8}$ and simplify.\nHere C(11,6) = 462.\nCheck: expression simplifies to $\\frac{526246875}{4294967296}$.\nFinal answer: $\\frac{526246875}{4294967296}$", "final_answer": "\\frac{526246875}{4294967296}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f0463fbbd5811e9a", "problem_id": "3e16788e7a868033", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A cyclist rides at 89 miles per hour for 3/2 hours, then at 62 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 89 × 3/2 = 267/2 miles.\nSecond leg distance = 62 × 3 = 186 miles.\nTotal distance = 267/2 + 186 = 639/2 miles.\nCheck: adding both legs gives 639/2.\nFinal answer: $\\frac{639}{2}$", "final_answer": "\\frac{639}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3a04c71827843ef5", "problem_id": "3e16788e7a868033", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "A cyclist rides at 89 miles per hour for 3/2 hours, then at 62 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 89 × 3/2 = 267/2 miles.\nSecond leg distance = 62 × 3 = 186 miles.\nTotal distance = 267/2 + 186 = 639/2 miles.\nFinal answer: $\\frac{639}{2}$", "final_answer": "\\frac{639}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7cd37d2af479e37b", "problem_id": "cfda32185878e889", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the equation of the tangent line to $y=- x^{2} + x + 2 \\sin{\\left(x \\right)} - 1$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 2 x + 2 \\cos{\\left(x \\right)} + 1$.\n2) Slope at x=2: m = f'(2) = $-3 + 2 \\cos{\\left(2 \\right)}$.\n3) Point on curve: (2, f(2)) = (2, $-3 + 2 \\sin{\\left(2 \\right)}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -3 + 2 \\sin{\\left(2 \\right)} = -3 + 2 \\cos{\\left(2 \\right)}(x − 2).\n6) Optionally expand: y = $- 3 x + 2 x \\cos{\\left(2 \\right)} - 4 \\cos{\\left(2 \\right)} + 2 \\sin{\\left(2 \\right)} + 3$.\nFinal answer: $y = - 3\\cdot x + 2\\cdot x\\cdot \\cos{\\left(2 \\right)} - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)} + 3$", "final_answer": "y = - 3\\cdot x + 2\\cdot x\\cdot \\cos{\\left(2 \\right)} - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)} + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "23f413a543124ffe", "problem_id": "fce51105bf7cf726", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{3} \\sin{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 x^{2} \\sin{\\left(x \\right)}) = 3 x^{2} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = 3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + 3 x^{2} \\cos{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 x^{2} \\sin{\\left(x \\right)}) = 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$", "final_answer": "3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8d4f43015b6309ad", "problem_id": "fce51105bf7cf726", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{3} \\sin{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $3 x^{3} \\sin{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} + 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} + 3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x\\cdot \\sin{\\left(x \\right)} + 3\\cdot x\\cdot \\cos{\\left(x \\right)} + 3\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} + 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} + 3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x\\cdot \\sin{\\left(x \\right)} + 3\\cdot x\\cdot \\cos{\\left(x \\right)} + 3\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f508b6e3152f5e81", "problem_id": "fce51105bf7cf726", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{3} \\sin{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $3 x^{3} \\sin{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} + 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} + 3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x\\cdot \\sin{\\left(x \\right)} + 3\\cdot x\\cdot \\cos{\\left(x \\right)} + 3\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} + 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} + 3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x\\cdot \\sin{\\left(x \\right)} + 3\\cdot x\\cdot \\cos{\\left(x \\right)} + 3\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0f25836a3718676f", "problem_id": "9226cf6badd482d3", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(- x^{2} - x\\right) \\cos{\\left(3 x^{2} - x + 3 \\right)}$.", "output": "1) Let f(x) = $\\left(x + 1\\right) e^{x} + \\left(- x^{2} - x\\right) \\cos{\\left(3 x^{2} - x + 3 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- x \\left(- x - 1\\right) \\left(6 x - 1\\right) \\sin{\\left(3 x^{2} - x + 3 \\right)} + \\left(- 2 x - 1\\right) \\cos{\\left(3 x^{2} - x + 3 \\right)} + \\left(x + 1\\right) e^{x} + e^{x}$.\nFinal answer: $- x\\cdot \\left(- x - 1\\right)\\cdot \\left(6\\cdot x - 1\\right)\\cdot \\sin{\\left(3\\cdot x^{2} - x + 3 \\right)} + \\left(- 2\\cdot x - 1\\right)\\cdot \\cos{\\left(3\\cdot x^{2} - x + 3 \\right)} + \\left(x + 1\\right)\\cdot e^{x} + e^{x}$", "final_answer": "- x\\cdot \\left(- x - 1\\right)\\cdot \\left(6\\cdot x - 1\\right)\\cdot \\sin{\\left(3\\cdot x^{2} - x + 3 \\right)} + \\left(- 2\\cdot x - 1\\right)\\cdot \\cos{\\left(3\\cdot x^{2} - x + 3 \\right)} + \\left(x + 1\\right)\\cdot e^{x} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a6337f388a82b24c", "problem_id": "9226cf6badd482d3", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(- x^{2} - x\\right) \\cos{\\left(3 x^{2} - x + 3 \\right)}$.", "output": "Let f(x) = $\\left(x + 1\\right) e^{x} + \\left(- x^{2} - x\\right) \\cos{\\left(3 x^{2} - x + 3 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- x \\left(- x - 1\\right) \\left(6 x - 1\\right) \\sin{\\left(3 x^{2} - x + 3 \\right)} + \\left(- 2 x - 1\\right) \\cos{\\left(3 x^{2} - x + 3 \\right)} + \\left(x + 1\\right) e^{x} + e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- x\\cdot \\left(- x - 1\\right)\\cdot \\left(6\\cdot x - 1\\right)\\cdot \\sin{\\left(3\\cdot x^{2} - x + 3 \\right)} + \\left(- 2\\cdot x - 1\\right)\\cdot \\cos{\\left(3\\cdot x^{2} - x + 3 \\right)} + \\left(x + 1\\right)\\cdot e^{x} + e^{x}$", "final_answer": "- x\\cdot \\left(- x - 1\\right)\\cdot \\left(6\\cdot x - 1\\right)\\cdot \\sin{\\left(3\\cdot x^{2} - x + 3 \\right)} + \\left(- 2\\cdot x - 1\\right)\\cdot \\cos{\\left(3\\cdot x^{2} - x + 3 \\right)} + \\left(x + 1\\right)\\cdot e^{x} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4ce9b4ae2b523774", "problem_id": "08c76d373dfd4e19", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡4 (mod 5), x≡8 (mod 11). Give the solution modulo 55.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 55.\nCompute the CRT solution to get x ≡ 19 (mod 55).\nCheck: 19 mod 5=4 and mod 11=8.\nFinal answer: $x\\equiv 19\\pmod{55}$", "final_answer": "x\\equiv 19\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2c8ddb6a8e8c4971", "problem_id": "08c76d373dfd4e19", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the congruences: x≡4 (mod 5), x≡8 (mod 11). Give the solution modulo 55.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 55.\n3) Compute the CRT solution to get x ≡ 19 (mod 55).\nFinal answer: $x\\equiv 19\\pmod{55}$", "final_answer": "x\\equiv 19\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8a2438143b6eacb3", "problem_id": "245a2ddbe6b2b2ae", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Two friends split $576 in the ratio 5:7. How much does each receive?", "output": "1) Total ratio parts = 5+7 = 12.\n2) Each part = 576 ÷ 12 = 48.\n3) First share = 5 × 48 = 240.\n4) Second share = 7 × 48 = 336.\nFinal answer: $336$", "final_answer": "336", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "30f909650d79152d", "problem_id": "c94653b55cacc193", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} \\sin{\\left(x \\right)} - x^{2} \\sin{\\left(x \\right)} - 3 x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x^{2} \\sin{\\left(x \\right)}) = - x^{2} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x^{2} \\sin{\\left(x \\right)}) = - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$", "final_answer": "- 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7c7a19a4ec0f7822", "problem_id": "57c0c15cccbdec53", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 22 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=22, k=3, p=$\\frac{4}{5}$ and simplify.\n4) Here C(22,3) = 1540.\nFinal answer: $\\frac{19712}{476837158203125}$", "final_answer": "\\frac{19712}{476837158203125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0f11329d867f5ab6", "problem_id": "57c0c15cccbdec53", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 22 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=22, k=3, p=$\\frac{4}{5}$ and simplify.\nFinal answer: $\\frac{19712}{476837158203125}$", "final_answer": "\\frac{19712}{476837158203125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d771742c581962a6", "problem_id": "f37b06ed3c377b52", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$- 6 y$ = $0$\n$7 x - 8 y$ = $42$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $7$: $- 42 y$ = $0$.\n3) Multiply the second equation by $0$: $0$ = $0$.\n4) Subtract to eliminate x: $- 42 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $6$.\nFinal answer: $(x,y)=\\left(6, 0\\right)$", "final_answer": "(x,y)=\\left(6, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "51ea5c2fcb4d4dbf", "problem_id": "6ca4b9c0c325f74b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} 15 - 3 x$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to 15 - 4 x.\n3) Then the limit is 31.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: 27.\nFinal answer: $27$", "final_answer": "27", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8f5b1fba7d268a6f", "problem_id": "945b9a93eafb0eba", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A store sells a backpack for $106. It is discounted by 40%. You buy 8 backpacks. What is the total cost?", "output": "Discounted price = 106 × (1 − 40/100) = 106 × 60/100 = 318/5.\nSubtotal = 8 × 318/5 = 2544/5.\nCheck: recomputing the formula yields 2544/5.\nFinal answer: $\\frac{2544}{5}$", "final_answer": "\\frac{2544}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5539374075b8ac6d", "problem_id": "945b9a93eafb0eba", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A store sells a backpack for $106. It is discounted by 40%. You buy 8 backpacks. What is the total cost?", "output": "1) Discounted price = 106 × (1 − 40/100) = 106 × 60/100 = 318/5.\n2) Subtotal = 8 × 318/5 = 2544/5.\nFinal answer: $\\frac{2544}{5}$", "final_answer": "\\frac{2544}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5557ea81d27e7d0b", "problem_id": "e17f9bbada516a88", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Worker A finishes a job in 9 hours and Worker B finishes it in 7 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/9 job/hour. B's rate = 1/7 job/hour.\n2) Combined rate = 1/9 + 1/7 = 16/63 job/hour.\n3) Time = 1 ÷ (combined rate) = 63/16 hours.\nFinal answer: $\\frac{63}{16}$", "final_answer": "\\frac{63}{16}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "54b11158e7117096", "problem_id": "e17f9bbada516a88", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Worker A finishes a job in 9 hours and Worker B finishes it in 7 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/9 job/hour. B's rate = 1/7 job/hour.\nCombined rate = 1/9 + 1/7 = 16/63 job/hour.\nTime = 1 ÷ (combined rate) = 63/16 hours.\nCheck: (combined rate)×(time) = 16/63×63/16 = 1.\nFinal answer: $\\frac{63}{16}$", "final_answer": "\\frac{63}{16}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5c702f9e71795f4c", "problem_id": "57f112fa9f2a64e6", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{10} x + \\frac{x}{15} - \\frac{2 \\sqrt{10}}{5} - \\frac{2}{15}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{1}{15} - \\sqrt{10}$, c=$- \\frac{2 \\sqrt{10}}{5} - \\frac{2}{15}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{22 \\sqrt{10}}{15} + \\frac{2371}{225}$, so √Δ = $\\sqrt{\\frac{22 \\sqrt{10}}{15} + \\frac{2371}{225}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{1}{30} + \\frac{\\sqrt{10}}{2} + \\frac{\\sqrt{330 \\sqrt{10} + 2371}}{30}$.\n7) x2 = $- \\frac{\\sqrt{330 \\sqrt{10} + 2371}}{30} - \\frac{1}{30} + \\frac{\\sqrt{10}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{1}{30} + \\frac{\\sqrt{10}}{2} + \\frac{\\sqrt{330\\cdot \\sqrt{10} + 2371}}{30}, - \\frac{\\sqrt{330\\cdot \\sqrt{10} + 2371}}{30} - \\frac{1}{30} + \\frac{\\sqrt{10}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{1}{30} + \\frac{\\sqrt{10}}{2} + \\frac{\\sqrt{330\\cdot \\sqrt{10} + 2371}}{30}, - \\frac{\\sqrt{330\\cdot \\sqrt{10} + 2371}}{30} - \\frac{1}{30} + \\frac{\\sqrt{10}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c4454a32403e6242", "problem_id": "57f112fa9f2a64e6", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{10} x + \\frac{x}{15} - \\frac{2 \\sqrt{10}}{5} - \\frac{2}{15}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{1}{15} - \\sqrt{10}$, c=$- \\frac{2 \\sqrt{10}}{5} - \\frac{2}{15}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{22 \\sqrt{10}}{15} + \\frac{2371}{225}$, so √Δ = $\\sqrt{\\frac{22 \\sqrt{10}}{15} + \\frac{2371}{225}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{1}{30} + \\frac{\\sqrt{10}}{2} + \\frac{\\sqrt{330 \\sqrt{10} + 2371}}{30}$.\nx2 = $- \\frac{\\sqrt{330 \\sqrt{10} + 2371}}{30} - \\frac{1}{30} + \\frac{\\sqrt{10}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{1}{30} + \\frac{\\sqrt{10}}{2} + \\frac{\\sqrt{330\\cdot \\sqrt{10} + 2371}}{30}, - \\frac{\\sqrt{330\\cdot \\sqrt{10} + 2371}}{30} - \\frac{1}{30} + \\frac{\\sqrt{10}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{1}{30} + \\frac{\\sqrt{10}}{2} + \\frac{\\sqrt{330\\cdot \\sqrt{10} + 2371}}{30}, - \\frac{\\sqrt{330\\cdot \\sqrt{10} + 2371}}{30} - \\frac{1}{30} + \\frac{\\sqrt{10}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cef34a6ee6b7d105", "problem_id": "c2e16a9e0f158a7e", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{5}$. If it is flipped 9 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=9, k=6, p=$\\frac{3}{5}$ and simplify.\n4) Here C(9,6) = 84.\nFinal answer: $\\frac{489888}{1953125}$", "final_answer": "\\frac{489888}{1953125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "70b73b1aaf759651", "problem_id": "c2e16a9e0f158a7e", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{5}$. If it is flipped 9 times, what is the probability of exactly 6 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=9, k=6, p=$\\frac{3}{5}$ and simplify.\nHere C(9,6) = 84.\nCheck: expression simplifies to $\\frac{489888}{1953125}$.\nFinal answer: $\\frac{489888}{1953125}$", "final_answer": "\\frac{489888}{1953125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7b51f1302463125d", "problem_id": "59a75b8a0e9d1131", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 2 x - \\frac{15}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$2$, c=$- \\frac{15}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $64$, so √Δ = $8$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{3}{2}$.\n7) x2 = $- \\frac{5}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{3}{2}, - \\frac{5}{2}\\}$", "final_answer": "x \\in \\{\\frac{3}{2}, - \\frac{5}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7bc645cf2e5c5760", "problem_id": "59a75b8a0e9d1131", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 2 x - \\frac{15}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$2$, c=$- \\frac{15}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $64$, so √Δ = $8$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{3}{2}$.\nx2 = $- \\frac{5}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{3}{2}, - \\frac{5}{2}\\}$", "final_answer": "x \\in \\{\\frac{3}{2}, - \\frac{5}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1483475e20f347c7", "problem_id": "8c663cac98cda5da", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡9 (mod 11), x≡2 (mod 5). Give the solution modulo 55.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 55.\nCompute the CRT solution to get x ≡ 42 (mod 55).\nCheck: 42 mod 11=9 and mod 5=2.\nFinal answer: $x\\equiv 42\\pmod{55}$", "final_answer": "x\\equiv 42\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "75f7ab44be9583ce", "problem_id": "954dc9b1336df2a1", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(2 x^{2} - 1\\right) \\cos{\\left(3 x^{2} - 3 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\n3) So $f'(x) = - 12 x^{3} \\sin{\\left(3 x^{2} - 3 x \\right)} + 6 x^{2} \\sin{\\left(3 x^{2} - 3 x \\right)} + x e^{x} + 6 x \\sin{\\left(3 x^{2} - 3 x \\right)} + 4 x \\cos{\\left(3 x^{2} - 3 x \\right)} + 3 e^{x} - 3 \\sin{\\left(3 x^{2} - 3 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = 4 x \\cos{\\left(3 x^{2} - 3 x \\right)} + \\left(x + 2\\right) e^{x} - 3 \\left(2 x - 1\\right) \\left(2 x^{2} - 1\\right) \\sin{\\left(3 x^{2} - 3 x \\right)} + e^{x}$.\nFinal answer: $4 x \\cos{\\left(3 x^{2} - 3 x \\right)} + \\left(x + 2\\right) e^{x} - 3 \\left(2 x - 1\\right) \\left(2 x^{2} - 1\\right) \\sin{\\left(3 x^{2} - 3 x \\right)} + e^{x}$", "final_answer": "4 x \\cos{\\left(3 x^{2} - 3 x \\right)} + \\left(x + 2\\right) e^{x} - 3 \\left(2 x - 1\\right) \\left(2 x^{2} - 1\\right) \\sin{\\left(3 x^{2} - 3 x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "4244aef22565d826", "problem_id": "8c6027dbc851e8b9", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "You want 54 liters of a 30% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 54−x liters is 70% solution.\n2) Salt equation: 25x + 70(54−x) = 30·54.\n3) (25-70)x = (30-70)·54 ⇒ x = 48.\n4) Then 54−x = 6.\nFinal answer: $48 L of 25% and 6 L of 70%$", "final_answer": "48 L of 25% and 6 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "13b221bc9defd6bd", "problem_id": "8c6027dbc851e8b9", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "You want 54 liters of a 30% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 54−x liters is 70% solution.\nSalt equation: 25x + 70(54−x) = 30·54.\n(25-70)x = (30-70)·54 ⇒ x = 48.\nFinal answer: $48 L of 25% and 6 L of 70%$", "final_answer": "48 L of 25% and 6 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "af0bbc70fc461ef2", "problem_id": "9d2dec6f2cd355d4", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 12 times, what is the probability of exactly 7 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=12, k=7, p=$\\frac{1}{2}$ and simplify.\nHere C(12,7) = 792.\nCheck: expression simplifies to $\\frac{99}{512}$.\nFinal answer: $\\frac{99}{512}$", "final_answer": "\\frac{99}{512}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c9db1458f5b9d31c", "problem_id": "9d2dec6f2cd355d4", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 12 times, what is the probability of exactly 7 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=7, p=$\\frac{1}{2}$ and simplify.\n4) Here C(12,7) = 792.\nFinal answer: $\\frac{99}{512}$", "final_answer": "\\frac{99}{512}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "453260d48d1316b5", "problem_id": "74a7f8ac37ad977d", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 2$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $9 x^{2} + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\n2) Slope at x=-2: m = f'(-2) = $2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e} + 34$.\n3) Point on curve: (-2, f(-2)) = (-2, $-18 + \\cos{\\left(4 \\right)} + e^{-1}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -18 + \\cos{\\left(4 \\right)} + e^{-1} = 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e} + 34(x − -2).\n6) Optionally expand: y = $2 x \\sin{\\left(4 \\right)} + \\frac{x}{2 e} + 34 x + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 50$.\nFinal answer: $y = 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 34\\cdot x + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 50$", "final_answer": "y = 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 34\\cdot x + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 50", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "af08e22514863887", "problem_id": "2f98b98631d9deb2", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 2 x^{3} - 5 x^{2} + 2 x - 5$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} - 5 x^{2} + x - 5.\n3) Then the limit is -13.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: -14.\nFinal answer: $-14$", "final_answer": "-14", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b37afa56756cb9db", "problem_id": "57d5a9f83e8b38e6", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A store sells a backpack for $73. It is discounted by 20%. You buy 12 backpacks. What is the total cost?", "output": "1) Discounted price = 73 × (1 − 20/100) = 73 × 80/100 = 292/5.\n2) Subtotal = 12 × 292/5 = 3504/5.\nFinal answer: $\\frac{3504}{5}$", "final_answer": "\\frac{3504}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "236b7c2e8863f2a6", "problem_id": "57d5a9f83e8b38e6", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $73. It is discounted by 20%. You buy 12 backpacks. What is the total cost?", "output": "Discounted price = 73 × (1 − 20/100) = 73 × 80/100 = 292/5.\nSubtotal = 12 × 292/5 = 3504/5.\nCheck: recomputing the formula yields 3504/5.\nFinal answer: $\\frac{3504}{5}$", "final_answer": "\\frac{3504}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ef6d2887be45b35f", "problem_id": "34496964d84262b4", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(3 x^{3} - 3 x^{2} + x + 1\\right) \\cos{\\left(x^{2} - x + 1 \\right)}$.", "output": "1) Let f(x) = $\\left(x + 1\\right) e^{x} + \\left(3 x^{3} - 3 x^{2} + x + 1\\right) \\cos{\\left(x^{2} - x + 1 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\left(x + 1\\right) e^{x} - \\left(2 x - 1\\right) \\left(3 x^{3} - 3 x^{2} + x + 1\\right) \\sin{\\left(x^{2} - x + 1 \\right)} + \\left(9 x^{2} - 6 x + 1\\right) \\cos{\\left(x^{2} - x + 1 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 1\\right)\\cdot e^{x} - \\left(2\\cdot x - 1\\right)\\cdot \\left(3\\cdot x^{3} - 3\\cdot x^{2} + x + 1\\right)\\cdot \\sin{\\left(x^{2} - x + 1 \\right)} + \\left(9\\cdot x^{2} - 6\\cdot x + 1\\right)\\cdot \\cos{\\left(x^{2} - x + 1 \\right)} + e^{x}$", "final_answer": "\\left(x + 1\\right)\\cdot e^{x} - \\left(2\\cdot x - 1\\right)\\cdot \\left(3\\cdot x^{3} - 3\\cdot x^{2} + x + 1\\right)\\cdot \\sin{\\left(x^{2} - x + 1 \\right)} + \\left(9\\cdot x^{2} - 6\\cdot x + 1\\right)\\cdot \\cos{\\left(x^{2} - x + 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "531dc9ae629e9977", "problem_id": "34496964d84262b4", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(3 x^{3} - 3 x^{2} + x + 1\\right) \\cos{\\left(x^{2} - x + 1 \\right)}$.", "output": "Let f(x) = $\\left(x + 1\\right) e^{x} + \\left(3 x^{3} - 3 x^{2} + x + 1\\right) \\cos{\\left(x^{2} - x + 1 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\left(x + 1\\right) e^{x} - \\left(2 x - 1\\right) \\left(3 x^{3} - 3 x^{2} + x + 1\\right) \\sin{\\left(x^{2} - x + 1 \\right)} + \\left(9 x^{2} - 6 x + 1\\right) \\cos{\\left(x^{2} - x + 1 \\right)} + e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\left(x + 1\\right)\\cdot e^{x} - \\left(2\\cdot x - 1\\right)\\cdot \\left(3\\cdot x^{3} - 3\\cdot x^{2} + x + 1\\right)\\cdot \\sin{\\left(x^{2} - x + 1 \\right)} + \\left(9\\cdot x^{2} - 6\\cdot x + 1\\right)\\cdot \\cos{\\left(x^{2} - x + 1 \\right)} + e^{x}$", "final_answer": "\\left(x + 1\\right)\\cdot e^{x} - \\left(2\\cdot x - 1\\right)\\cdot \\left(3\\cdot x^{3} - 3\\cdot x^{2} + x + 1\\right)\\cdot \\sin{\\left(x^{2} - x + 1 \\right)} + \\left(9\\cdot x^{2} - 6\\cdot x + 1\\right)\\cdot \\cos{\\left(x^{2} - x + 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "697e69945e840f32", "problem_id": "5873fffb770b5b7f", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "You want 27 liters of a 40% salt solution. You have a 30% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 27−x liters is 70% solution.\n2) Salt equation: 30x + 70(27−x) = 40·27.\n3) (30-70)x = (40-70)·27 ⇒ x = 81/4.\n4) Then 27−x = 27/4.\nFinal answer: $81/4 L of 30% and 27/4 L of 70%$", "final_answer": "81/4 L of 30% and 27/4 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f3ec3bf6a627b5c", "problem_id": "5873fffb770b5b7f", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 27 liters of a 40% salt solution. You have a 30% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 27−x liters is 70% solution.\nSalt equation: 30x + 70(27−x) = 40·27.\n(30-70)x = (40-70)·27 ⇒ x = 81/4.\nThen 27−x = 27/4.\nCheck: 30·81/4 + 70·27/4 = 40·27.\nFinal answer: $81/4 L of 30% and 27/4 L of 70%$", "final_answer": "81/4 L of 30% and 27/4 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eacc79285802bf48", "problem_id": "041ae8420d7371f6", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the multiplicative inverse of 3 modulo 13.", "output": "We want x such that 3x ≡ 1 (mod 13).\nUse extended Euclid to find s,t with 3s + 13t = 1.\nOne Bezout identity is: 3·(-4) + 13·(1) = 1.\nReduce s modulo 13: x ≡ 9 (mod 13).\nCheck: 3·9 mod 13 = 1.\nFinal answer: $x\\equiv 9\\pmod{13}$", "final_answer": "x\\equiv 9\\pmod{13}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eb99af359142a22d", "problem_id": "041ae8420d7371f6", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the multiplicative inverse of 3 modulo 13.", "output": "1) We want x such that 3x ≡ 1 (mod 13).\n2) Use extended Euclid to find s,t with 3s + 13t = 1.\n3) One Bezout identity is: 3·(-4) + 13·(1) = 1.\n4) Reduce s modulo 13: x ≡ 9 (mod 13).\nFinal answer: $x\\equiv 9\\pmod{13}$", "final_answer": "x\\equiv 9\\pmod{13}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a20ac99a3840d91a", "problem_id": "8d1d8fed712e133f", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A store sells a backpack for $121. It is discounted by 40%. You buy 10 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 121 × (1 − 40/100) = 121 × 60/100 = 363/5.\n2) Subtotal = 10 × 363/5 = 726.\n3) Total with tax = 726 × 110/100 = 3993/5.\nFinal answer: $\\frac{3993}{5}$", "final_answer": "\\frac{3993}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1a537ab7a0a464dd", "problem_id": "8d1d8fed712e133f", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A store sells a backpack for $121. It is discounted by 40%. You buy 10 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 121 × (1 − 40/100) = 121 × 60/100 = 363/5.\nSubtotal = 10 × 363/5 = 726.\nTotal with tax = 726 × 110/100 = 3993/5.\nCheck: recomputing the formula yields 3993/5.\nFinal answer: $\\frac{3993}{5}$", "final_answer": "\\frac{3993}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a40ed926238fe33", "problem_id": "56a336a50535582f", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "You want 51 liters of a 25% salt solution. You have a 10% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 51−x liters is 40% solution.\n2) Salt equation: 10x + 40(51−x) = 25·51.\n3) (10-40)x = (25-40)·51 ⇒ x = 51/2.\n4) Then 51−x = 51/2.\nFinal answer: $51/2 L of 10% and 51/2 L of 40%$", "final_answer": "51/2 L of 10% and 51/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a3d427f22076fa9c", "problem_id": "56a336a50535582f", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "You want 51 liters of a 25% salt solution. You have a 10% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 51−x liters is 40% solution.\nSalt equation: 10x + 40(51−x) = 25·51.\n(10-40)x = (25-40)·51 ⇒ x = 51/2.\nThen 51−x = 51/2.\nCheck: 10·51/2 + 40·51/2 = 25·51.\nFinal answer: $51/2 L of 10% and 51/2 L of 40%$", "final_answer": "51/2 L of 10% and 51/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a0c7324ebccd986", "problem_id": "e46790b6e69835f7", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 3 x^{3} + 3 x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{3} + 3 x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $- 9 \\pi - 4 + 9 \\pi^{3} + \\frac{45 \\pi^{4}}{4}$.\n4) Simplify the result.\nFinal answer: $- 9\\cdot \\pi - 4 + 9\\cdot \\pi^{3} + \\frac{45\\cdot \\pi^{4}}{4}$", "final_answer": "- 9\\cdot \\pi - 4 + 9\\cdot \\pi^{3} + \\frac{45\\cdot \\pi^{4}}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "481ef709145e7a19", "problem_id": "ae5ebb9642461a0e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} 3 x^{3} - 2 x - 2$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} - 3 x - 2.\n3) Then the limit is -2.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: -1.\nFinal answer: $-1$", "final_answer": "-1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ac22447c01a5ddaf", "problem_id": "b7afef273be53723", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 21 red balls and 16 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(37,9).\n3) Favorable samples: choose k reds and n−k blues: C(21,1)·C(16,8).\n4) Compute totals: C(37,9) = 124403620, and favorable = 270270.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{2457}{1130942}$", "final_answer": "\\frac{2457}{1130942}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "afbfec78151e3f87", "problem_id": "b7afef273be53723", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 21 red balls and 16 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(37,9).\nFavorable samples: choose k reds and n−k blues: C(21,1)·C(16,8).\nCompute totals: C(37,9) = 124403620, and favorable = 270270.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{2457}{1130942}$.\nFinal answer: $\\frac{2457}{1130942}$", "final_answer": "\\frac{2457}{1130942}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3bd361dcdf0ce969", "problem_id": "a7ac547347f56767", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_5=27$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_5=12.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=12 and k=5.\n4) Compute C(16,4) = 1820.\nFinal answer: $1820$", "final_answer": "1820", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "915cc6499113f754", "problem_id": "a7ac547347f56767", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_5=27$?", "output": "Let y_i=x_i-3, then y_i≥0 and y_1+…+y_5=12.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=12 and k=5.\nCompute C(16,4) = 1820.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $1820$", "final_answer": "1820", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0ef1930efb74101d", "problem_id": "85ca3950dd70f277", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Two friends split $564 in the ratio 7:5. How much does each receive?", "output": "Total ratio parts = 7+5 = 12.\nEach part = 564 ÷ 12 = 47.\nFirst share = 7 × 47 = 329.\nSecond share = 5 × 47 = 235.\nCheck: 329+235=564.\nFinal answer: $235$", "final_answer": "235", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9b749a9ba67f0d39", "problem_id": "85ca3950dd70f277", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Two friends split $564 in the ratio 7:5. How much does each receive?", "output": "1) Total ratio parts = 7+5 = 12.\n2) Each part = 564 ÷ 12 = 47.\n3) First share = 7 × 47 = 329.\n4) Second share = 5 × 47 = 235.\nFinal answer: $235$", "final_answer": "235", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "41cf77435ec0c6f6", "problem_id": "6ccf0246994147d6", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{3} - 2 x^{2} + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 3 x^{3} - 2 x^{2} + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}$.\n3) Compute an antiderivative: F(x) = $- \\frac{3 x^{4}}{4} - \\frac{2 x^{3}}{3} - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{3\\cdot x^{4}}{4} - \\frac{2\\cdot x^{3}}{3} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{3\\cdot x^{4}}{4} - \\frac{2\\cdot x^{3}}{3} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d3f732783d7efae", "problem_id": "6ccf0246994147d6", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{3} - 2 x^{2} + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 3 x^{3} - 2 x^{2} + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}$.\nCompute an antiderivative: F(x) = $- \\frac{3 x^{4}}{4} - \\frac{2 x^{3}}{3} - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $C - \\frac{3\\cdot x^{4}}{4} - \\frac{2\\cdot x^{3}}{3} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{3\\cdot x^{4}}{4} - \\frac{2\\cdot x^{3}}{3} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6f1d5831015b5f1b", "problem_id": "5f8b6c773dc9fdc7", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A store sells a backpack for $150. It is discounted by 35%. You buy 4 backpacks. What is the total cost?", "output": "1) Discounted price = 150 × (1 − 35/100) = 150 × 65/100 = 195/2.\n2) Subtotal = 4 × 195/2 = 390.\nFinal answer: $390$", "final_answer": "390", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d62e8732fb0a6f30", "problem_id": "5f8b6c773dc9fdc7", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $150. It is discounted by 35%. You buy 4 backpacks. What is the total cost?", "output": "Discounted price = 150 × (1 − 35/100) = 150 × 65/100 = 195/2.\nSubtotal = 4 × 195/2 = 390.\nCheck: recomputing the formula yields 390.\nFinal answer: $390$", "final_answer": "390", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "06795ae6fbe85f4d", "problem_id": "73bd502fe8e267f5", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{7} x}{2} + \\frac{11 x}{4} - \\frac{27 \\sqrt{7}}{8} + \\frac{9}{8}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{11}{4} - \\frac{3 \\sqrt{7}}{2}$, c=$\\frac{9}{8} - \\frac{27 \\sqrt{7}}{8}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{21 \\sqrt{7}}{4} + \\frac{301}{16}$, so √Δ = $\\sqrt{\\frac{21 \\sqrt{7}}{4} + \\frac{301}{16}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{11}{8} + \\frac{3 \\sqrt{7}}{4} + \\frac{\\sqrt{84 \\sqrt{7} + 301}}{8}$.\nx2 = $- \\frac{\\sqrt{84 \\sqrt{7} + 301}}{8} - \\frac{11}{8} + \\frac{3 \\sqrt{7}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{11}{8} + \\frac{3\\cdot \\sqrt{7}}{4} + \\frac{\\sqrt{84\\cdot \\sqrt{7} + 301}}{8}, - \\frac{\\sqrt{84\\cdot \\sqrt{7} + 301}}{8} - \\frac{11}{8} + \\frac{3\\cdot \\sqrt{7}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{11}{8} + \\frac{3\\cdot \\sqrt{7}}{4} + \\frac{\\sqrt{84\\cdot \\sqrt{7} + 301}}{8}, - \\frac{\\sqrt{84\\cdot \\sqrt{7} + 301}}{8} - \\frac{11}{8} + \\frac{3\\cdot \\sqrt{7}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1c60cb43acbd0c95", "problem_id": "73bd502fe8e267f5", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{7} x}{2} + \\frac{11 x}{4} - \\frac{27 \\sqrt{7}}{8} + \\frac{9}{8}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{11}{4} - \\frac{3 \\sqrt{7}}{2}$, c=$\\frac{9}{8} - \\frac{27 \\sqrt{7}}{8}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{21 \\sqrt{7}}{4} + \\frac{301}{16}$, so √Δ = $\\sqrt{\\frac{21 \\sqrt{7}}{4} + \\frac{301}{16}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{11}{8} + \\frac{3 \\sqrt{7}}{4} + \\frac{\\sqrt{84 \\sqrt{7} + 301}}{8}$.\n7) x2 = $- \\frac{\\sqrt{84 \\sqrt{7} + 301}}{8} - \\frac{11}{8} + \\frac{3 \\sqrt{7}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{11}{8} + \\frac{3\\cdot \\sqrt{7}}{4} + \\frac{\\sqrt{84\\cdot \\sqrt{7} + 301}}{8}, - \\frac{\\sqrt{84\\cdot \\sqrt{7} + 301}}{8} - \\frac{11}{8} + \\frac{3\\cdot \\sqrt{7}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{11}{8} + \\frac{3\\cdot \\sqrt{7}}{4} + \\frac{\\sqrt{84\\cdot \\sqrt{7} + 301}}{8}, - \\frac{\\sqrt{84\\cdot \\sqrt{7} + 301}}{8} - \\frac{11}{8} + \\frac{3\\cdot \\sqrt{7}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "509f9fbf2bb1025a", "problem_id": "ceae5bdabb11e2ee", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{3} - 4 x - 1\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $2 x^{3} - 4 x - 1$.\n3) Compute an antiderivative: F(x) = $\\frac{x^{4}}{2} - 2 x^{2} - x$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{x^{4}}{2} - 2\\cdot x^{2} - x$", "final_answer": "C + \\frac{x^{4}}{2} - 2\\cdot x^{2} - x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "024a044a8084127b", "problem_id": "ceae5bdabb11e2ee", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{3} - 4 x - 1\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $2 x^{3} - 4 x - 1$.\nCompute an antiderivative: F(x) = $\\frac{x^{4}}{2} - 2 x^{2} - x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{x^{4}}{2} - 2\\cdot x^{2} - x$", "final_answer": "C + \\frac{x^{4}}{2} - 2\\cdot x^{2} - x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "990060a1c33ebd15", "problem_id": "e60051aae1bc7315", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡10 (mod 11), x≡1 (mod 7). Give the solution modulo 77.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 77.\n3) Compute the CRT solution to get x ≡ 43 (mod 77).\nFinal answer: $x\\equiv 43\\pmod{77}$", "final_answer": "x\\equiv 43\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ef84a7225bae1c6a", "problem_id": "a1f650c53ce275bf", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 x^{2} - 9 x + 21$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{2} - 10 x + 21.\n3) Then the limit is 53.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 51.\nFinal answer: $51$", "final_answer": "51", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3ca05ee703ea4230", "problem_id": "b0b622da04c22044", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int x^{3} + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $x^{3} + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2$.\n3) Compute an antiderivative: F(x) = $\\frac{x^{4}}{4} + 2 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{x^{4}}{4} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{4}}{4} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4caf3f5248b11584", "problem_id": "b0b622da04c22044", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int x^{3} + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $x^{3} + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2$.\nCompute an antiderivative: F(x) = $\\frac{x^{4}}{4} + 2 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{x^{4}}{4} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{4}}{4} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "61097fd1ef8c44a8", "problem_id": "c28dd4ca9cfff975", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A store sells a backpack for $195. It is discounted by 20%. You buy 8 backpacks. What is the total cost?", "output": "Discounted price = 195 × (1 − 20/100) = 195 × 80/100 = 156.\nSubtotal = 8 × 156 = 1248.\nFinal answer: $1248$", "final_answer": "1248", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "31cdc377ef233c8d", "problem_id": "c28dd4ca9cfff975", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A store sells a backpack for $195. It is discounted by 20%. You buy 8 backpacks. What is the total cost?", "output": "1) Discounted price = 195 × (1 − 20/100) = 195 × 80/100 = 156.\n2) Subtotal = 8 × 156 = 1248.\nFinal answer: $1248$", "final_answer": "1248", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "754e0e34b0c5303b", "problem_id": "9078a693041d9ffc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} x^{2} - 4 x + 7$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to x^{2} - 5 x + 7.\n3) Then the limit is 21.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 19.\nFinal answer: $19$", "final_answer": "19", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "bac482f4fddc7902", "problem_id": "a9e7158fd69cc6ab", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "You want 39 liters of a 30% salt solution. You have a 20% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 39−x liters is 40% solution.\n2) Salt equation: 20x + 40(39−x) = 30·39.\n3) (20-40)x = (30-40)·39 ⇒ x = 39/2.\n4) Then 39−x = 39/2.\nFinal answer: $39/2 L of 20% and 39/2 L of 40%$", "final_answer": "39/2 L of 20% and 39/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "545575ace7a035ac", "problem_id": "9fc7ba85cc26bb6d", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 30 x + 27$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} + 30 x + 27$ = $3 \\left(x + 1\\right) \\left(x + 9\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-1$.\n5) From the second factor: x = $-9$.\nFinal answer: $x \\in \\{-1, -9\\}$", "final_answer": "x \\in \\{-1, -9\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0d5a5a1aa7ef7dc7", "problem_id": "9fc7ba85cc26bb6d", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 30 x + 27$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} + 30 x + 27$ = $3 \\left(x + 1\\right) \\left(x + 9\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $-1$.\nFrom the second factor: x = $-9$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-1, -9\\}$", "final_answer": "x \\in \\{-1, -9\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "803b52676721051d", "problem_id": "b55e1a673ba060bc", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - 3 x^{3} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{3} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{765 \\pi^{4}}{64} - 2 + 5 \\pi + \\frac{15 \\pi^{2}}{8}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{765\\cdot \\pi^{4}}{64} - 2 + 5\\cdot \\pi + \\frac{15\\cdot \\pi^{2}}{8}$", "final_answer": "- \\frac{765\\cdot \\pi^{4}}{64} - 2 + 5\\cdot \\pi + \\frac{15\\cdot \\pi^{2}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3a4dd55cbafb5e40", "problem_id": "b55e1a673ba060bc", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - 3 x^{3} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{3} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{765 \\pi^{4}}{64} - 2 + 5 \\pi + \\frac{15 \\pi^{2}}{8}$.\n4) Simplify the result.\nFinal answer: $- \\frac{765\\cdot \\pi^{4}}{64} - 2 + 5\\cdot \\pi + \\frac{15\\cdot \\pi^{2}}{8}$", "final_answer": "- \\frac{765\\cdot \\pi^{4}}{64} - 2 + 5\\cdot \\pi + \\frac{15\\cdot \\pi^{2}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e8322c19732f92b4", "problem_id": "5d2ce5fb9d74cd6a", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡6 (mod 7), x≡1 (mod 9). Give the solution modulo 63.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 63.\n3) Compute the CRT solution to get x ≡ 55 (mod 63).\nFinal answer: $x\\equiv 55\\pmod{63}$", "final_answer": "x\\equiv 55\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4c32022a2e6c8d29", "problem_id": "5d2ce5fb9d74cd6a", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡6 (mod 7), x≡1 (mod 9). Give the solution modulo 63.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 63.\nCompute the CRT solution to get x ≡ 55 (mod 63).\nCheck: 55 mod 7=6 and mod 9=1.\nFinal answer: $x\\equiv 55\\pmod{63}$", "final_answer": "x\\equiv 55\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9c9b9c1369804fb7", "problem_id": "ee3a0c80ccc9b95b", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many binary strings of length 12 contain exactly 2 ones with no two ones adjacent?", "output": "1) Reserve 1 mandatory zeros between the 2 ones.\n2) This leaves n−k+1 = 11 valid slots for the ones.\n3) Choose the slots: C(11,2) = 55.\nFinal answer: $55$", "final_answer": "55", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a7c425b0608f07f", "problem_id": "ee3a0c80ccc9b95b", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many binary strings of length 12 contain exactly 2 ones with no two ones adjacent?", "output": "Reserve 1 mandatory zeros between the 2 ones.\nThis leaves n−k+1 = 11 valid slots for the ones.\nChoose the slots: C(11,2) = 55.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $55$", "final_answer": "55", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a474fe48b16feb6", "problem_id": "677ed07a6018d436", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 19 red balls and 26 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(45,11).\nFavorable samples: choose k reds and n−k blues: C(19,1)·C(26,10).\nCompute totals: C(45,11) = 10150595910, and favorable = 100922965.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{81719}{8219106}$.\nFinal answer: $\\frac{81719}{8219106}$", "final_answer": "\\frac{81719}{8219106}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a7e586b3245118b", "problem_id": "677ed07a6018d436", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "An urn has 19 red balls and 26 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(45,11).\nFavorable samples: choose k reds and n−k blues: C(19,1)·C(26,10).\nFinal answer: $\\frac{81719}{8219106}$", "final_answer": "\\frac{81719}{8219106}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "80cee16bcccc6184", "problem_id": "46e369ebc133d49b", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 4 x^{4} + 2 x^{3} - 2 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $4 x^{4} + 2 x^{3} - 2 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $\\frac{3 \\pi \\left(- 20 \\pi^{2} + 20 + 15 \\pi + 25 \\pi^{3} + 88 \\pi^{4}\\right)}{10}$.\n4) Simplify the result.\nFinal answer: $\\frac{3\\cdot \\pi\\cdot \\left(- 20\\cdot \\pi^{2} + 20 + 15\\cdot \\pi + 25\\cdot \\pi^{3} + 88\\cdot \\pi^{4}\\right)}{10}$", "final_answer": "\\frac{3\\cdot \\pi\\cdot \\left(- 20\\cdot \\pi^{2} + 20 + 15\\cdot \\pi + 25\\cdot \\pi^{3} + 88\\cdot \\pi^{4}\\right)}{10}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a18b741ddb8671a", "problem_id": "46e369ebc133d49b", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 4 x^{4} + 2 x^{3} - 2 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $4 x^{4} + 2 x^{3} - 2 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $\\frac{3 \\pi \\left(- 20 \\pi^{2} + 20 + 15 \\pi + 25 \\pi^{3} + 88 \\pi^{4}\\right)}{10}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{3\\cdot \\pi\\cdot \\left(- 20\\cdot \\pi^{2} + 20 + 15\\cdot \\pi + 25\\cdot \\pi^{3} + 88\\cdot \\pi^{4}\\right)}{10}$", "final_answer": "\\frac{3\\cdot \\pi\\cdot \\left(- 20\\cdot \\pi^{2} + 20 + 15\\cdot \\pi + 25\\cdot \\pi^{3} + 88\\cdot \\pi^{4}\\right)}{10}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f8df1f445a706113", "problem_id": "7f0d9dc95dbad082", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(- x^{2} - 3\\right) \\cos{\\left(- 3 x^{2} + 3 x + 2 \\right)}$.", "output": "1) Let f(x) = $2 \\left(x - 3\\right) e^{x} + \\left(- x^{2} - 3\\right) \\cos{\\left(- 3 x^{2} + 3 x + 2 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- 2 x \\cos{\\left(- 3 x^{2} + 3 x + 2 \\right)} - 3 \\left(1 - 2 x\\right) \\left(- x^{2} - 3\\right) \\sin{\\left(- 3 x^{2} + 3 x + 2 \\right)} + 2 \\left(x - 3\\right) e^{x} + 2 e^{x}$.\nFinal answer: $- 2\\cdot x\\cdot \\cos{\\left(- 3\\cdot x^{2} + 3\\cdot x + 2 \\right)} - \\left(3 - 6\\cdot x\\right)\\cdot \\left(- x^{2} - 3\\right)\\cdot \\sin{\\left(- 3\\cdot x^{2} + 3\\cdot x + 2 \\right)} + \\left(2\\cdot x - 6\\right)\\cdot e^{x} + 2\\cdot e^{x}$", "final_answer": "- 2\\cdot x\\cdot \\cos{\\left(- 3\\cdot x^{2} + 3\\cdot x + 2 \\right)} - \\left(3 - 6\\cdot x\\right)\\cdot \\left(- x^{2} - 3\\right)\\cdot \\sin{\\left(- 3\\cdot x^{2} + 3\\cdot x + 2 \\right)} + \\left(2\\cdot x - 6\\right)\\cdot e^{x} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "13b2eda4ed2b1535", "problem_id": "7f0d9dc95dbad082", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(- x^{2} - 3\\right) \\cos{\\left(- 3 x^{2} + 3 x + 2 \\right)}$.", "output": "Let f(x) = $2 \\left(x - 3\\right) e^{x} + \\left(- x^{2} - 3\\right) \\cos{\\left(- 3 x^{2} + 3 x + 2 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- 2 x \\cos{\\left(- 3 x^{2} + 3 x + 2 \\right)} - 3 \\left(1 - 2 x\\right) \\left(- x^{2} - 3\\right) \\sin{\\left(- 3 x^{2} + 3 x + 2 \\right)} + 2 \\left(x - 3\\right) e^{x} + 2 e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 2\\cdot x\\cdot \\cos{\\left(- 3\\cdot x^{2} + 3\\cdot x + 2 \\right)} - \\left(3 - 6\\cdot x\\right)\\cdot \\left(- x^{2} - 3\\right)\\cdot \\sin{\\left(- 3\\cdot x^{2} + 3\\cdot x + 2 \\right)} + \\left(2\\cdot x - 6\\right)\\cdot e^{x} + 2\\cdot e^{x}$", "final_answer": "- 2\\cdot x\\cdot \\cos{\\left(- 3\\cdot x^{2} + 3\\cdot x + 2 \\right)} - \\left(3 - 6\\cdot x\\right)\\cdot \\left(- x^{2} - 3\\right)\\cdot \\sin{\\left(- 3\\cdot x^{2} + 3\\cdot x + 2 \\right)} + \\left(2\\cdot x - 6\\right)\\cdot e^{x} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "243a1ffd38333fd4", "problem_id": "d35295f480db05dd", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A cyclist rides at 29 miles per hour for 3/2 hours, then at 74 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 29 × 3/2 = 87/2 miles.\n2) Second leg distance = 74 × 5/2 = 185 miles.\n3) Total distance = 87/2 + 185 = 457/2 miles.\nFinal answer: $\\frac{457}{2}$", "final_answer": "\\frac{457}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f4c437efb90f5cdc", "problem_id": "d35295f480db05dd", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 29 miles per hour for 3/2 hours, then at 74 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 29 × 3/2 = 87/2 miles.\nSecond leg distance = 74 × 5/2 = 185 miles.\nTotal distance = 87/2 + 185 = 457/2 miles.\nCheck: adding both legs gives 457/2.\nFinal answer: $\\frac{457}{2}$", "final_answer": "\\frac{457}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4aed9f9665327edb", "problem_id": "4ccc399a3792c605", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the equation of the tangent line to $y=3 x^{3} - 2 x^{2} - x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=2$.", "output": "1) Differentiate: f'(x) = $9 x^{2} - 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 1$.\n2) Slope at x=2: m = f'(2) = $\\frac{e}{2} - 2 \\sin{\\left(4 \\right)} + 27$.\n3) Point on curve: (2, f(2)) = (2, $\\cos{\\left(4 \\right)} + e + 15$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(4 \\right)} + e + 15 = \\frac{e}{2} - 2 \\sin{\\left(4 \\right)} + 27(x − 2).\n6) Optionally expand: y = $\\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 27 x - 39 + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$.\nFinal answer: $y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 27\\cdot x - 39 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$", "final_answer": "y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 27\\cdot x - 39 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3ff58adafc1620d4", "problem_id": "4ccc399a3792c605", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} - 2 x^{2} - x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=2$.", "output": "Differentiate: f'(x) = $9 x^{2} - 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 1$.\nSlope at x=2: m = f'(2) = $\\frac{e}{2} - 2 \\sin{\\left(4 \\right)} + 27$.\nPoint on curve: (2, f(2)) = (2, $\\cos{\\left(4 \\right)} + e + 15$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − \\cos{\\left(4 \\right)} + e + 15 = \\frac{e}{2} - 2 \\sin{\\left(4 \\right)} + 27(x − 2).\nOptionally expand: y = $\\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 27 x - 39 + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 27\\cdot x - 39 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$", "final_answer": "y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 27\\cdot x - 39 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c90b54ab7efa38a3", "problem_id": "bc17865cbde7e1d6", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A store sells a backpack for $855. It is discounted by 35%. You buy 2 backpacks. What is the total cost?", "output": "Discounted price = 855 × (1 − 35/100) = 855 × 65/100 = 2223/4.\nSubtotal = 2 × 2223/4 = 2223/2.\nCheck: recomputing the formula yields 2223/2.\nFinal answer: $\\frac{2223}{2}$", "final_answer": "\\frac{2223}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "37ed99574687e554", "problem_id": "bc17865cbde7e1d6", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A store sells a backpack for $855. It is discounted by 35%. You buy 2 backpacks. What is the total cost?", "output": "1) Discounted price = 855 × (1 − 35/100) = 855 × 65/100 = 2223/4.\n2) Subtotal = 2 × 2223/4 = 2223/2.\nFinal answer: $\\frac{2223}{2}$", "final_answer": "\\frac{2223}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "95e909a6471d950b", "problem_id": "9c896fd62602fe4f", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Two friends split $480 in the ratio 5:5. How much does each receive?", "output": "Total ratio parts = 5+5 = 10.\nEach part = 480 ÷ 10 = 48.\nFirst share = 5 × 48 = 240.\nSecond share = 5 × 48 = 240.\nCheck: 240+240=480.\nFinal answer: $240$", "final_answer": "240", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9c886664984e7e8c", "problem_id": "9c896fd62602fe4f", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Two friends split $480 in the ratio 5:5. How much does each receive?", "output": "1) Total ratio parts = 5+5 = 10.\n2) Each part = 480 ÷ 10 = 48.\n3) First share = 5 × 48 = 240.\n4) Second share = 5 × 48 = 240.\nFinal answer: $240$", "final_answer": "240", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "27c7d44630005d83", "problem_id": "4f48e55cbb1a32da", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 22 x + 36$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} + 22 x + 36$ = $2 \\left(x + 2\\right) \\left(x + 9\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-2$.\n5) From the second factor: x = $-9$.\nFinal answer: $x \\in \\{-2, -9\\}$", "final_answer": "x \\in \\{-2, -9\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d3d30654634756dc", "problem_id": "192c30ef456c4825", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 x \\sin{\\left(x \\right)}) = 3 x \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 x \\sin{\\left(x \\right)}) = 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$", "final_answer": "- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f6f62d092efdbafd", "problem_id": "301fd6302e2c5752", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A cyclist rides at 59 miles per hour for 3/2 hours, then at 42 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 59 × 3/2 = 177/2 miles.\nSecond leg distance = 42 × 3 = 126 miles.\nTotal distance = 177/2 + 126 = 429/2 miles.\nCheck: adding both legs gives 429/2.\nFinal answer: $\\frac{429}{2}$", "final_answer": "\\frac{429}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1c12e5ee7a28fe7f", "problem_id": "301fd6302e2c5752", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A cyclist rides at 59 miles per hour for 3/2 hours, then at 42 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 59 × 3/2 = 177/2 miles.\n2) Second leg distance = 42 × 3 = 126 miles.\n3) Total distance = 177/2 + 126 = 429/2 miles.\nFinal answer: $\\frac{429}{2}$", "final_answer": "\\frac{429}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "07cccec7a8f1d3cb", "problem_id": "fdea728e332a4bd3", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} 4 x^{4} + 4 x^{3} + 2 x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $4 x^{4} + 4 x^{3} + 2 x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{3 \\pi^{2}}{2} - \\frac{3 \\pi}{2} - 1 + \\frac{3 \\pi^{3}}{4} + \\frac{15 \\pi^{4}}{16} + \\frac{33 \\pi^{5}}{40}$.\n4) Simplify the result.\nFinal answer: $- \\frac{3\\cdot \\pi^{2}}{2} - \\frac{3\\cdot \\pi}{2} - 1 + \\frac{3\\cdot \\pi^{3}}{4} + \\frac{15\\cdot \\pi^{4}}{16} + \\frac{33\\cdot \\pi^{5}}{40}$", "final_answer": "- \\frac{3\\cdot \\pi^{2}}{2} - \\frac{3\\cdot \\pi}{2} - 1 + \\frac{3\\cdot \\pi^{3}}{4} + \\frac{15\\cdot \\pi^{4}}{16} + \\frac{33\\cdot \\pi^{5}}{40}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6892f9601d0711ef", "problem_id": "fdea728e332a4bd3", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} 4 x^{4} + 4 x^{3} + 2 x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $4 x^{4} + 4 x^{3} + 2 x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1$.\nEvaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{3 \\pi^{2}}{2} - \\frac{3 \\pi}{2} - 1 + \\frac{3 \\pi^{3}}{4} + \\frac{15 \\pi^{4}}{16} + \\frac{33 \\pi^{5}}{40}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{3\\cdot \\pi^{2}}{2} - \\frac{3\\cdot \\pi}{2} - 1 + \\frac{3\\cdot \\pi^{3}}{4} + \\frac{15\\cdot \\pi^{4}}{16} + \\frac{33\\cdot \\pi^{5}}{40}$", "final_answer": "- \\frac{3\\cdot \\pi^{2}}{2} - \\frac{3\\cdot \\pi}{2} - 1 + \\frac{3\\cdot \\pi^{3}}{4} + \\frac{15\\cdot \\pi^{4}}{16} + \\frac{33\\cdot \\pi^{5}}{40}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "54bdb0a53c584b57", "problem_id": "fd51ea26a762d025", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the linear system:\n$- 6 x + 6 y$ = $-6$\n$- 4 x - 8 y$ = $-28$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $24 x - 24 y$ = $24$.\n3) Multiply the second equation by $-6$: $24 x + 48 y$ = $168$.\n4) Subtract to eliminate x: $- 72 y$ = $-144$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, 2\\right)$", "final_answer": "(x,y)=\\left(3, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e033af39a8a49803", "problem_id": "fd51ea26a762d025", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$- 6 x + 6 y$ = $-6$\n$- 4 x - 8 y$ = $-28$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $24 x - 24 y$ = $24$.\nMultiply the second equation by $-6$: $24 x + 48 y$ = $168$.\nSubtract to eliminate x: $- 72 y$ = $-144$.\nSolve for y: y = $2$.\nSubstitute back to find x: x = $3$.\nCheck: plug (x,y)=(3,2) into both equations.\nFinal answer: $(x,y)=\\left(3, 2\\right)$", "final_answer": "(x,y)=\\left(3, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "98e694c0ba78c21f", "problem_id": "64d4c2f2f046cdfc", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x^{2} - 4 x - 7$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{3} - 2 x^{2} + x - 2$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 3 x^{2} - 4 x + 1$.\n5) Evaluate at a: p'(2) = $-19$.\nFinal answer: $-19$", "final_answer": "-19", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a2fb95eb48c44358", "problem_id": "64d4c2f2f046cdfc", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x^{2} - 4 x - 7$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{3} - 2 x^{2} + x - 2$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 3 x^{2} - 4 x + 1$.\nEvaluate at a: p'(2) = $-19$.\nCheck: the identity gives $-19$.\nFinal answer: $-19$", "final_answer": "-19", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e5e6fb9f096ae10a", "problem_id": "a3ec19a9d00409e3", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Worker A finishes a job in 15 hours and Worker B finishes it in 22 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/15 job/hour. B's rate = 1/22 job/hour.\n2) Combined rate = 1/15 + 1/22 = 37/330 job/hour.\n3) Time = 1 ÷ (combined rate) = 330/37 hours.\nFinal answer: $\\frac{330}{37}$", "final_answer": "\\frac{330}{37}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d6a207292769201b", "problem_id": "a3ec19a9d00409e3", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Worker A finishes a job in 15 hours and Worker B finishes it in 22 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/15 job/hour. B's rate = 1/22 job/hour.\nCombined rate = 1/15 + 1/22 = 37/330 job/hour.\nTime = 1 ÷ (combined rate) = 330/37 hours.\nCheck: (combined rate)×(time) = 37/330×330/37 = 1.\nFinal answer: $\\frac{330}{37}$", "final_answer": "\\frac{330}{37}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ae597c1369ed742c", "problem_id": "e0e986325353d379", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "You want 58 liters of a 40% salt solution. You have a 15% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 58−x liters is 60% solution.\n2) Salt equation: 15x + 60(58−x) = 40·58.\n3) (15-60)x = (40-60)·58 ⇒ x = 232/9.\n4) Then 58−x = 290/9.\nFinal answer: $232/9 L of 15% and 290/9 L of 60%$", "final_answer": "232/9 L of 15% and 290/9 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e9090cddb986feba", "problem_id": "32d8d20afd531b76", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{3} + x^{2} - \\frac{x}{2} + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - \\frac{1}{2}}{x - 4}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}) = \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4}$.\n3) So $f'(x) = \\frac{4 x^{3}}{2 x^{2} - 16 x + 32} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{22 x^{2}}{2 x^{2} - 16 x + 32} - \\frac{8 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{16 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{16 x}{2 x^{2} - 16 x + 32} + \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 8 x + 16} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 8} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} - \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} + \\frac{16 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{32 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{5}{2 x^{2} - 16 x + 32} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}) = \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x - 4\\right)} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x - 4\\right)^{2}} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 2 x^{3} - 2 x^{2} + x + \\left(x - 4\\right) \\left(6 x^{2} + 4 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{2 \\left(x - 4\\right)^{2}}$.\nFinal answer: $\\frac{- 2 x^{3} - 2 x^{2} + x + \\left(x - 4\\right) \\left(6 x^{2} + 4 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{2 \\left(x - 4\\right)^{2}}$", "final_answer": "\\frac{- 2 x^{3} - 2 x^{2} + x + \\left(x - 4\\right) \\left(6 x^{2} + 4 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{2 \\left(x - 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "afb767e65a5668b4", "problem_id": "1cbfc7f56339c243", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the multiplicative inverse of 22 modulo 31.", "output": "1) We want x such that 22x ≡ 1 (mod 31).\n2) Use extended Euclid to find s,t with 22s + 31t = 1.\n3) One Bezout identity is: 22·(-7) + 31·(5) = 1.\n4) Reduce s modulo 31: x ≡ 24 (mod 31).\nFinal answer: $x\\equiv 24\\pmod{31}$", "final_answer": "x\\equiv 24\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "badfde2e3238ac51", "problem_id": "1cbfc7f56339c243", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the multiplicative inverse of 22 modulo 31.", "output": "We want x such that 22x ≡ 1 (mod 31).\nUse extended Euclid to find s,t with 22s + 31t = 1.\nOne Bezout identity is: 22·(-7) + 31·(5) = 1.\nReduce s modulo 31: x ≡ 24 (mod 31).\nCheck: 22·24 mod 31 = 1.\nFinal answer: $x\\equiv 24\\pmod{31}$", "final_answer": "x\\equiv 24\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d41fcbe8152aa653", "problem_id": "1dcf8d5443b2f279", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "You want 61 liters of a 30% salt solution. You have a 15% solution and a 60% solution. How many liters of each should you mix?", "output": "122/3 L of 15% and 61/3 L of 60%", "final_answer": "122/3 L of 15% and 61/3 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "954eadb3b33e4730", "problem_id": "1dcf8d5443b2f279", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "You want 61 liters of a 30% salt solution. You have a 15% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 61−x liters is 60% solution.\nSalt equation: 15x + 60(61−x) = 30·61.\n(15-60)x = (30-60)·61 ⇒ x = 122/3.\nThen 61−x = 61/3.\nCheck: 15·122/3 + 60·61/3 = 30·61.\nFinal answer: $122/3 L of 15% and 61/3 L of 60%$", "final_answer": "122/3 L of 15% and 61/3 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db43049d3b9d428d", "problem_id": "24f93353b522ffee", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A cyclist rides at 39 miles per hour for 2 hours, then at 83 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 39 × 2 = 78 miles.\n2) Second leg distance = 83 × 3 = 249 miles.\n3) Total distance = 78 + 249 = 327 miles.\nFinal answer: $327$", "final_answer": "327", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec15c1469de90750", "problem_id": "24f93353b522ffee", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "A cyclist rides at 39 miles per hour for 2 hours, then at 83 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 39 × 2 = 78 miles.\nSecond leg distance = 83 × 3 = 249 miles.\nTotal distance = 78 + 249 = 327 miles.\nFinal answer: $327$", "final_answer": "327", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "791d2771c71d1421", "problem_id": "686bb2946a7217ea", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} + 2 x - 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(2 x^{2}) = 4 x^{2}$.\n3) So $f'(x) = 4 x^{2} + 2$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(2 x^{2}) = 4 x$.\nTherefore the correct derivative is $f'(x) = 2 \\left(2 x + 1\\right)$.\nFinal answer: $2 \\left(2 x + 1\\right)$", "final_answer": "2 \\left(2 x + 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "206876699d75b86d", "problem_id": "616a3015fb8f1721", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the linear system:\n$- 7 x - 5 y$ = $-12$\n$- 2 x - 6 y$ = $24$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $14 x + 10 y$ = $24$.\nMultiply the second equation by $-7$: $14 x + 42 y$ = $-168$.\nFinal answer: $(x,y)=\\left(6, -6\\right)$", "final_answer": "(x,y)=\\left(6, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a64867bad059ba2", "problem_id": "a09d0456308ef28e", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} x^{3} - x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $x^{3} - x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\pi).\n3) Subtract: F(\\pi) − F(- \\pi) = $\\frac{2 \\pi \\left(12 - \\pi^{2}\\right)}{3}$.\n4) Simplify the result.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(12 - \\pi^{2}\\right)}{3}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(12 - \\pi^{2}\\right)}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5f5f68ede065cc03", "problem_id": "a09d0456308ef28e", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} x^{3} - x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $x^{3} - x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4$.\nEvaluate at the bounds: F(\\pi) and F(- \\pi).\nSubtract: F(\\pi) − F(- \\pi) = $\\frac{2 \\pi \\left(12 - \\pi^{2}\\right)}{3}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(12 - \\pi^{2}\\right)}{3}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(12 - \\pi^{2}\\right)}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "23ac8fa96576c7f9", "problem_id": "798f313b2c21b5a1", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡1 (mod 5), x≡5 (mod 13). Give the solution modulo 65.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 65.\n3) Compute the CRT solution to get x ≡ 31 (mod 65).\nFinal answer: $x\\equiv 31\\pmod{65}$", "final_answer": "x\\equiv 31\\pmod{65}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "47482553f792998c", "problem_id": "4f060fc8a978793b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(3 x^{2} + 2 x + 2\\right) \\cos{\\left(x - 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x e^{x}) = x e^{x}$.\n3) So $f'(x) = - 3 x^{2} \\sin{\\left(x - 3 \\right)} + x e^{x} - 2 x \\sin{\\left(x - 3 \\right)} + 6 x \\cos{\\left(x - 3 \\right)} + 2 e^{x} - 2 \\sin{\\left(x - 3 \\right)} + 2 \\cos{\\left(x - 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x e^{x}) = x e^{x} + e^{x}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 2\\right) e^{x} + 2 \\left(3 x + 1\\right) \\cos{\\left(x - 3 \\right)} - \\left(x \\left(3 x + 2\\right) + 2\\right) \\sin{\\left(x - 3 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 2\\right) e^{x} + 2 \\left(3 x + 1\\right) \\cos{\\left(x - 3 \\right)} - \\left(x \\left(3 x + 2\\right) + 2\\right) \\sin{\\left(x - 3 \\right)} + e^{x}$", "final_answer": "\\left(x + 2\\right) e^{x} + 2 \\left(3 x + 1\\right) \\cos{\\left(x - 3 \\right)} - \\left(x \\left(3 x + 2\\right) + 2\\right) \\sin{\\left(x - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0f8d63224e67d69f", "problem_id": "4bcd3b1cb4f49563", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$- 6 x - 5 y$ = $45$\n$5 x - 8 y$ = $-1$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $5$: $- 30 x - 25 y$ = $225$.\n3) Multiply the second equation by $-6$: $- 30 x + 48 y$ = $6$.\n4) Subtract to eliminate x: $- 73 y$ = $219$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, -3\\right)$", "final_answer": "(x,y)=\\left(-5, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7f87bf067de0e2b0", "problem_id": "4bcd3b1cb4f49563", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- 6 x - 5 y$ = $45$\n$5 x - 8 y$ = $-1$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $5$: $- 30 x - 25 y$ = $225$.\nMultiply the second equation by $-6$: $- 30 x + 48 y$ = $6$.\nSubtract to eliminate x: $- 73 y$ = $219$.\nSolve for y: y = $-3$.\nSubstitute back to find x: x = $-5$.\nCheck: plug (x,y)=(-5,-3) into both equations.\nFinal answer: $(x,y)=\\left(-5, -3\\right)$", "final_answer": "(x,y)=\\left(-5, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cae9705a8e81e816", "problem_id": "3eac81025339902d", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 12 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=5, p=$\\frac{3}{8}$ and simplify.\n4) Here C(12,5) = 792.\nFinal answer: $\\frac{1879453125}{8589934592}$", "final_answer": "\\frac{1879453125}{8589934592}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d2914ba00ee37a8f", "problem_id": "3eac81025339902d", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 12 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=12, k=5, p=$\\frac{3}{8}$ and simplify.\nFinal answer: $\\frac{1879453125}{8589934592}$", "final_answer": "\\frac{1879453125}{8589934592}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "efec3d4e9f696578", "problem_id": "bf0b3483a8c2efb8", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡7 (mod 8), x≡8 (mod 11). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 63 (mod 88).\nFinal answer: $x\\equiv 63\\pmod{88}$", "final_answer": "x\\equiv 63\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b3026d2fba7b576d", "problem_id": "bf0b3483a8c2efb8", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Solve the congruences: x≡7 (mod 8), x≡8 (mod 11). Give the solution modulo 88.", "output": "$x\\equiv 63\\pmod{88}$", "final_answer": "x\\equiv 63\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5c7f53170ee9fa05", "problem_id": "24ab67fa1824729c", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$- 3 x + 3 y$ = $-9$\n$6 x - y$ = $33$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $6$: $- 18 x + 18 y$ = $-54$.\n3) Multiply the second equation by $-3$: $- 18 x + 3 y$ = $-99$.\n4) Subtract to eliminate x: $15 y$ = $45$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $6$.\nFinal answer: $(x,y)=\\left(6, 3\\right)$", "final_answer": "(x,y)=\\left(6, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9dca94b545aa4414", "problem_id": "24ab67fa1824729c", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- 3 x + 3 y$ = $-9$\n$6 x - y$ = $33$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $6$: $- 18 x + 18 y$ = $-54$.\nMultiply the second equation by $-3$: $- 18 x + 3 y$ = $-99$.\nSubtract to eliminate x: $15 y$ = $45$.\nSolve for y: y = $3$.\nSubstitute back to find x: x = $6$.\nCheck: plug (x,y)=(6,3) into both equations.\nFinal answer: $(x,y)=\\left(6, 3\\right)$", "final_answer": "(x,y)=\\left(6, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0359c77cffd6afcf", "problem_id": "45a199b89ea9943a", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - 3 x^{2} - 10 x - 32$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 3 x^{3} - x^{2} - 2 x - 1$ and a = 3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 9 x^{2} - 2 x - 2$.\n5) Evaluate at a: p'(3) = $-89$.\nFinal answer: $-89$", "final_answer": "-89", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "92a87ddca223acac", "problem_id": "45a199b89ea9943a", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - 3 x^{2} - 10 x - 32$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 3 x^{3} - x^{2} - 2 x - 1$ and a = 3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-89$", "final_answer": "-89", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c21bee5e201c033f", "problem_id": "67d0c83ea7e04621", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{2} x}{2} + \\frac{7 x}{4} - \\frac{15}{4} + \\frac{15 \\sqrt{2}}{8}$ = 0.", "output": "$x \\in \\{- \\frac{7}{8} + \\frac{3\\cdot \\sqrt{2}}{4} + \\frac{\\sqrt{361 - 204\\cdot \\sqrt{2}}}{8}, - \\frac{\\sqrt{361 - 204\\cdot \\sqrt{2}}}{8} - \\frac{7}{8} + \\frac{3\\cdot \\sqrt{2}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{7}{8} + \\frac{3\\cdot \\sqrt{2}}{4} + \\frac{\\sqrt{361 - 204\\cdot \\sqrt{2}}}{8}, - \\frac{\\sqrt{361 - 204\\cdot \\sqrt{2}}}{8} - \\frac{7}{8} + \\frac{3\\cdot \\sqrt{2}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "556d78f4e6087df4", "problem_id": "67d0c83ea7e04621", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{2} x}{2} + \\frac{7 x}{4} - \\frac{15}{4} + \\frac{15 \\sqrt{2}}{8}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{7}{4} - \\frac{3 \\sqrt{2}}{2}$, c=$- \\frac{15}{4} + \\frac{15 \\sqrt{2}}{8}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{361}{16} - \\frac{51 \\sqrt{2}}{4}$, so √Δ = $\\sqrt{\\frac{361}{16} - \\frac{51 \\sqrt{2}}{4}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{7}{8} + \\frac{3 \\sqrt{2}}{4} + \\frac{\\sqrt{361 - 204 \\sqrt{2}}}{8}$.\n7) x2 = $- \\frac{\\sqrt{361 - 204 \\sqrt{2}}}{8} - \\frac{7}{8} + \\frac{3 \\sqrt{2}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{7}{8} + \\frac{3\\cdot \\sqrt{2}}{4} + \\frac{\\sqrt{361 - 204\\cdot \\sqrt{2}}}{8}, - \\frac{\\sqrt{361 - 204\\cdot \\sqrt{2}}}{8} - \\frac{7}{8} + \\frac{3\\cdot \\sqrt{2}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{7}{8} + \\frac{3\\cdot \\sqrt{2}}{4} + \\frac{\\sqrt{361 - 204\\cdot \\sqrt{2}}}{8}, - \\frac{\\sqrt{361 - 204\\cdot \\sqrt{2}}}{8} - \\frac{7}{8} + \\frac{3\\cdot \\sqrt{2}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4152249259a6e9c2", "problem_id": "e6bfbdd17a0da636", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A cyclist rides at 82 miles per hour for 3/2 hours, then at 69 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 82 × 3/2 = 123 miles.\n2) Second leg distance = 69 × 1 = 69 miles.\n3) Total distance = 123 + 69 = 192 miles.\nFinal answer: $192$", "final_answer": "192", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f2af3bcbefee4c82", "problem_id": "8c2b46771a27cbde", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} \\sin{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 \\cos{\\left(2 x \\right)}) = - 2 \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 \\cos{\\left(2 x \\right)}) = - 4 \\sin{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.\nFinal answer: $x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$", "final_answer": "x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e7d6e9441ceaf272", "problem_id": "eec4837b07a0f6bf", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the equation of the tangent line to $y=2 x^{3} - x^{2} + 4 x$ at $x=0$.", "output": "1) Differentiate: f'(x) = $6 x^{2} - 2 x + 4$.\n2) Slope at x=0: m = f'(0) = $4$.\n3) Point on curve: (0, f(0)) = (0, $0$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 0 = 4(x − 0).\n6) Optionally expand: y = $4 x$.\nFinal answer: $y = 4\\cdot x$", "final_answer": "y = 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "597fd3fcd33220fb", "problem_id": "eec4837b07a0f6bf", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} - x^{2} + 4 x$ at $x=0$.", "output": "Differentiate: f'(x) = $6 x^{2} - 2 x + 4$.\nSlope at x=0: m = f'(0) = $4$.\nPoint on curve: (0, f(0)) = (0, $0$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 0 = 4(x − 0).\nOptionally expand: y = $4 x$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 4\\cdot x$", "final_answer": "y = 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6e3ea701450d60d7", "problem_id": "a53bad13f9097225", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{3} + 3 x^{2} - 4 x$.", "output": "1) Differentiate term-by-term:\n2) d/dx(- 4 x) = -4.\n3) d/dx(- 3 x^{3}) = - 9 x^{2}.\n4) d/dx(3 x^{2}) = 6 x.\n5) Combine: f'(x) = - 9 x^{2} + 6 x - 4.\nFinal answer: $- 9\\cdot x^{2} + 6\\cdot x - 4$", "final_answer": "- 9\\cdot x^{2} + 6\\cdot x - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d4f7afd6c59030d6", "problem_id": "a53bad13f9097225", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{3} + 3 x^{2} - 4 x$.", "output": "Differentiate term-by-term:\nd/dx(- 4 x) = -4.\nd/dx(- 3 x^{3}) = - 9 x^{2}.\nd/dx(3 x^{2}) = 6 x.\nCombine: f'(x) = - 9 x^{2} + 6 x - 4.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 9\\cdot x^{2} + 6\\cdot x - 4$", "final_answer": "- 9\\cdot x^{2} + 6\\cdot x - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c2a38cfd429b3803", "problem_id": "a9fbf785c2dd00bc", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡7 (mod 11). Give the solution modulo 33.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 33.\nCompute the CRT solution to get x ≡ 7 (mod 33).\nCheck: 7 mod 3=1 and mod 11=7.\nFinal answer: $x\\equiv 7\\pmod{33}$", "final_answer": "x\\equiv 7\\pmod{33}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "85faff10d4f25be8", "problem_id": "a9fbf785c2dd00bc", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡7 (mod 11). Give the solution modulo 33.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 33.\n3) Compute the CRT solution to get x ≡ 7 (mod 33).\nFinal answer: $x\\equiv 7\\pmod{33}$", "final_answer": "x\\equiv 7\\pmod{33}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4fd593b2c9948ceb", "problem_id": "e3a0bc1b8fd624ab", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A store sells a backpack for $44. It is discounted by 15%. You buy 6 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 44 × (1 − 15/100) = 44 × 85/100 = 187/5.\n2) Subtotal = 6 × 187/5 = 1122/5.\n3) Total with tax = 1122/5 × 107/100 = 60027/250.\nFinal answer: $\\frac{60027}{250}$", "final_answer": "\\frac{60027}{250}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8ccf69d4e09a1ae2", "problem_id": "e3a0bc1b8fd624ab", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $44. It is discounted by 15%. You buy 6 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 44 × (1 − 15/100) = 44 × 85/100 = 187/5.\nSubtotal = 6 × 187/5 = 1122/5.\nTotal with tax = 1122/5 × 107/100 = 60027/250.\nCheck: recomputing the formula yields 60027/250.\nFinal answer: $\\frac{60027}{250}$", "final_answer": "\\frac{60027}{250}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "82e8c3f87c096f2b", "problem_id": "9383260f3166ef13", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "How many binary strings of length 13 contain exactly 6 ones with no two ones adjacent?", "output": "1) Reserve 5 mandatory zeros between the 6 ones.\n2) This leaves n−k+1 = 8 valid slots for the ones.\n3) Choose the slots: C(8,6) = 28.\nFinal answer: $28$", "final_answer": "28", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d3769a78ac437eca", "problem_id": "9383260f3166ef13", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many binary strings of length 13 contain exactly 6 ones with no two ones adjacent?", "output": "Reserve 5 mandatory zeros between the 6 ones.\nThis leaves n−k+1 = 8 valid slots for the ones.\nChoose the slots: C(8,6) = 28.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $28$", "final_answer": "28", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5e167e29268bc3ec", "problem_id": "2385d88659fcc2d7", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "A cyclist rides at 72 miles per hour for 1/2 hours, then at 82 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 72 × 1/2 = 36 miles.\nSecond leg distance = 82 × 1 = 82 miles.\nTotal distance = 36 + 82 = 118 miles.\nFinal answer: $118$", "final_answer": "118", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "113b0ac29fec850c", "problem_id": "2385d88659fcc2d7", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A cyclist rides at 72 miles per hour for 1/2 hours, then at 82 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 72 × 1/2 = 36 miles.\n2) Second leg distance = 82 × 1 = 82 miles.\n3) Total distance = 36 + 82 = 118 miles.\nFinal answer: $118$", "final_answer": "118", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "456c8b893771331a", "problem_id": "840f5b00fa794293", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} + 3 x^{2} + 2 \\sin{\\left(x \\right)}$ at $x=0$.", "output": "1) Differentiate: f'(x) = $6 x^{2} + 6 x + 2 \\cos{\\left(x \\right)}$.\n2) Slope at x=0: m = f'(0) = $2$.\n3) Point on curve: (0, f(0)) = (0, $0$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 0 = 2(x − 0).\n6) Optionally expand: y = $2 x$.\nFinal answer: $y = 2\\cdot x$", "final_answer": "y = 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d504506fda34b8e", "problem_id": "9fd93e05f75ea9dd", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(2 x^{2} + 2 x - 3\\right) \\cos{\\left(3 x^{2} + 2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x \\cos{\\left(3 x^{2} + 2 x \\right)}) = - 2 x \\sin{\\left(x \\left(3 x + 2\\right) \\right)}$.\n3) So $f'(x) = - 12 x^{3} \\sin{\\left(3 x^{2} + 2 x \\right)} - 4 x^{2} \\sin{\\left(3 x^{2} + 2 x \\right)} + x e^{x} - 2 x \\sin{\\left(3 x^{2} + 2 x \\right)} + 18 x \\sin{\\left(3 x^{2} + 2 x \\right)} + 4 x \\cos{\\left(3 x^{2} + 2 x \\right)} + 2 e^{x} + 6 \\sin{\\left(3 x^{2} + 2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x \\cos{\\left(3 x^{2} + 2 x \\right)}) = - 2 x \\left(6 x + 2\\right) \\sin{\\left(3 x^{2} + 2 x \\right)} + 2 \\cos{\\left(3 x^{2} + 2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 1\\right) e^{x} + 2 \\left(2 x + 1\\right) \\cos{\\left(3 x^{2} + 2 x \\right)} - 2 \\left(3 x + 1\\right) \\left(2 x \\left(x + 1\\right) - 3\\right) \\sin{\\left(3 x^{2} + 2 x \\right)} + e^{x}$.\nFinal answer: $\\left(x + 1\\right) e^{x} + 2 \\left(2 x + 1\\right) \\cos{\\left(3 x^{2} + 2 x \\right)} - 2 \\left(3 x + 1\\right) \\left(2 x \\left(x + 1\\right) - 3\\right) \\sin{\\left(3 x^{2} + 2 x \\right)} + e^{x}$", "final_answer": "\\left(x + 1\\right) e^{x} + 2 \\left(2 x + 1\\right) \\cos{\\left(3 x^{2} + 2 x \\right)} - 2 \\left(3 x + 1\\right) \\left(2 x \\left(x + 1\\right) - 3\\right) \\sin{\\left(3 x^{2} + 2 x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "eaea8f2134b6ab85", "problem_id": "514b551d20f212c4", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A cyclist rides at 55 miles per hour for 1/2 hours, then at 88 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 55 × 1/2 = 55/2 miles.\n2) Second leg distance = 88 × 3/2 = 132 miles.\n3) Total distance = 55/2 + 132 = 319/2 miles.\nFinal answer: $\\frac{319}{2}$", "final_answer": "\\frac{319}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b301e73b07d78326", "problem_id": "514b551d20f212c4", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "A cyclist rides at 55 miles per hour for 1/2 hours, then at 88 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 55 × 1/2 = 55/2 miles.\nSecond leg distance = 88 × 3/2 = 132 miles.\nTotal distance = 55/2 + 132 = 319/2 miles.\nFinal answer: $\\frac{319}{2}$", "final_answer": "\\frac{319}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b6cc133f6724ff49", "problem_id": "88a793bd78eec3b7", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the congruences: x≡4 (mod 7), x≡8 (mod 11). Give the solution modulo 77.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 77.\nCompute the CRT solution to get x ≡ 74 (mod 77).\nFinal answer: $x\\equiv 74\\pmod{77}$", "final_answer": "x\\equiv 74\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8ff3816320b54bb7", "problem_id": "88a793bd78eec3b7", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡4 (mod 7), x≡8 (mod 11). Give the solution modulo 77.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 77.\n3) Compute the CRT solution to get x ≡ 74 (mod 77).\nFinal answer: $x\\equiv 74\\pmod{77}$", "final_answer": "x\\equiv 74\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d32dd02e9fef6f05", "problem_id": "641f58c4ccfc7ef2", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3$.\nCompute an antiderivative: F(x) = $x^{3} + x^{2} - 3 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + x^{3} + x^{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + x^{3} + x^{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8ce9d063999309ed", "problem_id": "641f58c4ccfc7ef2", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3$.\n3) Compute an antiderivative: F(x) = $x^{3} + x^{2} - 3 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + x^{3} + x^{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + x^{3} + x^{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "305e7124cadd616e", "problem_id": "21ac539ddd4fae05", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} - 2 x^{4} - 4 x^{3} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{4} - 4 x^{3} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\nEvaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{33 \\pi^{5}}{80} - \\frac{15 \\pi^{4}}{16} - \\frac{9 \\pi}{2} - 1 + \\frac{3 \\pi^{2}}{2}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{33\\cdot \\pi^{5}}{80} - \\frac{15\\cdot \\pi^{4}}{16} - \\frac{9\\cdot \\pi}{2} - 1 + \\frac{3\\cdot \\pi^{2}}{2}$", "final_answer": "- \\frac{33\\cdot \\pi^{5}}{80} - \\frac{15\\cdot \\pi^{4}}{16} - \\frac{9\\cdot \\pi}{2} - 1 + \\frac{3\\cdot \\pi^{2}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "012ba88916d4fa08", "problem_id": "21ac539ddd4fae05", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} - 2 x^{4} - 4 x^{3} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{4} - 4 x^{3} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{33 \\pi^{5}}{80} - \\frac{15 \\pi^{4}}{16} - \\frac{9 \\pi}{2} - 1 + \\frac{3 \\pi^{2}}{2}$.\n4) Simplify the result.\nFinal answer: $- \\frac{33\\cdot \\pi^{5}}{80} - \\frac{15\\cdot \\pi^{4}}{16} - \\frac{9\\cdot \\pi}{2} - 1 + \\frac{3\\cdot \\pi^{2}}{2}$", "final_answer": "- \\frac{33\\cdot \\pi^{5}}{80} - \\frac{15\\cdot \\pi^{4}}{16} - \\frac{9\\cdot \\pi}{2} - 1 + \\frac{3\\cdot \\pi^{2}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6043c0c227f5c4d6", "problem_id": "feab8184270c736e", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many binary strings of length 23 contain exactly 6 ones with no two ones adjacent?", "output": "1) Reserve 5 mandatory zeros between the 6 ones.\n2) This leaves n−k+1 = 18 valid slots for the ones.\n3) Choose the slots: C(18,6) = 18564.\nFinal answer: $18564$", "final_answer": "18564", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e1ba6fd52898511a", "problem_id": "feab8184270c736e", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many binary strings of length 23 contain exactly 6 ones with no two ones adjacent?", "output": "Reserve 5 mandatory zeros between the 6 ones.\nThis leaves n−k+1 = 18 valid slots for the ones.\nChoose the slots: C(18,6) = 18564.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $18564$", "final_answer": "18564", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5e6bfbeb22c80ae1", "problem_id": "c42b7feffb10bcfe", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 7 x + 12$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} - 7 x + 12$ = $\\left(x - 4\\right) \\left(x - 3\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $4$.\n5) From the second factor: x = $3$.\nFinal answer: $x \\in \\{4, 3\\}$", "final_answer": "x \\in \\{4, 3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "790012df27d01ceb", "problem_id": "c42b7feffb10bcfe", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 7 x + 12$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} - 7 x + 12$ = $\\left(x - 4\\right) \\left(x - 3\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $4$.\nFrom the second factor: x = $3$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{4, 3\\}$", "final_answer": "x \\in \\{4, 3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "65a227fe7ba44ee0", "problem_id": "4154d331602d7bf5", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} 3 x^{2} + 4 x + 3$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{2} + 3 x + 3.\n3) Then the limit is 9.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: 10.\nFinal answer: $10$", "final_answer": "10", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "61aca20b50acf71d", "problem_id": "c0eac820ed13437c", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} - 3 x^{2} + 10 x - 40$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 3 x^{3} - 2 x^{2} - 3$ and a = -4.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 9 x^{2} - 4 x$.\nEvaluate at a: p'(-4) = $-128$.\nCheck: the identity gives $-128$.\nFinal answer: $-128$", "final_answer": "-128", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6c662345ceb74f71", "problem_id": "e5c5e56c8a3df964", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A cyclist rides at 69 miles per hour for 3/2 hours, then at 89 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 69 × 3/2 = 207/2 miles.\n2) Second leg distance = 89 × 2 = 178 miles.\n3) Total distance = 207/2 + 178 = 563/2 miles.\nFinal answer: $\\frac{563}{2}$", "final_answer": "\\frac{563}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db1d8de79155dffd", "problem_id": "e5c5e56c8a3df964", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A cyclist rides at 69 miles per hour for 3/2 hours, then at 89 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 69 × 3/2 = 207/2 miles.\nSecond leg distance = 89 × 2 = 178 miles.\nTotal distance = 207/2 + 178 = 563/2 miles.\nCheck: adding both legs gives 563/2.\nFinal answer: $\\frac{563}{2}$", "final_answer": "\\frac{563}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "65baf0fa81a4a28e", "problem_id": "a6a606518c323b8d", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "You want 50 liters of a 27% salt solution. You have a 15% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 50−x liters is 40% solution.\n2) Salt equation: 15x + 40(50−x) = 27·50.\n3) (15-40)x = (27-40)·50 ⇒ x = 26.\n4) Then 50−x = 24.\nFinal answer: $26 L of 15% and 24 L of 40%$", "final_answer": "26 L of 15% and 24 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2c433eb79eefc023", "problem_id": "a6a606518c323b8d", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "You want 50 liters of a 27% salt solution. You have a 15% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 50−x liters is 40% solution.\nSalt equation: 15x + 40(50−x) = 27·50.\n(15-40)x = (27-40)·50 ⇒ x = 26.\nFinal answer: $26 L of 15% and 24 L of 40%$", "final_answer": "26 L of 15% and 24 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "33910f441586c7bd", "problem_id": "cd27b84d77044cfb", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A store sells a backpack for $64. It is discounted by 35%. You buy 14 backpacks. What is the total cost?", "output": "1) Discounted price = 64 × (1 − 35/100) = 64 × 65/100 = 208/5.\n2) Subtotal = 14 × 208/5 = 2912/5.\nFinal answer: $\\frac{2912}{5}$", "final_answer": "\\frac{2912}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4249e5de4058ada2", "problem_id": "cd27b84d77044cfb", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $64. It is discounted by 35%. You buy 14 backpacks. What is the total cost?", "output": "Discounted price = 64 × (1 − 35/100) = 64 × 65/100 = 208/5.\nSubtotal = 14 × 208/5 = 2912/5.\nCheck: recomputing the formula yields 2912/5.\nFinal answer: $\\frac{2912}{5}$", "final_answer": "\\frac{2912}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "56be9178008f4b92", "problem_id": "9c7c6fa3ef3bab50", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the multiplicative inverse of 16 modulo 37.", "output": "1) We want x such that 16x ≡ 1 (mod 37).\n2) Use extended Euclid to find s,t with 16s + 37t = 1.\n3) One Bezout identity is: 16·(7) + 37·(-3) = 1.\n4) Reduce s modulo 37: x ≡ 7 (mod 37).\nFinal answer: $x\\equiv 7\\pmod{37}$", "final_answer": "x\\equiv 7\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7e823580fdce44dc", "problem_id": "9c7c6fa3ef3bab50", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the multiplicative inverse of 16 modulo 37.", "output": "We want x such that 16x ≡ 1 (mod 37).\nUse extended Euclid to find s,t with 16s + 37t = 1.\nOne Bezout identity is: 16·(7) + 37·(-3) = 1.\nReduce s modulo 37: x ≡ 7 (mod 37).\nCheck: 16·7 mod 37 = 1.\nFinal answer: $x\\equiv 7\\pmod{37}$", "final_answer": "x\\equiv 7\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "054b35850175de49", "problem_id": "c2516c611d74dc4f", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{3} + x^{2}$.", "output": "Differentiate term-by-term:\nd/dx(x^{2}) = 2 x.\nd/dx(4 x^{3}) = 12 x^{2}.\nFinal answer: $2\\cdot x\\cdot \\left(6\\cdot x + 1\\right)$", "final_answer": "2\\cdot x\\cdot \\left(6\\cdot x + 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dca62d13cd7a38cf", "problem_id": "c2516c611d74dc4f", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{3} + x^{2}$.", "output": "Differentiate term-by-term:\nd/dx(x^{2}) = 2 x.\nd/dx(4 x^{3}) = 12 x^{2}.\nCombine: f'(x) = 2 x \\left(6 x + 1\\right).\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $2\\cdot x\\cdot \\left(6\\cdot x + 1\\right)$", "final_answer": "2\\cdot x\\cdot \\left(6\\cdot x + 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff8482a28efe9761", "problem_id": "77059cff0e0e9a9a", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} x^{2} + 2$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{3} - x^{2} + 2 x - 2$ and a = 1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $3 x^{2} - 2 x + 2$.\n5) Evaluate at a: p'(1) = $3$.\nFinal answer: $3$", "final_answer": "3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a0dc3e8692dbe58", "problem_id": "77059cff0e0e9a9a", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} x^{2} + 2$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $x^{3} - x^{2} + 2 x - 2$ and a = 1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $3$", "final_answer": "3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dbd3f5245ef97166", "problem_id": "b8464a35d11fd0f2", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$4 x - 4 y$ = $-28$\n$- 3 x + 4 y$ = $24$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $- 12 x + 12 y$ = $84$.\n3) Multiply the second equation by $4$: $- 12 x + 16 y$ = $96$.\n4) Subtract to eliminate x: $- 4 y$ = $-12$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $-4$.\nFinal answer: $(x,y)=\\left(-4, 3\\right)$", "final_answer": "(x,y)=\\left(-4, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "31cb8fb89e69bf75", "problem_id": "a2834fb9e51a07b3", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - x^{3} + x + 5$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to 5 - x^{3}.\n3) Then the limit is -22.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: -19.\nFinal answer: $-19$", "final_answer": "-19", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ce1a2dc8cb89d133", "problem_id": "3328f8856b597562", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A cyclist rides at 42 miles per hour for 1 hours, then at 89 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 42 × 1 = 42 miles.\n2) Second leg distance = 89 × 2 = 178 miles.\n3) Total distance = 42 + 178 = 220 miles.\nFinal answer: $220$", "final_answer": "220", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "51836d7b648439d5", "problem_id": "3328f8856b597562", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A cyclist rides at 42 miles per hour for 1 hours, then at 89 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 42 × 1 = 42 miles.\nSecond leg distance = 89 × 2 = 178 miles.\nTotal distance = 42 + 178 = 220 miles.\nCheck: adding both legs gives 220.\nFinal answer: $220$", "final_answer": "220", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cb10e5b7e82934cb", "problem_id": "cc810424ab1a13a1", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A cyclist rides at 89 miles per hour for 1/2 hours, then at 51 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 89 × 1/2 = 89/2 miles.\n2) Second leg distance = 51 × 3 = 153 miles.\n3) Total distance = 89/2 + 153 = 395/2 miles.\nFinal answer: $\\frac{395}{2}$", "final_answer": "\\frac{395}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4f294c0a6bbe7ebb", "problem_id": "cc810424ab1a13a1", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A cyclist rides at 89 miles per hour for 1/2 hours, then at 51 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 89 × 1/2 = 89/2 miles.\nSecond leg distance = 51 × 3 = 153 miles.\nTotal distance = 89/2 + 153 = 395/2 miles.\nCheck: adding both legs gives 395/2.\nFinal answer: $\\frac{395}{2}$", "final_answer": "\\frac{395}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "65a227fe7ba44ee0", "problem_id": "4154d331602d7bf5", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} 3 x^{2} + 4 x + 3$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{2} + 3 x + 3.\n3) Then the limit is 9.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: 10.\nFinal answer: $10$", "final_answer": "10", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "9d16c15460f32535", "problem_id": "66a9764113887ef3", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{7} x + \\frac{3 x}{10} - \\frac{4 \\sqrt{7}}{5} - \\frac{2}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{3}{10} - \\sqrt{7}$, c=$- \\frac{4 \\sqrt{7}}{5} - \\frac{2}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{13 \\sqrt{7}}{5} + \\frac{869}{100}$, so √Δ = $\\sqrt{\\frac{13 \\sqrt{7}}{5} + \\frac{869}{100}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{3}{20} + \\frac{\\sqrt{7}}{2} + \\frac{\\sqrt{260 \\sqrt{7} + 869}}{20}$.\n7) x2 = $- \\frac{\\sqrt{260 \\sqrt{7} + 869}}{20} - \\frac{3}{20} + \\frac{\\sqrt{7}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{3}{20} + \\frac{\\sqrt{7}}{2} + \\frac{\\sqrt{260\\cdot \\sqrt{7} + 869}}{20}, - \\frac{\\sqrt{260\\cdot \\sqrt{7} + 869}}{20} - \\frac{3}{20} + \\frac{\\sqrt{7}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{3}{20} + \\frac{\\sqrt{7}}{2} + \\frac{\\sqrt{260\\cdot \\sqrt{7} + 869}}{20}, - \\frac{\\sqrt{260\\cdot \\sqrt{7} + 869}}{20} - \\frac{3}{20} + \\frac{\\sqrt{7}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b36e325d2e4386fb", "problem_id": "d987c1b18c55fcf3", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 12 times, what is the probability of exactly 9 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=9, p=$\\frac{1}{2}$ and simplify.\n4) Here C(12,9) = 220.\nFinal answer: $\\frac{55}{1024}$", "final_answer": "\\frac{55}{1024}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bc6f6389f9d060d7", "problem_id": "d987c1b18c55fcf3", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 12 times, what is the probability of exactly 9 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=12, k=9, p=$\\frac{1}{2}$ and simplify.\nHere C(12,9) = 220.\nCheck: expression simplifies to $\\frac{55}{1024}$.\nFinal answer: $\\frac{55}{1024}$", "final_answer": "\\frac{55}{1024}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "158964974687a579", "problem_id": "94cc202cfa9e6cc9", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Worker A finishes a job in 10 hours and Worker B finishes it in 17 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/10 job/hour. B's rate = 1/17 job/hour.\nCombined rate = 1/10 + 1/17 = 27/170 job/hour.\nTime = 1 ÷ (combined rate) = 170/27 hours.\nCheck: (combined rate)×(time) = 27/170×170/27 = 1.\nFinal answer: $\\frac{170}{27}$", "final_answer": "\\frac{170}{27}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c12ab7fd1af1ca45", "problem_id": "94cc202cfa9e6cc9", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Worker A finishes a job in 10 hours and Worker B finishes it in 17 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/10 job/hour. B's rate = 1/17 job/hour.\n2) Combined rate = 1/10 + 1/17 = 27/170 job/hour.\n3) Time = 1 ÷ (combined rate) = 170/27 hours.\nFinal answer: $\\frac{170}{27}$", "final_answer": "\\frac{170}{27}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0fa96e9bf3624c54", "problem_id": "038c4cc970d93461", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A store sells a backpack for $93. It is discounted by 5%. You buy 4 backpacks. What is the total cost?", "output": "1) Discounted price = 93 × (1 − 5/100) = 93 × 95/100 = 1767/20.\n2) Subtotal = 4 × 1767/20 = 1767/5.\nFinal answer: $\\frac{1767}{5}$", "final_answer": "\\frac{1767}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e91407b190149f1c", "problem_id": "038c4cc970d93461", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A store sells a backpack for $93. It is discounted by 5%. You buy 4 backpacks. What is the total cost?", "output": "Discounted price = 93 × (1 − 5/100) = 93 × 95/100 = 1767/20.\nSubtotal = 4 × 1767/20 = 1767/5.\nCheck: recomputing the formula yields 1767/5.\nFinal answer: $\\frac{1767}{5}$", "final_answer": "\\frac{1767}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6ce2804d2bbb4648", "problem_id": "f2af85909ca93b55", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the linear system:\n$- 4 y$ = $20$\n$- 3 x + y$ = $7$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $12 y$ = $-60$.\n3) Multiply the second equation by $0$: $0$ = $0$.\n4) Subtract to eliminate x: $12 y$ = $-60$.\n5) Solve for y: y = $-5$.\n6) Substitute back to find x: x = $-4$.\nFinal answer: $(x,y)=\\left(-4, -5\\right)$", "final_answer": "(x,y)=\\left(-4, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3074f86d73f55fc3", "problem_id": "f2af85909ca93b55", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the linear system:\n$- 4 y$ = $20$\n$- 3 x + y$ = $7$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $12 y$ = $-60$.\nMultiply the second equation by $0$: $0$ = $0$.\nFinal answer: $(x,y)=\\left(-4, -5\\right)$", "final_answer": "(x,y)=\\left(-4, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2072de94bd179f65", "problem_id": "2e20399cc565a8b0", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the equation of the tangent line to $y=x^{3} + 2 x^{2} - 3 x - 4$ at $x=2$.", "output": "1) Differentiate: f'(x) = $3 x^{2} + 4 x - 3$.\n2) Slope at x=2: m = f'(2) = $17$.\n3) Point on curve: (2, f(2)) = (2, $6$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 6 = 17(x − 2).\n6) Optionally expand: y = $17 x - 28$.\nFinal answer: $y = 17\\cdot x - 28$", "final_answer": "y = 17\\cdot x - 28", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0601de419fba831b", "problem_id": "2e20399cc565a8b0", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Find the equation of the tangent line to $y=x^{3} + 2 x^{2} - 3 x - 4$ at $x=2$.", "output": "$y = 17\\cdot x - 28$", "final_answer": "y = 17\\cdot x - 28", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ae166767c0f90772", "problem_id": "b7765bffbe816811", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - 2 x^{3} + 3 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{3} + 3 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{255 \\pi^{4}}{32} - \\frac{5 \\pi}{2} - 1 + \\frac{15 \\pi^{2}}{8} + \\frac{65 \\pi^{3}}{8}$.\nFinal answer: $- \\frac{255\\cdot \\pi^{4}}{32} - \\frac{5\\cdot \\pi}{2} - 1 + \\frac{15\\cdot \\pi^{2}}{8} + \\frac{65\\cdot \\pi^{3}}{8}$", "final_answer": "- \\frac{255\\cdot \\pi^{4}}{32} - \\frac{5\\cdot \\pi}{2} - 1 + \\frac{15\\cdot \\pi^{2}}{8} + \\frac{65\\cdot \\pi^{3}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db49badc51587ff2", "problem_id": "b7765bffbe816811", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - 2 x^{3} + 3 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{3} + 3 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{255 \\pi^{4}}{32} - \\frac{5 \\pi}{2} - 1 + \\frac{15 \\pi^{2}}{8} + \\frac{65 \\pi^{3}}{8}$.\n4) Simplify the result.\nFinal answer: $- \\frac{255\\cdot \\pi^{4}}{32} - \\frac{5\\cdot \\pi}{2} - 1 + \\frac{15\\cdot \\pi^{2}}{8} + \\frac{65\\cdot \\pi^{3}}{8}$", "final_answer": "- \\frac{255\\cdot \\pi^{4}}{32} - \\frac{5\\cdot \\pi}{2} - 1 + \\frac{15\\cdot \\pi^{2}}{8} + \\frac{65\\cdot \\pi^{3}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9bf3de0ec2c1f9f1", "problem_id": "00aefb28f0918cca", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many binary strings of length 20 contain exactly 7 ones with no two ones adjacent?", "output": "1) Reserve 6 mandatory zeros between the 7 ones.\n2) This leaves n−k+1 = 14 valid slots for the ones.\n3) Choose the slots: C(14,7) = 3432.\nFinal answer: $3432$", "final_answer": "3432", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0f9b2b7f564fabb4", "problem_id": "00aefb28f0918cca", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many binary strings of length 20 contain exactly 7 ones with no two ones adjacent?", "output": "Reserve 6 mandatory zeros between the 7 ones.\nThis leaves n−k+1 = 14 valid slots for the ones.\nChoose the slots: C(14,7) = 3432.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $3432$", "final_answer": "3432", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c67fca5e186b0775", "problem_id": "c8c636c28643c4dd", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "How many binary strings of length 18 contain exactly 3 ones with no two ones adjacent?", "output": "1) Reserve 2 mandatory zeros between the 3 ones.\n2) This leaves n−k+1 = 16 valid slots for the ones.\n3) Choose the slots: C(16,3) = 560.\nFinal answer: $560$", "final_answer": "560", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "29ac1881f2fcb339", "problem_id": "c8c636c28643c4dd", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many binary strings of length 18 contain exactly 3 ones with no two ones adjacent?", "output": "Reserve 2 mandatory zeros between the 3 ones.\nThis leaves n−k+1 = 16 valid slots for the ones.\nChoose the slots: C(16,3) = 560.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $560$", "final_answer": "560", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4f072fd4a4c7a2f4", "problem_id": "29376acc3fb537aa", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} x^{3} - x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $x^{3} - x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $\\frac{3 \\pi \\left(- 4 \\pi^{2} - 2 \\pi + 4 + 5 \\pi^{3}\\right)}{4}$.\nFinal answer: $\\frac{3\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} - 2\\cdot \\pi + 4 + 5\\cdot \\pi^{3}\\right)}{4}$", "final_answer": "\\frac{3\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} - 2\\cdot \\pi + 4 + 5\\cdot \\pi^{3}\\right)}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "30731933fcbcab6e", "problem_id": "29376acc3fb537aa", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} x^{3} - x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $x^{3} - x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $\\frac{3 \\pi \\left(- 4 \\pi^{2} - 2 \\pi + 4 + 5 \\pi^{3}\\right)}{4}$.\n4) Simplify the result.\nFinal answer: $\\frac{3\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} - 2\\cdot \\pi + 4 + 5\\cdot \\pi^{3}\\right)}{4}$", "final_answer": "\\frac{3\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} - 2\\cdot \\pi + 4 + 5\\cdot \\pi^{3}\\right)}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "00e571417590a412", "problem_id": "5ac93fcc353a21d2", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "You want 53 liters of a 25% salt solution. You have a 15% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 53−x liters is 40% solution.\n2) Salt equation: 15x + 40(53−x) = 25·53.\n3) (15-40)x = (25-40)·53 ⇒ x = 159/5.\n4) Then 53−x = 106/5.\nFinal answer: $159/5 L of 15% and 106/5 L of 40%$", "final_answer": "159/5 L of 15% and 106/5 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e41200ecb4fddec", "problem_id": "5ac93fcc353a21d2", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "You want 53 liters of a 25% salt solution. You have a 15% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 53−x liters is 40% solution.\nSalt equation: 15x + 40(53−x) = 25·53.\n(15-40)x = (25-40)·53 ⇒ x = 159/5.\nThen 53−x = 106/5.\nCheck: 15·159/5 + 40·106/5 = 25·53.\nFinal answer: $159/5 L of 15% and 106/5 L of 40%$", "final_answer": "159/5 L of 15% and 106/5 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aa08856e8a31a880", "problem_id": "e749c64b4c9524c1", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} - 2 x^{3} - x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "$\\frac{\\pi\\cdot \\left(-36 - \\pi^{2}\\right)}{12}$", "final_answer": "\\frac{\\pi\\cdot \\left(-36 - \\pi^{2}\\right)}{12}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "29cee9bab2b059a2", "problem_id": "e749c64b4c9524c1", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} - 2 x^{3} - x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{3} - x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $\\frac{\\pi \\left(-36 - \\pi^{2}\\right)}{12}$.\n4) Simplify the result.\nFinal answer: $\\frac{\\pi\\cdot \\left(-36 - \\pi^{2}\\right)}{12}$", "final_answer": "\\frac{\\pi\\cdot \\left(-36 - \\pi^{2}\\right)}{12}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "151713cae4bd0b1a", "problem_id": "6034589e77e802f5", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A disease has prevalence $\\frac{1}{200}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{27}{500}$.\nCompute P(D|+) = $\\frac{17}{216}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{17}{216}$", "final_answer": "\\frac{17}{216}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5af865e8367fdf54", "problem_id": "6034589e77e802f5", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A disease has prevalence $\\frac{1}{200}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{27}{500}$.\n3) Compute P(D|+) = $\\frac{17}{216}$.\nFinal answer: $\\frac{17}{216}$", "final_answer": "\\frac{17}{216}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d69e0b3df03fbc2a", "problem_id": "5f1d52bcce33dad5", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 \\sin{\\left(x \\right)}) = 2 \\cos{\\left(x \\right)}$.\n3) So $f'(x) = 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 \\sin{\\left(x \\right)}) = 2 \\cos{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$", "final_answer": "3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f203378d79b23a9a", "problem_id": "867f40197fabebcd", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{2} + x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $2 x^{2} + x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 1$.\nCompute an antiderivative: F(x) = $\\frac{2 x^{3}}{3} + \\frac{x^{2}}{2} + x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{2\\cdot x^{3}}{3} + \\frac{x^{2}}{2} + x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{2\\cdot x^{3}}{3} + \\frac{x^{2}}{2} + x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3121d85099947862", "problem_id": "0fc8045e26442222", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{4} - 4 x^{3} - x^{2} - x + 1$.", "output": "1) Differentiate term-by-term:\n2) d/dx(1) = 0.\n3) d/dx(- x) = -1.\n4) d/dx(- x^{2}) = - 2 x.\n5) d/dx(- 4 x^{3}) = - 12 x^{2}.\n6) d/dx(- 2 x^{4}) = - 8 x^{3}.\n7) Combine: f'(x) = - 8 x^{3} - 12 x^{2} - 2 x - 1.\nFinal answer: $- 8\\cdot x^{3} - 12\\cdot x^{2} - 2\\cdot x - 1$", "final_answer": "- 8\\cdot x^{3} - 12\\cdot x^{2} - 2\\cdot x - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "135ef94e78d0cdfb", "problem_id": "0fc8045e26442222", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{4} - 4 x^{3} - x^{2} - x + 1$.", "output": "Differentiate term-by-term:\nd/dx(1) = 0.\nd/dx(- x) = -1.\nFinal answer: $- 8\\cdot x^{3} - 12\\cdot x^{2} - 2\\cdot x - 1$", "final_answer": "- 8\\cdot x^{3} - 12\\cdot x^{2} - 2\\cdot x - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "40a262466b5b4cef", "problem_id": "2734c1e71f879866", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "You want 36 liters of a 35% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 36−x liters is 70% solution.\n2) Salt equation: 20x + 70(36−x) = 35·36.\n3) (20-70)x = (35-70)·36 ⇒ x = 126/5.\n4) Then 36−x = 54/5.\nFinal answer: $126/5 L of 20% and 54/5 L of 70%$", "final_answer": "126/5 L of 20% and 54/5 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f1ae01fc17f28bc9", "problem_id": "2734c1e71f879866", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "You want 36 liters of a 35% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 36−x liters is 70% solution.\nSalt equation: 20x + 70(36−x) = 35·36.\n(20-70)x = (35-70)·36 ⇒ x = 126/5.\nThen 36−x = 54/5.\nCheck: 20·126/5 + 70·54/5 = 35·36.\nFinal answer: $126/5 L of 20% and 54/5 L of 70%$", "final_answer": "126/5 L of 20% and 54/5 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a4bc29f30d94aa14", "problem_id": "409b9bdda778c70d", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Two friends split $595 in the ratio 1:6. How much does each receive?", "output": "Total ratio parts = 1+6 = 7.\nEach part = 595 ÷ 7 = 85.\nFirst share = 1 × 85 = 85.\nSecond share = 6 × 85 = 510.\nCheck: 85+510=595.\nFinal answer: $510$", "final_answer": "510", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "30d5d62921216b3f", "problem_id": "409b9bdda778c70d", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Two friends split $595 in the ratio 1:6. How much does each receive?", "output": "Total ratio parts = 1+6 = 7.\nEach part = 595 ÷ 7 = 85.\nFirst share = 1 × 85 = 85.\nFinal answer: $510$", "final_answer": "510", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a682fcbb3230904", "problem_id": "f018355b7ee08b26", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 3 x^{2} - 1$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} - x - 1.\n3) Then the limit is -3.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: -4.\nFinal answer: $-4$", "final_answer": "-4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "9480ad590ecdfc9b", "problem_id": "ca40ba2151cf369e", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 22 x + 48$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} + 22 x + 48$ = $2 \\left(x + 3\\right) \\left(x + 8\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $-3$.\nFrom the second factor: x = $-8$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-3, -8\\}$", "final_answer": "x \\in \\{-3, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f382d9dc017b75ce", "problem_id": "b2fc421f387185e6", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{10} x}{2} - \\frac{9 x}{2} + 5 + 3 \\sqrt{10}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{3 \\sqrt{10}}{2} - \\frac{9}{2}$, c=$5 + 3 \\sqrt{10}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{3 \\sqrt{10}}{2} + \\frac{91}{4}$, so √Δ = $\\sqrt{\\frac{3 \\sqrt{10}}{2} + \\frac{91}{4}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{9}{4} + \\frac{3 \\sqrt{10}}{4} + \\frac{\\sqrt{6 \\sqrt{10} + 91}}{4}$.\n7) x2 = $- \\frac{\\sqrt{6 \\sqrt{10} + 91}}{4} + \\frac{9}{4} + \\frac{3 \\sqrt{10}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{9}{4} + \\frac{3\\cdot \\sqrt{10}}{4} + \\frac{\\sqrt{6\\cdot \\sqrt{10} + 91}}{4}, - \\frac{\\sqrt{6\\cdot \\sqrt{10} + 91}}{4} + \\frac{9}{4} + \\frac{3\\cdot \\sqrt{10}}{4}\\}$", "final_answer": "x \\in \\{\\frac{9}{4} + \\frac{3\\cdot \\sqrt{10}}{4} + \\frac{\\sqrt{6\\cdot \\sqrt{10} + 91}}{4}, - \\frac{\\sqrt{6\\cdot \\sqrt{10} + 91}}{4} + \\frac{9}{4} + \\frac{3\\cdot \\sqrt{10}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4f30d892c8af4a6d", "problem_id": "b2fc421f387185e6", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{10} x}{2} - \\frac{9 x}{2} + 5 + 3 \\sqrt{10}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{3 \\sqrt{10}}{2} - \\frac{9}{2}$, c=$5 + 3 \\sqrt{10}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{3 \\sqrt{10}}{2} + \\frac{91}{4}$, so √Δ = $\\sqrt{\\frac{3 \\sqrt{10}}{2} + \\frac{91}{4}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{9}{4} + \\frac{3 \\sqrt{10}}{4} + \\frac{\\sqrt{6 \\sqrt{10} + 91}}{4}$.\nx2 = $- \\frac{\\sqrt{6 \\sqrt{10} + 91}}{4} + \\frac{9}{4} + \\frac{3 \\sqrt{10}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{9}{4} + \\frac{3\\cdot \\sqrt{10}}{4} + \\frac{\\sqrt{6\\cdot \\sqrt{10} + 91}}{4}, - \\frac{\\sqrt{6\\cdot \\sqrt{10} + 91}}{4} + \\frac{9}{4} + \\frac{3\\cdot \\sqrt{10}}{4}\\}$", "final_answer": "x \\in \\{\\frac{9}{4} + \\frac{3\\cdot \\sqrt{10}}{4} + \\frac{\\sqrt{6\\cdot \\sqrt{10} + 91}}{4}, - \\frac{\\sqrt{6\\cdot \\sqrt{10} + 91}}{4} + \\frac{9}{4} + \\frac{3\\cdot \\sqrt{10}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "56ea5efc5f327d49", "problem_id": "19d9151e5849447e", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_5=25$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_5=15.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=15 and k=5.\n4) Compute C(19,4) = 3876.\nFinal answer: $3876$", "final_answer": "3876", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b2df19d578e26999", "problem_id": "3451b6d8ff262c62", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\cos{\\left(2 x \\right)}) = - \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = - x^{3} \\cos{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\cos{\\left(2 x \\right)}) = - 2 \\sin{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = - x^{3} \\cos{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.\nFinal answer: $- x^{3} \\cos{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$", "final_answer": "- x^{3} \\cos{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b66997fcd274fe37", "problem_id": "dbac351134c8e764", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "An urn has 11 red balls and 24 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(35,11).\nFavorable samples: choose k reds and n−k blues: C(11,5)·C(24,6).\nFinal answer: $\\frac{740278}{4966975}$", "final_answer": "\\frac{740278}{4966975}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6dbcc867655631cb", "problem_id": "dbac351134c8e764", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 11 red balls and 24 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(35,11).\nFavorable samples: choose k reds and n−k blues: C(11,5)·C(24,6).\nCompute totals: C(35,11) = 417225900, and favorable = 62183352.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{740278}{4966975}$.\nFinal answer: $\\frac{740278}{4966975}$", "final_answer": "\\frac{740278}{4966975}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "99e3a47ebccf8574", "problem_id": "2d0a9a6fbaac81f2", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "How many binary strings of length 19 contain exactly 5 ones with no two ones adjacent?", "output": "1) Reserve 4 mandatory zeros between the 5 ones.\n2) This leaves n−k+1 = 15 valid slots for the ones.\n3) Choose the slots: C(15,5) = 3003.\nFinal answer: $3003$", "final_answer": "3003", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1277c4098a555c88", "problem_id": "2d0a9a6fbaac81f2", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "How many binary strings of length 19 contain exactly 5 ones with no two ones adjacent?", "output": "Reserve 4 mandatory zeros between the 5 ones.\nThis leaves n−k+1 = 15 valid slots for the ones.\nChoose the slots: C(15,5) = 3003.\nFinal answer: $3003$", "final_answer": "3003", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "68b6f382ddf90679", "problem_id": "a2303695e381522b", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{3} x}{2} + 2 x - \\frac{3 \\sqrt{3}}{2} + 1$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$2 - \\frac{3 \\sqrt{3}}{2}$, c=$1 - \\frac{3 \\sqrt{3}}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{27}{4}$, so √Δ = $\\frac{3 \\sqrt{3}}{2}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $-1 + \\frac{3 \\sqrt{3}}{2}$.\n7) x2 = $-1$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{-1 + \\frac{3\\cdot \\sqrt{3}}{2}, -1\\}$", "final_answer": "x \\in \\{-1 + \\frac{3\\cdot \\sqrt{3}}{2}, -1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "06f7e1d90587037b", "problem_id": "9ef4bedea90b0247", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(2 x^{2} - 3\\right) \\cos{\\left(x^{2} + x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x e^{x}) = 2 x e^{x}$.\n3) So $f'(x) = - 4 x^{3} \\sin{\\left(\\left(x^{2} + x\\right) + 3 \\right)} - 2 x^{2} \\sin{\\left(\\left(x^{2} + x\\right) + 3 \\right)} + 2 x e^{x} + 6 x \\sin{\\left(\\left(x^{2} + x\\right) + 3 \\right)} + 4 x \\cos{\\left(\\left(x^{2} + x\\right) + 3 \\right)} - 6 e^{x} + 3 \\sin{\\left(\\left(x^{2} + x\\right) + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x e^{x}) = 2 x e^{x} + 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = 4 x \\cos{\\left(\\left(x^{2} + x\\right) + 3 \\right)} + 2 \\left(x - 3\\right) e^{x} - \\left(2 x + 1\\right) \\left(2 x^{2} - 3\\right) \\sin{\\left(\\left(x^{2} + x\\right) + 3 \\right)} + 2 e^{x}$.\nFinal answer: $4 x \\cos{\\left(\\left(x^{2} + x\\right) + 3 \\right)} + 2 \\left(x - 3\\right) e^{x} - \\left(2 x + 1\\right) \\left(2 x^{2} - 3\\right) \\sin{\\left(\\left(x^{2} + x\\right) + 3 \\right)} + 2 e^{x}$", "final_answer": "4 x \\cos{\\left(\\left(x^{2} + x\\right) + 3 \\right)} + 2 \\left(x - 3\\right) e^{x} - \\left(2 x + 1\\right) \\left(2 x^{2} - 3\\right) \\sin{\\left(\\left(x^{2} + x\\right) + 3 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2435dc636f60ef09", "problem_id": "011a26c16e7d77af", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} 2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\n2) Evaluate at the bounds: F(\\pi) and F(0).\n3) Subtract: F(\\pi) − F(0) = $4 + 2 \\pi + \\pi^{2} + \\frac{2 \\pi^{3}}{3}$.\n4) Simplify the result.\nFinal answer: $4 + 2\\cdot \\pi + \\pi^{2} + \\frac{2\\cdot \\pi^{3}}{3}$", "final_answer": "4 + 2\\cdot \\pi + \\pi^{2} + \\frac{2\\cdot \\pi^{3}}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "18daf9053453656d", "problem_id": "011a26c16e7d77af", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} 2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\nEvaluate at the bounds: F(\\pi) and F(0).\nSubtract: F(\\pi) − F(0) = $4 + 2 \\pi + \\pi^{2} + \\frac{2 \\pi^{3}}{3}$.\nFinal answer: $4 + 2\\cdot \\pi + \\pi^{2} + \\frac{2\\cdot \\pi^{3}}{3}$", "final_answer": "4 + 2\\cdot \\pi + \\pi^{2} + \\frac{2\\cdot \\pi^{3}}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "59a6b92edb943c6e", "problem_id": "ea53852f9029395b", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $310. It is discounted by 20%. You buy 3 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 310 × (1 − 20/100) = 310 × 80/100 = 248.\nSubtotal = 3 × 248 = 744.\nTotal with tax = 744 × 110/100 = 4092/5.\nCheck: recomputing the formula yields 4092/5.\nFinal answer: $\\frac{4092}{5}$", "final_answer": "\\frac{4092}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9215ccbb3aa1b396", "problem_id": "ea53852f9029395b", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A store sells a backpack for $310. It is discounted by 20%. You buy 3 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 310 × (1 − 20/100) = 310 × 80/100 = 248.\n2) Subtotal = 3 × 248 = 744.\n3) Total with tax = 744 × 110/100 = 4092/5.\nFinal answer: $\\frac{4092}{5}$", "final_answer": "\\frac{4092}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7e92fd1a16f8597a", "problem_id": "f50329376ce2f04f", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 2 x^{3} - 4 x^{2} + 9 x - 16$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{4} + x^{2} + 2 x - 1$ and a = -2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $8 x^{3} + 2 x + 2$.\nEvaluate at a: p'(-2) = $-66$.\nCheck: the identity gives $-66$.\nFinal answer: $-66$", "final_answer": "-66", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4fdd37e9ded987fb", "problem_id": "f50329376ce2f04f", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 2 x^{3} - 4 x^{2} + 9 x - 16$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $2 x^{4} + x^{2} + 2 x - 1$ and a = -2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $8 x^{3} + 2 x + 2$.\n5) Evaluate at a: p'(-2) = $-66$.\nFinal answer: $-66$", "final_answer": "-66", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "90b0c1dd9995e2ae", "problem_id": "9f67e0511bef456e", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 68 liters of a 30% salt solution. You have a 10% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 68−x liters is 40% solution.\nSalt equation: 10x + 40(68−x) = 30·68.\n(10-40)x = (30-40)·68 ⇒ x = 68/3.\nThen 68−x = 136/3.\nCheck: 10·68/3 + 40·136/3 = 30·68.\nFinal answer: $68/3 L of 10% and 136/3 L of 40%$", "final_answer": "68/3 L of 10% and 136/3 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "606342c3e44572b6", "problem_id": "9f67e0511bef456e", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "You want 68 liters of a 30% salt solution. You have a 10% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 68−x liters is 40% solution.\n2) Salt equation: 10x + 40(68−x) = 30·68.\n3) (10-40)x = (30-40)·68 ⇒ x = 68/3.\n4) Then 68−x = 136/3.\nFinal answer: $68/3 L of 10% and 136/3 L of 40%$", "final_answer": "68/3 L of 10% and 136/3 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bad0d2b2107504e3", "problem_id": "d688b93afaef0031", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - 2$ at $x=0$.", "output": "1) Differentiate: f'(x) = $- 4 x + 2 \\cos{\\left(x \\right)} - 3$.\n2) Slope at x=0: m = f'(0) = $-1$.\n3) Point on curve: (0, f(0)) = (0, $-2$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -2 = -1(x − 0).\n6) Optionally expand: y = $- x - 2$.\nFinal answer: $y = - x - 2$", "final_answer": "y = - x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "60b77f55a68b2d21", "problem_id": "d688b93afaef0031", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - 2$ at $x=0$.", "output": "Differentiate: f'(x) = $- 4 x + 2 \\cos{\\left(x \\right)} - 3$.\nSlope at x=0: m = f'(0) = $-1$.\nPoint on curve: (0, f(0)) = (0, $-2$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -2 = -1(x − 0).\nOptionally expand: y = $- x - 2$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - x - 2$", "final_answer": "y = - x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d3970352b1d0ee86", "problem_id": "d6892d7b5e4db7f0", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} - 2 x^{2} + 11 x - 41$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{2} + 10 x - 41.\n3) Then the limit is -113.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: -117.\nFinal answer: $-117$", "final_answer": "-117", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c2b303def1ce2a95", "problem_id": "696e0e29fcaf3d8a", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Two friends split $1336 in the ratio 4:4. How much does each receive?", "output": "1) Total ratio parts = 4+4 = 8.\n2) Each part = 1336 ÷ 8 = 167.\n3) First share = 4 × 167 = 668.\n4) Second share = 4 × 167 = 668.\nFinal answer: $668$", "final_answer": "668", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3d3dcdab9d6808be", "problem_id": "06363675a0d028d6", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Two friends split $432 in the ratio 7:2. How much does each receive?", "output": "1) Total ratio parts = 7+2 = 9.\n2) Each part = 432 ÷ 9 = 48.\n3) First share = 7 × 48 = 336.\n4) Second share = 2 × 48 = 96.\nFinal answer: $96$", "final_answer": "96", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ab7ddb3fc7eaaef9", "problem_id": "06363675a0d028d6", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $432 in the ratio 7:2. How much does each receive?", "output": "Total ratio parts = 7+2 = 9.\nEach part = 432 ÷ 9 = 48.\nFirst share = 7 × 48 = 336.\nSecond share = 2 × 48 = 96.\nCheck: 336+96=432.\nFinal answer: $96$", "final_answer": "96", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "501dd4209f1b4efe", "problem_id": "bff14fef2e36aa36", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=21$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=21 and k=7.\nFinal answer: $296010$", "final_answer": "296010", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4115690eaec5f79b", "problem_id": "bff14fef2e36aa36", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=21$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=21 and k=7.\n4) Compute C(27,6) = 296010.\nFinal answer: $296010$", "final_answer": "296010", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eb263b6b3d476eda", "problem_id": "53724ca39b4bac6d", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 14 red balls and 16 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(30,8).\nFavorable samples: choose k reds and n−k blues: C(14,1)·C(16,7).\nCompute totals: C(30,8) = 5852925, and favorable = 160160.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{2464}{90045}$.\nFinal answer: $\\frac{2464}{90045}$", "final_answer": "\\frac{2464}{90045}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "502062a36fc26e66", "problem_id": "53724ca39b4bac6d", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 14 red balls and 16 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(30,8).\n3) Favorable samples: choose k reds and n−k blues: C(14,1)·C(16,7).\n4) Compute totals: C(30,8) = 5852925, and favorable = 160160.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{2464}{90045}$", "final_answer": "\\frac{2464}{90045}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7c654815f2215b9b", "problem_id": "4c54b689b793ddd9", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Two friends split $520 in the ratio 9:4. How much does each receive?", "output": "Total ratio parts = 9+4 = 13.\nEach part = 520 ÷ 13 = 40.\nFirst share = 9 × 40 = 360.\nFinal answer: $160$", "final_answer": "160", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8bc312205bbeecac", "problem_id": "4c54b689b793ddd9", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Two friends split $520 in the ratio 9:4. How much does each receive?", "output": "1) Total ratio parts = 9+4 = 13.\n2) Each part = 520 ÷ 13 = 40.\n3) First share = 9 × 40 = 360.\n4) Second share = 4 × 40 = 160.\nFinal answer: $160$", "final_answer": "160", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b02247e295cb2fe3", "problem_id": "c2374fd5ec3f4f68", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Two friends split $608 in the ratio 9:7. How much does each receive?", "output": "1) Total ratio parts = 9+7 = 16.\n2) Each part = 608 ÷ 16 = 38.\n3) First share = 9 × 38 = 342.\n4) Second share = 7 × 38 = 266.\nFinal answer: $266$", "final_answer": "266", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a09bdd2bfbdfdecc", "problem_id": "c2374fd5ec3f4f68", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Two friends split $608 in the ratio 9:7. How much does each receive?", "output": "$266$", "final_answer": "266", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d6b5fcd802aad25e", "problem_id": "d60af95fb3680e69", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡0 (mod 7), x≡5 (mod 8). Give the solution modulo 56.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 56.\n3) Compute the CRT solution to get x ≡ 21 (mod 56).\nFinal answer: $x\\equiv 21\\pmod{56}$", "final_answer": "x\\equiv 21\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ca68a87fcb564a54", "problem_id": "d60af95fb3680e69", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡0 (mod 7), x≡5 (mod 8). Give the solution modulo 56.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 56.\nCompute the CRT solution to get x ≡ 21 (mod 56).\nCheck: 21 mod 7=0 and mod 8=5.\nFinal answer: $x\\equiv 21\\pmod{56}$", "final_answer": "x\\equiv 21\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ea88a83dfbc63fab", "problem_id": "b3f1451a25f18fbd", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the linear system:\n$7 x - y$ = $51$\n$2 x - 8 y$ = $30$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $2$: $14 x - 2 y$ = $102$.\nMultiply the second equation by $7$: $14 x - 56 y$ = $210$.\nFinal answer: $(x,y)=\\left(7, -2\\right)$", "final_answer": "(x,y)=\\left(7, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d385961b1a4847b1", "problem_id": "b3f1451a25f18fbd", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$7 x - y$ = $51$\n$2 x - 8 y$ = $30$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $14 x - 2 y$ = $102$.\n3) Multiply the second equation by $7$: $14 x - 56 y$ = $210$.\n4) Subtract to eliminate x: $54 y$ = $-108$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $7$.\nFinal answer: $(x,y)=\\left(7, -2\\right)$", "final_answer": "(x,y)=\\left(7, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b36d2003e068a7ec", "problem_id": "d903813baeed4420", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A cyclist rides at 78 miles per hour for 3/2 hours, then at 85 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 78 × 3/2 = 117 miles.\n2) Second leg distance = 85 × 2 = 170 miles.\n3) Total distance = 117 + 170 = 287 miles.\nFinal answer: $287$", "final_answer": "287", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ef0192bb681a076f", "problem_id": "0e94698edb22bdb2", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 2 x - 7$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to x - 7.\n3) Then the limit is -10.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -13.\nFinal answer: $-13$", "final_answer": "-13", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e6509f79b1307f9e", "problem_id": "85b293d7e2f7dcbe", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Worker A finishes a job in 15 hours and Worker B finishes it in 9 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/15 job/hour. B's rate = 1/9 job/hour.\n2) Combined rate = 1/15 + 1/9 = 8/45 job/hour.\n3) Time = 1 ÷ (combined rate) = 45/8 hours.\nFinal answer: $\\frac{45}{8}$", "final_answer": "\\frac{45}{8}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6ba41c4670ee1a3b", "problem_id": "85b293d7e2f7dcbe", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Worker A finishes a job in 15 hours and Worker B finishes it in 9 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/15 job/hour. B's rate = 1/9 job/hour.\nCombined rate = 1/15 + 1/9 = 8/45 job/hour.\nTime = 1 ÷ (combined rate) = 45/8 hours.\nCheck: (combined rate)×(time) = 8/45×45/8 = 1.\nFinal answer: $\\frac{45}{8}$", "final_answer": "\\frac{45}{8}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "201320ba75dde386", "problem_id": "311b78c2567c2ab3", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡4 (mod 5), x≡10 (mod 11). Give the solution modulo 55.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 55.\n3) Compute the CRT solution to get x ≡ 54 (mod 55).\nFinal answer: $x\\equiv 54\\pmod{55}$", "final_answer": "x\\equiv 54\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d5d85ec70750789f", "problem_id": "311b78c2567c2ab3", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡4 (mod 5), x≡10 (mod 11). Give the solution modulo 55.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 55.\nCompute the CRT solution to get x ≡ 54 (mod 55).\nCheck: 54 mod 5=4 and mod 11=10.\nFinal answer: $x\\equiv 54\\pmod{55}$", "final_answer": "x\\equiv 54\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e6e5c81bdbc1ac83", "problem_id": "3f1e9432dc50d3b3", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A store sells a backpack for $350. It is discounted by 35%. You buy 2 backpacks. What is the total cost?", "output": "Discounted price = 350 × (1 − 35/100) = 350 × 65/100 = 455/2.\nSubtotal = 2 × 455/2 = 455.\nFinal answer: $455$", "final_answer": "455", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "33b300d81f605905", "problem_id": "3f1e9432dc50d3b3", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A store sells a backpack for $350. It is discounted by 35%. You buy 2 backpacks. What is the total cost?", "output": "1) Discounted price = 350 × (1 − 35/100) = 350 × 65/100 = 455/2.\n2) Subtotal = 2 × 455/2 = 455.\nFinal answer: $455$", "final_answer": "455", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e35662aed3501c5b", "problem_id": "b82e99176332d679", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} x \\left(- x - 2\\right)$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} - 3 x.\n3) Then the limit is 2.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: 1.\nFinal answer: $1$", "final_answer": "1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "397fc5189ab24741", "problem_id": "1aa31f825244b92a", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 20 times, what is the probability of exactly 10 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=20, k=10, p=$\\frac{2}{5}$ and simplify.\n4) Here C(20,10) = 184756.\nFinal answer: $\\frac{11171488813056}{95367431640625}$", "final_answer": "\\frac{11171488813056}{95367431640625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d6b2c4ad631ffaef", "problem_id": "1dd70a2144326e8b", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Worker A finishes a job in 18 hours and Worker B finishes it in 8 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/18 job/hour. B's rate = 1/8 job/hour.\n2) Combined rate = 1/18 + 1/8 = 13/72 job/hour.\n3) Time = 1 ÷ (combined rate) = 72/13 hours.\nFinal answer: $\\frac{72}{13}$", "final_answer": "\\frac{72}{13}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "194ad71bce9929f2", "problem_id": "1dd70a2144326e8b", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Worker A finishes a job in 18 hours and Worker B finishes it in 8 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/18 job/hour. B's rate = 1/8 job/hour.\nCombined rate = 1/18 + 1/8 = 13/72 job/hour.\nTime = 1 ÷ (combined rate) = 72/13 hours.\nCheck: (combined rate)×(time) = 13/72×72/13 = 1.\nFinal answer: $\\frac{72}{13}$", "final_answer": "\\frac{72}{13}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4c5d752386c0e8d0", "problem_id": "5d6146d3e3d49489", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $2 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $2 x^{2} \\cos{\\left(x \\right)} + 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$.\nFinal answer: $2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 4\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - 3\\cdot \\cos{\\left(x \\right)}$", "final_answer": "2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 4\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - 3\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "31ad689c917dd8b6", "problem_id": "5d6146d3e3d49489", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $2 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $2 x^{2} \\cos{\\left(x \\right)} + 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 4\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - 3\\cdot \\cos{\\left(x \\right)}$", "final_answer": "2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 4\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - 3\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0cd32c230a723cd4", "problem_id": "f60ee93e8036ad4e", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Two friends split $384 in the ratio 3:9. How much does each receive?", "output": "Total ratio parts = 3+9 = 12.\nEach part = 384 ÷ 12 = 32.\nFirst share = 3 × 32 = 96.\nSecond share = 9 × 32 = 288.\nCheck: 96+288=384.\nFinal answer: $288$", "final_answer": "288", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b7b4e4f800977f50", "problem_id": "9226cf6badd482d3", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(- x^{2} - x\\right) \\cos{\\left(3 x^{2} - x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x e^{x}) = x e^{x}$.\n3) So $f'(x) = 6 x^{3} \\sin{\\left(\\left(3 x^{2} - x\\right) + 3 \\right)} + 5 x^{2} \\sin{\\left(\\left(3 x^{2} - x\\right) + 3 \\right)} + x e^{x} - x \\sin{\\left(\\left(3 x^{2} - x\\right) + 3 \\right)} - 2 x \\cos{\\left(\\left(3 x^{2} - x\\right) + 3 \\right)} + e^{x} - \\cos{\\left(\\left(3 x^{2} - x\\right) + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x e^{x}) = x e^{x} + e^{x}$.\nTherefore the correct derivative is $f'(x) = - x \\left(- x - 1\\right) \\left(6 x - 1\\right) \\sin{\\left(\\left(3 x^{2} - x\\right) + 3 \\right)} + \\left(- 2 x - 1\\right) \\cos{\\left(\\left(3 x^{2} - x\\right) + 3 \\right)} + \\left(x + 1\\right) e^{x} + e^{x}$.\nFinal answer: $- x \\left(- x - 1\\right) \\left(6 x - 1\\right) \\sin{\\left(\\left(3 x^{2} - x\\right) + 3 \\right)} + \\left(- 2 x - 1\\right) \\cos{\\left(\\left(3 x^{2} - x\\right) + 3 \\right)} + \\left(x + 1\\right) e^{x} + e^{x}$", "final_answer": "- x \\left(- x - 1\\right) \\left(6 x - 1\\right) \\sin{\\left(\\left(3 x^{2} - x\\right) + 3 \\right)} + \\left(- 2 x - 1\\right) \\cos{\\left(\\left(3 x^{2} - x\\right) + 3 \\right)} + \\left(x + 1\\right) e^{x} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6c30ff64b331ef2b", "problem_id": "bbf88bec4181317c", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{10} x + \\frac{2 x}{5} - \\frac{7 \\sqrt{10}}{5} - \\frac{7}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{2}{5} - \\sqrt{10}$, c=$- \\frac{7 \\sqrt{10}}{5} - \\frac{7}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{24 \\sqrt{10}}{5} + \\frac{394}{25}$, so √Δ = $\\sqrt{\\frac{24 \\sqrt{10}}{5} + \\frac{394}{25}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{1}{5} + \\frac{\\sqrt{10}}{2} + \\frac{\\sqrt{120 \\sqrt{10} + 394}}{10}$.\n7) x2 = $- \\frac{\\sqrt{120 \\sqrt{10} + 394}}{10} - \\frac{1}{5} + \\frac{\\sqrt{10}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{1}{5} + \\frac{\\sqrt{10}}{2} + \\frac{\\sqrt{120\\cdot \\sqrt{10} + 394}}{10}, - \\frac{\\sqrt{120\\cdot \\sqrt{10} + 394}}{10} - \\frac{1}{5} + \\frac{\\sqrt{10}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{1}{5} + \\frac{\\sqrt{10}}{2} + \\frac{\\sqrt{120\\cdot \\sqrt{10} + 394}}{10}, - \\frac{\\sqrt{120\\cdot \\sqrt{10} + 394}}{10} - \\frac{1}{5} + \\frac{\\sqrt{10}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "63b4bc7757b6ceb3", "problem_id": "1098642163774e02", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 \\sin{\\left(x \\right)}) = 2 \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 \\sin{\\left(x \\right)}) = 2 \\cos{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}\\right)$.\nFinal answer: $2 \\left(- x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}\\right)$", "final_answer": "2 \\left(- x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "4f90a7b640975a93", "problem_id": "8f622d9a1b2c7d78", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{2} x}{2} - 2 x - \\frac{5}{4} - \\frac{3 \\sqrt{2}}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{3 \\sqrt{2}}{2} - 2$, c=$- \\frac{5}{4} - \\frac{3 \\sqrt{2}}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $9 \\sqrt{2} + \\frac{27}{2}$, so √Δ = $\\sqrt{9 \\sqrt{2} + \\frac{27}{2}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $1 + \\frac{3 \\sqrt{2}}{4} + \\frac{3 \\sqrt{4 \\sqrt{2} + 6}}{4}$.\n7) x2 = $- \\frac{3 \\sqrt{4 \\sqrt{2} + 6}}{4} + 1 + \\frac{3 \\sqrt{2}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{1 + \\frac{3\\cdot \\sqrt{2}}{4} + \\frac{3\\cdot \\sqrt{4\\cdot \\sqrt{2} + 6}}{4}, - \\frac{3\\cdot \\sqrt{4\\cdot \\sqrt{2} + 6}}{4} + 1 + \\frac{3\\cdot \\sqrt{2}}{4}\\}$", "final_answer": "x \\in \\{1 + \\frac{3\\cdot \\sqrt{2}}{4} + \\frac{3\\cdot \\sqrt{4\\cdot \\sqrt{2} + 6}}{4}, - \\frac{3\\cdot \\sqrt{4\\cdot \\sqrt{2} + 6}}{4} + 1 + \\frac{3\\cdot \\sqrt{2}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9241eb077346a352", "problem_id": "8f622d9a1b2c7d78", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{2} x}{2} - 2 x - \\frac{5}{4} - \\frac{3 \\sqrt{2}}{4}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{3 \\sqrt{2}}{2} - 2$, c=$- \\frac{5}{4} - \\frac{3 \\sqrt{2}}{4}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $9 \\sqrt{2} + \\frac{27}{2}$, so √Δ = $\\sqrt{9 \\sqrt{2} + \\frac{27}{2}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $1 + \\frac{3 \\sqrt{2}}{4} + \\frac{3 \\sqrt{4 \\sqrt{2} + 6}}{4}$.\nx2 = $- \\frac{3 \\sqrt{4 \\sqrt{2} + 6}}{4} + 1 + \\frac{3 \\sqrt{2}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{1 + \\frac{3\\cdot \\sqrt{2}}{4} + \\frac{3\\cdot \\sqrt{4\\cdot \\sqrt{2} + 6}}{4}, - \\frac{3\\cdot \\sqrt{4\\cdot \\sqrt{2} + 6}}{4} + 1 + \\frac{3\\cdot \\sqrt{2}}{4}\\}$", "final_answer": "x \\in \\{1 + \\frac{3\\cdot \\sqrt{2}}{4} + \\frac{3\\cdot \\sqrt{4\\cdot \\sqrt{2} + 6}}{4}, - \\frac{3\\cdot \\sqrt{4\\cdot \\sqrt{2} + 6}}{4} + 1 + \\frac{3\\cdot \\sqrt{2}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b3ddf5981e6e8412", "problem_id": "593ae8bcdc62cfea", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A store sells a backpack for $102. It is discounted by 35%. You buy 11 backpacks. What is the total cost?", "output": "1) Discounted price = 102 × (1 − 35/100) = 102 × 65/100 = 663/10.\n2) Subtotal = 11 × 663/10 = 7293/10.\nFinal answer: $\\frac{7293}{10}$", "final_answer": "\\frac{7293}{10}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "90694d8d33af645a", "problem_id": "593ae8bcdc62cfea", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "A store sells a backpack for $102. It is discounted by 35%. You buy 11 backpacks. What is the total cost?", "output": "$\\frac{7293}{10}$", "final_answer": "\\frac{7293}{10}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3a32be7ba189afd2", "problem_id": "9f0c8da878bf0139", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the linear system:\n$2 x + 3 y$ = $10$\n$- 3 x + 4 y$ = $-15$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $- 6 x - 9 y$ = $-30$.\nMultiply the second equation by $2$: $- 6 x + 8 y$ = $-30$.\nFinal answer: $(x,y)=\\left(5, 0\\right)$", "final_answer": "(x,y)=\\left(5, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5611b8acf5801d62", "problem_id": "9f0c8da878bf0139", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$2 x + 3 y$ = $10$\n$- 3 x + 4 y$ = $-15$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $- 6 x - 9 y$ = $-30$.\n3) Multiply the second equation by $2$: $- 6 x + 8 y$ = $-30$.\n4) Subtract to eliminate x: $- 17 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, 0\\right)$", "final_answer": "(x,y)=\\left(5, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "47655303cc3c0d7f", "problem_id": "22a44e77ea838bb5", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡8 (mod 11), x≡1 (mod 8). Give the solution modulo 88.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 88.\nCompute the CRT solution to get x ≡ 41 (mod 88).\nCheck: 41 mod 11=8 and mod 8=1.\nFinal answer: $x\\equiv 41\\pmod{88}$", "final_answer": "x\\equiv 41\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fed71d63cc267498", "problem_id": "22a44e77ea838bb5", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡8 (mod 11), x≡1 (mod 8). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 41 (mod 88).\nFinal answer: $x\\equiv 41\\pmod{88}$", "final_answer": "x\\equiv 41\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1ed98d8a1e01db56", "problem_id": "658bcaee3f338f53", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 11 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=11, k=3, p=$\\frac{1}{4}$ and simplify.\nHere C(11,3) = 165.\nCheck: expression simplifies to $\\frac{1082565}{4194304}$.\nFinal answer: $\\frac{1082565}{4194304}$", "final_answer": "\\frac{1082565}{4194304}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f4b0563b4bbdc50c", "problem_id": "658bcaee3f338f53", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 11 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=11, k=3, p=$\\frac{1}{4}$ and simplify.\nFinal answer: $\\frac{1082565}{4194304}$", "final_answer": "\\frac{1082565}{4194304}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "617b0265c4b28fb1", "problem_id": "7a17d9d04ef75132", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 13 red balls and 11 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(24,7).\n3) Favorable samples: choose k reds and n−k blues: C(13,2)·C(11,5).\n4) Compute totals: C(24,7) = 346104, and favorable = 36036.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{91}{874}$", "final_answer": "\\frac{91}{874}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bd4e940abb894359", "problem_id": "7a17d9d04ef75132", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "An urn has 13 red balls and 11 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(24,7).\nFavorable samples: choose k reds and n−k blues: C(13,2)·C(11,5).\nFinal answer: $\\frac{91}{874}$", "final_answer": "\\frac{91}{874}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "63b4bc7757b6ceb3", "problem_id": "1098642163774e02", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 \\sin{\\left(x \\right)}) = 2 \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 \\sin{\\left(x \\right)}) = 2 \\cos{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}\\right)$.\nFinal answer: $2 \\left(- x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}\\right)$", "final_answer": "2 \\left(- x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "af6b1cef8754f01a", "problem_id": "3aea0f5efd8c4821", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 13 x + 30$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} + 13 x + 30$ = $\\left(x + 3\\right) \\left(x + 10\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $-3$.\nFrom the second factor: x = $-10$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-3, -10\\}$", "final_answer": "x \\in \\{-3, -10\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6f1a8b4124f4aa4e", "problem_id": "e184d12a4d7e980e", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the multiplicative inverse of 2 modulo 37.", "output": "We want x such that 2x ≡ 1 (mod 37).\nUse extended Euclid to find s,t with 2s + 37t = 1.\nOne Bezout identity is: 2·(-18) + 37·(1) = 1.\nReduce s modulo 37: x ≡ 19 (mod 37).\nCheck: 2·19 mod 37 = 1.\nFinal answer: $x\\equiv 19\\pmod{37}$", "final_answer": "x\\equiv 19\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "36bd085ef0525818", "problem_id": "e184d12a4d7e980e", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the multiplicative inverse of 2 modulo 37.", "output": "1) We want x such that 2x ≡ 1 (mod 37).\n2) Use extended Euclid to find s,t with 2s + 37t = 1.\n3) One Bezout identity is: 2·(-18) + 37·(1) = 1.\n4) Reduce s modulo 37: x ≡ 19 (mod 37).\nFinal answer: $x\\equiv 19\\pmod{37}$", "final_answer": "x\\equiv 19\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "278e195a64f1dd16", "problem_id": "b8b65eda4bfdf578", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 7 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=7, k=2, p=$\\frac{4}{5}$ and simplify.\n4) Here C(7,2) = 21.\nFinal answer: $\\frac{336}{78125}$", "final_answer": "\\frac{336}{78125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ddc7542ce38bf6ef", "problem_id": "b8b65eda4bfdf578", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 7 times, what is the probability of exactly 2 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=7, k=2, p=$\\frac{4}{5}$ and simplify.\nFinal answer: $\\frac{336}{78125}$", "final_answer": "\\frac{336}{78125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cfc7175b5d60a080", "problem_id": "e451ee6b2e0527c4", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_4=18$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=18 and k=4.\n4) Compute C(21,3) = 1330.\nFinal answer: $1330$", "final_answer": "1330", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5c2ec7a195b356ac", "problem_id": "e451ee6b2e0527c4", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_4=18$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=18 and k=4.\nCompute C(21,3) = 1330.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $1330$", "final_answer": "1330", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6ffdf741ecc83af8", "problem_id": "0e9644396fa9b88b", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Two friends split $476 in the ratio 3:1. How much does each receive?", "output": "1) Total ratio parts = 3+1 = 4.\n2) Each part = 476 ÷ 4 = 119.\n3) First share = 3 × 119 = 357.\n4) Second share = 1 × 119 = 119.\nFinal answer: $119$", "final_answer": "119", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9aee5aaab4ae6aa4", "problem_id": "0e9644396fa9b88b", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $476 in the ratio 3:1. How much does each receive?", "output": "Total ratio parts = 3+1 = 4.\nEach part = 476 ÷ 4 = 119.\nFirst share = 3 × 119 = 357.\nSecond share = 1 × 119 = 119.\nCheck: 357+119=476.\nFinal answer: $119$", "final_answer": "119", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e3eebbd28ceb6a9e", "problem_id": "47096950d58f8de4", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $760. It is discounted by 30%. You buy 14 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 760 × (1 − 30/100) = 760 × 70/100 = 532.\nSubtotal = 14 × 532 = 7448.\nTotal with tax = 7448 × 109/100 = 202958/25.\nCheck: recomputing the formula yields 202958/25.\nFinal answer: $\\frac{202958}{25}$", "final_answer": "\\frac{202958}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "37d96ad3347b6e2d", "problem_id": "47096950d58f8de4", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A store sells a backpack for $760. It is discounted by 30%. You buy 14 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 760 × (1 − 30/100) = 760 × 70/100 = 532.\n2) Subtotal = 14 × 532 = 7448.\n3) Total with tax = 7448 × 109/100 = 202958/25.\nFinal answer: $\\frac{202958}{25}$", "final_answer": "\\frac{202958}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f5ca2ba66f5823bc", "problem_id": "b708677091c5e5b4", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(2 x^{3} - x^{2} + x - 2\\right) \\cos{\\left(2 x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(4 e^{x}) = 4 e^{x}$.\n3) So $f'(x) = - 4 x^{3} \\sin{\\left(2 x + 3 \\right)} + 2 x^{2} \\sin{\\left(2 x + 3 \\right)} + 6 x^{2} \\cos{\\left(2 x + 3 \\right)} + 2 x e^{x} - 2 x \\sin{\\left(2 x + 3 \\right)} - 2 x \\cos{\\left(2 x + 3 \\right)} + 6 e^{x} + 4 \\sin{\\left(2 x + 3 \\right)} + \\cos{\\left(2 x + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(4 e^{x}) = 4 e^{x}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x + 2\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) + 1\\right) - 2\\right) \\sin{\\left(2 x + 3 \\right)} + \\left(6 x^{2} - 2 x + 1\\right) \\cos{\\left(2 x + 3 \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(x + 2\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) + 1\\right) - 2\\right) \\sin{\\left(2 x + 3 \\right)} + \\left(6 x^{2} - 2 x + 1\\right) \\cos{\\left(2 x + 3 \\right)} + 2 e^{x}$", "final_answer": "2 \\left(x + 2\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) + 1\\right) - 2\\right) \\sin{\\left(2 x + 3 \\right)} + \\left(6 x^{2} - 2 x + 1\\right) \\cos{\\left(2 x + 3 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "475eddb9bf227058", "problem_id": "7cf1fc2e6e2693e6", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_9=45$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_9=36.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=36 and k=9.\nCompute C(44,8) = 177232627.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $177232627$", "final_answer": "177232627", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fff2cb0bdef4106b", "problem_id": "ab321ff0aed5e463", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 60 liters of a 30% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 60−x liters is 70% solution.\nSalt equation: 25x + 70(60−x) = 30·60.\n(25-70)x = (30-70)·60 ⇒ x = 160/3.\nThen 60−x = 20/3.\nCheck: 25·160/3 + 70·20/3 = 30·60.\nFinal answer: $160/3 L of 25% and 20/3 L of 70%$", "final_answer": "160/3 L of 25% and 20/3 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3cebcfc8a192980a", "problem_id": "8becc4ad099fc928", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 3 x^{2} + 12 x - 35$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} + 11 x - 35.\n3) Then the limit is -95.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -98.\nFinal answer: $-98$", "final_answer": "-98", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "21e8278833242bbe", "problem_id": "defc8517e97fcd50", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the multiplicative inverse of 36 modulo 47.", "output": "1) We want x such that 36x ≡ 1 (mod 47).\n2) Use extended Euclid to find s,t with 36s + 47t = 1.\n3) One Bezout identity is: 36·(17) + 47·(-13) = 1.\n4) Reduce s modulo 47: x ≡ 17 (mod 47).\nFinal answer: $x\\equiv 17\\pmod{47}$", "final_answer": "x\\equiv 17\\pmod{47}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e12f43cf054c4c4", "problem_id": "defc8517e97fcd50", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Find the multiplicative inverse of 36 modulo 47.", "output": "We want x such that 36x ≡ 1 (mod 47).\nUse extended Euclid to find s,t with 36s + 47t = 1.\nOne Bezout identity is: 36·(17) + 47·(-13) = 1.\nFinal answer: $x\\equiv 17\\pmod{47}$", "final_answer": "x\\equiv 17\\pmod{47}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1159fba9b8c7e619", "problem_id": "1900de58e4d0b17a", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{3} + 4 x^{2} - 2 x + 4\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $3 x^{3} + 4 x^{2} - 2 x + 4$.\n3) Compute an antiderivative: F(x) = $\\frac{3 x^{4}}{4} + \\frac{4 x^{3}}{3} - x^{2} + 4 x$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{3\\cdot x^{4}}{4} + \\frac{4\\cdot x^{3}}{3} - x^{2} + 4\\cdot x$", "final_answer": "C + \\frac{3\\cdot x^{4}}{4} + \\frac{4\\cdot x^{3}}{3} - x^{2} + 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "85697bf540b1d072", "problem_id": "1900de58e4d0b17a", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{3} + 4 x^{2} - 2 x + 4\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $3 x^{3} + 4 x^{2} - 2 x + 4$.\nCompute an antiderivative: F(x) = $\\frac{3 x^{4}}{4} + \\frac{4 x^{3}}{3} - x^{2} + 4 x$.\nFinal answer: $C + \\frac{3\\cdot x^{4}}{4} + \\frac{4\\cdot x^{3}}{3} - x^{2} + 4\\cdot x$", "final_answer": "C + \\frac{3\\cdot x^{4}}{4} + \\frac{4\\cdot x^{3}}{3} - x^{2} + 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6517faa4ede9048b", "problem_id": "24b03dfde8923f15", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A store sells a backpack for $110. It is discounted by 15%. You buy 14 backpacks. What is the total cost?", "output": "Discounted price = 110 × (1 − 15/100) = 110 × 85/100 = 187/2.\nSubtotal = 14 × 187/2 = 1309.\nCheck: recomputing the formula yields 1309.\nFinal answer: $1309$", "final_answer": "1309", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c3743af2fa017354", "problem_id": "24b03dfde8923f15", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A store sells a backpack for $110. It is discounted by 15%. You buy 14 backpacks. What is the total cost?", "output": "1) Discounted price = 110 × (1 − 15/100) = 110 × 85/100 = 187/2.\n2) Subtotal = 14 × 187/2 = 1309.\nFinal answer: $1309$", "final_answer": "1309", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "805235036a657337", "problem_id": "12e813b5733d1079", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 16 x + 64$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} + 16 x + 64$ = $\\left(x + 8\\right)^{2}$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $-8$.\nFrom the second factor: x = $-8$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-8, -8\\}$", "final_answer": "x \\in \\{-8, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bed4207e5bc19081", "problem_id": "12e813b5733d1079", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 16 x + 64$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} + 16 x + 64$ = $\\left(x + 8\\right)^{2}$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-8$.\n5) From the second factor: x = $-8$.\nFinal answer: $x \\in \\{-8, -8\\}$", "final_answer": "x \\in \\{-8, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "84862d2c175c7bf0", "problem_id": "424077a7f0490092", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Two friends split $1328 in the ratio 7:1. How much does each receive?", "output": "1) Total ratio parts = 7+1 = 8.\n2) Each part = 1328 ÷ 8 = 166.\n3) First share = 7 × 166 = 1162.\n4) Second share = 1 × 166 = 166.\nFinal answer: $166$", "final_answer": "166", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "947b352252fbe918", "problem_id": "424077a7f0490092", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $1328 in the ratio 7:1. How much does each receive?", "output": "Total ratio parts = 7+1 = 8.\nEach part = 1328 ÷ 8 = 166.\nFirst share = 7 × 166 = 1162.\nSecond share = 1 × 166 = 166.\nCheck: 1162+166=1328.\nFinal answer: $166$", "final_answer": "166", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dc83fb0ff00897de", "problem_id": "d5f952b98a5d0a5a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A cyclist rides at 90 miles per hour for 1 hours, then at 77 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 90 × 1 = 90 miles.\nSecond leg distance = 77 × 1 = 77 miles.\nTotal distance = 90 + 77 = 167 miles.\nCheck: adding both legs gives 167.\nFinal answer: $167$", "final_answer": "167", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3a6cb77f0a90a87e", "problem_id": "d5f952b98a5d0a5a", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A cyclist rides at 90 miles per hour for 1 hours, then at 77 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 90 × 1 = 90 miles.\n2) Second leg distance = 77 × 1 = 77 miles.\n3) Total distance = 90 + 77 = 167 miles.\nFinal answer: $167$", "final_answer": "167", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5c231f85fa548a3f", "problem_id": "2f98b98631d9deb2", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 2 x^{3} - 5 x^{2} + 2 x - 5$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $2 x^{4} - 3 x^{3} - 3 x^{2} - 3 x - 1$ and a = -1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $8 x^{3} - 9 x^{2} - 6 x - 3$.\n5) Evaluate at a: p'(-1) = $-14$.\nFinal answer: $-14$", "final_answer": "-14", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "204c246f94cacf28", "problem_id": "2f98b98631d9deb2", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 2 x^{3} - 5 x^{2} + 2 x - 5$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{4} - 3 x^{3} - 3 x^{2} - 3 x - 1$ and a = -1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $8 x^{3} - 9 x^{2} - 6 x - 3$.\nEvaluate at a: p'(-1) = $-14$.\nCheck: the identity gives $-14$.\nFinal answer: $-14$", "final_answer": "-14", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4f1f1e98691a35f8", "problem_id": "24fb70fbb766f88f", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} - 4 x^{3} + 3 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 4 x^{3} + 3 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $- 2 \\pi + 4 + \\frac{\\pi^{3}}{4}$.\n4) Simplify the result.\nFinal answer: $- 2\\cdot \\pi + 4 + \\frac{\\pi^{3}}{4}$", "final_answer": "- 2\\cdot \\pi + 4 + \\frac{\\pi^{3}}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2c8b14d6d14e465d", "problem_id": "24fb70fbb766f88f", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} - 4 x^{3} + 3 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 4 x^{3} + 3 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $- 2 \\pi + 4 + \\frac{\\pi^{3}}{4}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- 2\\cdot \\pi + 4 + \\frac{\\pi^{3}}{4}$", "final_answer": "- 2\\cdot \\pi + 4 + \\frac{\\pi^{3}}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8f00c0e979fd5d6a", "problem_id": "9d91723910ff0c97", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} - x^{2} - 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $9 x^{2} - 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 3$.\n2) Slope at x=-2: m = f'(-2) = $2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e} + 37$.\n3) Point on curve: (-2, f(-2)) = (-2, $-23 + \\cos{\\left(4 \\right)} + e^{-1}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -23 + \\cos{\\left(4 \\right)} + e^{-1} = 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e} + 37(x − -2).\n6) Optionally expand: y = $2 x \\sin{\\left(4 \\right)} + \\frac{x}{2 e} + 37 x + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 51$.\nFinal answer: $y = 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 37\\cdot x + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 51$", "final_answer": "y = 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 37\\cdot x + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 51", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5214ba60bae39fb3", "problem_id": "9d91723910ff0c97", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} - x^{2} - 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=-2$.", "output": "Differentiate: f'(x) = $9 x^{2} - 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 3$.\nSlope at x=-2: m = f'(-2) = $2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e} + 37$.\nPoint on curve: (-2, f(-2)) = (-2, $-23 + \\cos{\\left(4 \\right)} + e^{-1}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -23 + \\cos{\\left(4 \\right)} + e^{-1} = 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e} + 37(x − -2).\nOptionally expand: y = $2 x \\sin{\\left(4 \\right)} + \\frac{x}{2 e} + 37 x + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 51$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 37\\cdot x + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 51$", "final_answer": "y = 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 37\\cdot x + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 51", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "52864b7d6250b17e", "problem_id": "d2ccb0de15a7dca8", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $315. It is discounted by 10%. You buy 4 backpacks. What is the total cost?", "output": "Discounted price = 315 × (1 − 10/100) = 315 × 90/100 = 567/2.\nSubtotal = 4 × 567/2 = 1134.\nCheck: recomputing the formula yields 1134.\nFinal answer: $1134$", "final_answer": "1134", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9133e9614cf33237", "problem_id": "d2ccb0de15a7dca8", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A store sells a backpack for $315. It is discounted by 10%. You buy 4 backpacks. What is the total cost?", "output": "1) Discounted price = 315 × (1 − 10/100) = 315 × 90/100 = 567/2.\n2) Subtotal = 4 × 567/2 = 1134.\nFinal answer: $1134$", "final_answer": "1134", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f61f5121f31778bb", "problem_id": "4508c3d19dc61614", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 16 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=16, k=8, p=$\\frac{1}{5}$ and simplify.\nFinal answer: $\\frac{168689664}{30517578125}$", "final_answer": "\\frac{168689664}{30517578125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e5586563dd896c9c", "problem_id": "4508c3d19dc61614", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 16 times, what is the probability of exactly 8 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=16, k=8, p=$\\frac{1}{5}$ and simplify.\n4) Here C(16,8) = 12870.\nFinal answer: $\\frac{168689664}{30517578125}$", "final_answer": "\\frac{168689664}{30517578125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a8d3b92bdcbb16fc", "problem_id": "e0394646c116eeec", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 13 red balls and 9 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(22,8).\n3) Favorable samples: choose k reds and n−k blues: C(13,2)·C(9,6).\n4) Compute totals: C(22,8) = 319770, and favorable = 6552.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{364}{17765}$", "final_answer": "\\frac{364}{17765}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "12f59e88ac755438", "problem_id": "e0394646c116eeec", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 13 red balls and 9 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(22,8).\nFavorable samples: choose k reds and n−k blues: C(13,2)·C(9,6).\nCompute totals: C(22,8) = 319770, and favorable = 6552.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{364}{17765}$.\nFinal answer: $\\frac{364}{17765}$", "final_answer": "\\frac{364}{17765}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a14299903f892f7b", "problem_id": "87de0311ffc16592", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=25$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=25 and k=5.\nCompute C(29,4) = 23751.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $23751$", "final_answer": "23751", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "75e5905801e44377", "problem_id": "87de0311ffc16592", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=25$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=25 and k=5.\n4) Compute C(29,4) = 23751.\nFinal answer: $23751$", "final_answer": "23751", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "46cfc9d8614c1a01", "problem_id": "846deecb9de432dc", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(x^{3} + 3 x^{2} - x - 1\\right) \\cos{\\left(x^{2} - 3 \\right)}$.", "output": "1) Let f(x) = $\\left(x + 1\\right) e^{x} + \\left(x^{3} + 3 x^{2} - x - 1\\right) \\cos{\\left(x^{2} - 3 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- 2 x \\left(x^{3} + 3 x^{2} - x - 1\\right) \\sin{\\left(x^{2} - 3 \\right)} + \\left(x + 1\\right) e^{x} + \\left(3 x^{2} + 6 x - 1\\right) \\cos{\\left(x^{2} - 3 \\right)} + e^{x}$.\nFinal answer: $- 2\\cdot x\\cdot \\left(x^{3} + 3\\cdot x^{2} - x - 1\\right)\\cdot \\sin{\\left(x^{2} - 3 \\right)} + \\left(x + 1\\right)\\cdot e^{x} + \\left(3\\cdot x^{2} + 6\\cdot x - 1\\right)\\cdot \\cos{\\left(x^{2} - 3 \\right)} + e^{x}$", "final_answer": "- 2\\cdot x\\cdot \\left(x^{3} + 3\\cdot x^{2} - x - 1\\right)\\cdot \\sin{\\left(x^{2} - 3 \\right)} + \\left(x + 1\\right)\\cdot e^{x} + \\left(3\\cdot x^{2} + 6\\cdot x - 1\\right)\\cdot \\cos{\\left(x^{2} - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "127ef2f85ed65aa7", "problem_id": "846deecb9de432dc", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(x^{3} + 3 x^{2} - x - 1\\right) \\cos{\\left(x^{2} - 3 \\right)}$.", "output": "Let f(x) = $\\left(x + 1\\right) e^{x} + \\left(x^{3} + 3 x^{2} - x - 1\\right) \\cos{\\left(x^{2} - 3 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- 2 x \\left(x^{3} + 3 x^{2} - x - 1\\right) \\sin{\\left(x^{2} - 3 \\right)} + \\left(x + 1\\right) e^{x} + \\left(3 x^{2} + 6 x - 1\\right) \\cos{\\left(x^{2} - 3 \\right)} + e^{x}$.\nFinal answer: $- 2\\cdot x\\cdot \\left(x^{3} + 3\\cdot x^{2} - x - 1\\right)\\cdot \\sin{\\left(x^{2} - 3 \\right)} + \\left(x + 1\\right)\\cdot e^{x} + \\left(3\\cdot x^{2} + 6\\cdot x - 1\\right)\\cdot \\cos{\\left(x^{2} - 3 \\right)} + e^{x}$", "final_answer": "- 2\\cdot x\\cdot \\left(x^{3} + 3\\cdot x^{2} - x - 1\\right)\\cdot \\sin{\\left(x^{2} - 3 \\right)} + \\left(x + 1\\right)\\cdot e^{x} + \\left(3\\cdot x^{2} + 6\\cdot x - 1\\right)\\cdot \\cos{\\left(x^{2} - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "60da2bc7dd79e3fa", "problem_id": "93e08bc7a6c9189e", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{2} + 2 x + \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{3 x + 7}$.", "output": "1) Let f(x) = $\\frac{x^{2} + 2 x + \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{3 x + 7}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{- 6 x^{2} - 12 x + \\left(3 x + 7\\right) \\left(4 x + \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 6 \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 18}{2 \\left(3 x + 7\\right)^{2}}$.\nFinal answer: $\\frac{- 6\\cdot x^{2} - 12\\cdot x + \\left(3\\cdot x + 7\\right)\\cdot \\left(4\\cdot x + \\left(3\\cdot x + 7\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x + 14\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 4\\right) - \\left(18\\cdot x + 42\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 18}{2\\cdot \\left(3\\cdot x + 7\\right)^{2}}$", "final_answer": "\\frac{- 6\\cdot x^{2} - 12\\cdot x + \\left(3\\cdot x + 7\\right)\\cdot \\left(4\\cdot x + \\left(3\\cdot x + 7\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x + 14\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 4\\right) - \\left(18\\cdot x + 42\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 18}{2\\cdot \\left(3\\cdot x + 7\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b1ced8cf960b2f4e", "problem_id": "93e08bc7a6c9189e", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{2} + 2 x + \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{3 x + 7}$.", "output": "Let f(x) = $\\frac{x^{2} + 2 x + \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{3 x + 7}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{- 6 x^{2} - 12 x + \\left(3 x + 7\\right) \\left(4 x + \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 6 \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 18}{2 \\left(3 x + 7\\right)^{2}}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\frac{- 6\\cdot x^{2} - 12\\cdot x + \\left(3\\cdot x + 7\\right)\\cdot \\left(4\\cdot x + \\left(3\\cdot x + 7\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x + 14\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 4\\right) - \\left(18\\cdot x + 42\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 18}{2\\cdot \\left(3\\cdot x + 7\\right)^{2}}$", "final_answer": "\\frac{- 6\\cdot x^{2} - 12\\cdot x + \\left(3\\cdot x + 7\\right)\\cdot \\left(4\\cdot x + \\left(3\\cdot x + 7\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x + 14\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 4\\right) - \\left(18\\cdot x + 42\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 18}{2\\cdot \\left(3\\cdot x + 7\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "748c971eaeb8cc93", "problem_id": "8a67f09317357ee7", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} + x^{2} - x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 3$ at $x=1$.", "output": "Differentiate: f'(x) = $- 3 x^{2} + 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 1$.\nSlope at x=1: m = f'(1) = $-2 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}$.\nPoint on curve: (1, f(1)) = (1, $\\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} + 2$).\nFinal answer: $y = - 2\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 4$", "final_answer": "y = - 2\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "192242d2ca7bf6ea", "problem_id": "8a67f09317357ee7", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} + x^{2} - x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 3$ at $x=1$.", "output": "Differentiate: f'(x) = $- 3 x^{2} + 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 1$.\nSlope at x=1: m = f'(1) = $-2 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}$.\nPoint on curve: (1, f(1)) = (1, $\\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} + 2$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} + 2 = -2 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}(x − 1).\nOptionally expand: y = $- 2 x - 2 x \\sin{\\left(2 \\right)} + \\frac{x e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2 \\sin{\\left(2 \\right)} + 4$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 2\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 4$", "final_answer": "y = - 2\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8de00666d6ef1c10", "problem_id": "3787fad3e50f603c", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} + 4$.", "output": "1) Differentiate term-by-term:\n2) d/dx(4) = 0.\n3) d/dx(3 x^{2}) = 6 x.\n4) Combine: f'(x) = 6 x.\nFinal answer: $6\\cdot x$", "final_answer": "6\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d9004880940b02c2", "problem_id": "3787fad3e50f603c", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} + 4$.", "output": "Differentiate term-by-term:\nd/dx(4) = 0.\nd/dx(3 x^{2}) = 6 x.\nCombine: f'(x) = 6 x.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $6\\cdot x$", "final_answer": "6\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7255021530f6fa57", "problem_id": "794b91e86dacdb7d", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "You want 30 liters of a 20% salt solution. You have a 15% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 30−x liters is 70% solution.\n2) Salt equation: 15x + 70(30−x) = 20·30.\n3) (15-70)x = (20-70)·30 ⇒ x = 300/11.\n4) Then 30−x = 30/11.\nFinal answer: $300/11 L of 15% and 30/11 L of 70%$", "final_answer": "300/11 L of 15% and 30/11 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "334bb99829609b6a", "problem_id": "9d3d72eaf058d2a7", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} 3 x^{3} + 6 x^{2} + 15 x + 48$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} + 6 x^{2} + 14 x + 48.\n3) Then the limit is 225.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: 228.\nFinal answer: $228$", "final_answer": "228", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f417edb99747fa26", "problem_id": "07e2207304920762", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$6 x + 6 y$ = $30$\n$6 x + 3 y$ = $33$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $6$: $36 x + 36 y$ = $180$.\n3) Multiply the second equation by $6$: $36 x + 18 y$ = $198$.\n4) Subtract to eliminate x: $18 y$ = $-18$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $6$.\nFinal answer: $(x,y)=\\left(6, -1\\right)$", "final_answer": "(x,y)=\\left(6, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "753c0488d2082507", "problem_id": "07e2207304920762", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$6 x + 6 y$ = $30$\n$6 x + 3 y$ = $33$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $6$: $36 x + 36 y$ = $180$.\nMultiply the second equation by $6$: $36 x + 18 y$ = $198$.\nSubtract to eliminate x: $18 y$ = $-18$.\nSolve for y: y = $-1$.\nSubstitute back to find x: x = $6$.\nCheck: plug (x,y)=(6,-1) into both equations.\nFinal answer: $(x,y)=\\left(6, -1\\right)$", "final_answer": "(x,y)=\\left(6, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "af7419642391f58d", "problem_id": "455b6c1723bd9de1", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$4 x + 8 y$ = $84$\n$8 x - 7 y$ = $7$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $8$: $32 x + 64 y$ = $672$.\n3) Multiply the second equation by $4$: $32 x - 28 y$ = $28$.\n4) Subtract to eliminate x: $92 y$ = $644$.\n5) Solve for y: y = $7$.\n6) Substitute back to find x: x = $7$.\nFinal answer: $(x,y)=\\left(7, 7\\right)$", "final_answer": "(x,y)=\\left(7, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "faf6e800c329e8df", "problem_id": "455b6c1723bd9de1", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$4 x + 8 y$ = $84$\n$8 x - 7 y$ = $7$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $8$: $32 x + 64 y$ = $672$.\nMultiply the second equation by $4$: $32 x - 28 y$ = $28$.\nSubtract to eliminate x: $92 y$ = $644$.\nSolve for y: y = $7$.\nSubstitute back to find x: x = $7$.\nCheck: plug (x,y)=(7,7) into both equations.\nFinal answer: $(x,y)=\\left(7, 7\\right)$", "final_answer": "(x,y)=\\left(7, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "71bb8c427cfa50df", "problem_id": "0ca1d77451398f7c", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 11 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=4, p=$\\frac{1}{5}$ and simplify.\n4) Here C(11,4) = 330.\nFinal answer: $\\frac{1081344}{9765625}$", "final_answer": "\\frac{1081344}{9765625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1e7787e4703462cc", "problem_id": "0ca1d77451398f7c", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 11 times, what is the probability of exactly 4 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=11, k=4, p=$\\frac{1}{5}$ and simplify.\nFinal answer: $\\frac{1081344}{9765625}$", "final_answer": "\\frac{1081344}{9765625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9a784bceba90655f", "problem_id": "fadf7809e1ddb476", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 17 times, what is the probability of exactly 7 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=17, k=7, p=$\\frac{1}{4}$ and simplify.\n4) Here C(17,7) = 19448.\nFinal answer: $\\frac{143548119}{2147483648}$", "final_answer": "\\frac{143548119}{2147483648}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "03a7dbbe2422ddbd", "problem_id": "fadf7809e1ddb476", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 17 times, what is the probability of exactly 7 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=17, k=7, p=$\\frac{1}{4}$ and simplify.\nFinal answer: $\\frac{143548119}{2147483648}$", "final_answer": "\\frac{143548119}{2147483648}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b91731801ccbca87", "problem_id": "34b4ef7c1790a954", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 11 times, what is the probability of exactly 9 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=9, p=$\\frac{1}{8}$ and simplify.\n4) Here C(11,9) = 55.\nFinal answer: $\\frac{2695}{8589934592}$", "final_answer": "\\frac{2695}{8589934592}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "132b09e94e392ff6", "problem_id": "34b4ef7c1790a954", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 11 times, what is the probability of exactly 9 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=11, k=9, p=$\\frac{1}{8}$ and simplify.\nFinal answer: $\\frac{2695}{8589934592}$", "final_answer": "\\frac{2695}{8589934592}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d446d3c4b7b278b4", "problem_id": "ddd96130fed1a992", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Worker A finishes a job in 8 hours and Worker B finishes it in 12 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/8 job/hour. B's rate = 1/12 job/hour.\n2) Combined rate = 1/8 + 1/12 = 5/24 job/hour.\n3) Time = 1 ÷ (combined rate) = 24/5 hours.\nFinal answer: $\\frac{24}{5}$", "final_answer": "\\frac{24}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "07f622a66d46e91b", "problem_id": "ddd96130fed1a992", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Worker A finishes a job in 8 hours and Worker B finishes it in 12 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/8 job/hour. B's rate = 1/12 job/hour.\nCombined rate = 1/8 + 1/12 = 5/24 job/hour.\nTime = 1 ÷ (combined rate) = 24/5 hours.\nFinal answer: $\\frac{24}{5}$", "final_answer": "\\frac{24}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b50dc43420815dbb", "problem_id": "4c4729f24bcd6931", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "How many binary strings of length 26 contain exactly 9 ones with no two ones adjacent?", "output": "1) Reserve 8 mandatory zeros between the 9 ones.\n2) This leaves n−k+1 = 18 valid slots for the ones.\n3) Choose the slots: C(18,9) = 48620.\nFinal answer: $48620$", "final_answer": "48620", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6353219dc32c76ab", "problem_id": "4c4729f24bcd6931", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many binary strings of length 26 contain exactly 9 ones with no two ones adjacent?", "output": "Reserve 8 mandatory zeros between the 9 ones.\nThis leaves n−k+1 = 18 valid slots for the ones.\nChoose the slots: C(18,9) = 48620.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $48620$", "final_answer": "48620", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "04d82a16b7be5360", "problem_id": "49882216d0f286e6", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - x^{3} - 7 x^{2} - 25 x - 99$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to - x^{3} - 7 x^{2} - 26 x - 99.\n3) Then the limit is -379.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: -375.\nFinal answer: $-375$", "final_answer": "-375", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "540c679244a217ca", "problem_id": "d8bf114373514c85", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 9 red balls and 6 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(15,7).\n3) Favorable samples: choose k reds and n−k blues: C(9,1)·C(6,6).\n4) Compute totals: C(15,7) = 6435, and favorable = 9.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1}{715}$", "final_answer": "\\frac{1}{715}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b6b0e9b953c94aca", "problem_id": "9a277675d764e5ef", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} + x^{2} - x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 3$ at $x=2$.", "output": "1) Differentiate: f'(x) = $6 x^{2} + 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 1$.\n2) Slope at x=2: m = f'(2) = $\\frac{e}{2} - 2 \\sin{\\left(4 \\right)} + 27$.\n3) Point on curve: (2, f(2)) = (2, $\\cos{\\left(4 \\right)} + e + 21$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(4 \\right)} + e + 21 = \\frac{e}{2} - 2 \\sin{\\left(4 \\right)} + 27(x − 2).\n6) Optionally expand: y = $\\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 27 x - 33 + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$.\nFinal answer: $y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 27\\cdot x - 33 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$", "final_answer": "y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 27\\cdot x - 33 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cad1a0963ff30a9f", "problem_id": "9a277675d764e5ef", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} + x^{2} - x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 3$ at $x=2$.", "output": "Differentiate: f'(x) = $6 x^{2} + 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 1$.\nSlope at x=2: m = f'(2) = $\\frac{e}{2} - 2 \\sin{\\left(4 \\right)} + 27$.\nPoint on curve: (2, f(2)) = (2, $\\cos{\\left(4 \\right)} + e + 21$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − \\cos{\\left(4 \\right)} + e + 21 = \\frac{e}{2} - 2 \\sin{\\left(4 \\right)} + 27(x − 2).\nOptionally expand: y = $\\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 27 x - 33 + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 27\\cdot x - 33 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$", "final_answer": "y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 27\\cdot x - 33 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff6411596f7764bc", "problem_id": "1e0272ff59b3377e", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "You want 63 liters of a 35% salt solution. You have a 30% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 63−x liters is 60% solution.\n2) Salt equation: 30x + 60(63−x) = 35·63.\n3) (30-60)x = (35-60)·63 ⇒ x = 105/2.\n4) Then 63−x = 21/2.\nFinal answer: $105/2 L of 30% and 21/2 L of 60%$", "final_answer": "105/2 L of 30% and 21/2 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cebd25610bb2cac4", "problem_id": "1e0272ff59b3377e", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 63 liters of a 35% salt solution. You have a 30% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 63−x liters is 60% solution.\nSalt equation: 30x + 60(63−x) = 35·63.\n(30-60)x = (35-60)·63 ⇒ x = 105/2.\nThen 63−x = 21/2.\nCheck: 30·105/2 + 60·21/2 = 35·63.\nFinal answer: $105/2 L of 30% and 21/2 L of 60%$", "final_answer": "105/2 L of 30% and 21/2 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "372018fe2c7f4e20", "problem_id": "207626080385f49d", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_6=41$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_6=35.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=35 and k=6.\n4) Compute C(40,5) = 658008.\nFinal answer: $658008$", "final_answer": "658008", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "00089dc19f2e1737", "problem_id": "207626080385f49d", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_6=41$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_6=35.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=35 and k=6.\nCompute C(40,5) = 658008.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $658008$", "final_answer": "658008", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "36c220636338cb20", "problem_id": "c5b3070c81c0b39e", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$x + 3 y$ = $5$\n$x - y$ = $5$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $x + 3 y$ = $5$.\nMultiply the second equation by $1$: $x - y$ = $5$.\nSubtract to eliminate x: $4 y$ = $0$.\nSolve for y: y = $0$.\nSubstitute back to find x: x = $5$.\nCheck: plug (x,y)=(5,0) into both equations.\nFinal answer: $(x,y)=\\left(5, 0\\right)$", "final_answer": "(x,y)=\\left(5, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2507a1c63ea63778", "problem_id": "c5b3070c81c0b39e", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the linear system:\n$x + 3 y$ = $5$\n$x - y$ = $5$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $x + 3 y$ = $5$.\nMultiply the second equation by $1$: $x - y$ = $5$.\nFinal answer: $(x,y)=\\left(5, 0\\right)$", "final_answer": "(x,y)=\\left(5, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c709a728f881906d", "problem_id": "1727594bdbf2745d", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 5 x - 12$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$-5$, c=$-12$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $121$, so √Δ = $11$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $4$.\nx2 = $- \\frac{3}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{4, - \\frac{3}{2}\\}$", "final_answer": "x \\in \\{4, - \\frac{3}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9f8770e172b0cce8", "problem_id": "1727594bdbf2745d", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 5 x - 12$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$-5$, c=$-12$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $121$, so √Δ = $11$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $4$.\n7) x2 = $- \\frac{3}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{4, - \\frac{3}{2}\\}$", "final_answer": "x \\in \\{4, - \\frac{3}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e468b0729f69bd7", "problem_id": "0ba53fdbdb1a9d9c", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} 4 x^{4} + x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $4 x^{4} + x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $-1 + \\frac{5 \\pi}{2} + \\frac{15 \\pi^{2}}{8} + \\frac{65 \\pi^{3}}{24} + \\frac{205 \\pi^{5}}{8}$.\n4) Simplify the result.\nFinal answer: $-1 + \\frac{5\\cdot \\pi}{2} + \\frac{15\\cdot \\pi^{2}}{8} + \\frac{65\\cdot \\pi^{3}}{24} + \\frac{205\\cdot \\pi^{5}}{8}$", "final_answer": "-1 + \\frac{5\\cdot \\pi}{2} + \\frac{15\\cdot \\pi^{2}}{8} + \\frac{65\\cdot \\pi^{3}}{24} + \\frac{205\\cdot \\pi^{5}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9e700146f4fbb7dd", "problem_id": "0ba53fdbdb1a9d9c", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} 4 x^{4} + x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $4 x^{4} + x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $-1 + \\frac{5 \\pi}{2} + \\frac{15 \\pi^{2}}{8} + \\frac{65 \\pi^{3}}{24} + \\frac{205 \\pi^{5}}{8}$.\nFinal answer: $-1 + \\frac{5\\cdot \\pi}{2} + \\frac{15\\cdot \\pi^{2}}{8} + \\frac{65\\cdot \\pi^{3}}{24} + \\frac{205\\cdot \\pi^{5}}{8}$", "final_answer": "-1 + \\frac{5\\cdot \\pi}{2} + \\frac{15\\cdot \\pi^{2}}{8} + \\frac{65\\cdot \\pi^{3}}{24} + \\frac{205\\cdot \\pi^{5}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "15bc10769ac9549d", "problem_id": "aede9159c7e75813", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} - 3 x \\sin{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $- 2 x^{2} \\sin{\\left(x \\right)} - 3 x \\sin{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 3\\cdot x\\cdot \\cos{\\left(x \\right)} - 3\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - 3\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 3\\cdot x\\cdot \\cos{\\left(x \\right)} - 3\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - 3\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b422436e5c7ef2f8", "problem_id": "aede9159c7e75813", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} - 3 x \\sin{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $- 2 x^{2} \\sin{\\left(x \\right)} - 3 x \\sin{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 3\\cdot x\\cdot \\cos{\\left(x \\right)} - 3\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - 3\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 3\\cdot x\\cdot \\cos{\\left(x \\right)} - 3\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - 3\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "994813ca99d0056b", "problem_id": "30c94b6ba3bd1dd3", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡3 (mod 11), x≡2 (mod 8). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 58 (mod 88).\nFinal answer: $x\\equiv 58\\pmod{88}$", "final_answer": "x\\equiv 58\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "763809ed0bb5c862", "problem_id": "30c94b6ba3bd1dd3", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the congruences: x≡3 (mod 11), x≡2 (mod 8). Give the solution modulo 88.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 88.\nCompute the CRT solution to get x ≡ 58 (mod 88).\nFinal answer: $x\\equiv 58\\pmod{88}$", "final_answer": "x\\equiv 58\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "21440a9a99af5e51", "problem_id": "f859b743037329d6", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Two friends split $480 in the ratio 9:7. How much does each receive?", "output": "1) Total ratio parts = 9+7 = 16.\n2) Each part = 480 ÷ 16 = 30.\n3) First share = 9 × 30 = 270.\n4) Second share = 7 × 30 = 210.\nFinal answer: $210$", "final_answer": "210", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3c781a935d6c3081", "problem_id": "f859b743037329d6", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $480 in the ratio 9:7. How much does each receive?", "output": "Total ratio parts = 9+7 = 16.\nEach part = 480 ÷ 16 = 30.\nFirst share = 9 × 30 = 270.\nSecond share = 7 × 30 = 210.\nCheck: 270+210=480.\nFinal answer: $210$", "final_answer": "210", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "731170a72f2f454a", "problem_id": "c3a1eac95158d038", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$- 4 x + y$ = $18$\n$- 6 x + 8 y$ = $14$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-6$: $24 x - 6 y$ = $-108$.\n3) Multiply the second equation by $-4$: $24 x - 32 y$ = $-56$.\n4) Subtract to eliminate x: $26 y$ = $-52$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, -2\\right)$", "final_answer": "(x,y)=\\left(-5, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4e9869a5e193e038", "problem_id": "c3a1eac95158d038", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the linear system:\n$- 4 x + y$ = $18$\n$- 6 x + 8 y$ = $14$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-6$: $24 x - 6 y$ = $-108$.\nMultiply the second equation by $-4$: $24 x - 32 y$ = $-56$.\nFinal answer: $(x,y)=\\left(-5, -2\\right)$", "final_answer": "(x,y)=\\left(-5, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "653b10ae85281f07", "problem_id": "68788f63f9642bc8", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} 2 x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{15 \\pi}{2} - 2 + \\frac{65 \\pi^{3}}{12}$.\n4) Simplify the result.\nFinal answer: $- \\frac{15\\cdot \\pi}{2} - 2 + \\frac{65\\cdot \\pi^{3}}{12}$", "final_answer": "- \\frac{15\\cdot \\pi}{2} - 2 + \\frac{65\\cdot \\pi^{3}}{12}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6eeb3df7d6c04f8e", "problem_id": "68788f63f9642bc8", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} 2 x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $2 x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{15 \\pi}{2} - 2 + \\frac{65 \\pi^{3}}{12}$.\nFinal answer: $- \\frac{15\\cdot \\pi}{2} - 2 + \\frac{65\\cdot \\pi^{3}}{12}$", "final_answer": "- \\frac{15\\cdot \\pi}{2} - 2 + \\frac{65\\cdot \\pi^{3}}{12}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e28690ce6d902242", "problem_id": "df768cff59b9db9c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(3 x^{2} - x + 1\\right) \\cos{\\left(x^{2} + x + 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 3 e^{x}) = - 3 e^{x}$.\n3) So $f'(x) = - 6 x^{3} \\sin{\\left(\\left(x^{2} + x\\right) + 2 \\right)} - x^{2} \\sin{\\left(\\left(x^{2} + x\\right) + 2 \\right)} + x e^{x} - x \\sin{\\left(\\left(x^{2} + x\\right) + 2 \\right)} + 6 x \\cos{\\left(\\left(x^{2} + x\\right) + 2 \\right)} - 2 e^{x} - \\sin{\\left(\\left(x^{2} + x\\right) + 2 \\right)} - \\cos{\\left(\\left(x^{2} + x\\right) + 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 3 e^{x}) = - 3 e^{x}$.\nTherefore the correct derivative is $f'(x) = \\left(x - 3\\right) e^{x} - \\left(2 x + 1\\right) \\left(x \\left(3 x - 1\\right) + 1\\right) \\sin{\\left(\\left(x^{2} + x\\right) + 2 \\right)} + \\left(6 x - 1\\right) \\cos{\\left(\\left(x^{2} + x\\right) + 2 \\right)} + e^{x}$.\nFinal answer: $\\left(x - 3\\right) e^{x} - \\left(2 x + 1\\right) \\left(x \\left(3 x - 1\\right) + 1\\right) \\sin{\\left(\\left(x^{2} + x\\right) + 2 \\right)} + \\left(6 x - 1\\right) \\cos{\\left(\\left(x^{2} + x\\right) + 2 \\right)} + e^{x}$", "final_answer": "\\left(x - 3\\right) e^{x} - \\left(2 x + 1\\right) \\left(x \\left(3 x - 1\\right) + 1\\right) \\sin{\\left(\\left(x^{2} + x\\right) + 2 \\right)} + \\left(6 x - 1\\right) \\cos{\\left(\\left(x^{2} + x\\right) + 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c7a327d6f726dd71", "problem_id": "75a30a635f0831db", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A store sells a backpack for $700. It is discounted by 15%. You buy 7 backpacks. What is the total cost?", "output": "1) Discounted price = 700 × (1 − 15/100) = 700 × 85/100 = 595.\n2) Subtotal = 7 × 595 = 4165.\nFinal answer: $4165$", "final_answer": "4165", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "629ab7ffe1e1fc4e", "problem_id": "82f49cd34d98cb42", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x^{3} - x^{2} - 3 x - 3$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - x^{3} - x^{2} - 4 x - 3.\n3) Then the limit is -23.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -21.\nFinal answer: $-21$", "final_answer": "-21", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7cb828c9d2572b7d", "problem_id": "151dfb95194dad80", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡4 (mod 7), x≡10 (mod 11). Give the solution modulo 77.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 77.\n3) Compute the CRT solution to get x ≡ 32 (mod 77).\nFinal answer: $x\\equiv 32\\pmod{77}$", "final_answer": "x\\equiv 32\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0ad1b3dd325208c9", "problem_id": "a4fd8fe5fe59b118", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A store sells a backpack for $119. It is discounted by 15%. You buy 8 backpacks. What is the total cost?", "output": "1) Discounted price = 119 × (1 − 15/100) = 119 × 85/100 = 2023/20.\n2) Subtotal = 8 × 2023/20 = 4046/5.\nFinal answer: $\\frac{4046}{5}$", "final_answer": "\\frac{4046}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4bec7f1149661378", "problem_id": "a4fd8fe5fe59b118", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "A store sells a backpack for $119. It is discounted by 15%. You buy 8 backpacks. What is the total cost?", "output": "Discounted price = 119 × (1 − 15/100) = 119 × 85/100 = 2023/20.\nSubtotal = 8 × 2023/20 = 4046/5.\nFinal answer: $\\frac{4046}{5}$", "final_answer": "\\frac{4046}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "75f4accde4e8c0bf", "problem_id": "4719412a483a0cc7", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A store sells a backpack for $190. It is discounted by 40%. You buy 8 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 190 × (1 − 40/100) = 190 × 60/100 = 114.\nSubtotal = 8 × 114 = 912.\nTotal with tax = 912 × 110/100 = 5016/5.\nCheck: recomputing the formula yields 5016/5.\nFinal answer: $\\frac{5016}{5}$", "final_answer": "\\frac{5016}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db880a371d28e843", "problem_id": "4719412a483a0cc7", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A store sells a backpack for $190. It is discounted by 40%. You buy 8 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 190 × (1 − 40/100) = 190 × 60/100 = 114.\n2) Subtotal = 8 × 114 = 912.\n3) Total with tax = 912 × 110/100 = 5016/5.\nFinal answer: $\\frac{5016}{5}$", "final_answer": "\\frac{5016}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ebc9dd108c564721", "problem_id": "46d4d0f3dc9f72f2", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the multiplicative inverse of 3 modulo 19.", "output": "1) We want x such that 3x ≡ 1 (mod 19).\n2) Use extended Euclid to find s,t with 3s + 19t = 1.\n3) One Bezout identity is: 3·(-6) + 19·(1) = 1.\n4) Reduce s modulo 19: x ≡ 13 (mod 19).\nFinal answer: $x\\equiv 13\\pmod{19}$", "final_answer": "x\\equiv 13\\pmod{19}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3ea45c27977aca86", "problem_id": "46d4d0f3dc9f72f2", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the multiplicative inverse of 3 modulo 19.", "output": "We want x such that 3x ≡ 1 (mod 19).\nUse extended Euclid to find s,t with 3s + 19t = 1.\nOne Bezout identity is: 3·(-6) + 19·(1) = 1.\nReduce s modulo 19: x ≡ 13 (mod 19).\nCheck: 3·13 mod 19 = 1.\nFinal answer: $x\\equiv 13\\pmod{19}$", "final_answer": "x\\equiv 13\\pmod{19}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e1223f8fdf091b39", "problem_id": "b40e7b417be46014", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$4 x + 4 y$ = $-8$\n$4 x - 4 y$ = $-32$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $16 x + 16 y$ = $-32$.\n3) Multiply the second equation by $4$: $16 x - 16 y$ = $-128$.\n4) Subtract to eliminate x: $32 y$ = $96$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, 3\\right)$", "final_answer": "(x,y)=\\left(-5, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e96446175b78aee8", "problem_id": "b40e7b417be46014", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$4 x + 4 y$ = $-8$\n$4 x - 4 y$ = $-32$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $16 x + 16 y$ = $-32$.\nMultiply the second equation by $4$: $16 x - 16 y$ = $-128$.\nSubtract to eliminate x: $32 y$ = $96$.\nSolve for y: y = $3$.\nSubstitute back to find x: x = $-5$.\nCheck: plug (x,y)=(-5,3) into both equations.\nFinal answer: $(x,y)=\\left(-5, 3\\right)$", "final_answer": "(x,y)=\\left(-5, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "497e2f8e20eda988", "problem_id": "07133fe752a54018", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{5} x - \\frac{11 x}{2} + \\frac{9}{2} + \\frac{27 \\sqrt{5}}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- 3 \\sqrt{5} - \\frac{11}{2}$, c=$\\frac{9}{2} + \\frac{27 \\sqrt{5}}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{229}{4} - 21 \\sqrt{5}$, so √Δ = $\\sqrt{\\frac{229}{4} - 21 \\sqrt{5}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{229 - 84 \\sqrt{5}}}{4} + \\frac{11}{4} + \\frac{3 \\sqrt{5}}{2}$.\nx2 = $- \\frac{\\sqrt{229 - 84 \\sqrt{5}}}{4} + \\frac{11}{4} + \\frac{3 \\sqrt{5}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{229 - 84\\cdot \\sqrt{5}}}{4} + \\frac{11}{4} + \\frac{3\\cdot \\sqrt{5}}{2}, - \\frac{\\sqrt{229 - 84\\cdot \\sqrt{5}}}{4} + \\frac{11}{4} + \\frac{3\\cdot \\sqrt{5}}{2}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{229 - 84\\cdot \\sqrt{5}}}{4} + \\frac{11}{4} + \\frac{3\\cdot \\sqrt{5}}{2}, - \\frac{\\sqrt{229 - 84\\cdot \\sqrt{5}}}{4} + \\frac{11}{4} + \\frac{3\\cdot \\sqrt{5}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4741509cd3958705", "problem_id": "07133fe752a54018", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{5} x - \\frac{11 x}{2} + \\frac{9}{2} + \\frac{27 \\sqrt{5}}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- 3 \\sqrt{5} - \\frac{11}{2}$, c=$\\frac{9}{2} + \\frac{27 \\sqrt{5}}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{229}{4} - 21 \\sqrt{5}$, so √Δ = $\\sqrt{\\frac{229}{4} - 21 \\sqrt{5}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{229 - 84 \\sqrt{5}}}{4} + \\frac{11}{4} + \\frac{3 \\sqrt{5}}{2}$.\n7) x2 = $- \\frac{\\sqrt{229 - 84 \\sqrt{5}}}{4} + \\frac{11}{4} + \\frac{3 \\sqrt{5}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{229 - 84\\cdot \\sqrt{5}}}{4} + \\frac{11}{4} + \\frac{3\\cdot \\sqrt{5}}{2}, - \\frac{\\sqrt{229 - 84\\cdot \\sqrt{5}}}{4} + \\frac{11}{4} + \\frac{3\\cdot \\sqrt{5}}{2}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{229 - 84\\cdot \\sqrt{5}}}{4} + \\frac{11}{4} + \\frac{3\\cdot \\sqrt{5}}{2}, - \\frac{\\sqrt{229 - 84\\cdot \\sqrt{5}}}{4} + \\frac{11}{4} + \\frac{3\\cdot \\sqrt{5}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "de58077399b44233", "problem_id": "1a715bf0cd64dd85", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Find the multiplicative inverse of 21 modulo 37.", "output": "1) We want x such that 21x ≡ 1 (mod 37).\n2) Use extended Euclid to find s,t with 21s + 37t = 1.\n3) One Bezout identity is: 21·(-7) + 37·(4) = 1.\n4) Reduce s modulo 37: x ≡ 30 (mod 37).\nFinal answer: $x\\equiv 30\\pmod{37}$", "final_answer": "x\\equiv 30\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3342006fc8833cef", "problem_id": "1a715bf0cd64dd85", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the multiplicative inverse of 21 modulo 37.", "output": "We want x such that 21x ≡ 1 (mod 37).\nUse extended Euclid to find s,t with 21s + 37t = 1.\nOne Bezout identity is: 21·(-7) + 37·(4) = 1.\nReduce s modulo 37: x ≡ 30 (mod 37).\nCheck: 21·30 mod 37 = 1.\nFinal answer: $x\\equiv 30\\pmod{37}$", "final_answer": "x\\equiv 30\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "92c3931d708895e1", "problem_id": "f449fcd77b43a7bd", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡5 (mod 8), x≡1 (mod 7). Give the solution modulo 56.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 56.\n3) Compute the CRT solution to get x ≡ 29 (mod 56).\nFinal answer: $x\\equiv 29\\pmod{56}$", "final_answer": "x\\equiv 29\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d2a69608f3dae421", "problem_id": "6ffbb966a04e4bba", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x + 3$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 3.\n3) Then the limit is 3.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 5.\nFinal answer: $5$", "final_answer": "5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b911640e30d84e4a", "problem_id": "4a590b2d11ab6d2c", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the equation of the tangent line to $y=- 2 x^{2} + 3 x - 4$ at $x=2$.", "output": "1) Differentiate: f'(x) = $3 - 4 x$.\n2) Slope at x=2: m = f'(2) = $-5$.\n3) Point on curve: (2, f(2)) = (2, $-6$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -6 = -5(x − 2).\n6) Optionally expand: y = $4 - 5 x$.\nFinal answer: $y = 4 - 5\\cdot x$", "final_answer": "y = 4 - 5\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "334bb99829609b6a", "problem_id": "9d3d72eaf058d2a7", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} 3 x^{3} + 6 x^{2} + 15 x + 48$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} + 6 x^{2} + 14 x + 48.\n3) Then the limit is 225.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: 228.\nFinal answer: $228$", "final_answer": "228", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8d62bb2757d7c35f", "problem_id": "7c7fbe543cff6c01", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Two friends split $638 in the ratio 6:5. How much does each receive?", "output": "1) Total ratio parts = 6+5 = 11.\n2) Each part = 638 ÷ 11 = 58.\n3) First share = 6 × 58 = 348.\n4) Second share = 5 × 58 = 290.\nFinal answer: $290$", "final_answer": "290", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "67813c3a8ada2f67", "problem_id": "fa06e5968b2870a6", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} 3 x^{3} + 2 x^{2} + 4 x + 4$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} + 2 x^{2} + 3 x + 4.\n3) Then the limit is 12.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: 13.\nFinal answer: $13$", "final_answer": "13", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "34d75cb124e0bede", "problem_id": "f08b8b0a94126bc9", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$4 x^{2} - 4 \\sqrt{10} x - 3 x + 3 \\sqrt{10}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$4$, b=$- 4 \\sqrt{10} - 3$, c=$3 \\sqrt{10}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $169 - 24 \\sqrt{10}$, so √Δ = $\\sqrt{169 - 24 \\sqrt{10}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{3}{8} + \\frac{\\sqrt{169 - 24 \\sqrt{10}}}{8} + \\frac{\\sqrt{10}}{2}$.\nx2 = $- \\frac{\\sqrt{169 - 24 \\sqrt{10}}}{8} + \\frac{3}{8} + \\frac{\\sqrt{10}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{3}{8} + \\frac{\\sqrt{169 - 24\\cdot \\sqrt{10}}}{8} + \\frac{\\sqrt{10}}{2}, - \\frac{\\sqrt{169 - 24\\cdot \\sqrt{10}}}{8} + \\frac{3}{8} + \\frac{\\sqrt{10}}{2}\\}$", "final_answer": "x \\in \\{\\frac{3}{8} + \\frac{\\sqrt{169 - 24\\cdot \\sqrt{10}}}{8} + \\frac{\\sqrt{10}}{2}, - \\frac{\\sqrt{169 - 24\\cdot \\sqrt{10}}}{8} + \\frac{3}{8} + \\frac{\\sqrt{10}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0157ef6b400fa00b", "problem_id": "d5161946143d352a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{2 x^{2} + 3 x + \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{x + 2}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 2}) = \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 2}$.\n3) So $f'(x) = \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 8 x + 8} + \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + 8 x + 8} + \\frac{4 x^{2}}{2 x^{2} + 8 x + 8} + \\frac{4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 8 x + 8} + \\frac{8 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + 8 x + 8} + \\frac{16 x}{2 x^{2} + 8 x + 8} + \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 4 x + 4} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x + 4} - \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 2} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 2} + \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 8 x + 8} + \\frac{8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + 8 x + 8} + \\frac{8}{2 x^{2} + 8 x + 8} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 2}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 2}) = \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x + 2\\right)} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 2} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x + 2\\right)^{2}} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 2}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 4 x^{2} - 6 x + \\left(x + 2\\right) \\left(8 x + \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4}{2 \\left(x + 2\\right)^{2}}$.\nFinal answer: $\\frac{- 4 x^{2} - 6 x + \\left(x + 2\\right) \\left(8 x + \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4}{2 \\left(x + 2\\right)^{2}}$", "final_answer": "\\frac{- 4 x^{2} - 6 x + \\left(x + 2\\right) \\left(8 x + \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4}{2 \\left(x + 2\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "4dab0cfb02ccbb6d", "problem_id": "86ad5e3e1f9c98d9", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - x^{2} + 5 x - 10$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} + 4 x - 10.\n3) Then the limit is -22.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -24.\nFinal answer: $-24$", "final_answer": "-24", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d7e8e13c395ebe4b", "problem_id": "736b2aeefba253d7", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_5=38$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_5=28.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=28 and k=5.\nCompute C(32,4) = 35960.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $35960$", "final_answer": "35960", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0a0102b795708aff", "problem_id": "76dc0cfa8cac0569", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1$.\n3) Compute an antiderivative: F(x) = $\\frac{x^{3}}{3} - x^{2} - x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{x^{3}}{3} - x^{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{3}}{3} - x^{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f7e3e667be97bb55", "problem_id": "76dc0cfa8cac0569", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1$.\nCompute an antiderivative: F(x) = $\\frac{x^{3}}{3} - x^{2} - x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $C + \\frac{x^{3}}{3} - x^{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{3}}{3} - x^{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b6c843552adba59c", "problem_id": "8a278a2b47869cf0", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡7 (mod 8), x≡1 (mod 11). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 23 (mod 88).\nFinal answer: $x\\equiv 23\\pmod{88}$", "final_answer": "x\\equiv 23\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b435d5a3b56ebcaa", "problem_id": "39a036e8e4b3525b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 3 x - 12$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to 2 x - 12.\n3) Then the limit is -18.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -21.\nFinal answer: $-21$", "final_answer": "-21", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "31ce96d14e327523", "problem_id": "c06a22f2f769cade", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{13 x}{3} + 2$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$- \\frac{13}{3}$, c=$2$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{25}{9}$, so √Δ = $\\frac{5}{3}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{3}{2}$.\n7) x2 = $\\frac{2}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{3}{2}, \\frac{2}{3}\\}$", "final_answer": "x \\in \\{\\frac{3}{2}, \\frac{2}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b573e23da5bddc29", "problem_id": "c06a22f2f769cade", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{13 x}{3} + 2$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$- \\frac{13}{3}$, c=$2$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{25}{9}$, so √Δ = $\\frac{5}{3}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{3}{2}$.\nx2 = $\\frac{2}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{3}{2}, \\frac{2}{3}\\}$", "final_answer": "x \\in \\{\\frac{3}{2}, \\frac{2}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b1ff1caf1e7edf2d", "problem_id": "a30e95f3ee449584", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $304. It is discounted by 40%. You buy 6 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 304 × (1 − 40/100) = 304 × 60/100 = 912/5.\nSubtotal = 6 × 912/5 = 5472/5.\nTotal with tax = 5472/5 × 110/100 = 30096/25.\nCheck: recomputing the formula yields 30096/25.\nFinal answer: $\\frac{30096}{25}$", "final_answer": "\\frac{30096}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5e4a3c15cd9af439", "problem_id": "a30e95f3ee449584", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A store sells a backpack for $304. It is discounted by 40%. You buy 6 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 304 × (1 − 40/100) = 304 × 60/100 = 912/5.\n2) Subtotal = 6 × 912/5 = 5472/5.\n3) Total with tax = 5472/5 × 110/100 = 30096/25.\nFinal answer: $\\frac{30096}{25}$", "final_answer": "\\frac{30096}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d2739a4d1347af6d", "problem_id": "5a15bf5231c1ca39", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $510 in the ratio 1:2. How much does each receive?", "output": "1) Total ratio parts = 1+2 = 3.\n2) Each part = 510 ÷ 3 = 170.\n3) First share = 1 × 170 = 170.\n4) Second share = 2 × 170 = 340.\nFinal answer: $340$", "final_answer": "340", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "392635ee5b99f70f", "problem_id": "5a15bf5231c1ca39", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Two friends split $510 in the ratio 1:2. How much does each receive?", "output": "Total ratio parts = 1+2 = 3.\nEach part = 510 ÷ 3 = 170.\nFirst share = 1 × 170 = 170.\nSecond share = 2 × 170 = 340.\nCheck: 170+340=510.\nFinal answer: $340$", "final_answer": "340", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7a0f922c32d8792c", "problem_id": "3431e49fda1e3c92", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "You want 69 liters of a 45% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 69−x liters is 70% solution.\n2) Salt equation: 20x + 70(69−x) = 45·69.\n3) (20-70)x = (45-70)·69 ⇒ x = 69/2.\n4) Then 69−x = 69/2.\nFinal answer: $69/2 L of 20% and 69/2 L of 70%$", "final_answer": "69/2 L of 20% and 69/2 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "daeaa52f4973487f", "problem_id": "3431e49fda1e3c92", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "You want 69 liters of a 45% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 69−x liters is 70% solution.\nSalt equation: 20x + 70(69−x) = 45·69.\n(20-70)x = (45-70)·69 ⇒ x = 69/2.\nFinal answer: $69/2 L of 20% and 69/2 L of 70%$", "final_answer": "69/2 L of 20% and 69/2 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e4106cd90b2da9bf", "problem_id": "14fcfcbda5659fa2", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\nEvaluate at the bounds: F(2 \\pi) and F(0).\nSubtract: F(2 \\pi) − F(0) = $4 \\pi \\left(-1 + \\pi + 2 \\pi^{2}\\right)$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $4\\cdot \\pi\\cdot \\left(-1 + \\pi + 2\\cdot \\pi^{2}\\right)$", "final_answer": "4\\cdot \\pi\\cdot \\left(-1 + \\pi + 2\\cdot \\pi^{2}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "034ba0b924a61571", "problem_id": "14fcfcbda5659fa2", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} 3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(0).\n3) Subtract: F(2 \\pi) − F(0) = $4 \\pi \\left(-1 + \\pi + 2 \\pi^{2}\\right)$.\n4) Simplify the result.\nFinal answer: $4\\cdot \\pi\\cdot \\left(-1 + \\pi + 2\\cdot \\pi^{2}\\right)$", "final_answer": "4\\cdot \\pi\\cdot \\left(-1 + \\pi + 2\\cdot \\pi^{2}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0bfbd4de9fbfe0b8", "problem_id": "e039ff0cb7a4262d", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$8 x - 11 y$ = $-8$\n$6 y$ = $0$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $8$: $48 y$ = $0$.\n4) Subtract to eliminate x: $- 48 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, 0\\right)$", "final_answer": "(x,y)=\\left(-1, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "342d51a31627224b", "problem_id": "2412cdcdf3201af4", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x^{3} - 5 x^{2} - 12 x - 22$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{3} - 5 x^{2} - 13 x - 22.\n3) Then the limit is -84.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -82.\nFinal answer: $-82$", "final_answer": "-82", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "eb7fb95ff455611a", "problem_id": "c6cd40d0f75426c8", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{2} - 3 x - 4\\,dx$.", "output": "$C + \\frac{2\\cdot x^{3}}{3} - \\frac{3\\cdot x^{2}}{2} - 4\\cdot x$", "final_answer": "C + \\frac{2\\cdot x^{3}}{3} - \\frac{3\\cdot x^{2}}{2} - 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d6fc4d224ffc24eb", "problem_id": "c6cd40d0f75426c8", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{2} - 3 x - 4\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $2 x^{2} - 3 x - 4$.\nCompute an antiderivative: F(x) = $\\frac{2 x^{3}}{3} - \\frac{3 x^{2}}{2} - 4 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{2\\cdot x^{3}}{3} - \\frac{3\\cdot x^{2}}{2} - 4\\cdot x$", "final_answer": "C + \\frac{2\\cdot x^{3}}{3} - \\frac{3\\cdot x^{2}}{2} - 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f610b0a33b4a4e4", "problem_id": "639146ed12b5794d", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the multiplicative inverse of 7 modulo 19.", "output": "We want x such that 7x ≡ 1 (mod 19).\nUse extended Euclid to find s,t with 7s + 19t = 1.\nOne Bezout identity is: 7·(-8) + 19·(3) = 1.\nReduce s modulo 19: x ≡ 11 (mod 19).\nCheck: 7·11 mod 19 = 1.\nFinal answer: $x\\equiv 11\\pmod{19}$", "final_answer": "x\\equiv 11\\pmod{19}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6169cd3c6573a2a5", "problem_id": "639146ed12b5794d", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the multiplicative inverse of 7 modulo 19.", "output": "1) We want x such that 7x ≡ 1 (mod 19).\n2) Use extended Euclid to find s,t with 7s + 19t = 1.\n3) One Bezout identity is: 7·(-8) + 19·(3) = 1.\n4) Reduce s modulo 19: x ≡ 11 (mod 19).\nFinal answer: $x\\equiv 11\\pmod{19}$", "final_answer": "x\\equiv 11\\pmod{19}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e5219556696e3202", "problem_id": "ef0d8c78bfc473e1", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "You want 32 liters of a 40% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 32−x liters is 70% solution.\nSalt equation: 20x + 70(32−x) = 40·32.\n(20-70)x = (40-70)·32 ⇒ x = 96/5.\nThen 32−x = 64/5.\nCheck: 20·96/5 + 70·64/5 = 40·32.\nFinal answer: $96/5 L of 20% and 64/5 L of 70%$", "final_answer": "96/5 L of 20% and 64/5 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eac95358b56550d4", "problem_id": "ef0d8c78bfc473e1", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "You want 32 liters of a 40% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 32−x liters is 70% solution.\n2) Salt equation: 20x + 70(32−x) = 40·32.\n3) (20-70)x = (40-70)·32 ⇒ x = 96/5.\n4) Then 32−x = 64/5.\nFinal answer: $96/5 L of 20% and 64/5 L of 70%$", "final_answer": "96/5 L of 20% and 64/5 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0a68b77936da3628", "problem_id": "a8a47290df2d0099", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "A store sells a backpack for $55. It is discounted by 35%. You buy 11 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 55 × (1 − 35/100) = 55 × 65/100 = 143/4.\nSubtotal = 11 × 143/4 = 1573/4.\nTotal with tax = 1573/4 × 109/100 = 171457/400.\nFinal answer: $\\frac{171457}{400}$", "final_answer": "\\frac{171457}{400}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "108377d859a01589", "problem_id": "a8a47290df2d0099", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A store sells a backpack for $55. It is discounted by 35%. You buy 11 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 55 × (1 − 35/100) = 55 × 65/100 = 143/4.\n2) Subtotal = 11 × 143/4 = 1573/4.\n3) Total with tax = 1573/4 × 109/100 = 171457/400.\nFinal answer: $\\frac{171457}{400}$", "final_answer": "\\frac{171457}{400}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c9f51b672c4287c7", "problem_id": "cacb96b5ed3ef339", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Worker A finishes a job in 12 hours and Worker B finishes it in 5 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/12 job/hour. B's rate = 1/5 job/hour.\nCombined rate = 1/12 + 1/5 = 17/60 job/hour.\nTime = 1 ÷ (combined rate) = 60/17 hours.\nCheck: (combined rate)×(time) = 17/60×60/17 = 1.\nFinal answer: $\\frac{60}{17}$", "final_answer": "\\frac{60}{17}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9a1f9e37690be66f", "problem_id": "cacb96b5ed3ef339", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Worker A finishes a job in 12 hours and Worker B finishes it in 5 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/12 job/hour. B's rate = 1/5 job/hour.\n2) Combined rate = 1/12 + 1/5 = 17/60 job/hour.\n3) Time = 1 ÷ (combined rate) = 60/17 hours.\nFinal answer: $\\frac{60}{17}$", "final_answer": "\\frac{60}{17}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "68caf3e8b64b6e64", "problem_id": "7a8593616b62b908", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{3} + 4 x^{2} + x + 4\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $3 x^{3} + 4 x^{2} + x + 4$.\n3) Compute an antiderivative: F(x) = $\\frac{3 x^{4}}{4} + \\frac{4 x^{3}}{3} + \\frac{x^{2}}{2} + 4 x$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{3\\cdot x^{4}}{4} + \\frac{4\\cdot x^{3}}{3} + \\frac{x^{2}}{2} + 4\\cdot x$", "final_answer": "C + \\frac{3\\cdot x^{4}}{4} + \\frac{4\\cdot x^{3}}{3} + \\frac{x^{2}}{2} + 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "94279c7877f66450", "problem_id": "7a8593616b62b908", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{3} + 4 x^{2} + x + 4\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $3 x^{3} + 4 x^{2} + x + 4$.\nCompute an antiderivative: F(x) = $\\frac{3 x^{4}}{4} + \\frac{4 x^{3}}{3} + \\frac{x^{2}}{2} + 4 x$.\nFinal answer: $C + \\frac{3\\cdot x^{4}}{4} + \\frac{4\\cdot x^{3}}{3} + \\frac{x^{2}}{2} + 4\\cdot x$", "final_answer": "C + \\frac{3\\cdot x^{4}}{4} + \\frac{4\\cdot x^{3}}{3} + \\frac{x^{2}}{2} + 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aa31459267e08b00", "problem_id": "a2514a0e264608f1", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Two friends split $407 in the ratio 8:3. How much does each receive?", "output": "1) Total ratio parts = 8+3 = 11.\n2) Each part = 407 ÷ 11 = 37.\n3) First share = 8 × 37 = 296.\n4) Second share = 3 × 37 = 111.\nFinal answer: $111$", "final_answer": "111", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c9acb7eb29cf9d93", "problem_id": "93e08bc7a6c9189e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{2} + 2 x + \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{3 x + 7}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{7 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 7}) = \\frac{7 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 7}$.\n3) So $f'(x) = \\frac{9 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} + 84 x + 98} + \\frac{18 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} + 84 x + 98} + \\frac{6 x^{2}}{18 x^{2} + 84 x + 98} + \\frac{42 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} + 84 x + 98} + \\frac{84 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} + 84 x + 98} + \\frac{28 x}{18 x^{2} + 84 x + 98} + \\frac{49 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} + 84 x + 98} + \\frac{98 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} + 84 x + 98} + \\frac{46}{18 x^{2} + 84 x + 98} + \\frac{21 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{2} + 42 x + 49} - \\frac{7 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x + 14} + \\frac{7 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 7} - \\frac{7 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 7}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{7 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 7}) = \\frac{7 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(3 x + 7\\right)} + \\frac{7 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 7} - \\frac{21 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(3 x + 7\\right)^{2}}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 6 x^{2} - 12 x + \\left(3 x + 7\\right) \\left(4 x + \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 6 \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 18}{2 \\left(3 x + 7\\right)^{2}}$.\nFinal answer: $\\frac{- 6 x^{2} - 12 x + \\left(3 x + 7\\right) \\left(4 x + \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 6 \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 18}{2 \\left(3 x + 7\\right)^{2}}$", "final_answer": "\\frac{- 6 x^{2} - 12 x + \\left(3 x + 7\\right) \\left(4 x + \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 6 \\left(3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 18}{2 \\left(3 x + 7\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d1c86bc36abb988c", "problem_id": "7b7b3f566c34cc58", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 24 red balls and 30 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(54,12).\n3) Favorable samples: choose k reds and n−k blues: C(24,4)·C(30,8).\n4) Compute totals: C(54,12) = 343006888770, and favorable = 62193181050.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{330165}{1820921}$", "final_answer": "\\frac{330165}{1820921}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c2778f57a9d37414", "problem_id": "7b7b3f566c34cc58", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "An urn has 24 red balls and 30 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(54,12).\nFavorable samples: choose k reds and n−k blues: C(24,4)·C(30,8).\nFinal answer: $\\frac{330165}{1820921}$", "final_answer": "\\frac{330165}{1820921}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "779bf191c157f432", "problem_id": "763c60e03ed75c2a", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A cyclist rides at 62 miles per hour for 1 hours, then at 26 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 62 × 1 = 62 miles.\n2) Second leg distance = 26 × 2 = 52 miles.\n3) Total distance = 62 + 52 = 114 miles.\nFinal answer: $114$", "final_answer": "114", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "18e1ea47b1265f5d", "problem_id": "763c60e03ed75c2a", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A cyclist rides at 62 miles per hour for 1 hours, then at 26 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 62 × 1 = 62 miles.\nSecond leg distance = 26 × 2 = 52 miles.\nTotal distance = 62 + 52 = 114 miles.\nCheck: adding both legs gives 114.\nFinal answer: $114$", "final_answer": "114", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3d7b83e0e036587d", "problem_id": "694ac8f6f8da0488", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} - x^{2} + 7 x - 26$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} + 6 x - 26.\n3) Then the limit is -66.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: -70.\nFinal answer: $-70$", "final_answer": "-70", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e410f449e8730152", "problem_id": "5aa67b94b96c0948", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{5} x}{3} + \\frac{5 x}{6} - \\frac{2}{3} + \\frac{\\sqrt{5}}{6}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{5}{6} - \\frac{\\sqrt{5}}{3}$, c=$- \\frac{2}{3} + \\frac{\\sqrt{5}}{6}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{47}{12} - \\frac{11 \\sqrt{5}}{9}$, so √Δ = $\\sqrt{\\frac{47}{12} - \\frac{11 \\sqrt{5}}{9}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{5}{12} + \\frac{\\sqrt{5}}{6} + \\frac{\\sqrt{141 - 44 \\sqrt{5}}}{12}$.\n7) x2 = $- \\frac{\\sqrt{141 - 44 \\sqrt{5}}}{12} - \\frac{5}{12} + \\frac{\\sqrt{5}}{6}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{5}{12} + \\frac{\\sqrt{5}}{6} + \\frac{\\sqrt{141 - 44\\cdot \\sqrt{5}}}{12}, - \\frac{\\sqrt{141 - 44\\cdot \\sqrt{5}}}{12} - \\frac{5}{12} + \\frac{\\sqrt{5}}{6}\\}$", "final_answer": "x \\in \\{- \\frac{5}{12} + \\frac{\\sqrt{5}}{6} + \\frac{\\sqrt{141 - 44\\cdot \\sqrt{5}}}{12}, - \\frac{\\sqrt{141 - 44\\cdot \\sqrt{5}}}{12} - \\frac{5}{12} + \\frac{\\sqrt{5}}{6}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "17d97262c1f3c2c0", "problem_id": "31b89b02e76cefb1", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "How many binary strings of length 26 contain exactly 5 ones with no two ones adjacent?", "output": "1) Reserve 4 mandatory zeros between the 5 ones.\n2) This leaves n−k+1 = 22 valid slots for the ones.\n3) Choose the slots: C(22,5) = 26334.\nFinal answer: $26334$", "final_answer": "26334", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "25ff075c0e3e8cb0", "problem_id": "31b89b02e76cefb1", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "How many binary strings of length 26 contain exactly 5 ones with no two ones adjacent?", "output": "Reserve 4 mandatory zeros between the 5 ones.\nThis leaves n−k+1 = 22 valid slots for the ones.\nChoose the slots: C(22,5) = 26334.\nFinal answer: $26334$", "final_answer": "26334", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1c610c8ca8bd51b7", "problem_id": "e4113fec9cad4d90", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=18$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=18 and k=7.\n4) Compute C(24,6) = 134596.\nFinal answer: $134596$", "final_answer": "134596", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "65ccc2deeec61b30", "problem_id": "e4113fec9cad4d90", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=18$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=18 and k=7.\nCompute C(24,6) = 134596.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $134596$", "final_answer": "134596", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "af37f25d993b0d05", "problem_id": "8b69f0e2800b5db2", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Two friends split $1508 in the ratio 5:8. How much does each receive?", "output": "1) Total ratio parts = 5+8 = 13.\n2) Each part = 1508 ÷ 13 = 116.\n3) First share = 5 × 116 = 580.\n4) Second share = 8 × 116 = 928.\nFinal answer: $928$", "final_answer": "928", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "702ea81163953729", "problem_id": "beaf7b6780904fb7", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - x^{2} + 5 x - 8$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} + 4 x - 8.\n3) Then the limit is -20.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -22.\nFinal answer: $-22$", "final_answer": "-22", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "dfd9d54fbcf27602", "problem_id": "591d5640dea6d5b5", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 13 red balls and 17 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(30,11).\n3) Favorable samples: choose k reds and n−k blues: C(13,5)·C(17,6).\n4) Compute totals: C(30,11) = 54627300, and favorable = 15927912.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{4862}{16675}$", "final_answer": "\\frac{4862}{16675}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4c17c16dbd6c025d", "problem_id": "591d5640dea6d5b5", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 13 red balls and 17 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(30,11).\nFavorable samples: choose k reds and n−k blues: C(13,5)·C(17,6).\nCompute totals: C(30,11) = 54627300, and favorable = 15927912.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{4862}{16675}$.\nFinal answer: $\\frac{4862}{16675}$", "final_answer": "\\frac{4862}{16675}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e724c0843aae825a", "problem_id": "26e9853dcb83fb74", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{4} - 2 x^{3} - x^{2} + x - 1$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-1) = 0.\n3) d/dx(x) = 1.\n4) d/dx(- x^{2}) = - 2 x.\n5) d/dx(- 2 x^{3}) = - 6 x^{2}.\n6) d/dx(3 x^{4}) = 12 x^{3}.\n7) Combine: f'(x) = 12 x^{3} - 6 x^{2} - 2 x + 1.\nFinal answer: $12\\cdot x^{3} - 6\\cdot x^{2} - 2\\cdot x + 1$", "final_answer": "12\\cdot x^{3} - 6\\cdot x^{2} - 2\\cdot x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "35e02a7d9196f5af", "problem_id": "26e9853dcb83fb74", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{4} - 2 x^{3} - x^{2} + x - 1$.", "output": "Differentiate term-by-term:\nd/dx(-1) = 0.\nd/dx(x) = 1.\nd/dx(- x^{2}) = - 2 x.\nd/dx(- 2 x^{3}) = - 6 x^{2}.\nd/dx(3 x^{4}) = 12 x^{3}.\nCombine: f'(x) = 12 x^{3} - 6 x^{2} - 2 x + 1.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $12\\cdot x^{3} - 6\\cdot x^{2} - 2\\cdot x + 1$", "final_answer": "12\\cdot x^{3} - 6\\cdot x^{2} - 2\\cdot x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9b0105f10ce2c241", "problem_id": "3e52bc4ea3c56672", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$4 x + 2 y$ = $-4$\n$- 2 x - 4 y$ = $-10$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $- 8 x - 4 y$ = $8$.\n3) Multiply the second equation by $4$: $- 8 x - 16 y$ = $-40$.\n4) Subtract to eliminate x: $12 y$ = $48$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, 4\\right)$", "final_answer": "(x,y)=\\left(-3, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "68b4802067b5f0ba", "problem_id": "331742ad7366406c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 3 x^{2} - 5 x - 12$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} - 6 x - 12.\n3) Then the limit is -36.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -34.\nFinal answer: $-34$", "final_answer": "-34", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8c61d455dd783c47", "problem_id": "7be7a4ba453c9c94", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- 2 x - 2 y$ = $2$\n$3 x + y$ = $3$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $- 6 x - 6 y$ = $6$.\nMultiply the second equation by $-2$: $- 6 x - 2 y$ = $-6$.\nSubtract to eliminate x: $- 4 y$ = $12$.\nSolve for y: y = $-3$.\nSubstitute back to find x: x = $2$.\nCheck: plug (x,y)=(2,-3) into both equations.\nFinal answer: $(x,y)=\\left(2, -3\\right)$", "final_answer": "(x,y)=\\left(2, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dd85c817b5b08737", "problem_id": "7be7a4ba453c9c94", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$- 2 x - 2 y$ = $2$\n$3 x + y$ = $3$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $3$: $- 6 x - 6 y$ = $6$.\n3) Multiply the second equation by $-2$: $- 6 x - 2 y$ = $-6$.\n4) Subtract to eliminate x: $- 4 y$ = $12$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $2$.\nFinal answer: $(x,y)=\\left(2, -3\\right)$", "final_answer": "(x,y)=\\left(2, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3a1a03fdcc736dba", "problem_id": "bb3ffb773bd8150f", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{7} x - \\frac{23 x}{4} + \\frac{15}{4} + \\frac{9 \\sqrt{7}}{4}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- 3 \\sqrt{7} - \\frac{23}{4}$, c=$\\frac{15}{4} + \\frac{9 \\sqrt{7}}{4}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{51 \\sqrt{7}}{2} + \\frac{1297}{16}$, so √Δ = $\\sqrt{\\frac{51 \\sqrt{7}}{2} + \\frac{1297}{16}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{23}{8} + \\frac{3 \\sqrt{7}}{2} + \\frac{\\sqrt{408 \\sqrt{7} + 1297}}{8}$.\nx2 = $- \\frac{\\sqrt{408 \\sqrt{7} + 1297}}{8} + \\frac{23}{8} + \\frac{3 \\sqrt{7}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{23}{8} + \\frac{3\\cdot \\sqrt{7}}{2} + \\frac{\\sqrt{408\\cdot \\sqrt{7} + 1297}}{8}, - \\frac{\\sqrt{408\\cdot \\sqrt{7} + 1297}}{8} + \\frac{23}{8} + \\frac{3\\cdot \\sqrt{7}}{2}\\}$", "final_answer": "x \\in \\{\\frac{23}{8} + \\frac{3\\cdot \\sqrt{7}}{2} + \\frac{\\sqrt{408\\cdot \\sqrt{7} + 1297}}{8}, - \\frac{\\sqrt{408\\cdot \\sqrt{7} + 1297}}{8} + \\frac{23}{8} + \\frac{3\\cdot \\sqrt{7}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7f2c6ec3687bd222", "problem_id": "bb3ffb773bd8150f", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{7} x - \\frac{23 x}{4} + \\frac{15}{4} + \\frac{9 \\sqrt{7}}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- 3 \\sqrt{7} - \\frac{23}{4}$, c=$\\frac{15}{4} + \\frac{9 \\sqrt{7}}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{51 \\sqrt{7}}{2} + \\frac{1297}{16}$, so √Δ = $\\sqrt{\\frac{51 \\sqrt{7}}{2} + \\frac{1297}{16}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{23}{8} + \\frac{3 \\sqrt{7}}{2} + \\frac{\\sqrt{408 \\sqrt{7} + 1297}}{8}$.\n7) x2 = $- \\frac{\\sqrt{408 \\sqrt{7} + 1297}}{8} + \\frac{23}{8} + \\frac{3 \\sqrt{7}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{23}{8} + \\frac{3\\cdot \\sqrt{7}}{2} + \\frac{\\sqrt{408\\cdot \\sqrt{7} + 1297}}{8}, - \\frac{\\sqrt{408\\cdot \\sqrt{7} + 1297}}{8} + \\frac{23}{8} + \\frac{3\\cdot \\sqrt{7}}{2}\\}$", "final_answer": "x \\in \\{\\frac{23}{8} + \\frac{3\\cdot \\sqrt{7}}{2} + \\frac{\\sqrt{408\\cdot \\sqrt{7} + 1297}}{8}, - \\frac{\\sqrt{408\\cdot \\sqrt{7} + 1297}}{8} + \\frac{23}{8} + \\frac{3\\cdot \\sqrt{7}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "acedcaebf3dcebc9", "problem_id": "2f64fe5b41cb180d", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Two friends split $63 in the ratio 6:1. How much does each receive?", "output": "Total ratio parts = 6+1 = 7.\nEach part = 63 ÷ 7 = 9.\nFirst share = 6 × 9 = 54.\nFinal answer: $9$", "final_answer": "9", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3a99e897a9219e95", "problem_id": "2f64fe5b41cb180d", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Two friends split $63 in the ratio 6:1. How much does each receive?", "output": "1) Total ratio parts = 6+1 = 7.\n2) Each part = 63 ÷ 7 = 9.\n3) First share = 6 × 9 = 54.\n4) Second share = 1 × 9 = 9.\nFinal answer: $9$", "final_answer": "9", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "02c2c0f549a85d24", "problem_id": "5ed6fc7e3100e9df", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$7 x - 7 y$ = $14$\n$6 x$ = $0$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $6$: $42 x - 42 y$ = $84$.\n3) Multiply the second equation by $7$: $42 x$ = $0$.\n4) Subtract to eliminate x: $- 42 y$ = $84$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, -2\\right)$", "final_answer": "(x,y)=\\left(0, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ce3c40db03db2010", "problem_id": "5ed6fc7e3100e9df", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$7 x - 7 y$ = $14$\n$6 x$ = $0$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $6$: $42 x - 42 y$ = $84$.\nMultiply the second equation by $7$: $42 x$ = $0$.\nSubtract to eliminate x: $- 42 y$ = $84$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $0$.\nCheck: plug (x,y)=(0,-2) into both equations.\nFinal answer: $(x,y)=\\left(0, -2\\right)$", "final_answer": "(x,y)=\\left(0, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "52d00b831f7ca6b1", "problem_id": "e61ac4f9fab06dc5", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=36$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=36 and k=10.\n4) Compute C(45,9) = 886163135.\nFinal answer: $886163135$", "final_answer": "886163135", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ed9ed7e5e8a1647c", "problem_id": "e61ac4f9fab06dc5", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=36$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=36 and k=10.\nCompute C(45,9) = 886163135.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $886163135$", "final_answer": "886163135", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0ad7e5c6518803ff", "problem_id": "1348b128c0344707", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the equation of the tangent line to $y=3 - x^{2}$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 2 x$.\n2) Slope at x=2: m = f'(2) = $-4$.\n3) Point on curve: (2, f(2)) = (2, $-1$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -1 = -4(x − 2).\n6) Optionally expand: y = $7 - 4 x$.\nFinal answer: $y = 7 - 4\\cdot x$", "final_answer": "y = 7 - 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "96c465e9775bec5a", "problem_id": "1348b128c0344707", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the equation of the tangent line to $y=3 - x^{2}$ at $x=2$.", "output": "Differentiate: f'(x) = $- 2 x$.\nSlope at x=2: m = f'(2) = $-4$.\nPoint on curve: (2, f(2)) = (2, $-1$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -1 = -4(x − 2).\nOptionally expand: y = $7 - 4 x$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 7 - 4\\cdot x$", "final_answer": "y = 7 - 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d3970352b1d0ee86", "problem_id": "d6892d7b5e4db7f0", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} - 2 x^{2} + 11 x - 41$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{2} + 10 x - 41.\n3) Then the limit is -113.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: -117.\nFinal answer: $-117$", "final_answer": "-117", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3f7eae87c1b4609e", "problem_id": "17746b8e14158497", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Worker A finishes a job in 21 hours and Worker B finishes it in 4 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/21 job/hour. B's rate = 1/4 job/hour.\n2) Combined rate = 1/21 + 1/4 = 25/84 job/hour.\n3) Time = 1 ÷ (combined rate) = 84/25 hours.\nFinal answer: $\\frac{84}{25}$", "final_answer": "\\frac{84}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bf3ad664c08edd49", "problem_id": "bed96c227c94a5b1", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the linear system:\n$- 4 x + 4 y$ = $-40$\n$2 x$ = $10$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $- 8 x + 8 y$ = $-80$.\n3) Multiply the second equation by $-4$: $- 8 x$ = $-40$.\n4) Subtract to eliminate x: $8 y$ = $-40$.\n5) Solve for y: y = $-5$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, -5\\right)$", "final_answer": "(x,y)=\\left(5, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f78ad390f2b7c2d5", "problem_id": "bed96c227c94a5b1", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- 4 x + 4 y$ = $-40$\n$2 x$ = $10$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $2$: $- 8 x + 8 y$ = $-80$.\nMultiply the second equation by $-4$: $- 8 x$ = $-40$.\nSubtract to eliminate x: $8 y$ = $-40$.\nSolve for y: y = $-5$.\nSubstitute back to find x: x = $5$.\nCheck: plug (x,y)=(5,-5) into both equations.\nFinal answer: $(x,y)=\\left(5, -5\\right)$", "final_answer": "(x,y)=\\left(5, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff01a3796e68ce9a", "problem_id": "e9a6373ba51b85fe", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "You want 59 liters of a 40% salt solution. You have a 30% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 59−x liters is 50% solution.\n2) Salt equation: 30x + 50(59−x) = 40·59.\n3) (30-50)x = (40-50)·59 ⇒ x = 59/2.\n4) Then 59−x = 59/2.\nFinal answer: $59/2 L of 30% and 59/2 L of 50%$", "final_answer": "59/2 L of 30% and 59/2 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bbe42052b624c410", "problem_id": "e9a6373ba51b85fe", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "You want 59 liters of a 40% salt solution. You have a 30% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 59−x liters is 50% solution.\nSalt equation: 30x + 50(59−x) = 40·59.\n(30-50)x = (40-50)·59 ⇒ x = 59/2.\nThen 59−x = 59/2.\nCheck: 30·59/2 + 50·59/2 = 40·59.\nFinal answer: $59/2 L of 30% and 59/2 L of 50%$", "final_answer": "59/2 L of 30% and 59/2 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6f4053006bda585f", "problem_id": "e1ab32c27a6fa125", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$2 x$ = $-10$\n$- 4 x + 2 y$ = $12$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $- 8 x$ = $40$.\n3) Multiply the second equation by $2$: $- 8 x + 4 y$ = $24$.\n4) Subtract to eliminate x: $- 4 y$ = $16$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, -4\\right)$", "final_answer": "(x,y)=\\left(-5, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "80cfa442f54a0f10", "problem_id": "e1ab32c27a6fa125", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$2 x$ = $-10$\n$- 4 x + 2 y$ = $12$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $- 8 x$ = $40$.\nMultiply the second equation by $2$: $- 8 x + 4 y$ = $24$.\nSubtract to eliminate x: $- 4 y$ = $16$.\nSolve for y: y = $-4$.\nSubstitute back to find x: x = $-5$.\nCheck: plug (x,y)=(-5,-4) into both equations.\nFinal answer: $(x,y)=\\left(-5, -4\\right)$", "final_answer": "(x,y)=\\left(-5, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2c541826a758422e", "problem_id": "d15870ed05d8a6d9", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the congruences: x≡1 (mod 9), x≡1 (mod 11). Give the solution modulo 99.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 99.\nCompute the CRT solution to get x ≡ 1 (mod 99).\nFinal answer: $x\\equiv 1\\pmod{99}$", "final_answer": "x\\equiv 1\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "04a1f8194708da9d", "problem_id": "d15870ed05d8a6d9", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡1 (mod 9), x≡1 (mod 11). Give the solution modulo 99.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 99.\nCompute the CRT solution to get x ≡ 1 (mod 99).\nCheck: 1 mod 9=1 and mod 11=1.\nFinal answer: $x\\equiv 1\\pmod{99}$", "final_answer": "x\\equiv 1\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a1e052e490ddc2c0", "problem_id": "e0cbb6a0b350776a", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many binary strings of length 16 contain exactly 3 ones with no two ones adjacent?", "output": "1) Reserve 2 mandatory zeros between the 3 ones.\n2) This leaves n−k+1 = 14 valid slots for the ones.\n3) Choose the slots: C(14,3) = 364.\nFinal answer: $364$", "final_answer": "364", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "47ef28921613f4de", "problem_id": "e0cbb6a0b350776a", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "How many binary strings of length 16 contain exactly 3 ones with no two ones adjacent?", "output": "Reserve 2 mandatory zeros between the 3 ones.\nThis leaves n−k+1 = 14 valid slots for the ones.\nChoose the slots: C(14,3) = 364.\nFinal answer: $364$", "final_answer": "364", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ecc937deb96fe45f", "problem_id": "4017acd51d319e55", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 81 miles per hour for 1 hours, then at 63 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 81 × 1 = 81 miles.\nSecond leg distance = 63 × 1 = 63 miles.\nTotal distance = 81 + 63 = 144 miles.\nCheck: adding both legs gives 144.\nFinal answer: $144$", "final_answer": "144", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a4f2260ea8939eeb", "problem_id": "4017acd51d319e55", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A cyclist rides at 81 miles per hour for 1 hours, then at 63 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 81 × 1 = 81 miles.\n2) Second leg distance = 63 × 1 = 63 miles.\n3) Total distance = 81 + 63 = 144 miles.\nFinal answer: $144$", "final_answer": "144", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2b5ed80776aaa469", "problem_id": "59e0f5147ec2bb3a", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡2 (mod 9), x≡10 (mod 11). Give the solution modulo 99.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 99.\n3) Compute the CRT solution to get x ≡ 65 (mod 99).\nFinal answer: $x\\equiv 65\\pmod{99}$", "final_answer": "x\\equiv 65\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "da688227f99768df", "problem_id": "59e0f5147ec2bb3a", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 9), x≡10 (mod 11). Give the solution modulo 99.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 99.\nCompute the CRT solution to get x ≡ 65 (mod 99).\nFinal answer: $x\\equiv 65\\pmod{99}$", "final_answer": "x\\equiv 65\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cbb34c182556a775", "problem_id": "3b43bc825dfa612a", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡7 (mod 9), x≡8 (mod 11). Give the solution modulo 99.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 99.\n3) Compute the CRT solution to get x ≡ 52 (mod 99).\nFinal answer: $x\\equiv 52\\pmod{99}$", "final_answer": "x\\equiv 52\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "70268bc774ae89dd", "problem_id": "3b43bc825dfa612a", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡7 (mod 9), x≡8 (mod 11). Give the solution modulo 99.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 99.\nCompute the CRT solution to get x ≡ 52 (mod 99).\nCheck: 52 mod 9=7 and mod 11=8.\nFinal answer: $x\\equiv 52\\pmod{99}$", "final_answer": "x\\equiv 52\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "201fc6a223876e14", "problem_id": "8138dc84f0cf1adf", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 12 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=12, k=8, p=$\\frac{1}{2}$ and simplify.\nHere C(12,8) = 495.\nCheck: expression simplifies to $\\frac{495}{4096}$.\nFinal answer: $\\frac{495}{4096}$", "final_answer": "\\frac{495}{4096}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a997b2410b9930a3", "problem_id": "8138dc84f0cf1adf", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 12 times, what is the probability of exactly 8 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=8, p=$\\frac{1}{2}$ and simplify.\n4) Here C(12,8) = 495.\nFinal answer: $\\frac{495}{4096}$", "final_answer": "\\frac{495}{4096}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3ab30a33dee7c02a", "problem_id": "25244b326216bb4d", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{3} x - \\frac{x}{2} - \\frac{15}{2} + 3 \\sqrt{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\sqrt{3} - \\frac{1}{2}$, c=$- \\frac{15}{2} + 3 \\sqrt{3}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{1}{4} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{133 - 44\\cdot \\sqrt{3}}}{4}, - \\frac{\\sqrt{133 - 44\\cdot \\sqrt{3}}}{4} + \\frac{1}{4} + \\frac{\\sqrt{3}}{2}\\}$", "final_answer": "x \\in \\{\\frac{1}{4} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{133 - 44\\cdot \\sqrt{3}}}{4}, - \\frac{\\sqrt{133 - 44\\cdot \\sqrt{3}}}{4} + \\frac{1}{4} + \\frac{\\sqrt{3}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5964cf1f49c4d444", "problem_id": "25244b326216bb4d", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{3} x - \\frac{x}{2} - \\frac{15}{2} + 3 \\sqrt{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\sqrt{3} - \\frac{1}{2}$, c=$- \\frac{15}{2} + 3 \\sqrt{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{133}{4} - 11 \\sqrt{3}$, so √Δ = $\\sqrt{\\frac{133}{4} - 11 \\sqrt{3}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{1}{4} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{133 - 44 \\sqrt{3}}}{4}$.\n7) x2 = $- \\frac{\\sqrt{133 - 44 \\sqrt{3}}}{4} + \\frac{1}{4} + \\frac{\\sqrt{3}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{1}{4} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{133 - 44\\cdot \\sqrt{3}}}{4}, - \\frac{\\sqrt{133 - 44\\cdot \\sqrt{3}}}{4} + \\frac{1}{4} + \\frac{\\sqrt{3}}{2}\\}$", "final_answer": "x \\in \\{\\frac{1}{4} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{133 - 44\\cdot \\sqrt{3}}}{4}, - \\frac{\\sqrt{133 - 44\\cdot \\sqrt{3}}}{4} + \\frac{1}{4} + \\frac{\\sqrt{3}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "84cb0cef2ec7dc96", "problem_id": "58fde6efccf935ab", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Worker A finishes a job in 7 hours and Worker B finishes it in 11 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/7 job/hour. B's rate = 1/11 job/hour.\n2) Combined rate = 1/7 + 1/11 = 18/77 job/hour.\n3) Time = 1 ÷ (combined rate) = 77/18 hours.\nFinal answer: $\\frac{77}{18}$", "final_answer": "\\frac{77}{18}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "87acf837597dced4", "problem_id": "58fde6efccf935ab", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Worker A finishes a job in 7 hours and Worker B finishes it in 11 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/7 job/hour. B's rate = 1/11 job/hour.\nCombined rate = 1/7 + 1/11 = 18/77 job/hour.\nTime = 1 ÷ (combined rate) = 77/18 hours.\nFinal answer: $\\frac{77}{18}$", "final_answer": "\\frac{77}{18}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72138a9dd05b6e1c", "problem_id": "399ece830e72a8be", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - 2 x^{3} - 5 x^{2} - 21 x - 83$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{3} - 5 x^{2} - 22 x - 83.\n3) Then the limit is -379.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: -375.\nFinal answer: $-375$", "final_answer": "-375", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5c62a7538c9b3080", "problem_id": "e58416070c949525", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 11 times, what is the probability of exactly 8 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=8, p=$\\frac{1}{6}$ and simplify.\n4) Here C(11,8) = 165.\nFinal answer: $\\frac{6875}{120932352}$", "final_answer": "\\frac{6875}{120932352}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8a000f8fe283b995", "problem_id": "b692dc822d61e7b1", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_4=17$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=17 and k=4.\n4) Compute C(20,3) = 1140.\nFinal answer: $1140$", "final_answer": "1140", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cf1c3cf19fc8aa92", "problem_id": "b692dc822d61e7b1", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_4=17$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=17 and k=4.\nCompute C(20,3) = 1140.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $1140$", "final_answer": "1140", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c59c2b82a94a48ef", "problem_id": "913318ed96f8fd21", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡5 (mod 9), x≡4 (mod 13). Give the solution modulo 117.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 117.\nCompute the CRT solution to get x ≡ 95 (mod 117).\nCheck: 95 mod 9=5 and mod 13=4.\nFinal answer: $x\\equiv 95\\pmod{117}$", "final_answer": "x\\equiv 95\\pmod{117}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a179bc8da918a472", "problem_id": "913318ed96f8fd21", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡5 (mod 9), x≡4 (mod 13). Give the solution modulo 117.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 117.\n3) Compute the CRT solution to get x ≡ 95 (mod 117).\nFinal answer: $x\\equiv 95\\pmod{117}$", "final_answer": "x\\equiv 95\\pmod{117}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e350aff6de66f3f", "problem_id": "d8b1b1df9759d0d8", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 9 red balls and 13 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(22,7).\n3) Favorable samples: choose k reds and n−k blues: C(9,1)·C(13,6).\n4) Compute totals: C(22,7) = 170544, and favorable = 15444.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{117}{1292}$", "final_answer": "\\frac{117}{1292}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "422c49a241f02026", "problem_id": "d8b1b1df9759d0d8", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "An urn has 9 red balls and 13 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(22,7).\nFavorable samples: choose k reds and n−k blues: C(9,1)·C(13,6).\nFinal answer: $\\frac{117}{1292}$", "final_answer": "\\frac{117}{1292}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4a8ee0a4f881d48b", "problem_id": "deca8fd48c75c0c0", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$- 6 x + 7 y$ = $-26$\n$- 3 x - 4 y$ = $2$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $18 x - 21 y$ = $78$.\nMultiply the second equation by $-6$: $18 x + 24 y$ = $-12$.\nSubtract to eliminate x: $- 45 y$ = $90$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $2$.\nCheck: plug (x,y)=(2,-2) into both equations.\nFinal answer: $(x,y)=\\left(2, -2\\right)$", "final_answer": "(x,y)=\\left(2, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9e819d16179b4e2b", "problem_id": "deca8fd48c75c0c0", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the linear system:\n$- 6 x + 7 y$ = $-26$\n$- 3 x - 4 y$ = $2$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $18 x - 21 y$ = $78$.\n3) Multiply the second equation by $-6$: $18 x + 24 y$ = $-12$.\n4) Subtract to eliminate x: $- 45 y$ = $90$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $2$.\nFinal answer: $(x,y)=\\left(2, -2\\right)$", "final_answer": "(x,y)=\\left(2, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a65d807e8ad1878", "problem_id": "b87846c4b1289fe4", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 20 red balls and 12 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(32,12).\nFavorable samples: choose k reds and n−k blues: C(20,3)·C(12,9).\nCompute totals: C(32,12) = 225792840, and favorable = 250800.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{2090}{1881607}$.\nFinal answer: $\\frac{2090}{1881607}$", "final_answer": "\\frac{2090}{1881607}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "82d8d29bdf09da4e", "problem_id": "b87846c4b1289fe4", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "An urn has 20 red balls and 12 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(32,12).\n3) Favorable samples: choose k reds and n−k blues: C(20,3)·C(12,9).\n4) Compute totals: C(32,12) = 225792840, and favorable = 250800.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{2090}{1881607}$", "final_answer": "\\frac{2090}{1881607}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2c7666c2cb23a4ba", "problem_id": "a628ebee75d1ed22", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡5 (mod 7), x≡3 (mod 10). Give the solution modulo 70.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 70.\n3) Compute the CRT solution to get x ≡ 33 (mod 70).\nFinal answer: $x\\equiv 33\\pmod{70}$", "final_answer": "x\\equiv 33\\pmod{70}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e266b289b8d85c11", "problem_id": "a628ebee75d1ed22", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the congruences: x≡5 (mod 7), x≡3 (mod 10). Give the solution modulo 70.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 70.\nCompute the CRT solution to get x ≡ 33 (mod 70).\nFinal answer: $x\\equiv 33\\pmod{70}$", "final_answer": "x\\equiv 33\\pmod{70}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "54dc0c4d1c38eda3", "problem_id": "6e3f5c7a0d673a9d", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_5=41$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_5=31.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=31 and k=5.\nCompute C(35,4) = 52360.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $52360$", "final_answer": "52360", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "61a0c67a77377e29", "problem_id": "6e3f5c7a0d673a9d", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_5=41$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_5=31.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=31 and k=5.\n4) Compute C(35,4) = 52360.\nFinal answer: $52360$", "final_answer": "52360", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9fa8d62d9bf0fdcd", "problem_id": "ace749ef4ef4cded", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A store sells a backpack for $950. It is discounted by 30%. You buy 5 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 950 × (1 − 30/100) = 950 × 70/100 = 665.\n2) Subtotal = 5 × 665 = 3325.\n3) Total with tax = 3325 × 109/100 = 14497/4.\nFinal answer: $\\frac{14497}{4}$", "final_answer": "\\frac{14497}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dad4aaca937eca8e", "problem_id": "ace749ef4ef4cded", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $950. It is discounted by 30%. You buy 5 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 950 × (1 − 30/100) = 950 × 70/100 = 665.\nSubtotal = 5 × 665 = 3325.\nTotal with tax = 3325 × 109/100 = 14497/4.\nCheck: recomputing the formula yields 14497/4.\nFinal answer: $\\frac{14497}{4}$", "final_answer": "\\frac{14497}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8a5570db9559f167", "problem_id": "8a02941847c2af3a", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$9 x - 2 y$ = $-63$\n$- 2 x - 12 y$ = $-98$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $- 18 x + 4 y$ = $126$.\n3) Multiply the second equation by $9$: $- 18 x - 108 y$ = $-882$.\n4) Subtract to eliminate x: $112 y$ = $1008$.\n5) Solve for y: y = $9$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, 9\\right)$", "final_answer": "(x,y)=\\left(-5, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fce9221c97f84fba", "problem_id": "8a02941847c2af3a", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$9 x - 2 y$ = $-63$\n$- 2 x - 12 y$ = $-98$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $- 18 x + 4 y$ = $126$.\nMultiply the second equation by $9$: $- 18 x - 108 y$ = $-882$.\nSubtract to eliminate x: $112 y$ = $1008$.\nSolve for y: y = $9$.\nSubstitute back to find x: x = $-5$.\nCheck: plug (x,y)=(-5,9) into both equations.\nFinal answer: $(x,y)=\\left(-5, 9\\right)$", "final_answer": "(x,y)=\\left(-5, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3637a1a1355d6e4b", "problem_id": "7ad602b7d6879177", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- 3 x^{2} - 3 x - 3\\right) \\cos{\\left(2 x + 1 \\right)}$.", "output": "Let f(x) = $2 \\left(x + 1\\right) e^{x} + \\left(- 3 x^{2} - 3 x - 3\\right) \\cos{\\left(2 x + 1 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $3 \\left(- 2 x - 1\\right) \\cos{\\left(2 x + 1 \\right)} + 2 \\left(x + 1\\right) e^{x} - 6 \\left(- x^{2} - x - 1\\right) \\sin{\\left(2 x + 1 \\right)} + 2 e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\left(- 6\\cdot x - 3\\right)\\cdot \\cos{\\left(2\\cdot x + 1 \\right)} + \\left(2\\cdot x + 2\\right)\\cdot e^{x} - \\left(- 6\\cdot x^{2} - 6\\cdot x - 6\\right)\\cdot \\sin{\\left(2\\cdot x + 1 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(- 6\\cdot x - 3\\right)\\cdot \\cos{\\left(2\\cdot x + 1 \\right)} + \\left(2\\cdot x + 2\\right)\\cdot e^{x} - \\left(- 6\\cdot x^{2} - 6\\cdot x - 6\\right)\\cdot \\sin{\\left(2\\cdot x + 1 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f7073f58c5bd47c", "problem_id": "7ad602b7d6879177", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- 3 x^{2} - 3 x - 3\\right) \\cos{\\left(2 x + 1 \\right)}$.", "output": "1) Let f(x) = $2 \\left(x + 1\\right) e^{x} + \\left(- 3 x^{2} - 3 x - 3\\right) \\cos{\\left(2 x + 1 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $3 \\left(- 2 x - 1\\right) \\cos{\\left(2 x + 1 \\right)} + 2 \\left(x + 1\\right) e^{x} - 6 \\left(- x^{2} - x - 1\\right) \\sin{\\left(2 x + 1 \\right)} + 2 e^{x}$.\nFinal answer: $\\left(- 6\\cdot x - 3\\right)\\cdot \\cos{\\left(2\\cdot x + 1 \\right)} + \\left(2\\cdot x + 2\\right)\\cdot e^{x} - \\left(- 6\\cdot x^{2} - 6\\cdot x - 6\\right)\\cdot \\sin{\\left(2\\cdot x + 1 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(- 6\\cdot x - 3\\right)\\cdot \\cos{\\left(2\\cdot x + 1 \\right)} + \\left(2\\cdot x + 2\\right)\\cdot e^{x} - \\left(- 6\\cdot x^{2} - 6\\cdot x - 6\\right)\\cdot \\sin{\\left(2\\cdot x + 1 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9e340c175f57a011", "problem_id": "0398c0b744502010", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 2 x^{3} - 3 x^{2} + 8 x - 27$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $2 x^{4} + 3 x^{3} - x^{2} - 3 x + 3$ and a = -3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $8 x^{3} + 9 x^{2} - 2 x - 3$.\n5) Evaluate at a: p'(-3) = $-132$.\nFinal answer: $-132$", "final_answer": "-132", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d8a79a2bf0913edd", "problem_id": "414fde83bfa8e3ed", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 2 x^{2} + x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}) = - \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} - \\frac{6 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} - \\frac{12 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} - \\frac{2 x^{2}}{2 x^{4} - 12 x^{2} + 18} + \\frac{24 x}{2 x^{4} - 12 x^{2} + 18} - \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 6 x^{2} + 9} + \\frac{9 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} + \\frac{18 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} - \\frac{6}{2 x^{4} - 12 x^{2} + 18} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 6} + \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3} - \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}) = \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - 3\\right)^{2}} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x^{2} - 3\\right)} - \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 4 x \\left(- 2 x^{2} + x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right) \\left(4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8 x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{2 \\left(x^{2} - 3\\right)^{2}}$.\nFinal answer: $\\frac{- 4 x \\left(- 2 x^{2} + x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right) \\left(4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8 x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{2 \\left(x^{2} - 3\\right)^{2}}$", "final_answer": "\\frac{- 4 x \\left(- 2 x^{2} + x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right) \\left(4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8 x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{2 \\left(x^{2} - 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d5c23cfbeacb057e", "problem_id": "129c7644e90252d8", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$6 x$ = $30$\n$- 4 x + 7 y$ = $-69$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $- 24 x$ = $-120$.\n3) Multiply the second equation by $6$: $- 24 x + 42 y$ = $-414$.\n4) Subtract to eliminate x: $- 42 y$ = $294$.\n5) Solve for y: y = $-7$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, -7\\right)$", "final_answer": "(x,y)=\\left(5, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2e27128aa410b9fe", "problem_id": "1b3279da9aef3ca3", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{3} + 2 x^{2} + 2 x + 3 \\left(x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 \\left(x - 1\\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 3}) = \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 1}$.\n3) So $f'(x) = \\frac{4 x^{3}}{6 x^{2} - 12 x + 6} + \\frac{3 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{2} - 12 x + 6} + \\frac{6 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{2} - 12 x + 6} - \\frac{2 x^{2}}{6 x^{2} - 12 x + 6} + \\frac{9 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{2} - 18 x + 9} - \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{2} - 12 x + 6} - \\frac{12 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{2} - 12 x + 6} - \\frac{8 x}{6 x^{2} - 12 x + 6} - \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x - 6} - \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 3} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 1} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{2} - 12 x + 6} + \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{6 x^{2} - 12 x + 6} - \\frac{4}{6 x^{2} - 12 x + 6} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 3}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 3}) = \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(3 x - 3\\right)} + \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 3} - \\frac{9 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(3 x - 3\\right)^{2}} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 3}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 2 x^{3} - 4 x^{2} - 4 x + \\left(x - 1\\right) \\left(6 x^{2} + 8 x + 3 \\left(x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x - 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 6 \\left(x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 \\left(x - 1\\right)^{2}}$.\nFinal answer: $\\frac{- 2 x^{3} - 4 x^{2} - 4 x + \\left(x - 1\\right) \\left(6 x^{2} + 8 x + 3 \\left(x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x - 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 6 \\left(x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 \\left(x - 1\\right)^{2}}$", "final_answer": "\\frac{- 2 x^{3} - 4 x^{2} - 4 x + \\left(x - 1\\right) \\left(6 x^{2} + 8 x + 3 \\left(x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x - 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 6 \\left(x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 \\left(x - 1\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "26d2ad4c753b6a13", "problem_id": "80cd2d9d70c1bc48", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{3} \\sin{\\left(x \\right)} - x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $2 x^{3} \\sin{\\left(x \\right)} - x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)}$.\nFinal answer: $2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - x^{2}\\cdot \\cos{\\left(x \\right)} - 2\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)}$", "final_answer": "2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - x^{2}\\cdot \\cos{\\left(x \\right)} - 2\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7d12d12a63c8c186", "problem_id": "884ffe89a210d9f4", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - x^{2} + 2 x - 7$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} + x - 7.\n3) Then the limit is -19.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -22.\nFinal answer: $-22$", "final_answer": "-22", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "81164e91f6c4cc17", "problem_id": "6e158675a77f84e0", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡2 (mod 3), x≡2 (mod 10). Give the solution modulo 30.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 30.\n3) Compute the CRT solution to get x ≡ 2 (mod 30).\nFinal answer: $x\\equiv 2\\pmod{30}$", "final_answer": "x\\equiv 2\\pmod{30}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b55927378352852b", "problem_id": "6e158675a77f84e0", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 3), x≡2 (mod 10). Give the solution modulo 30.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 30.\nCompute the CRT solution to get x ≡ 2 (mod 30).\nCheck: 2 mod 3=2 and mod 10=2.\nFinal answer: $x\\equiv 2\\pmod{30}$", "final_answer": "x\\equiv 2\\pmod{30}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "322a284acc961d51", "problem_id": "4e57eee42728e35d", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the multiplicative inverse of 6 modulo 47.", "output": "1) We want x such that 6x ≡ 1 (mod 47).\n2) Use extended Euclid to find s,t with 6s + 47t = 1.\n3) One Bezout identity is: 6·(8) + 47·(-1) = 1.\n4) Reduce s modulo 47: x ≡ 8 (mod 47).\nFinal answer: $x\\equiv 8\\pmod{47}$", "final_answer": "x\\equiv 8\\pmod{47}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "658b746cedcbaec4", "problem_id": "4e57eee42728e35d", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the multiplicative inverse of 6 modulo 47.", "output": "We want x such that 6x ≡ 1 (mod 47).\nUse extended Euclid to find s,t with 6s + 47t = 1.\nOne Bezout identity is: 6·(8) + 47·(-1) = 1.\nReduce s modulo 47: x ≡ 8 (mod 47).\nCheck: 6·8 mod 47 = 1.\nFinal answer: $x\\equiv 8\\pmod{47}$", "final_answer": "x\\equiv 8\\pmod{47}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "56cbfceaab71ab69", "problem_id": "752a034058b35906", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the multiplicative inverse of 4 modulo 29.", "output": "1) We want x such that 4x ≡ 1 (mod 29).\n2) Use extended Euclid to find s,t with 4s + 29t = 1.\n3) One Bezout identity is: 4·(-7) + 29·(1) = 1.\n4) Reduce s modulo 29: x ≡ 22 (mod 29).\nFinal answer: $x\\equiv 22\\pmod{29}$", "final_answer": "x\\equiv 22\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3cb3fee5c9dafa3e", "problem_id": "752a034058b35906", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the multiplicative inverse of 4 modulo 29.", "output": "We want x such that 4x ≡ 1 (mod 29).\nUse extended Euclid to find s,t with 4s + 29t = 1.\nOne Bezout identity is: 4·(-7) + 29·(1) = 1.\nReduce s modulo 29: x ≡ 22 (mod 29).\nCheck: 4·22 mod 29 = 1.\nFinal answer: $x\\equiv 22\\pmod{29}$", "final_answer": "x\\equiv 22\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "56cb4af1fec58af8", "problem_id": "330d679579ff5a9d", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{8 x}{3} - \\frac{64}{9}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$- \\frac{8}{3}$, c=$- \\frac{64}{9}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $64$, so √Δ = $8$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{8}{3}$.\n7) x2 = $- \\frac{4}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{8}{3}, - \\frac{4}{3}\\}$", "final_answer": "x \\in \\{\\frac{8}{3}, - \\frac{4}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "20977643e772d15e", "problem_id": "330d679579ff5a9d", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{8 x}{3} - \\frac{64}{9}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$- \\frac{8}{3}$, c=$- \\frac{64}{9}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{8}{3}, - \\frac{4}{3}\\}$", "final_answer": "x \\in \\{\\frac{8}{3}, - \\frac{4}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "030b66003a67cda2", "problem_id": "b2a63c58bc6ef813", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Two friends split $559 in the ratio 6:7. How much does each receive?", "output": "1) Total ratio parts = 6+7 = 13.\n2) Each part = 559 ÷ 13 = 43.\n3) First share = 6 × 43 = 258.\n4) Second share = 7 × 43 = 301.\nFinal answer: $301$", "final_answer": "301", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7c35e4ed2e172534", "problem_id": "b2a63c58bc6ef813", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Two friends split $559 in the ratio 6:7. How much does each receive?", "output": "Total ratio parts = 6+7 = 13.\nEach part = 559 ÷ 13 = 43.\nFirst share = 6 × 43 = 258.\nSecond share = 7 × 43 = 301.\nCheck: 258+301=559.\nFinal answer: $301$", "final_answer": "301", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d6f3ad0edbc82f72", "problem_id": "8746c0bac862159a", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A cyclist rides at 56 miles per hour for 2 hours, then at 40 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 56 × 2 = 112 miles.\nSecond leg distance = 40 × 2 = 80 miles.\nTotal distance = 112 + 80 = 192 miles.\nCheck: adding both legs gives 192.\nFinal answer: $192$", "final_answer": "192", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8f2306b08ab54e80", "problem_id": "8746c0bac862159a", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A cyclist rides at 56 miles per hour for 2 hours, then at 40 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 56 × 2 = 112 miles.\n2) Second leg distance = 40 × 2 = 80 miles.\n3) Total distance = 112 + 80 = 192 miles.\nFinal answer: $192$", "final_answer": "192", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3469e2422f0c8a00", "problem_id": "079b730b94dcd66a", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the linear system:\n$5 x + 2 y$ = $-23$\n$- 8 x - 5 y$ = $35$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-8$: $- 40 x - 16 y$ = $184$.\n3) Multiply the second equation by $5$: $- 40 x - 25 y$ = $175$.\n4) Subtract to eliminate x: $9 y$ = $9$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, 1\\right)$", "final_answer": "(x,y)=\\left(-5, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f841e0837361e124", "problem_id": "079b730b94dcd66a", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$5 x + 2 y$ = $-23$\n$- 8 x - 5 y$ = $35$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-8$: $- 40 x - 16 y$ = $184$.\nMultiply the second equation by $5$: $- 40 x - 25 y$ = $175$.\nSubtract to eliminate x: $9 y$ = $9$.\nSolve for y: y = $1$.\nSubstitute back to find x: x = $-5$.\nCheck: plug (x,y)=(-5,1) into both equations.\nFinal answer: $(x,y)=\\left(-5, 1\\right)$", "final_answer": "(x,y)=\\left(-5, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b566cc661f63bbb2", "problem_id": "24486170e4b37c0e", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A cyclist rides at 58 miles per hour for 2 hours, then at 77 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 58 × 2 = 116 miles.\n2) Second leg distance = 77 × 2 = 154 miles.\n3) Total distance = 116 + 154 = 270 miles.\nFinal answer: $270$", "final_answer": "270", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff6b85d035e3d58d", "problem_id": "24486170e4b37c0e", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A cyclist rides at 58 miles per hour for 2 hours, then at 77 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 58 × 2 = 116 miles.\nSecond leg distance = 77 × 2 = 154 miles.\nTotal distance = 116 + 154 = 270 miles.\nCheck: adding both legs gives 270.\nFinal answer: $270$", "final_answer": "270", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ebfa0e8d90f6ba95", "problem_id": "f0cace33ed62a217", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 2 x^{3} - 3 x^{2} + 3 x - 5$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{4} - x^{3} - 2 x + 1$ and a = -1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $8 x^{3} - 3 x^{2} - 2$.\nEvaluate at a: p'(-1) = $-13$.\nCheck: the identity gives $-13$.\nFinal answer: $-13$", "final_answer": "-13", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7e127b6bc4d75411", "problem_id": "f0cace33ed62a217", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 2 x^{3} - 3 x^{2} + 3 x - 5$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{4} - x^{3} - 2 x + 1$ and a = -1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-13$", "final_answer": "-13", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "63c5676081781ca7", "problem_id": "c892c2f20fdde140", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 4 x^{4} - 3 x^{3} + x^{2} - 2 x - 2\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $4 x^{4} - 3 x^{3} + x^{2} - 2 x - 2$.\n3) Compute an antiderivative: F(x) = $\\frac{4 x^{5}}{5} - \\frac{3 x^{4}}{4} + \\frac{x^{3}}{3} - x^{2} - 2 x$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{4\\cdot x^{5}}{5} - \\frac{3\\cdot x^{4}}{4} + \\frac{x^{3}}{3} - x^{2} - 2\\cdot x$", "final_answer": "C + \\frac{4\\cdot x^{5}}{5} - \\frac{3\\cdot x^{4}}{4} + \\frac{x^{3}}{3} - x^{2} - 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "58d55a8a455cb6dc", "problem_id": "c892c2f20fdde140", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 4 x^{4} - 3 x^{3} + x^{2} - 2 x - 2\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $4 x^{4} - 3 x^{3} + x^{2} - 2 x - 2$.\nCompute an antiderivative: F(x) = $\\frac{4 x^{5}}{5} - \\frac{3 x^{4}}{4} + \\frac{x^{3}}{3} - x^{2} - 2 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{4\\cdot x^{5}}{5} - \\frac{3\\cdot x^{4}}{4} + \\frac{x^{3}}{3} - x^{2} - 2\\cdot x$", "final_answer": "C + \\frac{4\\cdot x^{5}}{5} - \\frac{3\\cdot x^{4}}{4} + \\frac{x^{3}}{3} - x^{2} - 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c5e8f637226d439c", "problem_id": "da68ca0158e4bdcc", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=- x^{4} - 2 x^{2} - 2 x + 3$.", "output": "1) Differentiate term-by-term:\n2) d/dx(3) = 0.\n3) d/dx(- x^{4}) = - 4 x^{3}.\n4) d/dx(- 2 x) = -2.\n5) d/dx(- 2 x^{2}) = - 4 x.\n6) Combine: f'(x) = 2 \\left(- 2 x^{3} - 2 x - 1\\right).\nFinal answer: $- 4\\cdot x^{3} - 4\\cdot x - 2$", "final_answer": "- 4\\cdot x^{3} - 4\\cdot x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "17cd7cdfcbd770ef", "problem_id": "da68ca0158e4bdcc", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{4} - 2 x^{2} - 2 x + 3$.", "output": "Differentiate term-by-term:\nd/dx(3) = 0.\nd/dx(- x^{4}) = - 4 x^{3}.\nd/dx(- 2 x) = -2.\nd/dx(- 2 x^{2}) = - 4 x.\nCombine: f'(x) = 2 \\left(- 2 x^{3} - 2 x - 1\\right).\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 4\\cdot x^{3} - 4\\cdot x - 2$", "final_answer": "- 4\\cdot x^{3} - 4\\cdot x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e6a7e9e28a3d8981", "problem_id": "46594408354beab2", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{3} + x^{2} - 3 x + 2\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $3 x^{3} + x^{2} - 3 x + 2$.\n3) Compute an antiderivative: F(x) = $\\frac{3 x^{4}}{4} + \\frac{x^{3}}{3} - \\frac{3 x^{2}}{2} + 2 x$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{3\\cdot x^{4}}{4} + \\frac{x^{3}}{3} - \\frac{3\\cdot x^{2}}{2} + 2\\cdot x$", "final_answer": "C + \\frac{3\\cdot x^{4}}{4} + \\frac{x^{3}}{3} - \\frac{3\\cdot x^{2}}{2} + 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "500b392ad35c6abc", "problem_id": "46594408354beab2", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{3} + x^{2} - 3 x + 2\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $3 x^{3} + x^{2} - 3 x + 2$.\nCompute an antiderivative: F(x) = $\\frac{3 x^{4}}{4} + \\frac{x^{3}}{3} - \\frac{3 x^{2}}{2} + 2 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{3\\cdot x^{4}}{4} + \\frac{x^{3}}{3} - \\frac{3\\cdot x^{2}}{2} + 2\\cdot x$", "final_answer": "C + \\frac{3\\cdot x^{4}}{4} + \\frac{x^{3}}{3} - \\frac{3\\cdot x^{2}}{2} + 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1c4ab1479006eccd", "problem_id": "16e77d370a8597d6", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the congruences: x≡3 (mod 5), x≡6 (mod 8). Give the solution modulo 40.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 40.\n3) Compute the CRT solution to get x ≡ 38 (mod 40).\nFinal answer: $x\\equiv 38\\pmod{40}$", "final_answer": "x\\equiv 38\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "751b33d2db28141c", "problem_id": "16e77d370a8597d6", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the congruences: x≡3 (mod 5), x≡6 (mod 8). Give the solution modulo 40.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 40.\nCompute the CRT solution to get x ≡ 38 (mod 40).\nFinal answer: $x\\equiv 38\\pmod{40}$", "final_answer": "x\\equiv 38\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f7a3a643443b57da", "problem_id": "abd0f1065a9349a2", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} - 3 x^{4} - x^{3} + x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{4} - x^{3} + x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\nSubtract: F(\\frac{\\pi}{2}) − F(0) = $- 2 \\pi - \\frac{3 \\pi^{5}}{160} - \\frac{\\pi^{4}}{64} - \\frac{\\pi^{2}}{8} + \\frac{\\pi^{3}}{24} + 5$.\nFinal answer: $- 2\\cdot \\pi - \\frac{3\\cdot \\pi^{5}}{160} - \\frac{\\pi^{4}}{64} - \\frac{\\pi^{2}}{8} + \\frac{\\pi^{3}}{24} + 5$", "final_answer": "- 2\\cdot \\pi - \\frac{3\\cdot \\pi^{5}}{160} - \\frac{\\pi^{4}}{64} - \\frac{\\pi^{2}}{8} + \\frac{\\pi^{3}}{24} + 5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f48bbe73ca2d9472", "problem_id": "abd0f1065a9349a2", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} - 3 x^{4} - x^{3} + x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{4} - x^{3} + x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\n3) Subtract: F(\\frac{\\pi}{2}) − F(0) = $- 2 \\pi - \\frac{3 \\pi^{5}}{160} - \\frac{\\pi^{4}}{64} - \\frac{\\pi^{2}}{8} + \\frac{\\pi^{3}}{24} + 5$.\n4) Simplify the result.\nFinal answer: $- 2\\cdot \\pi - \\frac{3\\cdot \\pi^{5}}{160} - \\frac{\\pi^{4}}{64} - \\frac{\\pi^{2}}{8} + \\frac{\\pi^{3}}{24} + 5$", "final_answer": "- 2\\cdot \\pi - \\frac{3\\cdot \\pi^{5}}{160} - \\frac{\\pi^{4}}{64} - \\frac{\\pi^{2}}{8} + \\frac{\\pi^{3}}{24} + 5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3d3607a7a0a556c0", "problem_id": "4cd6accdf6a45f73", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "How many binary strings of length 16 contain exactly 6 ones with no two ones adjacent?", "output": "1) Reserve 5 mandatory zeros between the 6 ones.\n2) This leaves n−k+1 = 11 valid slots for the ones.\n3) Choose the slots: C(11,6) = 462.\nFinal answer: $462$", "final_answer": "462", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "60588b03b3be929f", "problem_id": "5baffeeac5a4af65", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{2} - 4$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- x^{2}) = - 2 x^{2}$.\n3) So $f'(x) = - 2 x^{2}$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- x^{2}) = - 2 x$.\nTherefore the correct derivative is $f'(x) = - 2 x$.\nFinal answer: $- 2 x$", "final_answer": "- 2 x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "61cf6274ad23f31a", "problem_id": "00e0062b807db502", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_9=31$?", "output": "Let y_i=x_i-3, then y_i≥0 and y_1+…+y_9=4.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=4 and k=9.\nFinal answer: $495$", "final_answer": "495", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "af85dadcc66767a8", "problem_id": "00e0062b807db502", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_9=31$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_9=4.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=4 and k=9.\n4) Compute C(12,8) = 495.\nFinal answer: $495$", "final_answer": "495", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dc25a6f998619c81", "problem_id": "49837cfc3bf25074", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - 2 x - 6$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to - 3 x - 6.\n3) Then the limit is -15.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: -12.\nFinal answer: $-12$", "final_answer": "-12", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "761cef294783738a", "problem_id": "4e5e51f53bf24457", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} 3 x^{3} + 10 x^{2} + 39 x + 153$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{4} - 2 x^{3} - x^{2} - 3 x - 2$ and a = 4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $12 x^{3} - 6 x^{2} - 2 x - 3$.\n5) Evaluate at a: p'(4) = $661$.\nFinal answer: $661$", "final_answer": "661", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "62e11f58a8c6729e", "problem_id": "59d27f1c36e7fef1", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 13 red balls and 11 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(24,9).\nFavorable samples: choose k reds and n−k blues: C(13,2)·C(11,7).\nCompute totals: C(24,9) = 1307504, and favorable = 25740.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{585}{29716}$.\nFinal answer: $\\frac{585}{29716}$", "final_answer": "\\frac{585}{29716}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4716fb1e3de49bc0", "problem_id": "b7addf236b39dd38", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} - x + 2$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(2 x^{2}) = 4 x^{2}$.\n3) So $f'(x) = 4 x^{2} - 1$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(2 x^{2}) = 4 x$.\nTherefore the correct derivative is $f'(x) = 4 x - 1$.\nFinal answer: $4 x - 1$", "final_answer": "4 x - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "be0418a31cda55c5", "problem_id": "b9ccd93f332e6bec", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Two friends split $570 in the ratio 6:4. How much does each receive?", "output": "1) Total ratio parts = 6+4 = 10.\n2) Each part = 570 ÷ 10 = 57.\n3) First share = 6 × 57 = 342.\n4) Second share = 4 × 57 = 228.\nFinal answer: $228$", "final_answer": "228", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9d00c7eedbd56fad", "problem_id": "02e8074d46070dff", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 3 x^{2} + 6 x + 14$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{2} + 5 x + 14.\n3) Then the limit is 36.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 38.\nFinal answer: $38$", "final_answer": "38", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "da7fddabca3c6bea", "problem_id": "a3e30ac850d7a850", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{2 \\sqrt{3} x}{3} + \\frac{23 x}{12} - \\frac{5 \\sqrt{3}}{6} + \\frac{5}{6}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{23}{12} - \\frac{2 \\sqrt{3}}{3}$, c=$\\frac{5}{6} - \\frac{5 \\sqrt{3}}{6}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{7 \\sqrt{3}}{9} + \\frac{241}{144}$, so √Δ = $\\sqrt{\\frac{7 \\sqrt{3}}{9} + \\frac{241}{144}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{23}{24} + \\frac{\\sqrt{3}}{3} + \\frac{\\sqrt{112 \\sqrt{3} + 241}}{24}$.\n7) x2 = $- \\frac{23}{24} - \\frac{\\sqrt{112 \\sqrt{3} + 241}}{24} + \\frac{\\sqrt{3}}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{23}{24} + \\frac{\\sqrt{3}}{3} + \\frac{\\sqrt{112\\cdot \\sqrt{3} + 241}}{24}, - \\frac{23}{24} - \\frac{\\sqrt{112\\cdot \\sqrt{3} + 241}}{24} + \\frac{\\sqrt{3}}{3}\\}$", "final_answer": "x \\in \\{- \\frac{23}{24} + \\frac{\\sqrt{3}}{3} + \\frac{\\sqrt{112\\cdot \\sqrt{3} + 241}}{24}, - \\frac{23}{24} - \\frac{\\sqrt{112\\cdot \\sqrt{3} + 241}}{24} + \\frac{\\sqrt{3}}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ab923649ed386b6a", "problem_id": "a3e30ac850d7a850", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{2 \\sqrt{3} x}{3} + \\frac{23 x}{12} - \\frac{5 \\sqrt{3}}{6} + \\frac{5}{6}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{23}{12} - \\frac{2 \\sqrt{3}}{3}$, c=$\\frac{5}{6} - \\frac{5 \\sqrt{3}}{6}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{7 \\sqrt{3}}{9} + \\frac{241}{144}$, so √Δ = $\\sqrt{\\frac{7 \\sqrt{3}}{9} + \\frac{241}{144}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{23}{24} + \\frac{\\sqrt{3}}{3} + \\frac{\\sqrt{112 \\sqrt{3} + 241}}{24}$.\nx2 = $- \\frac{23}{24} - \\frac{\\sqrt{112 \\sqrt{3} + 241}}{24} + \\frac{\\sqrt{3}}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{23}{24} + \\frac{\\sqrt{3}}{3} + \\frac{\\sqrt{112\\cdot \\sqrt{3} + 241}}{24}, - \\frac{23}{24} - \\frac{\\sqrt{112\\cdot \\sqrt{3} + 241}}{24} + \\frac{\\sqrt{3}}{3}\\}$", "final_answer": "x \\in \\{- \\frac{23}{24} + \\frac{\\sqrt{3}}{3} + \\frac{\\sqrt{112\\cdot \\sqrt{3} + 241}}{24}, - \\frac{23}{24} - \\frac{\\sqrt{112\\cdot \\sqrt{3} + 241}}{24} + \\frac{\\sqrt{3}}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4daa3218c25d7d8e", "problem_id": "76ab315e03eaba53", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 2 \\sqrt{2} x + \\frac{3 x}{2} - \\frac{9}{2} + 3 \\sqrt{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{3}{2} - 2 \\sqrt{2}$, c=$- \\frac{9}{2} + 3 \\sqrt{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{113}{4} - 18 \\sqrt{2}$, so √Δ = $\\sqrt{\\frac{113}{4} - 18 \\sqrt{2}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{3}{4} + \\frac{\\sqrt{113 - 72 \\sqrt{2}}}{4} + \\sqrt{2}$.\n7) x2 = $- \\frac{\\sqrt{113 - 72 \\sqrt{2}}}{4} - \\frac{3}{4} + \\sqrt{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{3}{4} + \\frac{\\sqrt{113 - 72\\cdot \\sqrt{2}}}{4} + \\sqrt{2}, - \\frac{\\sqrt{113 - 72\\cdot \\sqrt{2}}}{4} - \\frac{3}{4} + \\sqrt{2}\\}$", "final_answer": "x \\in \\{- \\frac{3}{4} + \\frac{\\sqrt{113 - 72\\cdot \\sqrt{2}}}{4} + \\sqrt{2}, - \\frac{\\sqrt{113 - 72\\cdot \\sqrt{2}}}{4} - \\frac{3}{4} + \\sqrt{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "00efbfef3e341336", "problem_id": "76ab315e03eaba53", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 2 \\sqrt{2} x + \\frac{3 x}{2} - \\frac{9}{2} + 3 \\sqrt{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{3}{2} - 2 \\sqrt{2}$, c=$- \\frac{9}{2} + 3 \\sqrt{2}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{113}{4} - 18 \\sqrt{2}$, so √Δ = $\\sqrt{\\frac{113}{4} - 18 \\sqrt{2}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{3}{4} + \\frac{\\sqrt{113 - 72 \\sqrt{2}}}{4} + \\sqrt{2}$.\nx2 = $- \\frac{\\sqrt{113 - 72 \\sqrt{2}}}{4} - \\frac{3}{4} + \\sqrt{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{3}{4} + \\frac{\\sqrt{113 - 72\\cdot \\sqrt{2}}}{4} + \\sqrt{2}, - \\frac{\\sqrt{113 - 72\\cdot \\sqrt{2}}}{4} - \\frac{3}{4} + \\sqrt{2}\\}$", "final_answer": "x \\in \\{- \\frac{3}{4} + \\frac{\\sqrt{113 - 72\\cdot \\sqrt{2}}}{4} + \\sqrt{2}, - \\frac{\\sqrt{113 - 72\\cdot \\sqrt{2}}}{4} - \\frac{3}{4} + \\sqrt{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a3a20aa8f740ae51", "problem_id": "d6bb2ee279f828cb", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} + x + 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(3 x^{2}) = 6 x^{2}$.\n3) So $f'(x) = 6 x^{2} + 1$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(3 x^{2}) = 6 x$.\nTherefore the correct derivative is $f'(x) = 6 x + 1$.\nFinal answer: $6 x + 1$", "final_answer": "6 x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "02813d1f22636807", "problem_id": "7b97324addb772cd", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "An urn has 7 red balls and 15 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(22,6).\n3) Favorable samples: choose k reds and n−k blues: C(7,2)·C(15,4).\n4) Compute totals: C(22,6) = 74613, and favorable = 28665.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1365}{3553}$", "final_answer": "\\frac{1365}{3553}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "041ca4d8878e0ffb", "problem_id": "6c2e320dcb5b8efc", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 20 x + 18$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 20 x + 18$ = $2 \\left(x - 9\\right) \\left(x - 1\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $9$.\n5) From the second factor: x = $1$.\nFinal answer: $x \\in \\{9, 1\\}$", "final_answer": "x \\in \\{9, 1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "50d4f93b3273ef49", "problem_id": "6c2e320dcb5b8efc", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 20 x + 18$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} - 20 x + 18$ = $2 \\left(x - 9\\right) \\left(x - 1\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $9$.\nFrom the second factor: x = $1$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{9, 1\\}$", "final_answer": "x \\in \\{9, 1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f57f5952d9fce4fa", "problem_id": "c0fbe97de884fdee", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} + 4 x + 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x^{3}) = 3 x^{3}$.\n3) So $f'(x) = 3 x^{3} + 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x^{3}) = 3 x^{2}$.\nTherefore the correct derivative is $f'(x) = 3 x^{2} + 4$.\nFinal answer: $3 x^{2} + 4$", "final_answer": "3 x^{2} + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6b1583f679282c03", "problem_id": "a1b2eef8eadbb9cc", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Two friends split $240 in the ratio 1:3. How much does each receive?", "output": "$180$", "final_answer": "180", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "88c67cb081a847cc", "problem_id": "603a39b5f51d8fdd", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the linear system:\n$- 7 x + 8 y$ = $-90$\n$- 6 x + 7 y$ = $-78$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-6$: $42 x - 48 y$ = $540$.\nMultiply the second equation by $-7$: $42 x - 49 y$ = $546$.\nFinal answer: $(x,y)=\\left(6, -6\\right)$", "final_answer": "(x,y)=\\left(6, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b1d07f087edba76b", "problem_id": "603a39b5f51d8fdd", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the linear system:\n$- 7 x + 8 y$ = $-90$\n$- 6 x + 7 y$ = $-78$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-6$: $42 x - 48 y$ = $540$.\n3) Multiply the second equation by $-7$: $42 x - 49 y$ = $546$.\n4) Subtract to eliminate x: $y$ = $-6$.\n5) Solve for y: y = $-6$.\n6) Substitute back to find x: x = $6$.\nFinal answer: $(x,y)=\\left(6, -6\\right)$", "final_answer": "(x,y)=\\left(6, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "105a1e0e22a38671", "problem_id": "5d81f8e1ccf78ec9", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$- 4 x - y$ = $8$\n$4 x - 4 y$ = $-28$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $- 16 x - 4 y$ = $32$.\n3) Multiply the second equation by $-4$: $- 16 x + 16 y$ = $112$.\n4) Subtract to eliminate x: $- 20 y$ = $-80$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, 4\\right)$", "final_answer": "(x,y)=\\left(-3, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "714d54652bc70fce", "problem_id": "5d81f8e1ccf78ec9", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- 4 x - y$ = $8$\n$4 x - 4 y$ = $-28$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $- 16 x - 4 y$ = $32$.\nMultiply the second equation by $-4$: $- 16 x + 16 y$ = $112$.\nSubtract to eliminate x: $- 20 y$ = $-80$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $-3$.\nCheck: plug (x,y)=(-3,4) into both equations.\nFinal answer: $(x,y)=\\left(-3, 4\\right)$", "final_answer": "(x,y)=\\left(-3, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "56bf23419fef4317", "problem_id": "1d8c52e486dd31b4", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{11 x}{2} - \\frac{\\sqrt{7} x}{3} + \\frac{3 \\sqrt{7}}{2} + \\frac{9}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{11}{2} - \\frac{\\sqrt{7}}{3}$, c=$\\frac{3 \\sqrt{7}}{2} + \\frac{9}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{469}{36} - \\frac{7 \\sqrt{7}}{3}$, so √Δ = $\\sqrt{\\frac{469}{36} - \\frac{7 \\sqrt{7}}{3}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{7}}{6} + \\frac{\\sqrt{469 - 84 \\sqrt{7}}}{12} + \\frac{11}{4}$.\n7) x2 = $- \\frac{\\sqrt{469 - 84 \\sqrt{7}}}{12} + \\frac{\\sqrt{7}}{6} + \\frac{11}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{7}}{6} + \\frac{\\sqrt{469 - 84\\cdot \\sqrt{7}}}{12} + \\frac{11}{4}, - \\frac{\\sqrt{469 - 84\\cdot \\sqrt{7}}}{12} + \\frac{\\sqrt{7}}{6} + \\frac{11}{4}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{7}}{6} + \\frac{\\sqrt{469 - 84\\cdot \\sqrt{7}}}{12} + \\frac{11}{4}, - \\frac{\\sqrt{469 - 84\\cdot \\sqrt{7}}}{12} + \\frac{\\sqrt{7}}{6} + \\frac{11}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "48bcbf866f8b69c4", "problem_id": "1d8c52e486dd31b4", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{11 x}{2} - \\frac{\\sqrt{7} x}{3} + \\frac{3 \\sqrt{7}}{2} + \\frac{9}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{11}{2} - \\frac{\\sqrt{7}}{3}$, c=$\\frac{3 \\sqrt{7}}{2} + \\frac{9}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{469}{36} - \\frac{7 \\sqrt{7}}{3}$, so √Δ = $\\sqrt{\\frac{469}{36} - \\frac{7 \\sqrt{7}}{3}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{7}}{6} + \\frac{\\sqrt{469 - 84 \\sqrt{7}}}{12} + \\frac{11}{4}$.\nx2 = $- \\frac{\\sqrt{469 - 84 \\sqrt{7}}}{12} + \\frac{\\sqrt{7}}{6} + \\frac{11}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{7}}{6} + \\frac{\\sqrt{469 - 84\\cdot \\sqrt{7}}}{12} + \\frac{11}{4}, - \\frac{\\sqrt{469 - 84\\cdot \\sqrt{7}}}{12} + \\frac{\\sqrt{7}}{6} + \\frac{11}{4}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{7}}{6} + \\frac{\\sqrt{469 - 84\\cdot \\sqrt{7}}}{12} + \\frac{11}{4}, - \\frac{\\sqrt{469 - 84\\cdot \\sqrt{7}}}{12} + \\frac{\\sqrt{7}}{6} + \\frac{11}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "68b4802067b5f0ba", "problem_id": "331742ad7366406c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 3 x^{2} - 5 x - 12$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} - 6 x - 12.\n3) Then the limit is -36.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -34.\nFinal answer: $-34$", "final_answer": "-34", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "bc9ab26172cc821e", "problem_id": "2c2b50380b1a9641", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{2} x + \\frac{19 x}{5} - \\frac{9 \\sqrt{2}}{5} + \\frac{18}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{19}{5} - \\sqrt{2}$, c=$\\frac{18}{5} - \\frac{9 \\sqrt{2}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{51}{25} - \\frac{2 \\sqrt{2}}{5}$, so √Δ = $\\sqrt{\\frac{51}{25} - \\frac{2 \\sqrt{2}}{5}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{19}{10} + \\frac{\\sqrt{51 - 10 \\sqrt{2}}}{10} + \\frac{\\sqrt{2}}{2}$.\n7) x2 = $- \\frac{19}{10} - \\frac{\\sqrt{51 - 10 \\sqrt{2}}}{10} + \\frac{\\sqrt{2}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{19}{10} + \\frac{\\sqrt{51 - 10\\cdot \\sqrt{2}}}{10} + \\frac{\\sqrt{2}}{2}, - \\frac{19}{10} - \\frac{\\sqrt{51 - 10\\cdot \\sqrt{2}}}{10} + \\frac{\\sqrt{2}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{19}{10} + \\frac{\\sqrt{51 - 10\\cdot \\sqrt{2}}}{10} + \\frac{\\sqrt{2}}{2}, - \\frac{19}{10} - \\frac{\\sqrt{51 - 10\\cdot \\sqrt{2}}}{10} + \\frac{\\sqrt{2}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a319f289ff6680c1", "problem_id": "52eab6bee47e47d6", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 2 x - 1$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$2$, c=$-1$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $16$, so √Δ = $4$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{1}{3}$.\n7) x2 = $-1$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{1}{3}, -1\\}$", "final_answer": "x \\in \\{\\frac{1}{3}, -1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "324e1e86d8b44b85", "problem_id": "52eab6bee47e47d6", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 2 x - 1$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$2$, c=$-1$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $16$, so √Δ = $4$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{1}{3}$.\nx2 = $-1$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{1}{3}, -1\\}$", "final_answer": "x \\in \\{\\frac{1}{3}, -1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a3d710a58c02d97c", "problem_id": "824d66b033a32758", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 12 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=5, p=$\\frac{2}{7}$ and simplify.\n4) Here C(12,5) = 792.\nFinal answer: $\\frac{1980000000}{13841287201}$", "final_answer": "\\frac{1980000000}{13841287201}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7270b585715e7683", "problem_id": "824d66b033a32758", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 12 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=12, k=5, p=$\\frac{2}{7}$ and simplify.\nHere C(12,5) = 792.\nCheck: expression simplifies to $\\frac{1980000000}{13841287201}$.\nFinal answer: $\\frac{1980000000}{13841287201}$", "final_answer": "\\frac{1980000000}{13841287201}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2e04289a358f609e", "problem_id": "63276b9a8b3a014e", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the linear system:\n$5 x - 7 y$ = $17$\n$- 2 x - 6 y$ = $46$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $- 10 x + 14 y$ = $-34$.\n3) Multiply the second equation by $5$: $- 10 x - 30 y$ = $230$.\n4) Subtract to eliminate x: $44 y$ = $-264$.\n5) Solve for y: y = $-6$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, -6\\right)$", "final_answer": "(x,y)=\\left(-5, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e06089213d59e3e", "problem_id": "63276b9a8b3a014e", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the linear system:\n$5 x - 7 y$ = $17$\n$- 2 x - 6 y$ = $46$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $- 10 x + 14 y$ = $-34$.\nMultiply the second equation by $5$: $- 10 x - 30 y$ = $230$.\nFinal answer: $(x,y)=\\left(-5, -6\\right)$", "final_answer": "(x,y)=\\left(-5, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a0b8161aea6fd811", "problem_id": "7f04d5f62b129297", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "You want 25 liters of a 30% salt solution. You have a 25% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 25−x liters is 40% solution.\n2) Salt equation: 25x + 40(25−x) = 30·25.\n3) (25-40)x = (30-40)·25 ⇒ x = 50/3.\n4) Then 25−x = 25/3.\nFinal answer: $50/3 L of 25% and 25/3 L of 40%$", "final_answer": "50/3 L of 25% and 25/3 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7a2f1dcb78312339", "problem_id": "7f04d5f62b129297", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 25 liters of a 30% salt solution. You have a 25% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 25−x liters is 40% solution.\nSalt equation: 25x + 40(25−x) = 30·25.\n(25-40)x = (30-40)·25 ⇒ x = 50/3.\nThen 25−x = 25/3.\nCheck: 25·50/3 + 40·25/3 = 30·25.\nFinal answer: $50/3 L of 25% and 25/3 L of 40%$", "final_answer": "50/3 L of 25% and 25/3 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7b48ed2dd45a7ad8", "problem_id": "4ccda9e8744cfe67", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "An urn has 9 red balls and 12 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(21,7).\n3) Favorable samples: choose k reds and n−k blues: C(9,2)·C(12,5).\n4) Compute totals: C(21,7) = 116280, and favorable = 28512.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{396}{1615}$", "final_answer": "\\frac{396}{1615}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72a97175e9054ee3", "problem_id": "4ccda9e8744cfe67", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 9 red balls and 12 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(21,7).\nFavorable samples: choose k reds and n−k blues: C(9,2)·C(12,5).\nCompute totals: C(21,7) = 116280, and favorable = 28512.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{396}{1615}$.\nFinal answer: $\\frac{396}{1615}$", "final_answer": "\\frac{396}{1615}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e9bb1a4a1c77f11d", "problem_id": "6cbc4c7a2f1b0b3b", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} 2 x^{3} - x^{2} - 3 x - 1$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $2 x^{4} - 3 x^{3} - 2 x^{2} + 2 x - 2$ and a = 1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $8 x^{3} - 9 x^{2} - 4 x + 2$.\n5) Evaluate at a: p'(1) = $-3$.\nFinal answer: $-3$", "final_answer": "-3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2726eb886d492f85", "problem_id": "6cbc4c7a2f1b0b3b", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} 2 x^{3} - x^{2} - 3 x - 1$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{4} - 3 x^{3} - 2 x^{2} + 2 x - 2$ and a = 1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $8 x^{3} - 9 x^{2} - 4 x + 2$.\nEvaluate at a: p'(1) = $-3$.\nCheck: the identity gives $-3$.\nFinal answer: $-3$", "final_answer": "-3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d796df1838f418bd", "problem_id": "074ec73094380b90", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_7=43$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_7=29.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=29 and k=7.\n4) Compute C(35,6) = 1623160.\nFinal answer: $1623160$", "final_answer": "1623160", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0486f176715423f4", "problem_id": "074ec73094380b90", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_7=43$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_7=29.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=29 and k=7.\nFinal answer: $1623160$", "final_answer": "1623160", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e8d20af9197e1674", "problem_id": "258527e3da38b0e7", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 10 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=4, p=$\\frac{3}{7}$ and simplify.\n4) Here C(10,4) = 210.\nFinal answer: $\\frac{9953280}{40353607}$", "final_answer": "\\frac{9953280}{40353607}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e7fe480e07a990af", "problem_id": "fcb9bd0865def3fa", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 3 x - 3$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to 2 x - 3.\n3) Then the limit is -5.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: -6.\nFinal answer: $-6$", "final_answer": "-6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a9e3caf00b9850d1", "problem_id": "e679f95f5ccf66b1", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Worker A finishes a job in 13 hours and Worker B finishes it in 14 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/13 job/hour. B's rate = 1/14 job/hour.\n2) Combined rate = 1/13 + 1/14 = 27/182 job/hour.\n3) Time = 1 ÷ (combined rate) = 182/27 hours.\nFinal answer: $\\frac{182}{27}$", "final_answer": "\\frac{182}{27}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d9c094d7a1930fe7", "problem_id": "e679f95f5ccf66b1", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Worker A finishes a job in 13 hours and Worker B finishes it in 14 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/13 job/hour. B's rate = 1/14 job/hour.\nCombined rate = 1/13 + 1/14 = 27/182 job/hour.\nTime = 1 ÷ (combined rate) = 182/27 hours.\nCheck: (combined rate)×(time) = 27/182×182/27 = 1.\nFinal answer: $\\frac{182}{27}$", "final_answer": "\\frac{182}{27}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d9d1bb2049fb4a6e", "problem_id": "bfa1adf437a9496f", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} + 2 x^{2} + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 2$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 6 x^{2} + 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)}$.\n2) Slope at x=2: m = f'(2) = $-16 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\n3) Point on curve: (2, f(2)) = (2, $-6 + \\cos{\\left(4 \\right)} + e$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -6 + \\cos{\\left(4 \\right)} + e = -16 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\n6) Optionally expand: y = $- 16 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 26$.\nFinal answer: $y = - 16\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 26$", "final_answer": "y = - 16\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 26", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b889eea1e62c0e3f", "problem_id": "bfa1adf437a9496f", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} + 2 x^{2} + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 2$ at $x=2$.", "output": "Differentiate: f'(x) = $- 6 x^{2} + 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)}$.\nSlope at x=2: m = f'(2) = $-16 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\nPoint on curve: (2, f(2)) = (2, $-6 + \\cos{\\left(4 \\right)} + e$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -6 + \\cos{\\left(4 \\right)} + e = -16 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\nOptionally expand: y = $- 16 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 26$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 16\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 26$", "final_answer": "y = - 16\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 26", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "036003a26ca7d90d", "problem_id": "12b4969b790ddd49", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_10=40$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_10=10.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=10 and k=10.\n4) Compute C(19,9) = 92378.\nFinal answer: $92378$", "final_answer": "92378", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "605f798f79e19985", "problem_id": "12b4969b790ddd49", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_10=40$?", "output": "Let y_i=x_i-3, then y_i≥0 and y_1+…+y_10=10.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=10 and k=10.\nCompute C(19,9) = 92378.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $92378$", "final_answer": "92378", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "70ea21a1d5b902d6", "problem_id": "e8608d8fa0c2bd05", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "You want 27 liters of a 45% salt solution. You have a 30% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 27−x liters is 60% solution.\n2) Salt equation: 30x + 60(27−x) = 45·27.\n3) (30-60)x = (45-60)·27 ⇒ x = 27/2.\n4) Then 27−x = 27/2.\nFinal answer: $27/2 L of 30% and 27/2 L of 60%$", "final_answer": "27/2 L of 30% and 27/2 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0dda772be87f6905", "problem_id": "e8608d8fa0c2bd05", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "You want 27 liters of a 45% salt solution. You have a 30% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 27−x liters is 60% solution.\nSalt equation: 30x + 60(27−x) = 45·27.\n(30-60)x = (45-60)·27 ⇒ x = 27/2.\nThen 27−x = 27/2.\nCheck: 30·27/2 + 60·27/2 = 45·27.\nFinal answer: $27/2 L of 30% and 27/2 L of 60%$", "final_answer": "27/2 L of 30% and 27/2 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "049ae0ddfca25811", "problem_id": "32819586899470f1", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{13 x}{3} + \\frac{5}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$- \\frac{13}{3}$, c=$\\frac{5}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{49}{9}$, so √Δ = $\\frac{7}{3}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{5}{3}$.\n7) x2 = $\\frac{1}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{5}{3}, \\frac{1}{2}\\}$", "final_answer": "x \\in \\{\\frac{5}{3}, \\frac{1}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e813b39028539e15", "problem_id": "32819586899470f1", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{13 x}{3} + \\frac{5}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$- \\frac{13}{3}$, c=$\\frac{5}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{49}{9}$, so √Δ = $\\frac{7}{3}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{5}{3}$.\nx2 = $\\frac{1}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{5}{3}, \\frac{1}{2}\\}$", "final_answer": "x \\in \\{\\frac{5}{3}, \\frac{1}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "81e7381b0138d927", "problem_id": "42c27c3f6b56ffbc", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $320. It is discounted by 20%. You buy 3 backpacks. What is the total cost?", "output": "Discounted price = 320 × (1 − 20/100) = 320 × 80/100 = 256.\nSubtotal = 3 × 256 = 768.\nCheck: recomputing the formula yields 768.\nFinal answer: $768$", "final_answer": "768", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "232b95b678c4db12", "problem_id": "42c27c3f6b56ffbc", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A store sells a backpack for $320. It is discounted by 20%. You buy 3 backpacks. What is the total cost?", "output": "1) Discounted price = 320 × (1 − 20/100) = 320 × 80/100 = 256.\n2) Subtotal = 3 × 256 = 768.\nFinal answer: $768$", "final_answer": "768", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "13ba0b82104aeee2", "problem_id": "6c11a4ec5fb44d0e", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$x + 3 y$ = $16$\n$- x - y$ = $-8$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $- x - 3 y$ = $-16$.\n3) Multiply the second equation by $1$: $- x - y$ = $-8$.\n4) Subtract to eliminate x: $- 2 y$ = $-8$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, 4\\right)$", "final_answer": "(x,y)=\\left(4, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f9286bdfbbc8563b", "problem_id": "6c11a4ec5fb44d0e", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the linear system:\n$x + 3 y$ = $16$\n$- x - y$ = $-8$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $- x - 3 y$ = $-16$.\nMultiply the second equation by $1$: $- x - y$ = $-8$.\nFinal answer: $(x,y)=\\left(4, 4\\right)$", "final_answer": "(x,y)=\\left(4, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b6e6d895e78064a8", "problem_id": "ea53b516f14bc027", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} x^{3} - x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $x^{3} - x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{65 \\pi^{3}}{24} - 1 + \\frac{5 \\pi}{2} + \\frac{15 \\pi^{2}}{8} + \\frac{255 \\pi^{4}}{64}$.\nFinal answer: $- \\frac{65\\cdot \\pi^{3}}{24} - 1 + \\frac{5\\cdot \\pi}{2} + \\frac{15\\cdot \\pi^{2}}{8} + \\frac{255\\cdot \\pi^{4}}{64}$", "final_answer": "- \\frac{65\\cdot \\pi^{3}}{24} - 1 + \\frac{5\\cdot \\pi}{2} + \\frac{15\\cdot \\pi^{2}}{8} + \\frac{255\\cdot \\pi^{4}}{64}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bac231a172825924", "problem_id": "ea53b516f14bc027", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} x^{3} - x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $x^{3} - x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{65 \\pi^{3}}{24} - 1 + \\frac{5 \\pi}{2} + \\frac{15 \\pi^{2}}{8} + \\frac{255 \\pi^{4}}{64}$.\n4) Simplify the result.\nFinal answer: $- \\frac{65\\cdot \\pi^{3}}{24} - 1 + \\frac{5\\cdot \\pi}{2} + \\frac{15\\cdot \\pi^{2}}{8} + \\frac{255\\cdot \\pi^{4}}{64}$", "final_answer": "- \\frac{65\\cdot \\pi^{3}}{24} - 1 + \\frac{5\\cdot \\pi}{2} + \\frac{15\\cdot \\pi^{2}}{8} + \\frac{255\\cdot \\pi^{4}}{64}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f2206613685a1065", "problem_id": "15d1a9bc20048c32", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 18 liters of a 45% salt solution. You have a 30% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 18−x liters is 60% solution.\nSalt equation: 30x + 60(18−x) = 45·18.\n(30-60)x = (45-60)·18 ⇒ x = 9.\nThen 18−x = 9.\nCheck: 30·9 + 60·9 = 45·18.\nFinal answer: $9 L of 30% and 9 L of 60%$", "final_answer": "9 L of 30% and 9 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f6b674f4b1055a18", "problem_id": "15d1a9bc20048c32", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "You want 18 liters of a 45% salt solution. You have a 30% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 18−x liters is 60% solution.\n2) Salt equation: 30x + 60(18−x) = 45·18.\n3) (30-60)x = (45-60)·18 ⇒ x = 9.\n4) Then 18−x = 9.\nFinal answer: $9 L of 30% and 9 L of 60%$", "final_answer": "9 L of 30% and 9 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0ead4eceb4989e36", "problem_id": "345a6508e6fb732d", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} x^{3} - x^{2} + 2$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $x^{4} - x^{2} + 2 x - 3$ and a = -1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $4 x^{3} - 2 x + 2$.\nEvaluate at a: p'(-1) = $0$.\nCheck: the identity gives $0$.\nFinal answer: $0$", "final_answer": "0", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "461130b0cdbf57c7", "problem_id": "345a6508e6fb732d", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} x^{3} - x^{2} + 2$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{4} - x^{2} + 2 x - 3$ and a = -1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $4 x^{3} - 2 x + 2$.\n5) Evaluate at a: p'(-1) = $0$.\nFinal answer: $0$", "final_answer": "0", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d5f57a971731bc90", "problem_id": "d698bfdbb20ce625", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Worker A finishes a job in 14 hours and Worker B finishes it in 11 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/14 job/hour. B's rate = 1/11 job/hour.\nCombined rate = 1/14 + 1/11 = 25/154 job/hour.\nTime = 1 ÷ (combined rate) = 154/25 hours.\nCheck: (combined rate)×(time) = 25/154×154/25 = 1.\nFinal answer: $\\frac{154}{25}$", "final_answer": "\\frac{154}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6bf896ed74b262e9", "problem_id": "d698bfdbb20ce625", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Worker A finishes a job in 14 hours and Worker B finishes it in 11 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/14 job/hour. B's rate = 1/11 job/hour.\nCombined rate = 1/14 + 1/11 = 25/154 job/hour.\nTime = 1 ÷ (combined rate) = 154/25 hours.\nFinal answer: $\\frac{154}{25}$", "final_answer": "\\frac{154}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0cd2fbbff7ee5ff4", "problem_id": "4fce3017af5944fb", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the linear system:\n$4 x - 2 y$ = $12$\n$3 x - y$ = $9$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $12 x - 6 y$ = $36$.\nMultiply the second equation by $4$: $12 x - 4 y$ = $36$.\nFinal answer: $(x,y)=\\left(3, 0\\right)$", "final_answer": "(x,y)=\\left(3, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eefb29034f7d54a7", "problem_id": "4fce3017af5944fb", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$4 x - 2 y$ = $12$\n$3 x - y$ = $9$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $3$: $12 x - 6 y$ = $36$.\n3) Multiply the second equation by $4$: $12 x - 4 y$ = $36$.\n4) Subtract to eliminate x: $- 2 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, 0\\right)$", "final_answer": "(x,y)=\\left(3, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7abc772c12104250", "problem_id": "5518e56a2d51be63", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} - 3 x^{3} + x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{3} + x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{45 \\pi^{4}}{64} - \\frac{9 \\pi}{2} + 2 + \\frac{3 \\pi^{2}}{4} + \\frac{3 \\pi^{3}}{8}$.\n4) Simplify the result.\nFinal answer: $- \\frac{45\\cdot \\pi^{4}}{64} - \\frac{9\\cdot \\pi}{2} + 2 + \\frac{3\\cdot \\pi^{2}}{4} + \\frac{3\\cdot \\pi^{3}}{8}$", "final_answer": "- \\frac{45\\cdot \\pi^{4}}{64} - \\frac{9\\cdot \\pi}{2} + 2 + \\frac{3\\cdot \\pi^{2}}{4} + \\frac{3\\cdot \\pi^{3}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "55fa2b3d9635e126", "problem_id": "5518e56a2d51be63", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} - 3 x^{3} + x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{3} + x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\nEvaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{45 \\pi^{4}}{64} - \\frac{9 \\pi}{2} + 2 + \\frac{3 \\pi^{2}}{4} + \\frac{3 \\pi^{3}}{8}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{45\\cdot \\pi^{4}}{64} - \\frac{9\\cdot \\pi}{2} + 2 + \\frac{3\\cdot \\pi^{2}}{4} + \\frac{3\\cdot \\pi^{3}}{8}$", "final_answer": "- \\frac{45\\cdot \\pi^{4}}{64} - \\frac{9\\cdot \\pi}{2} + 2 + \\frac{3\\cdot \\pi^{2}}{4} + \\frac{3\\cdot \\pi^{3}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7f2b9954a227d94d", "problem_id": "c9a59aef7af2f295", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Worker A finishes a job in 20 hours and Worker B finishes it in 4 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/20 job/hour. B's rate = 1/4 job/hour.\n2) Combined rate = 1/20 + 1/4 = 3/10 job/hour.\n3) Time = 1 ÷ (combined rate) = 10/3 hours.\nFinal answer: $\\frac{10}{3}$", "final_answer": "\\frac{10}{3}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d05e5b8e5cd46afe", "problem_id": "c9a59aef7af2f295", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Worker A finishes a job in 20 hours and Worker B finishes it in 4 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/20 job/hour. B's rate = 1/4 job/hour.\nCombined rate = 1/20 + 1/4 = 3/10 job/hour.\nTime = 1 ÷ (combined rate) = 10/3 hours.\nCheck: (combined rate)×(time) = 3/10×10/3 = 1.\nFinal answer: $\\frac{10}{3}$", "final_answer": "\\frac{10}{3}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "817c6b3b7ef20a8d", "problem_id": "661aedf6a837106e", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$2 y$ = $16$\n$5 x + y$ = $13$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $5$: $10 y$ = $80$.\n3) Multiply the second equation by $0$: $0$ = $0$.\n4) Subtract to eliminate x: $10 y$ = $80$.\n5) Solve for y: y = $8$.\n6) Substitute back to find x: x = $1$.\nFinal answer: $(x,y)=\\left(1, 8\\right)$", "final_answer": "(x,y)=\\left(1, 8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c19185061a355c4a", "problem_id": "94cffce7e5cccf76", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 2 x - 3$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to x - 3.\n3) Then the limit is -5.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -7.\nFinal answer: $-7$", "final_answer": "-7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7472f84914615fcc", "problem_id": "f4922f7148f44650", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡0 (mod 9), x≡0 (mod 7). Give the solution modulo 63.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 63.\n3) Compute the CRT solution to get x ≡ 0 (mod 63).\nFinal answer: $x\\equiv 0\\pmod{63}$", "final_answer": "x\\equiv 0\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0eff8d9c87907ef7", "problem_id": "90ce192073de0a89", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{4} - 2 x^{3} + x^{2} - 2 x$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x) = - 2 x$.\n3) So $f'(x) = 8 x^{3} - 6 x^{2}$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x) = -2$.\nTherefore the correct derivative is $f'(x) = 2 \\left(4 x^{3} - 3 x^{2} + x - 1\\right)$.\nFinal answer: $2 \\left(4 x^{3} - 3 x^{2} + x - 1\\right)$", "final_answer": "2 \\left(4 x^{3} - 3 x^{2} + x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "bb2061c5b9541b3f", "problem_id": "b72d33c917832bf4", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 7 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=7, k=5, p=$\\frac{2}{7}$ and simplify.\n4) Here C(7,5) = 21.\nFinal answer: $\\frac{2400}{117649}$", "final_answer": "\\frac{2400}{117649}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e93fbec666229d1d", "problem_id": "b72d33c917832bf4", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 7 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=7, k=5, p=$\\frac{2}{7}$ and simplify.\nHere C(7,5) = 21.\nCheck: expression simplifies to $\\frac{2400}{117649}$.\nFinal answer: $\\frac{2400}{117649}$", "final_answer": "\\frac{2400}{117649}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0bf031a6a8412dcb", "problem_id": "e1e1ac1d60cc40f5", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡1 (mod 7), x≡5 (mod 8). Give the solution modulo 56.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 56.\n3) Compute the CRT solution to get x ≡ 29 (mod 56).\nFinal answer: $x\\equiv 29\\pmod{56}$", "final_answer": "x\\equiv 29\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "05582209bc1b2479", "problem_id": "0afa354efc24e7f3", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - 3 x^{2} - 11 x - 43$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} - 12 x - 43.\n3) Then the limit is -139.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: -135.\nFinal answer: $-135$", "final_answer": "-135", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "154fa2a240a4c951", "problem_id": "5ff943fa5188d755", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_5=30$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_5=25.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=25 and k=5.\n4) Compute C(29,4) = 23751.\nFinal answer: $23751$", "final_answer": "23751", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "54e28ecabfeb9da4", "problem_id": "5ff943fa5188d755", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_5=30$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_5=25.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=25 and k=5.\nCompute C(29,4) = 23751.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $23751$", "final_answer": "23751", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3b46bf415f0fe5c4", "problem_id": "1dc9caf4991eef4a", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡2 (mod 7), x≡0 (mod 8). Give the solution modulo 56.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 56.\n3) Compute the CRT solution to get x ≡ 16 (mod 56).\nFinal answer: $x\\equiv 16\\pmod{56}$", "final_answer": "x\\equiv 16\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f51bd4199b753f0", "problem_id": "1dc9caf4991eef4a", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 7), x≡0 (mod 8). Give the solution modulo 56.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 56.\nCompute the CRT solution to get x ≡ 16 (mod 56).\nCheck: 16 mod 7=2 and mod 8=0.\nFinal answer: $x\\equiv 16\\pmod{56}$", "final_answer": "x\\equiv 16\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "34682655660bf06f", "problem_id": "cc08214473d284d9", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Find the multiplicative inverse of 5 modulo 41.", "output": "We want x such that 5x ≡ 1 (mod 41).\nUse extended Euclid to find s,t with 5s + 41t = 1.\nOne Bezout identity is: 5·(-8) + 41·(1) = 1.\nFinal answer: $x\\equiv 33\\pmod{41}$", "final_answer": "x\\equiv 33\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9cfeea24715be7d1", "problem_id": "cc08214473d284d9", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the multiplicative inverse of 5 modulo 41.", "output": "1) We want x such that 5x ≡ 1 (mod 41).\n2) Use extended Euclid to find s,t with 5s + 41t = 1.\n3) One Bezout identity is: 5·(-8) + 41·(1) = 1.\n4) Reduce s modulo 41: x ≡ 33 (mod 41).\nFinal answer: $x\\equiv 33\\pmod{41}$", "final_answer": "x\\equiv 33\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0f25d261551e0567", "problem_id": "bf5d5a73e8858dc2", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A disease has prevalence $\\frac{2}{25}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{57}{500}$.\n3) Compute P(D|+) = $\\frac{34}{57}$.\nFinal answer: $\\frac{34}{57}$", "final_answer": "\\frac{34}{57}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e683489bc30478d0", "problem_id": "bf5d5a73e8858dc2", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A disease has prevalence $\\frac{2}{25}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{57}{500}$.\nCompute P(D|+) = $\\frac{34}{57}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{34}{57}$", "final_answer": "\\frac{34}{57}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b1d0a6ed5c449a3a", "problem_id": "566eccae9f352026", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡10 (mod 11). Give the solution modulo 33.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 33.\n3) Compute the CRT solution to get x ≡ 10 (mod 33).\nFinal answer: $x\\equiv 10\\pmod{33}$", "final_answer": "x\\equiv 10\\pmod{33}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "af75e3ef5c2b88f6", "problem_id": "755a7e23efb2939a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} - x^{2} + 4 x - 16$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} + 3 x - 16.\n3) Then the limit is -44.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: -48.\nFinal answer: $-48$", "final_answer": "-48", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a88f504b485c0997", "problem_id": "ed7ab19a7254032b", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the linear system:\n$3 x - y$ = $9$\n$- 3 x - 9 y$ = $21$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $- 9 x + 3 y$ = $-27$.\n3) Multiply the second equation by $3$: $- 9 x - 27 y$ = $63$.\n4) Subtract to eliminate x: $30 y$ = $-90$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $2$.\nFinal answer: $(x,y)=\\left(2, -3\\right)$", "final_answer": "(x,y)=\\left(2, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "475cc78c267d0d72", "problem_id": "9d4cd9c05c99030c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x^{3} + 4 x^{2} + 7 x + 13$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to x^{3} + 4 x^{2} + 6 x + 13.\n3) Then the limit is 49.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 51.\nFinal answer: $51$", "final_answer": "51", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "4a14ce6437f253c1", "problem_id": "bfa05cd85d8d5b07", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 3 x^{3} - 8 x^{2} - 16 x - 30$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 3 x^{4} - 2 x^{3} + 2 x - 3$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 12 x^{3} - 6 x^{2} + 2$.\n5) Evaluate at a: p'(2) = $-118$.\nFinal answer: $-118$", "final_answer": "-118", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "581f15c04b6b8f80", "problem_id": "bfa05cd85d8d5b07", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 3 x^{3} - 8 x^{2} - 16 x - 30$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 3 x^{4} - 2 x^{3} + 2 x - 3$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-118$", "final_answer": "-118", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "603cf819002189d6", "problem_id": "00f955997000b891", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "An urn has 15 red balls and 14 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(29,10).\n3) Favorable samples: choose k reds and n−k blues: C(15,2)·C(14,8).\n4) Compute totals: C(29,10) = 20030010, and favorable = 315315.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{21}{1334}$", "final_answer": "\\frac{21}{1334}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "58b2d7b574dafee7", "problem_id": "00f955997000b891", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 15 red balls and 14 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(29,10).\nFavorable samples: choose k reds and n−k blues: C(15,2)·C(14,8).\nCompute totals: C(29,10) = 20030010, and favorable = 315315.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{21}{1334}$.\nFinal answer: $\\frac{21}{1334}$", "final_answer": "\\frac{21}{1334}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "19f2ac688fff3b78", "problem_id": "49837cfc3bf25074", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - 2 x - 6$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 2 x^{2} - 3$ and a = 3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 4 x$.\nEvaluate at a: p'(3) = $-12$.\nCheck: the identity gives $-12$.\nFinal answer: $-12$", "final_answer": "-12", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e6639e63ec796c06", "problem_id": "49837cfc3bf25074", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - 2 x - 6$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 2 x^{2} - 3$ and a = 3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 4 x$.\n5) Evaluate at a: p'(3) = $-12$.\nFinal answer: $-12$", "final_answer": "-12", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a8424ffd84b2fd00", "problem_id": "122901adf677e497", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A cyclist rides at 47 miles per hour for 1 hours, then at 68 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 47 × 1 = 47 miles.\n2) Second leg distance = 68 × 5/2 = 170 miles.\n3) Total distance = 47 + 170 = 217 miles.\nFinal answer: $217$", "final_answer": "217", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "675ddf540fc19f2d", "problem_id": "122901adf677e497", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 47 miles per hour for 1 hours, then at 68 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 47 × 1 = 47 miles.\nSecond leg distance = 68 × 5/2 = 170 miles.\nTotal distance = 47 + 170 = 217 miles.\nCheck: adding both legs gives 217.\nFinal answer: $217$", "final_answer": "217", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "32d5306d142626cb", "problem_id": "ee684811f8fb75ef", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A store sells a backpack for $66. It is discounted by 40%. You buy 8 backpacks. What is the total cost?", "output": "1) Discounted price = 66 × (1 − 40/100) = 66 × 60/100 = 198/5.\n2) Subtotal = 8 × 198/5 = 1584/5.\nFinal answer: $\\frac{1584}{5}$", "final_answer": "\\frac{1584}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d0f09b560631ce6a", "problem_id": "ee684811f8fb75ef", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A store sells a backpack for $66. It is discounted by 40%. You buy 8 backpacks. What is the total cost?", "output": "Discounted price = 66 × (1 − 40/100) = 66 × 60/100 = 198/5.\nSubtotal = 8 × 198/5 = 1584/5.\nFinal answer: $\\frac{1584}{5}$", "final_answer": "\\frac{1584}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a7e7fbcde6884bd6", "problem_id": "321df43b66d30b37", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 26 red balls and 14 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(40,9).\nFavorable samples: choose k reds and n−k blues: C(26,4)·C(14,5).\nCompute totals: C(40,9) = 273438880, and favorable = 29929900.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{10465}{95608}$.\nFinal answer: $\\frac{10465}{95608}$", "final_answer": "\\frac{10465}{95608}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "82fd6f5a2df66d53", "problem_id": "321df43b66d30b37", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 26 red balls and 14 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(40,9).\n3) Favorable samples: choose k reds and n−k blues: C(26,4)·C(14,5).\n4) Compute totals: C(40,9) = 273438880, and favorable = 29929900.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{10465}{95608}$", "final_answer": "\\frac{10465}{95608}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "35711fa222e04819", "problem_id": "df43a0685ce05e76", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{11} x}{3} + \\frac{11 x}{2} - \\frac{7 \\sqrt{11}}{6} + 7$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{11}{2} - \\frac{\\sqrt{11}}{3}$, c=$7 - \\frac{7 \\sqrt{11}}{6}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\sqrt{11} + \\frac{125}{36}$, so √Δ = $\\sqrt{\\sqrt{11} + \\frac{125}{36}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{11}{4} + \\frac{\\sqrt{11}}{6} + \\frac{\\sqrt{36 \\sqrt{11} + 125}}{12}$.\n7) x2 = $- \\frac{11}{4} - \\frac{\\sqrt{36 \\sqrt{11} + 125}}{12} + \\frac{\\sqrt{11}}{6}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{11}{4} + \\frac{\\sqrt{11}}{6} + \\frac{\\sqrt{36\\cdot \\sqrt{11} + 125}}{12}, - \\frac{11}{4} - \\frac{\\sqrt{36\\cdot \\sqrt{11} + 125}}{12} + \\frac{\\sqrt{11}}{6}\\}$", "final_answer": "x \\in \\{- \\frac{11}{4} + \\frac{\\sqrt{11}}{6} + \\frac{\\sqrt{36\\cdot \\sqrt{11} + 125}}{12}, - \\frac{11}{4} - \\frac{\\sqrt{36\\cdot \\sqrt{11} + 125}}{12} + \\frac{\\sqrt{11}}{6}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7d10fcca9a343dec", "problem_id": "a296c28adb8b96c6", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 4 - 3 x$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 4 - 4 x.\n3) Then the limit is 12.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 10.\nFinal answer: $10$", "final_answer": "10", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c24a25bd1b69225a", "problem_id": "182f8d6a5f6e5cee", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} + 3 x^{2} - 2$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x^{3}) = 3 x^{3}$.\n3) So $f'(x) = 3 x^{3} + 6 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x^{3}) = 3 x^{2}$.\nTherefore the correct derivative is $f'(x) = 3 x \\left(x + 2\\right)$.\nFinal answer: $3 x \\left(x + 2\\right)$", "final_answer": "3 x \\left(x + 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "782e33856787a12e", "problem_id": "06c5852e02073b09", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{1}{7}$. If it is flipped 14 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=14, k=3, p=$\\frac{1}{7}$ and simplify.\n4) Here C(14,3) = 364.\nFinal answer: $\\frac{18865446912}{96889010407}$", "final_answer": "\\frac{18865446912}{96889010407}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fc31edf497c7a3d1", "problem_id": "bb2c625674e3f88c", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - 2 x^{3} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{3} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{255 \\pi^{4}}{32} - \\frac{45 \\pi^{2}}{8} - \\frac{5 \\pi}{2} - 2$.\n4) Simplify the result.\nFinal answer: $- \\frac{255\\cdot \\pi^{4}}{32} - \\frac{45\\cdot \\pi^{2}}{8} - \\frac{5\\cdot \\pi}{2} - 2$", "final_answer": "- \\frac{255\\cdot \\pi^{4}}{32} - \\frac{45\\cdot \\pi^{2}}{8} - \\frac{5\\cdot \\pi}{2} - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b413427b3574d3c2", "problem_id": "bb2c625674e3f88c", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - 2 x^{3} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{3} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{255 \\pi^{4}}{32} - \\frac{45 \\pi^{2}}{8} - \\frac{5 \\pi}{2} - 2$.\nFinal answer: $- \\frac{255\\cdot \\pi^{4}}{32} - \\frac{45\\cdot \\pi^{2}}{8} - \\frac{5\\cdot \\pi}{2} - 2$", "final_answer": "- \\frac{255\\cdot \\pi^{4}}{32} - \\frac{45\\cdot \\pi^{2}}{8} - \\frac{5\\cdot \\pi}{2} - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4f37690e56d00a86", "problem_id": "4e450ea5e72b3c64", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{11} x + \\frac{3 x}{2} - \\frac{5}{2} + \\sqrt{11}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{3}{2} - \\sqrt{11}$, c=$- \\frac{5}{2} + \\sqrt{11}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{93}{4} - 7 \\sqrt{11}$, so √Δ = $\\sqrt{\\frac{93}{4} - 7 \\sqrt{11}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{3}{4} + \\frac{\\sqrt{93 - 28 \\sqrt{11}}}{4} + \\frac{\\sqrt{11}}{2}$.\n7) x2 = $- \\frac{3}{4} - \\frac{\\sqrt{93 - 28 \\sqrt{11}}}{4} + \\frac{\\sqrt{11}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{3}{4} + \\frac{\\sqrt{93 - 28\\cdot \\sqrt{11}}}{4} + \\frac{\\sqrt{11}}{2}, - \\frac{3}{4} - \\frac{\\sqrt{93 - 28\\cdot \\sqrt{11}}}{4} + \\frac{\\sqrt{11}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{3}{4} + \\frac{\\sqrt{93 - 28\\cdot \\sqrt{11}}}{4} + \\frac{\\sqrt{11}}{2}, - \\frac{3}{4} - \\frac{\\sqrt{93 - 28\\cdot \\sqrt{11}}}{4} + \\frac{\\sqrt{11}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "869960296b2c61c3", "problem_id": "4e450ea5e72b3c64", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{11} x + \\frac{3 x}{2} - \\frac{5}{2} + \\sqrt{11}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{3}{2} - \\sqrt{11}$, c=$- \\frac{5}{2} + \\sqrt{11}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{93}{4} - 7 \\sqrt{11}$, so √Δ = $\\sqrt{\\frac{93}{4} - 7 \\sqrt{11}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{3}{4} + \\frac{\\sqrt{93 - 28 \\sqrt{11}}}{4} + \\frac{\\sqrt{11}}{2}$.\nx2 = $- \\frac{3}{4} - \\frac{\\sqrt{93 - 28 \\sqrt{11}}}{4} + \\frac{\\sqrt{11}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{3}{4} + \\frac{\\sqrt{93 - 28\\cdot \\sqrt{11}}}{4} + \\frac{\\sqrt{11}}{2}, - \\frac{3}{4} - \\frac{\\sqrt{93 - 28\\cdot \\sqrt{11}}}{4} + \\frac{\\sqrt{11}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{3}{4} + \\frac{\\sqrt{93 - 28\\cdot \\sqrt{11}}}{4} + \\frac{\\sqrt{11}}{2}, - \\frac{3}{4} - \\frac{\\sqrt{93 - 28\\cdot \\sqrt{11}}}{4} + \\frac{\\sqrt{11}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c2574425d28d3b17", "problem_id": "ddaec76482e6fcda", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡2 (mod 8), x≡10 (mod 13). Give the solution modulo 104.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 104.\nCompute the CRT solution to get x ≡ 10 (mod 104).\nCheck: 10 mod 8=2 and mod 13=10.\nFinal answer: $x\\equiv 10\\pmod{104}$", "final_answer": "x\\equiv 10\\pmod{104}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "83327a9ab4438323", "problem_id": "ddaec76482e6fcda", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the congruences: x≡2 (mod 8), x≡10 (mod 13). Give the solution modulo 104.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 104.\n3) Compute the CRT solution to get x ≡ 10 (mod 104).\nFinal answer: $x\\equiv 10\\pmod{104}$", "final_answer": "x\\equiv 10\\pmod{104}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3c1c125fde6daf06", "problem_id": "b2f58f62a83d9bf8", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A store sells a backpack for $151. It is discounted by 35%. You buy 12 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 151 × (1 − 35/100) = 151 × 65/100 = 1963/20.\n2) Subtotal = 12 × 1963/20 = 5889/5.\n3) Total with tax = 5889/5 × 108/100 = 159003/125.\nFinal answer: $\\frac{159003}{125}$", "final_answer": "\\frac{159003}{125}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1fbc90b24aee957a", "problem_id": "b2f58f62a83d9bf8", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $151. It is discounted by 35%. You buy 12 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 151 × (1 − 35/100) = 151 × 65/100 = 1963/20.\nSubtotal = 12 × 1963/20 = 5889/5.\nTotal with tax = 5889/5 × 108/100 = 159003/125.\nCheck: recomputing the formula yields 159003/125.\nFinal answer: $\\frac{159003}{125}$", "final_answer": "\\frac{159003}{125}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0ff92520810a6ea8", "problem_id": "3c99d5b5c1d9847c", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "You want 49 liters of a 35% salt solution. You have a 25% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 49−x liters is 60% solution.\n2) Salt equation: 25x + 60(49−x) = 35·49.\n3) (25-60)x = (35-60)·49 ⇒ x = 35.\n4) Then 49−x = 14.\nFinal answer: $35 L of 25% and 14 L of 60%$", "final_answer": "35 L of 25% and 14 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aa34aabdc113ed9e", "problem_id": "3c99d5b5c1d9847c", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "You want 49 liters of a 35% salt solution. You have a 25% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 49−x liters is 60% solution.\nSalt equation: 25x + 60(49−x) = 35·49.\n(25-60)x = (35-60)·49 ⇒ x = 35.\nFinal answer: $35 L of 25% and 14 L of 60%$", "final_answer": "35 L of 25% and 14 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "df5e77f791525ece", "problem_id": "2f1583809e1498a2", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} + 2 x + 2 \\sin{\\left(x \\right)} + 3$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $9 x^{2} + 2 \\cos{\\left(x \\right)} + 2$.\n2) Slope at x=-1: m = f'(-1) = $2 \\cos{\\left(1 \\right)} + 11$.\n3) Point on curve: (-1, f(-1)) = (-1, $-2 - 2 \\sin{\\left(1 \\right)}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -2 - 2 \\sin{\\left(1 \\right)} = 2 \\cos{\\left(1 \\right)} + 11(x − -1).\n6) Optionally expand: y = $2 x \\cos{\\left(1 \\right)} + 11 x - 2 \\sin{\\left(1 \\right)} + 2 \\cos{\\left(1 \\right)} + 9$.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 11\\cdot x - 2\\cdot \\sin{\\left(1 \\right)} + 2\\cdot \\cos{\\left(1 \\right)} + 9$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 11\\cdot x - 2\\cdot \\sin{\\left(1 \\right)} + 2\\cdot \\cos{\\left(1 \\right)} + 9", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6f6ce033cff33b6c", "problem_id": "2f1583809e1498a2", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} + 2 x + 2 \\sin{\\left(x \\right)} + 3$ at $x=-1$.", "output": "Differentiate: f'(x) = $9 x^{2} + 2 \\cos{\\left(x \\right)} + 2$.\nSlope at x=-1: m = f'(-1) = $2 \\cos{\\left(1 \\right)} + 11$.\nPoint on curve: (-1, f(-1)) = (-1, $-2 - 2 \\sin{\\left(1 \\right)}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -2 - 2 \\sin{\\left(1 \\right)} = 2 \\cos{\\left(1 \\right)} + 11(x − -1).\nOptionally expand: y = $2 x \\cos{\\left(1 \\right)} + 11 x - 2 \\sin{\\left(1 \\right)} + 2 \\cos{\\left(1 \\right)} + 9$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 11\\cdot x - 2\\cdot \\sin{\\left(1 \\right)} + 2\\cdot \\cos{\\left(1 \\right)} + 9$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 11\\cdot x - 2\\cdot \\sin{\\left(1 \\right)} + 2\\cdot \\cos{\\left(1 \\right)} + 9", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c5c212bdb6bc3874", "problem_id": "a68544786ec9c4de", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A store sells a backpack for $170. It is discounted by 5%. You buy 11 backpacks. What is the total cost?", "output": "1) Discounted price = 170 × (1 − 5/100) = 170 × 95/100 = 323/2.\n2) Subtotal = 11 × 323/2 = 3553/2.\nFinal answer: $\\frac{3553}{2}$", "final_answer": "\\frac{3553}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3e940fc5339bd642", "problem_id": "a68544786ec9c4de", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A store sells a backpack for $170. It is discounted by 5%. You buy 11 backpacks. What is the total cost?", "output": "Discounted price = 170 × (1 − 5/100) = 170 × 95/100 = 323/2.\nSubtotal = 11 × 323/2 = 3553/2.\nCheck: recomputing the formula yields 3553/2.\nFinal answer: $\\frac{3553}{2}$", "final_answer": "\\frac{3553}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f580e88c1d2cc189", "problem_id": "c7bfb8dd758b5664", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "You want 66 liters of a 30% salt solution. You have a 20% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 66−x liters is 60% solution.\n2) Salt equation: 20x + 60(66−x) = 30·66.\n3) (20-60)x = (30-60)·66 ⇒ x = 99/2.\n4) Then 66−x = 33/2.\nFinal answer: $99/2 L of 20% and 33/2 L of 60%$", "final_answer": "99/2 L of 20% and 33/2 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "26e1bf9b8f2b1b2f", "problem_id": "c7bfb8dd758b5664", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "You want 66 liters of a 30% salt solution. You have a 20% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 66−x liters is 60% solution.\nSalt equation: 20x + 60(66−x) = 30·66.\n(20-60)x = (30-60)·66 ⇒ x = 99/2.\nThen 66−x = 33/2.\nCheck: 20·99/2 + 60·33/2 = 30·66.\nFinal answer: $99/2 L of 20% and 33/2 L of 60%$", "final_answer": "99/2 L of 20% and 33/2 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cb795fdd84708bfa", "problem_id": "64f49288910c747d", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the congruences: x≡0 (mod 9), x≡8 (mod 10). Give the solution modulo 90.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 90.\n3) Compute the CRT solution to get x ≡ 18 (mod 90).\nFinal answer: $x\\equiv 18\\pmod{90}$", "final_answer": "x\\equiv 18\\pmod{90}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b524489cd9bdd10b", "problem_id": "64f49288910c747d", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡0 (mod 9), x≡8 (mod 10). Give the solution modulo 90.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 90.\nCompute the CRT solution to get x ≡ 18 (mod 90).\nCheck: 18 mod 9=0 and mod 10=8.\nFinal answer: $x\\equiv 18\\pmod{90}$", "final_answer": "x\\equiv 18\\pmod{90}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7fc7a5d7e2cb44f6", "problem_id": "39b78257f5aca526", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} - x^{3} - 3 x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{3} - 3 x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $\\pi - \\frac{\\pi^{3}}{4}$.\n4) Simplify the result.\nFinal answer: $\\pi - \\frac{\\pi^{3}}{4}$", "final_answer": "\\pi - \\frac{\\pi^{3}}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1c4dc7dfe72d9c4a", "problem_id": "39b78257f5aca526", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} - x^{3} - 3 x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{3} - 3 x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $\\pi - \\frac{\\pi^{3}}{4}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\pi - \\frac{\\pi^{3}}{4}$", "final_answer": "\\pi - \\frac{\\pi^{3}}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "909817e4effc2883", "problem_id": "474f8314afc0204f", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Two friends split $2499 in the ratio 9:8. How much does each receive?", "output": "1) Total ratio parts = 9+8 = 17.\n2) Each part = 2499 ÷ 17 = 147.\n3) First share = 9 × 147 = 1323.\n4) Second share = 8 × 147 = 1176.\nFinal answer: $1176$", "final_answer": "1176", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "52f6736ed0fe6b36", "problem_id": "474f8314afc0204f", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Two friends split $2499 in the ratio 9:8. How much does each receive?", "output": "Total ratio parts = 9+8 = 17.\nEach part = 2499 ÷ 17 = 147.\nFirst share = 9 × 147 = 1323.\nSecond share = 8 × 147 = 1176.\nCheck: 1323+1176=2499.\nFinal answer: $1176$", "final_answer": "1176", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "78b5c57f5ef7c0ed", "problem_id": "f260ff5275f5152a", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 22 red balls and 10 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(32,7).\nFavorable samples: choose k reds and n−k blues: C(22,1)·C(10,6).\nCompute totals: C(32,7) = 3365856, and favorable = 4620.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{385}{280488}$.\nFinal answer: $\\frac{385}{280488}$", "final_answer": "\\frac{385}{280488}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e6f24c05fb769cc5", "problem_id": "f260ff5275f5152a", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "An urn has 22 red balls and 10 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(32,7).\n3) Favorable samples: choose k reds and n−k blues: C(22,1)·C(10,6).\n4) Compute totals: C(32,7) = 3365856, and favorable = 4620.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{385}{280488}$", "final_answer": "\\frac{385}{280488}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2694b6673c5f9ccc", "problem_id": "68e3c1c7b2923efa", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A store sells a backpack for $174. It is discounted by 35%. You buy 9 backpacks. What is the total cost?", "output": "1) Discounted price = 174 × (1 − 35/100) = 174 × 65/100 = 1131/10.\n2) Subtotal = 9 × 1131/10 = 10179/10.\nFinal answer: $\\frac{10179}{10}$", "final_answer": "\\frac{10179}{10}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d25345974a9882e6", "problem_id": "68e3c1c7b2923efa", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A store sells a backpack for $174. It is discounted by 35%. You buy 9 backpacks. What is the total cost?", "output": "Discounted price = 174 × (1 − 35/100) = 174 × 65/100 = 1131/10.\nSubtotal = 9 × 1131/10 = 10179/10.\nCheck: recomputing the formula yields 10179/10.\nFinal answer: $\\frac{10179}{10}$", "final_answer": "\\frac{10179}{10}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bf98c04815cb9761", "problem_id": "f3d3b4e037485f8c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(x^{2} - 3 x - 2\\right) \\cos{\\left(- 3 x^{2} + 3 x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x^{2} \\cos{\\left(\\left(- 3 x^{2} + 3 x\\right) + 3 \\right)}) = - x^{2} \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)}$.\n3) So $f'(x) = - 19 x^{2} \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} + x e^{x} - 3 x \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} + 3 e^{x} + 6 \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} - 3 \\cos{\\left(\\left(- 3 x^{2} + 3 x\\right) + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x^{2} \\cos{\\left(\\left(- 3 x^{2} + 3 x\\right) + 3 \\right)}) = - x^{2} \\left(3 - 6 x\\right) \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} + 2 x \\cos{\\left(\\left(- 3 x^{2} + 3 x\\right) + 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = - 3 \\left(1 - 2 x\\right) \\left(x \\left(x - 3\\right) - 2\\right) \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} + \\left(x + 2\\right) e^{x} + \\left(2 x - 3\\right) \\cos{\\left(\\left(- 3 x^{2} + 3 x\\right) + 3 \\right)} + e^{x}$.\nFinal answer: $- 3 \\left(1 - 2 x\\right) \\left(x \\left(x - 3\\right) - 2\\right) \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} + \\left(x + 2\\right) e^{x} + \\left(2 x - 3\\right) \\cos{\\left(\\left(- 3 x^{2} + 3 x\\right) + 3 \\right)} + e^{x}$", "final_answer": "- 3 \\left(1 - 2 x\\right) \\left(x \\left(x - 3\\right) - 2\\right) \\sin{\\left(- 3 x^{2} + 3 x + 3 \\right)} + \\left(x + 2\\right) e^{x} + \\left(2 x - 3\\right) \\cos{\\left(\\left(- 3 x^{2} + 3 x\\right) + 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7dec39403ad9d2f6", "problem_id": "64ff3512a7f7729e", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} 3 x^{4} + 3 x^{3} - 2 x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{4} + 3 x^{3} - 2 x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{45 \\pi^{4}}{64} - \\frac{3 \\pi^{3}}{4} - \\frac{9 \\pi}{2} + \\frac{3 \\pi^{2}}{8} + 5 + \\frac{99 \\pi^{5}}{160}$.\n4) Simplify the result.\nFinal answer: $- \\frac{45\\cdot \\pi^{4}}{64} - \\frac{3\\cdot \\pi^{3}}{4} - \\frac{9\\cdot \\pi}{2} + \\frac{3\\cdot \\pi^{2}}{8} + 5 + \\frac{99\\cdot \\pi^{5}}{160}$", "final_answer": "- \\frac{45\\cdot \\pi^{4}}{64} - \\frac{3\\cdot \\pi^{3}}{4} - \\frac{9\\cdot \\pi}{2} + \\frac{3\\cdot \\pi^{2}}{8} + 5 + \\frac{99\\cdot \\pi^{5}}{160}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "40135097f29cb131", "problem_id": "49bcbfb4c72e90fa", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "You want 33 liters of a 25% salt solution. You have a 10% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 33−x liters is 70% solution.\nSalt equation: 10x + 70(33−x) = 25·33.\n(10-70)x = (25-70)·33 ⇒ x = 99/4.\nFinal answer: $99/4 L of 10% and 33/4 L of 70%$", "final_answer": "99/4 L of 10% and 33/4 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "827faa8d29f57395", "problem_id": "49bcbfb4c72e90fa", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "You want 33 liters of a 25% salt solution. You have a 10% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 33−x liters is 70% solution.\n2) Salt equation: 10x + 70(33−x) = 25·33.\n3) (10-70)x = (25-70)·33 ⇒ x = 99/4.\n4) Then 33−x = 33/4.\nFinal answer: $99/4 L of 10% and 33/4 L of 70%$", "final_answer": "99/4 L of 10% and 33/4 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3cc3d9a7328f0ff0", "problem_id": "0f8379e99132d26b", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{11 x}{3} - 24$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$\\frac{11}{3}$, c=$-24$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{1849}{9}$, so √Δ = $\\frac{43}{3}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{8}{3}$.\n7) x2 = $- \\frac{9}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{8}{3}, - \\frac{9}{2}\\}$", "final_answer": "x \\in \\{\\frac{8}{3}, - \\frac{9}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "79e422241ef5bd42", "problem_id": "0f8379e99132d26b", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{11 x}{3} - 24$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$\\frac{11}{3}$, c=$-24$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{1849}{9}$, so √Δ = $\\frac{43}{3}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{8}{3}$.\nx2 = $- \\frac{9}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{8}{3}, - \\frac{9}{2}\\}$", "final_answer": "x \\in \\{\\frac{8}{3}, - \\frac{9}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a82b4994e7721a5", "problem_id": "3608e443d65b01e9", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Solve the congruences: x≡2 (mod 4), x≡6 (mod 13). Give the solution modulo 52.", "output": "$x\\equiv 6\\pmod{52}$", "final_answer": "x\\equiv 6\\pmod{52}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "32a14e9a9e4f8a3e", "problem_id": "3608e443d65b01e9", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡2 (mod 4), x≡6 (mod 13). Give the solution modulo 52.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 52.\n3) Compute the CRT solution to get x ≡ 6 (mod 52).\nFinal answer: $x\\equiv 6\\pmod{52}$", "final_answer": "x\\equiv 6\\pmod{52}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b4d5db0e5b753f3c", "problem_id": "c0733be60b977022", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} 2 x^{2} + 5 x + 19$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{2} + 4 x + 19.\n3) Then the limit is 67.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: 71.\nFinal answer: $71$", "final_answer": "71", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "30e1220882175347", "problem_id": "f94d5286b33ad42c", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Find the equation of the tangent line to $y=3 x^{2} + x - 1$ at $x=0$.", "output": "1) Differentiate: f'(x) = $6 x + 1$.\n2) Slope at x=0: m = f'(0) = $1$.\n3) Point on curve: (0, f(0)) = (0, $-1$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -1 = 1(x − 0).\n6) Optionally expand: y = $x - 1$.\nFinal answer: $y = x - 1$", "final_answer": "y = x - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "549594dc067d8002", "problem_id": "7c1c648e7d6d4707", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=33$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=33 and k=6.\nCompute C(38,5) = 501942.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $501942$", "final_answer": "501942", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "947bc32cd49af0fb", "problem_id": "7c1c648e7d6d4707", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=33$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=33 and k=6.\nFinal answer: $501942$", "final_answer": "501942", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5841ff223f81fbd7", "problem_id": "ab3931110d38ecc0", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡5 (mod 9), x≡10 (mod 11). Give the solution modulo 99.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 99.\nCompute the CRT solution to get x ≡ 32 (mod 99).\nCheck: 32 mod 9=5 and mod 11=10.\nFinal answer: $x\\equiv 32\\pmod{99}$", "final_answer": "x\\equiv 32\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d6ad2b5f7b131fdc", "problem_id": "ab3931110d38ecc0", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡5 (mod 9), x≡10 (mod 11). Give the solution modulo 99.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 99.\n3) Compute the CRT solution to get x ≡ 32 (mod 99).\nFinal answer: $x\\equiv 32\\pmod{99}$", "final_answer": "x\\equiv 32\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8e000f02d8efdd7e", "problem_id": "45095198baaac0d1", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡2 (mod 4), x≡10 (mod 11). Give the solution modulo 44.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 44.\n3) Compute the CRT solution to get x ≡ 10 (mod 44).\nFinal answer: $x\\equiv 10\\pmod{44}$", "final_answer": "x\\equiv 10\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6bba7090afefe154", "problem_id": "45095198baaac0d1", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 4), x≡10 (mod 11). Give the solution modulo 44.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 44.\nCompute the CRT solution to get x ≡ 10 (mod 44).\nCheck: 10 mod 4=2 and mod 11=10.\nFinal answer: $x\\equiv 10\\pmod{44}$", "final_answer": "x\\equiv 10\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9404947d32da660d", "problem_id": "57fccadd46b14a23", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} + 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)}$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 6 x^{2} + 6 x + 2 \\cos{\\left(x \\right)} + 3$.\n2) Slope at x=2: m = f'(2) = $-9 + 2 \\cos{\\left(2 \\right)}$.\n3) Point on curve: (2, f(2)) = (2, $2 \\sin{\\left(2 \\right)} + 2$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 2 \\sin{\\left(2 \\right)} + 2 = -9 + 2 \\cos{\\left(2 \\right)}(x − 2).\n6) Optionally expand: y = $- 9 x + 2 x \\cos{\\left(2 \\right)} - 4 \\cos{\\left(2 \\right)} + 2 \\sin{\\left(2 \\right)} + 20$.\nFinal answer: $y = - 9\\cdot x + 2\\cdot x\\cdot \\cos{\\left(2 \\right)} - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)} + 20$", "final_answer": "y = - 9\\cdot x + 2\\cdot x\\cdot \\cos{\\left(2 \\right)} - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)} + 20", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1e15df7fbb8df2aa", "problem_id": "57fccadd46b14a23", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} + 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)}$ at $x=2$.", "output": "Differentiate: f'(x) = $- 6 x^{2} + 6 x + 2 \\cos{\\left(x \\right)} + 3$.\nSlope at x=2: m = f'(2) = $-9 + 2 \\cos{\\left(2 \\right)}$.\nPoint on curve: (2, f(2)) = (2, $2 \\sin{\\left(2 \\right)} + 2$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 2 \\sin{\\left(2 \\right)} + 2 = -9 + 2 \\cos{\\left(2 \\right)}(x − 2).\nOptionally expand: y = $- 9 x + 2 x \\cos{\\left(2 \\right)} - 4 \\cos{\\left(2 \\right)} + 2 \\sin{\\left(2 \\right)} + 20$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 9\\cdot x + 2\\cdot x\\cdot \\cos{\\left(2 \\right)} - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)} + 20$", "final_answer": "y = - 9\\cdot x + 2\\cdot x\\cdot \\cos{\\left(2 \\right)} - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)} + 20", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1dbf95ac6eb21cc7", "problem_id": "028851bbb01636a1", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the congruences: x≡2 (mod 4), x≡4 (mod 11). Give the solution modulo 44.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 44.\nCompute the CRT solution to get x ≡ 26 (mod 44).\nFinal answer: $x\\equiv 26\\pmod{44}$", "final_answer": "x\\equiv 26\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "461b011ad3db34ac", "problem_id": "028851bbb01636a1", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡2 (mod 4), x≡4 (mod 11). Give the solution modulo 44.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 44.\n3) Compute the CRT solution to get x ≡ 26 (mod 44).\nFinal answer: $x\\equiv 26\\pmod{44}$", "final_answer": "x\\equiv 26\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d43c5b551908f798", "problem_id": "e64758d2eccda60d", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_6=24$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_6=6.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=6 and k=6.\n4) Compute C(11,5) = 462.\nFinal answer: $462$", "final_answer": "462", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a5c4d297a348775a", "problem_id": "e64758d2eccda60d", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_6=24$?", "output": "Let y_i=x_i-3, then y_i≥0 and y_1+…+y_6=6.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=6 and k=6.\nCompute C(11,5) = 462.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $462$", "final_answer": "462", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5037a7c303fe6d1e", "problem_id": "52c1227a7bc03751", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + x - 12$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} + x - 12$ = $\\left(x - 3\\right) \\left(x + 4\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $3$.\n5) From the second factor: x = $-4$.\nFinal answer: $x \\in \\{3, -4\\}$", "final_answer": "x \\in \\{3, -4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "883511375eccd5db", "problem_id": "52c1227a7bc03751", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + x - 12$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} + x - 12$ = $\\left(x - 3\\right) \\left(x + 4\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFinal answer: $x \\in \\{3, -4\\}$", "final_answer": "x \\in \\{3, -4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f6052fca4220c443", "problem_id": "e3366c32945b2377", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 10 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=4, p=$\\frac{1}{10}$ and simplify.\n4) Here C(10,4) = 210.\nFinal answer: $\\frac{11160261}{1000000000}$", "final_answer": "\\frac{11160261}{1000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e538a6a821434846", "problem_id": "e3366c32945b2377", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 10 times, what is the probability of exactly 4 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=4, p=$\\frac{1}{10}$ and simplify.\nHere C(10,4) = 210.\nCheck: expression simplifies to $\\frac{11160261}{1000000000}$.\nFinal answer: $\\frac{11160261}{1000000000}$", "final_answer": "\\frac{11160261}{1000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4b4157cd94df401d", "problem_id": "16bf85f26db9a2c4", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} - x^{2} - x + 2 \\sin{\\left(x \\right)} - 2$ at $x=1$.", "output": "Differentiate: f'(x) = $9 x^{2} - 2 x + 2 \\cos{\\left(x \\right)} - 1$.\nSlope at x=1: m = f'(1) = $2 \\cos{\\left(1 \\right)} + 6$.\nPoint on curve: (1, f(1)) = (1, $-1 + 2 \\sin{\\left(1 \\right)}$).\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 6\\cdot x - 7 - 2\\cdot \\cos{\\left(1 \\right)} + 2\\cdot \\sin{\\left(1 \\right)}$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 6\\cdot x - 7 - 2\\cdot \\cos{\\left(1 \\right)} + 2\\cdot \\sin{\\left(1 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "965c66b8d93b1692", "problem_id": "16bf85f26db9a2c4", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} - x^{2} - x + 2 \\sin{\\left(x \\right)} - 2$ at $x=1$.", "output": "Differentiate: f'(x) = $9 x^{2} - 2 x + 2 \\cos{\\left(x \\right)} - 1$.\nSlope at x=1: m = f'(1) = $2 \\cos{\\left(1 \\right)} + 6$.\nPoint on curve: (1, f(1)) = (1, $-1 + 2 \\sin{\\left(1 \\right)}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -1 + 2 \\sin{\\left(1 \\right)} = 2 \\cos{\\left(1 \\right)} + 6(x − 1).\nOptionally expand: y = $2 x \\cos{\\left(1 \\right)} + 6 x - 7 - 2 \\cos{\\left(1 \\right)} + 2 \\sin{\\left(1 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 6\\cdot x - 7 - 2\\cdot \\cos{\\left(1 \\right)} + 2\\cdot \\sin{\\left(1 \\right)}$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 6\\cdot x - 7 - 2\\cdot \\cos{\\left(1 \\right)} + 2\\cdot \\sin{\\left(1 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "01f535cccfa69814", "problem_id": "a223d00ecfb0dba4", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "A store sells a backpack for $386. It is discounted by 20%. You buy 12 backpacks. What is the total cost?", "output": "Discounted price = 386 × (1 − 20/100) = 386 × 80/100 = 1544/5.\nSubtotal = 12 × 1544/5 = 18528/5.\nFinal answer: $\\frac{18528}{5}$", "final_answer": "\\frac{18528}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2755e65589254e9f", "problem_id": "a223d00ecfb0dba4", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A store sells a backpack for $386. It is discounted by 20%. You buy 12 backpacks. What is the total cost?", "output": "1) Discounted price = 386 × (1 − 20/100) = 386 × 80/100 = 1544/5.\n2) Subtotal = 12 × 1544/5 = 18528/5.\nFinal answer: $\\frac{18528}{5}$", "final_answer": "\\frac{18528}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dbf4c4560407e6dd", "problem_id": "fb66368f2415621d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(- 3 x^{3} - x + 1\\right) \\cos{\\left(- x^{2} + x + 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x \\cos{\\left(\\left(- x^{2} + x\\right) + 2 \\right)}) = x \\sin{\\left(- x^{2} + x + 2 \\right)}$.\n3) So $f'(x) = - 6 x^{4} \\sin{\\left(- x^{2} + x + 2 \\right)} + 3 x^{3} \\sin{\\left(- x^{2} + x + 2 \\right)} - 9 x^{2} \\cos{\\left(\\left(- x^{2} + x\\right) + 2 \\right)} + 2 x e^{x} + 3 x \\sin{\\left(- x^{2} + x + 2 \\right)} + 6 e^{x} - \\sin{\\left(- x^{2} + x + 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x \\cos{\\left(\\left(- x^{2} + x\\right) + 2 \\right)}) = x \\left(1 - 2 x\\right) \\sin{\\left(- x^{2} + x + 2 \\right)} - \\cos{\\left(\\left(- x^{2} + x\\right) + 2 \\right)}$.\nTherefore the correct derivative is $f'(x) = - \\left(1 - 2 x\\right) \\left(x \\left(- 3 x^{2} - 1\\right) + 1\\right) \\sin{\\left(- x^{2} + x + 2 \\right)} + 2 \\left(x + 2\\right) e^{x} + \\left(- 9 x^{2} - 1\\right) \\cos{\\left(\\left(- x^{2} + x\\right) + 2 \\right)} + 2 e^{x}$.\nFinal answer: $- \\left(1 - 2 x\\right) \\left(x \\left(- 3 x^{2} - 1\\right) + 1\\right) \\sin{\\left(- x^{2} + x + 2 \\right)} + 2 \\left(x + 2\\right) e^{x} + \\left(- 9 x^{2} - 1\\right) \\cos{\\left(\\left(- x^{2} + x\\right) + 2 \\right)} + 2 e^{x}$", "final_answer": "- \\left(1 - 2 x\\right) \\left(x \\left(- 3 x^{2} - 1\\right) + 1\\right) \\sin{\\left(- x^{2} + x + 2 \\right)} + 2 \\left(x + 2\\right) e^{x} + \\left(- 9 x^{2} - 1\\right) \\cos{\\left(\\left(- x^{2} + x\\right) + 2 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f9a255c35b136a0a", "problem_id": "8bf13e6c84659b7b", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A cyclist rides at 45 miles per hour for 1/2 hours, then at 70 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 45 × 1/2 = 45/2 miles.\n2) Second leg distance = 70 × 2 = 140 miles.\n3) Total distance = 45/2 + 140 = 325/2 miles.\nFinal answer: $\\frac{325}{2}$", "final_answer": "\\frac{325}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "23752fb2211c9e15", "problem_id": "63d0dbe2bd6f431b", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the multiplicative inverse of 19 modulo 31.", "output": "1) We want x such that 19x ≡ 1 (mod 31).\n2) Use extended Euclid to find s,t with 19s + 31t = 1.\n3) One Bezout identity is: 19·(-13) + 31·(8) = 1.\n4) Reduce s modulo 31: x ≡ 18 (mod 31).\nFinal answer: $x\\equiv 18\\pmod{31}$", "final_answer": "x\\equiv 18\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "51522eb15d1c1d5b", "problem_id": "63d0dbe2bd6f431b", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the multiplicative inverse of 19 modulo 31.", "output": "We want x such that 19x ≡ 1 (mod 31).\nUse extended Euclid to find s,t with 19s + 31t = 1.\nOne Bezout identity is: 19·(-13) + 31·(8) = 1.\nReduce s modulo 31: x ≡ 18 (mod 31).\nCheck: 19·18 mod 31 = 1.\nFinal answer: $x\\equiv 18\\pmod{31}$", "final_answer": "x\\equiv 18\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "52c7165b3b4bb99f", "problem_id": "b7504cc7c76b3ae5", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡6 (mod 9), x≡0 (mod 8). Give the solution modulo 72.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 72.\n3) Compute the CRT solution to get x ≡ 24 (mod 72).\nFinal answer: $x\\equiv 24\\pmod{72}$", "final_answer": "x\\equiv 24\\pmod{72}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ac33a510812441f6", "problem_id": "b7504cc7c76b3ae5", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the congruences: x≡6 (mod 9), x≡0 (mod 8). Give the solution modulo 72.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 72.\nCompute the CRT solution to get x ≡ 24 (mod 72).\nFinal answer: $x\\equiv 24\\pmod{72}$", "final_answer": "x\\equiv 24\\pmod{72}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d95b48a6bfd4c0bb", "problem_id": "3c8d6cf74e88fe11", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$4 y$ = $4$\n$- x - 4 y$ = $-3$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $- 4 y$ = $-4$.\n3) Multiply the second equation by $0$: $0$ = $0$.\n4) Subtract to eliminate x: $- 4 y$ = $-4$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, 1\\right)$", "final_answer": "(x,y)=\\left(-1, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "81cc0a8d14d47419", "problem_id": "3c8d6cf74e88fe11", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the linear system:\n$4 y$ = $4$\n$- x - 4 y$ = $-3$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $- 4 y$ = $-4$.\nMultiply the second equation by $0$: $0$ = $0$.\nFinal answer: $(x,y)=\\left(-1, 1\\right)$", "final_answer": "(x,y)=\\left(-1, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2bab98efb9247fff", "problem_id": "8074404e38538415", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 10 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=5, p=$\\frac{2}{7}$ and simplify.\n4) Here C(10,5) = 252.\nFinal answer: $\\frac{3600000}{40353607}$", "final_answer": "\\frac{3600000}{40353607}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "42ec4dc92b56e57c", "problem_id": "8074404e38538415", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 10 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=5, p=$\\frac{2}{7}$ and simplify.\nHere C(10,5) = 252.\nCheck: expression simplifies to $\\frac{3600000}{40353607}$.\nFinal answer: $\\frac{3600000}{40353607}$", "final_answer": "\\frac{3600000}{40353607}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7b6ec9482fcf572e", "problem_id": "e49ed671a44a2edf", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Two friends split $1404 in the ratio 2:7. How much does each receive?", "output": "1) Total ratio parts = 2+7 = 9.\n2) Each part = 1404 ÷ 9 = 156.\n3) First share = 2 × 156 = 312.\n4) Second share = 7 × 156 = 1092.\nFinal answer: $1092$", "final_answer": "1092", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f6701be86fb3a45", "problem_id": "e49ed671a44a2edf", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $1404 in the ratio 2:7. How much does each receive?", "output": "Total ratio parts = 2+7 = 9.\nEach part = 1404 ÷ 9 = 156.\nFirst share = 2 × 156 = 312.\nSecond share = 7 × 156 = 1092.\nCheck: 312+1092=1404.\nFinal answer: $1092$", "final_answer": "1092", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f3456d748f123e71", "problem_id": "50c760bb555bbe67", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the congruences: x≡1 (mod 3), x≡2 (mod 10). Give the solution modulo 30.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 30.\n3) Compute the CRT solution to get x ≡ 22 (mod 30).\nFinal answer: $x\\equiv 22\\pmod{30}$", "final_answer": "x\\equiv 22\\pmod{30}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "656647d2ed6ef063", "problem_id": "ca1db2df9672f21b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x \\sin{\\left(x \\right)}) = x \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x \\sin{\\left(x \\right)}) = x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$.\nFinal answer: $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$", "final_answer": "- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "65a7b8a92ee954ec", "problem_id": "9c3745db3f163ff1", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$3 y$ = $-21$\n$8 x + y$ = $-39$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $8$: $24 y$ = $-168$.\nMultiply the second equation by $0$: $0$ = $0$.\nSubtract to eliminate x: $24 y$ = $-168$.\nSolve for y: y = $-7$.\nSubstitute back to find x: x = $-4$.\nCheck: plug (x,y)=(-4,-7) into both equations.\nFinal answer: $(x,y)=\\left(-4, -7\\right)$", "final_answer": "(x,y)=\\left(-4, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec3c32d4007b5f37", "problem_id": "9c3745db3f163ff1", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the linear system:\n$3 y$ = $-21$\n$8 x + y$ = $-39$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $8$: $24 y$ = $-168$.\nMultiply the second equation by $0$: $0$ = $0$.\nFinal answer: $(x,y)=\\left(-4, -7\\right)$", "final_answer": "(x,y)=\\left(-4, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bff437b685050b60", "problem_id": "8d450a317b120a28", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡7 (mod 11), x≡4 (mod 8). Give the solution modulo 88.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 88.\nCompute the CRT solution to get x ≡ 84 (mod 88).\nCheck: 84 mod 11=7 and mod 8=4.\nFinal answer: $x\\equiv 84\\pmod{88}$", "final_answer": "x\\equiv 84\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e423e634e883951f", "problem_id": "8d450a317b120a28", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡7 (mod 11), x≡4 (mod 8). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 84 (mod 88).\nFinal answer: $x\\equiv 84\\pmod{88}$", "final_answer": "x\\equiv 84\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d5172b359120edd", "problem_id": "faeb0d0963cbb625", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A store sells a backpack for $75. It is discounted by 30%. You buy 13 backpacks. What is the total cost?", "output": "Discounted price = 75 × (1 − 30/100) = 75 × 70/100 = 105/2.\nSubtotal = 13 × 105/2 = 1365/2.\nCheck: recomputing the formula yields 1365/2.\nFinal answer: $\\frac{1365}{2}$", "final_answer": "\\frac{1365}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "699dca322f6baa75", "problem_id": "faeb0d0963cbb625", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A store sells a backpack for $75. It is discounted by 30%. You buy 13 backpacks. What is the total cost?", "output": "1) Discounted price = 75 × (1 − 30/100) = 75 × 70/100 = 105/2.\n2) Subtotal = 13 × 105/2 = 1365/2.\nFinal answer: $\\frac{1365}{2}$", "final_answer": "\\frac{1365}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e39b7594bffef131", "problem_id": "8651ef6ee3cfa9c4", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A store sells a backpack for $350. It is discounted by 15%. You buy 8 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 350 × (1 − 15/100) = 350 × 85/100 = 595/2.\n2) Subtotal = 8 × 595/2 = 2380.\n3) Total with tax = 2380 × 108/100 = 12852/5.\nFinal answer: $\\frac{12852}{5}$", "final_answer": "\\frac{12852}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7469d2174ba462f6", "problem_id": "09726800bf81c01d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} + 4 x^{2} + 3 x$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x^{3}) = 3 x^{3}$.\n3) So $f'(x) = 3 x^{3} + 8 x + 3$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x^{3}) = 3 x^{2}$.\nTherefore the correct derivative is $f'(x) = 3 x^{2} + 8 x + 3$.\nFinal answer: $3 x^{2} + 8 x + 3$", "final_answer": "3 x^{2} + 8 x + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7a9bc8b30c4246fe", "problem_id": "33249a3d8a0a25e5", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "How many binary strings of length 22 contain exactly 10 ones with no two ones adjacent?", "output": "1) Reserve 9 mandatory zeros between the 10 ones.\n2) This leaves n−k+1 = 13 valid slots for the ones.\n3) Choose the slots: C(13,10) = 286.\nFinal answer: $286$", "final_answer": "286", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a9d4cafd06614bb6", "problem_id": "33249a3d8a0a25e5", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many binary strings of length 22 contain exactly 10 ones with no two ones adjacent?", "output": "Reserve 9 mandatory zeros between the 10 ones.\nThis leaves n−k+1 = 13 valid slots for the ones.\nChoose the slots: C(13,10) = 286.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $286$", "final_answer": "286", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1649ef9be9385600", "problem_id": "16775b2f17ca803a", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{17 x}{6} + 2$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{17}{6}$, c=$2$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{1}{36}$, so √Δ = $\\frac{1}{6}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{3}{2}$.\n7) x2 = $\\frac{4}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{3}{2}, \\frac{4}{3}\\}$", "final_answer": "x \\in \\{\\frac{3}{2}, \\frac{4}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4d4368d0105d8908", "problem_id": "16775b2f17ca803a", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{17 x}{6} + 2$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{17}{6}$, c=$2$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{1}{36}$, so √Δ = $\\frac{1}{6}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{3}{2}$.\nx2 = $\\frac{4}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{3}{2}, \\frac{4}{3}\\}$", "final_answer": "x \\in \\{\\frac{3}{2}, \\frac{4}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5164a1befc98c7c3", "problem_id": "e47051fc91b6cf5a", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 4 x - 7$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$-4$, c=$-7$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $100$, so √Δ = $10$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{7}{3}$.\n7) x2 = $-1$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{7}{3}, -1\\}$", "final_answer": "x \\in \\{\\frac{7}{3}, -1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c7a016c588cad1c6", "problem_id": "e47051fc91b6cf5a", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 4 x - 7$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$-4$, c=$-7$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $100$, so √Δ = $10$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{7}{3}$.\nx2 = $-1$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{7}{3}, -1\\}$", "final_answer": "x \\in \\{\\frac{7}{3}, -1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b1c773b61fb6672f", "problem_id": "4c7ca2ae9d45d420", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{14 x}{3} + \\frac{8}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$- \\frac{14}{3}$, c=$\\frac{8}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{4}{9}$, so √Δ = $\\frac{2}{3}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{4}{3}$.\nx2 = $1$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{4}{3}, 1\\}$", "final_answer": "x \\in \\{\\frac{4}{3}, 1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dd430b82292069d3", "problem_id": "4c7ca2ae9d45d420", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{14 x}{3} + \\frac{8}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$- \\frac{14}{3}$, c=$\\frac{8}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{4}{9}$, so √Δ = $\\frac{2}{3}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{4}{3}$.\n7) x2 = $1$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{4}{3}, 1\\}$", "final_answer": "x \\in \\{\\frac{4}{3}, 1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c625c93f6856abd4", "problem_id": "7b528cec9725fd2d", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the multiplicative inverse of 8 modulo 59.", "output": "1) We want x such that 8x ≡ 1 (mod 59).\n2) Use extended Euclid to find s,t with 8s + 59t = 1.\n3) One Bezout identity is: 8·(-22) + 59·(3) = 1.\n4) Reduce s modulo 59: x ≡ 37 (mod 59).\nFinal answer: $x\\equiv 37\\pmod{59}$", "final_answer": "x\\equiv 37\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4df59c277fe62666", "problem_id": "7b528cec9725fd2d", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the multiplicative inverse of 8 modulo 59.", "output": "We want x such that 8x ≡ 1 (mod 59).\nUse extended Euclid to find s,t with 8s + 59t = 1.\nOne Bezout identity is: 8·(-22) + 59·(3) = 1.\nReduce s modulo 59: x ≡ 37 (mod 59).\nCheck: 8·37 mod 59 = 1.\nFinal answer: $x\\equiv 37\\pmod{59}$", "final_answer": "x\\equiv 37\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d22f3416ad62e6be", "problem_id": "7a939442eaebb549", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$2 x - 2 y$ = $-14$\n$- 2 x - 3 y$ = $-11$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $- 4 x + 4 y$ = $28$.\nMultiply the second equation by $2$: $- 4 x - 6 y$ = $-22$.\nSubtract to eliminate x: $10 y$ = $50$.\nSolve for y: y = $5$.\nSubstitute back to find x: x = $-2$.\nCheck: plug (x,y)=(-2,5) into both equations.\nFinal answer: $(x,y)=\\left(-2, 5\\right)$", "final_answer": "(x,y)=\\left(-2, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a97bfd773d98673b", "problem_id": "8ca33e415b9c53cf", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} + 3 x + 4$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x^{2}) = - 4 x^{2}$.\n3) So $f'(x) = 3 - 4 x^{2}$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x^{2}) = - 4 x$.\nTherefore the correct derivative is $f'(x) = 3 - 4 x$.\nFinal answer: $3 - 4 x$", "final_answer": "3 - 4 x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "4d3915998b0ea0ac", "problem_id": "58e82bfc4e4ff392", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$- 5 x + 4 y$ = $-20$\n$- 3 x + 11 y$ = $31$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $15 x - 12 y$ = $60$.\nMultiply the second equation by $-5$: $15 x - 55 y$ = $-155$.\nSubtract to eliminate x: $43 y$ = $215$.\nSolve for y: y = $5$.\nSubstitute back to find x: x = $8$.\nCheck: plug (x,y)=(8,5) into both equations.\nFinal answer: $(x,y)=\\left(8, 5\\right)$", "final_answer": "(x,y)=\\left(8, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "015b5b4ca659f82e", "problem_id": "58e82bfc4e4ff392", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the linear system:\n$- 5 x + 4 y$ = $-20$\n$- 3 x + 11 y$ = $31$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $15 x - 12 y$ = $60$.\n3) Multiply the second equation by $-5$: $15 x - 55 y$ = $-155$.\n4) Subtract to eliminate x: $43 y$ = $215$.\n5) Solve for y: y = $5$.\n6) Substitute back to find x: x = $8$.\nFinal answer: $(x,y)=\\left(8, 5\\right)$", "final_answer": "(x,y)=\\left(8, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "729733ea306ee1e8", "problem_id": "e7382abbcd1f9809", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the equation of the tangent line to $y=- x^{2} - 2 x + 2$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 2 x - 2$.\n2) Slope at x=2: m = f'(2) = $-6$.\n3) Point on curve: (2, f(2)) = (2, $-6$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -6 = -6(x − 2).\n6) Optionally expand: y = $6 - 6 x$.\nFinal answer: $y = 6 - 6\\cdot x$", "final_answer": "y = 6 - 6\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "481c60fff925fa8f", "problem_id": "3f28178ada29b9c7", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 2 x^{2} \\sin{\\left(x \\right)}) = - 2 x^{2} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 2 x^{2} \\sin{\\left(x \\right)}) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$", "final_answer": "- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c1eae72ff3e1ecc1", "problem_id": "2358d6fdeaeb1766", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} 2 x^{3} - x^{2} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{3} - x^{2} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\n3) Subtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{\\pi^{3}}{24} + \\frac{\\pi}{2} + \\frac{\\pi^{4}}{32} + \\frac{\\pi^{2}}{2} + 5$.\n4) Simplify the result.\nFinal answer: $- \\frac{\\pi^{3}}{24} + \\frac{\\pi}{2} + \\frac{\\pi^{4}}{32} + \\frac{\\pi^{2}}{2} + 5$", "final_answer": "- \\frac{\\pi^{3}}{24} + \\frac{\\pi}{2} + \\frac{\\pi^{4}}{32} + \\frac{\\pi^{2}}{2} + 5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "12fc750101d36716", "problem_id": "2358d6fdeaeb1766", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} 2 x^{3} - x^{2} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $2 x^{3} - x^{2} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\nSubtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{\\pi^{3}}{24} + \\frac{\\pi}{2} + \\frac{\\pi^{4}}{32} + \\frac{\\pi^{2}}{2} + 5$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{\\pi^{3}}{24} + \\frac{\\pi}{2} + \\frac{\\pi^{4}}{32} + \\frac{\\pi^{2}}{2} + 5$", "final_answer": "- \\frac{\\pi^{3}}{24} + \\frac{\\pi}{2} + \\frac{\\pi^{4}}{32} + \\frac{\\pi^{2}}{2} + 5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "289967d93f071487", "problem_id": "04ae0047425cff72", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$- 2 x + 2 y$ = $-20$\n$3 x - 4 y$ = $36$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $- 6 x + 6 y$ = $-60$.\nMultiply the second equation by $-2$: $- 6 x + 8 y$ = $-72$.\nSubtract to eliminate x: $- 2 y$ = $12$.\nSolve for y: y = $-6$.\nSubstitute back to find x: x = $4$.\nCheck: plug (x,y)=(4,-6) into both equations.\nFinal answer: $(x,y)=\\left(4, -6\\right)$", "final_answer": "(x,y)=\\left(4, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1e7d62fe19247b0e", "problem_id": "04ae0047425cff72", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$- 2 x + 2 y$ = $-20$\n$3 x - 4 y$ = $36$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $3$: $- 6 x + 6 y$ = $-60$.\n3) Multiply the second equation by $-2$: $- 6 x + 8 y$ = $-72$.\n4) Subtract to eliminate x: $- 2 y$ = $12$.\n5) Solve for y: y = $-6$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, -6\\right)$", "final_answer": "(x,y)=\\left(4, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "acd32009d92a7276", "problem_id": "15425e5904155cf0", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A cyclist rides at 38 miles per hour for 1 hours, then at 23 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 38 × 1 = 38 miles.\n2) Second leg distance = 23 × 5/2 = 115/2 miles.\n3) Total distance = 38 + 115/2 = 191/2 miles.\nFinal answer: $\\frac{191}{2}$", "final_answer": "\\frac{191}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8ee1aee24c6b34f7", "problem_id": "15425e5904155cf0", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A cyclist rides at 38 miles per hour for 1 hours, then at 23 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 38 × 1 = 38 miles.\nSecond leg distance = 23 × 5/2 = 115/2 miles.\nTotal distance = 38 + 115/2 = 191/2 miles.\nFinal answer: $\\frac{191}{2}$", "final_answer": "\\frac{191}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "31cdb75fd78af96d", "problem_id": "9d115156c21e17b8", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the multiplicative inverse of 22 modulo 43.", "output": "We want x such that 22x ≡ 1 (mod 43).\nUse extended Euclid to find s,t with 22s + 43t = 1.\nOne Bezout identity is: 22·(2) + 43·(-1) = 1.\nReduce s modulo 43: x ≡ 2 (mod 43).\nCheck: 22·2 mod 43 = 1.\nFinal answer: $x\\equiv 2\\pmod{43}$", "final_answer": "x\\equiv 2\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fbdf104b41a502c2", "problem_id": "9d115156c21e17b8", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the multiplicative inverse of 22 modulo 43.", "output": "1) We want x such that 22x ≡ 1 (mod 43).\n2) Use extended Euclid to find s,t with 22s + 43t = 1.\n3) One Bezout identity is: 22·(2) + 43·(-1) = 1.\n4) Reduce s modulo 43: x ≡ 2 (mod 43).\nFinal answer: $x\\equiv 2\\pmod{43}$", "final_answer": "x\\equiv 2\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "067a3b2f018e38f2", "problem_id": "43199a59849d810a", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(3 x^{3} - 3 x^{2} + 2\\right) \\cos{\\left(x - 3 \\right)}$.", "output": "1) Let f(x) = $\\left(x + 2\\right) e^{x} + \\left(3 x^{3} - 3 x^{2} + 2\\right) \\cos{\\left(x - 3 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $3 x \\left(3 x - 2\\right) \\cos{\\left(x - 3 \\right)} + \\left(x + 2\\right) e^{x} - \\left(3 x^{3} - 3 x^{2} + 2\\right) \\sin{\\left(x - 3 \\right)} + e^{x}$.\nFinal answer: $3\\cdot x\\cdot \\left(3\\cdot x - 2\\right)\\cdot \\cos{\\left(x - 3 \\right)} + \\left(x + 2\\right)\\cdot e^{x} - \\left(3\\cdot x^{3} - 3\\cdot x^{2} + 2\\right)\\cdot \\sin{\\left(x - 3 \\right)} + e^{x}$", "final_answer": "3\\cdot x\\cdot \\left(3\\cdot x - 2\\right)\\cdot \\cos{\\left(x - 3 \\right)} + \\left(x + 2\\right)\\cdot e^{x} - \\left(3\\cdot x^{3} - 3\\cdot x^{2} + 2\\right)\\cdot \\sin{\\left(x - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fc945a42311867c9", "problem_id": "43199a59849d810a", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(3 x^{3} - 3 x^{2} + 2\\right) \\cos{\\left(x - 3 \\right)}$.", "output": "Let f(x) = $\\left(x + 2\\right) e^{x} + \\left(3 x^{3} - 3 x^{2} + 2\\right) \\cos{\\left(x - 3 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $3 x \\left(3 x - 2\\right) \\cos{\\left(x - 3 \\right)} + \\left(x + 2\\right) e^{x} - \\left(3 x^{3} - 3 x^{2} + 2\\right) \\sin{\\left(x - 3 \\right)} + e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $3\\cdot x\\cdot \\left(3\\cdot x - 2\\right)\\cdot \\cos{\\left(x - 3 \\right)} + \\left(x + 2\\right)\\cdot e^{x} - \\left(3\\cdot x^{3} - 3\\cdot x^{2} + 2\\right)\\cdot \\sin{\\left(x - 3 \\right)} + e^{x}$", "final_answer": "3\\cdot x\\cdot \\left(3\\cdot x - 2\\right)\\cdot \\cos{\\left(x - 3 \\right)} + \\left(x + 2\\right)\\cdot e^{x} - \\left(3\\cdot x^{3} - 3\\cdot x^{2} + 2\\right)\\cdot \\sin{\\left(x - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9dc3922432ef460a", "problem_id": "eccf7e9cd1c918fb", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$7 x + 3 y$ = $8$\n$- 7 x - 5 y$ = $10$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-7$: $- 49 x - 21 y$ = $-56$.\n3) Multiply the second equation by $7$: $- 49 x - 35 y$ = $70$.\n4) Subtract to eliminate x: $14 y$ = $-126$.\n5) Solve for y: y = $-9$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, -9\\right)$", "final_answer": "(x,y)=\\left(5, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2f01f55dd638b1f4", "problem_id": "49f21da7ba95c98b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(x^{2} - 2 x + 3\\right) \\cos{\\left(x^{2} + x - 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 \\cos{\\left(\\left(x^{2} + x\\right) - 3 \\right)}) = - 3 \\sin{\\left(x^{2} + x - 3 \\right)}$.\n3) So $f'(x) = - 2 x^{3} \\sin{\\left(x^{2} + x - 3 \\right)} + 3 x^{2} \\sin{\\left(x^{2} + x - 3 \\right)} + x e^{x} + 2 x \\sin{\\left(x^{2} + x - 3 \\right)} + 2 x \\cos{\\left(\\left(x^{2} + x\\right) - 3 \\right)} + 3 e^{x} - 3 \\sin{\\left(x^{2} + x - 3 \\right)} - 2 \\cos{\\left(\\left(x^{2} + x\\right) - 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 \\cos{\\left(\\left(x^{2} + x\\right) - 3 \\right)}) = - 3 \\left(2 x + 1\\right) \\sin{\\left(x^{2} + x - 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x - 1\\right) \\cos{\\left(\\left(x^{2} + x\\right) - 3 \\right)} + \\left(x + 2\\right) e^{x} - \\left(2 x + 1\\right) \\left(x \\left(x - 2\\right) + 3\\right) \\sin{\\left(x^{2} + x - 3 \\right)} + e^{x}$.\nFinal answer: $2 \\left(x - 1\\right) \\cos{\\left(\\left(x^{2} + x\\right) - 3 \\right)} + \\left(x + 2\\right) e^{x} - \\left(2 x + 1\\right) \\left(x \\left(x - 2\\right) + 3\\right) \\sin{\\left(x^{2} + x - 3 \\right)} + e^{x}$", "final_answer": "2 \\left(x - 1\\right) \\cos{\\left(\\left(x^{2} + x\\right) - 3 \\right)} + \\left(x + 2\\right) e^{x} - \\left(2 x + 1\\right) \\left(x \\left(x - 2\\right) + 3\\right) \\sin{\\left(x^{2} + x - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "63b571e88475f88e", "problem_id": "3f0283dbf60ea0d7", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 10 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=5, p=$\\frac{3}{7}$ and simplify.\nHere C(10,5) = 252.\nCheck: expression simplifies to $\\frac{8957952}{40353607}$.\nFinal answer: $\\frac{8957952}{40353607}$", "final_answer": "\\frac{8957952}{40353607}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ea58c1e22213acb7", "problem_id": "3f0283dbf60ea0d7", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 10 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=5, p=$\\frac{3}{7}$ and simplify.\n4) Here C(10,5) = 252.\nFinal answer: $\\frac{8957952}{40353607}$", "final_answer": "\\frac{8957952}{40353607}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1ffc0c5eee61f4ba", "problem_id": "c41a9284f30777d4", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Two friends split $2288 in the ratio 4:9. How much does each receive?", "output": "1) Total ratio parts = 4+9 = 13.\n2) Each part = 2288 ÷ 13 = 176.\n3) First share = 4 × 176 = 704.\n4) Second share = 9 × 176 = 1584.\nFinal answer: $1584$", "final_answer": "1584", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c8e6d6162ea3aac2", "problem_id": "c41a9284f30777d4", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Two friends split $2288 in the ratio 4:9. How much does each receive?", "output": "Total ratio parts = 4+9 = 13.\nEach part = 2288 ÷ 13 = 176.\nFirst share = 4 × 176 = 704.\nSecond share = 9 × 176 = 1584.\nCheck: 704+1584=2288.\nFinal answer: $1584$", "final_answer": "1584", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f54c848faf3d99c6", "problem_id": "0beccee8c876ee80", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $795 in the ratio 6:9. How much does each receive?", "output": "Total ratio parts = 6+9 = 15.\nEach part = 795 ÷ 15 = 53.\nFirst share = 6 × 53 = 318.\nSecond share = 9 × 53 = 477.\nCheck: 318+477=795.\nFinal answer: $477$", "final_answer": "477", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "94bde82fa941ae51", "problem_id": "0beccee8c876ee80", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Two friends split $795 in the ratio 6:9. How much does each receive?", "output": "1) Total ratio parts = 6+9 = 15.\n2) Each part = 795 ÷ 15 = 53.\n3) First share = 6 × 53 = 318.\n4) Second share = 9 × 53 = 477.\nFinal answer: $477$", "final_answer": "477", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8d8c9f437c9417f4", "problem_id": "d1c2e076a34be24e", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 4 x^{2} - 4 x - 3\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $4 x^{2} - 4 x - 3$.\nCompute an antiderivative: F(x) = $\\frac{4 x^{3}}{3} - 2 x^{2} - 3 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{4\\cdot x^{3}}{3} - 2\\cdot x^{2} - 3\\cdot x$", "final_answer": "C + \\frac{4\\cdot x^{3}}{3} - 2\\cdot x^{2} - 3\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aed44d8eff0995eb", "problem_id": "d1c2e076a34be24e", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 4 x^{2} - 4 x - 3\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $4 x^{2} - 4 x - 3$.\n3) Compute an antiderivative: F(x) = $\\frac{4 x^{3}}{3} - 2 x^{2} - 3 x$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{4\\cdot x^{3}}{3} - 2\\cdot x^{2} - 3\\cdot x$", "final_answer": "C + \\frac{4\\cdot x^{3}}{3} - 2\\cdot x^{2} - 3\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b37699dff89212c", "problem_id": "3d3bee11cb576bee", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 6 x + \\frac{28}{9}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$6$, c=$\\frac{28}{9}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{100}{9}$, so √Δ = $\\frac{10}{3}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{2}{3}$.\nx2 = $- \\frac{7}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{2}{3}, - \\frac{7}{3}\\}$", "final_answer": "x \\in \\{- \\frac{2}{3}, - \\frac{7}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fa32de5c7eff09d6", "problem_id": "3d3bee11cb576bee", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 6 x + \\frac{28}{9}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$6$, c=$\\frac{28}{9}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{100}{9}$, so √Δ = $\\frac{10}{3}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{2}{3}$.\n7) x2 = $- \\frac{7}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{2}{3}, - \\frac{7}{3}\\}$", "final_answer": "x \\in \\{- \\frac{2}{3}, - \\frac{7}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c57c7b4005c7b015", "problem_id": "13c84f4a4d7e434e", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many binary strings of length 18 contain exactly 5 ones with no two ones adjacent?", "output": "Reserve 4 mandatory zeros between the 5 ones.\nThis leaves n−k+1 = 14 valid slots for the ones.\nChoose the slots: C(14,5) = 2002.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $2002$", "final_answer": "2002", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b77f2cca08bf0e20", "problem_id": "13c84f4a4d7e434e", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many binary strings of length 18 contain exactly 5 ones with no two ones adjacent?", "output": "1) Reserve 4 mandatory zeros between the 5 ones.\n2) This leaves n−k+1 = 14 valid slots for the ones.\n3) Choose the slots: C(14,5) = 2002.\nFinal answer: $2002$", "final_answer": "2002", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "30bef2ab766fb235", "problem_id": "c51068deddb5d1e6", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡0 (mod 4), x≡1 (mod 7). Give the solution modulo 28.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 28.\n3) Compute the CRT solution to get x ≡ 8 (mod 28).\nFinal answer: $x\\equiv 8\\pmod{28}$", "final_answer": "x\\equiv 8\\pmod{28}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4e6f6c49ae55c167", "problem_id": "c51068deddb5d1e6", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡0 (mod 4), x≡1 (mod 7). Give the solution modulo 28.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 28.\nCompute the CRT solution to get x ≡ 8 (mod 28).\nCheck: 8 mod 4=0 and mod 7=1.\nFinal answer: $x\\equiv 8\\pmod{28}$", "final_answer": "x\\equiv 8\\pmod{28}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0945b8b2e5939eb3", "problem_id": "608fff4ecd21142c", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} - 4 x$.", "output": "1) Differentiate term-by-term:\n2) d/dx(- 4 x) = -4.\n3) d/dx(2 x^{2}) = 4 x.\n4) Combine: f'(x) = 4 \\left(x - 1\\right).\nFinal answer: $4\\cdot x - 4$", "final_answer": "4\\cdot x - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d3d30654634756dc", "problem_id": "192c30ef456c4825", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 x \\sin{\\left(x \\right)}) = 3 x \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 x \\sin{\\left(x \\right)}) = 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$", "final_answer": "- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "489e7d5641b53f46", "problem_id": "bfb6988a95ab898a", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "You want 48 liters of a 47% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 48−x liters is 70% solution.\n2) Salt equation: 25x + 70(48−x) = 47·48.\n3) (25-70)x = (47-70)·48 ⇒ x = 368/15.\n4) Then 48−x = 352/15.\nFinal answer: $368/15 L of 25% and 352/15 L of 70%$", "final_answer": "368/15 L of 25% and 352/15 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "af19fb1133fe76e5", "problem_id": "bfb6988a95ab898a", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 48 liters of a 47% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 48−x liters is 70% solution.\nSalt equation: 25x + 70(48−x) = 47·48.\n(25-70)x = (47-70)·48 ⇒ x = 368/15.\nThen 48−x = 352/15.\nCheck: 25·368/15 + 70·352/15 = 47·48.\nFinal answer: $368/15 L of 25% and 352/15 L of 70%$", "final_answer": "368/15 L of 25% and 352/15 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "27d5245450270c5c", "problem_id": "da4f9b4b91cf0b48", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 11 times, what is the probability of exactly 8 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=8, p=$\\frac{2}{5}$ and simplify.\n4) Here C(11,8) = 165.\nFinal answer: $\\frac{228096}{9765625}$", "final_answer": "\\frac{228096}{9765625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "85c15b7f851c5bea", "problem_id": "da4f9b4b91cf0b48", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 11 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=11, k=8, p=$\\frac{2}{5}$ and simplify.\nHere C(11,8) = 165.\nCheck: expression simplifies to $\\frac{228096}{9765625}$.\nFinal answer: $\\frac{228096}{9765625}$", "final_answer": "\\frac{228096}{9765625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dbe3cc2999b9172c", "problem_id": "9be8563bda41e6c8", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡6 (mod 7), x≡0 (mod 13). Give the solution modulo 91.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 91.\nCompute the CRT solution to get x ≡ 13 (mod 91).\nCheck: 13 mod 7=6 and mod 13=0.\nFinal answer: $x\\equiv 13\\pmod{91}$", "final_answer": "x\\equiv 13\\pmod{91}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ad253f3805e4f7c", "problem_id": "9be8563bda41e6c8", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡6 (mod 7), x≡0 (mod 13). Give the solution modulo 91.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 91.\n3) Compute the CRT solution to get x ≡ 13 (mod 91).\nFinal answer: $x\\equiv 13\\pmod{91}$", "final_answer": "x\\equiv 13\\pmod{91}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e462be170412a08f", "problem_id": "90ce192073de0a89", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{4} - 2 x^{3} + x^{2} - 2 x$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x^{2}) = 2 x^{2}$.\n3) So $f'(x) = 8 x^{3} - 4 x^{2} - 2$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x^{2}) = 2 x$.\nTherefore the correct derivative is $f'(x) = 2 \\left(4 x^{3} - 3 x^{2} + x - 1\\right)$.\nFinal answer: $2 \\left(4 x^{3} - 3 x^{2} + x - 1\\right)$", "final_answer": "2 \\left(4 x^{3} - 3 x^{2} + x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8886a4de7c7c3a0a", "problem_id": "e8d3d9a66179b3cc", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 12 times, what is the probability of exactly 10 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=10, p=$\\frac{1}{8}$ and simplify.\n4) Here C(12,10) = 66.\nFinal answer: $\\frac{1617}{34359738368}$", "final_answer": "\\frac{1617}{34359738368}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f74fc507da6fe0df", "problem_id": "49e27953c5c5c975", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the congruences: x≡6 (mod 8), x≡4 (mod 11). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 70 (mod 88).\nFinal answer: $x\\equiv 70\\pmod{88}$", "final_answer": "x\\equiv 70\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a405ec58dd524ae6", "problem_id": "89d6d184019b9d06", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(3 x^{2} + 1\\right) \\cos{\\left(3 x + 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x e^{x}) = x e^{x}$.\n3) So $f'(x) = - 9 x^{2} \\sin{\\left(3 x + 2 \\right)} + x e^{x} + 6 x \\cos{\\left(3 x + 2 \\right)} - 3 e^{x} - 3 \\sin{\\left(3 x + 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x e^{x}) = x e^{x} + e^{x}$.\nTherefore the correct derivative is $f'(x) = 6 x \\cos{\\left(3 x + 2 \\right)} + \\left(x - 3\\right) e^{x} - 3 \\left(3 x^{2} + 1\\right) \\sin{\\left(3 x + 2 \\right)} + e^{x}$.\nFinal answer: $6 x \\cos{\\left(3 x + 2 \\right)} + \\left(x - 3\\right) e^{x} - 3 \\left(3 x^{2} + 1\\right) \\sin{\\left(3 x + 2 \\right)} + e^{x}$", "final_answer": "6 x \\cos{\\left(3 x + 2 \\right)} + \\left(x - 3\\right) e^{x} - 3 \\left(3 x^{2} + 1\\right) \\sin{\\left(3 x + 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8652959ce06438ae", "problem_id": "3096fef20ed547b8", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A store sells a backpack for $184. It is discounted by 20%. You buy 8 backpacks. What is the total cost?", "output": "Discounted price = 184 × (1 − 20/100) = 184 × 80/100 = 736/5.\nSubtotal = 8 × 736/5 = 5888/5.\nFinal answer: $\\frac{5888}{5}$", "final_answer": "\\frac{5888}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1746e760d7632aec", "problem_id": "3096fef20ed547b8", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A store sells a backpack for $184. It is discounted by 20%. You buy 8 backpacks. What is the total cost?", "output": "1) Discounted price = 184 × (1 − 20/100) = 184 × 80/100 = 736/5.\n2) Subtotal = 8 × 736/5 = 5888/5.\nFinal answer: $\\frac{5888}{5}$", "final_answer": "\\frac{5888}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "79845dd30875b02f", "problem_id": "b227c85639ebb464", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - 4 x^{4} + 2 x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 4 x^{4} + 2 x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{205 \\pi^{5}}{8} - \\frac{15 \\pi^{2}}{8} - 1 + \\frac{5 \\pi}{2} + \\frac{65 \\pi^{3}}{12}$.\n4) Simplify the result.\nFinal answer: $- \\frac{205\\cdot \\pi^{5}}{8} - \\frac{15\\cdot \\pi^{2}}{8} - 1 + \\frac{5\\cdot \\pi}{2} + \\frac{65\\cdot \\pi^{3}}{12}$", "final_answer": "- \\frac{205\\cdot \\pi^{5}}{8} - \\frac{15\\cdot \\pi^{2}}{8} - 1 + \\frac{5\\cdot \\pi}{2} + \\frac{65\\cdot \\pi^{3}}{12}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "70557c6d41d55f46", "problem_id": "b227c85639ebb464", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - 4 x^{4} + 2 x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 4 x^{4} + 2 x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{205 \\pi^{5}}{8} - \\frac{15 \\pi^{2}}{8} - 1 + \\frac{5 \\pi}{2} + \\frac{65 \\pi^{3}}{12}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{205\\cdot \\pi^{5}}{8} - \\frac{15\\cdot \\pi^{2}}{8} - 1 + \\frac{5\\cdot \\pi}{2} + \\frac{65\\cdot \\pi^{3}}{12}$", "final_answer": "- \\frac{205\\cdot \\pi^{5}}{8} - \\frac{15\\cdot \\pi^{2}}{8} - 1 + \\frac{5\\cdot \\pi}{2} + \\frac{65\\cdot \\pi^{3}}{12}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b0c83332ea60c4ff", "problem_id": "091c0d5472be0676", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "You want 48 liters of a 35% salt solution. You have a 20% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 48−x liters is 50% solution.\n2) Salt equation: 20x + 50(48−x) = 35·48.\n3) (20-50)x = (35-50)·48 ⇒ x = 24.\n4) Then 48−x = 24.\nFinal answer: $24 L of 20% and 24 L of 50%$", "final_answer": "24 L of 20% and 24 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bfaf05c694f4c5a4", "problem_id": "091c0d5472be0676", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 48 liters of a 35% salt solution. You have a 20% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 48−x liters is 50% solution.\nSalt equation: 20x + 50(48−x) = 35·48.\n(20-50)x = (35-50)·48 ⇒ x = 24.\nThen 48−x = 24.\nCheck: 20·24 + 50·24 = 35·48.\nFinal answer: $24 L of 20% and 24 L of 50%$", "final_answer": "24 L of 20% and 24 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f63accd5c7064611", "problem_id": "a658831472d16244", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A cyclist rides at 51 miles per hour for 1/2 hours, then at 32 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 51 × 1/2 = 51/2 miles.\nSecond leg distance = 32 × 2 = 64 miles.\nTotal distance = 51/2 + 64 = 179/2 miles.\nCheck: adding both legs gives 179/2.\nFinal answer: $\\frac{179}{2}$", "final_answer": "\\frac{179}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "67a030772b3b2930", "problem_id": "a658831472d16244", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A cyclist rides at 51 miles per hour for 1/2 hours, then at 32 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 51 × 1/2 = 51/2 miles.\nSecond leg distance = 32 × 2 = 64 miles.\nTotal distance = 51/2 + 64 = 179/2 miles.\nFinal answer: $\\frac{179}{2}$", "final_answer": "\\frac{179}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "585a99813cc999a5", "problem_id": "4740014e8200bfa5", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the equation of the tangent line to $y=x^{3} + 3 x^{2} + x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=0$.", "output": "1) Differentiate: f'(x) = $3 x^{2} + 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 1$.\n2) Slope at x=0: m = f'(0) = $\\frac{3}{2}$.\n3) Point on curve: (0, f(0)) = (0, $1$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 1 = \\frac{3}{2}(x − 0).\n6) Optionally expand: y = $\\frac{3 x}{2} + 1$.\nFinal answer: $y = \\frac{3\\cdot x}{2} + 1$", "final_answer": "y = \\frac{3\\cdot x}{2} + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a1d344230070c0ed", "problem_id": "4740014e8200bfa5", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the equation of the tangent line to $y=x^{3} + 3 x^{2} + x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=0$.", "output": "Differentiate: f'(x) = $3 x^{2} + 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 1$.\nSlope at x=0: m = f'(0) = $\\frac{3}{2}$.\nPoint on curve: (0, f(0)) = (0, $1$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 1 = \\frac{3}{2}(x − 0).\nOptionally expand: y = $\\frac{3 x}{2} + 1$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = \\frac{3\\cdot x}{2} + 1$", "final_answer": "y = \\frac{3\\cdot x}{2} + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2b1e05f28dedb378", "problem_id": "5dec35dc5dde310b", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$- 4 x$ = $-20$\n$- 5 x - 3 y$ = $-34$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-5$: $20 x$ = $100$.\n3) Multiply the second equation by $-4$: $20 x + 12 y$ = $136$.\n4) Subtract to eliminate x: $- 12 y$ = $-36$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, 3\\right)$", "final_answer": "(x,y)=\\left(5, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c562710a9d765d4f", "problem_id": "5dec35dc5dde310b", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$- 4 x$ = $-20$\n$- 5 x - 3 y$ = $-34$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-5$: $20 x$ = $100$.\nMultiply the second equation by $-4$: $20 x + 12 y$ = $136$.\nSubtract to eliminate x: $- 12 y$ = $-36$.\nSolve for y: y = $3$.\nSubstitute back to find x: x = $5$.\nCheck: plug (x,y)=(5,3) into both equations.\nFinal answer: $(x,y)=\\left(5, 3\\right)$", "final_answer": "(x,y)=\\left(5, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "66664fcaf4d02e4c", "problem_id": "8a628b6d98557572", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 29 red balls and 29 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(58,8).\nFavorable samples: choose k reds and n−k blues: C(29,2)·C(29,6).\nCompute totals: C(58,8) = 1916797311, and favorable = 192858120.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{56840}{564927}$.\nFinal answer: $\\frac{56840}{564927}$", "final_answer": "\\frac{56840}{564927}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e35662aed3501c5b", "problem_id": "b82e99176332d679", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} x \\left(- x - 2\\right)$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} - 3 x.\n3) Then the limit is 2.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: 1.\nFinal answer: $1$", "final_answer": "1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "fd5e079f8eaaa7fb", "problem_id": "c447e87aed5d3644", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 50 miles per hour for 1/2 hours, then at 78 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 50 × 1/2 = 25 miles.\nSecond leg distance = 78 × 5/2 = 195 miles.\nTotal distance = 25 + 195 = 220 miles.\nCheck: adding both legs gives 220.\nFinal answer: $220$", "final_answer": "220", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "461e04bb007bbadb", "problem_id": "c447e87aed5d3644", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A cyclist rides at 50 miles per hour for 1/2 hours, then at 78 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 50 × 1/2 = 25 miles.\n2) Second leg distance = 78 × 5/2 = 195 miles.\n3) Total distance = 25 + 195 = 220 miles.\nFinal answer: $220$", "final_answer": "220", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ec815b40cc7c9f4", "problem_id": "535cc338788e531e", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A store sells a backpack for $278. It is discounted by 25%. You buy 5 backpacks. What is the total cost?", "output": "1) Discounted price = 278 × (1 − 25/100) = 278 × 75/100 = 417/2.\n2) Subtotal = 5 × 417/2 = 2085/2.\nFinal answer: $\\frac{2085}{2}$", "final_answer": "\\frac{2085}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "92fce90bb2fd844d", "problem_id": "535cc338788e531e", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $278. It is discounted by 25%. You buy 5 backpacks. What is the total cost?", "output": "Discounted price = 278 × (1 − 25/100) = 278 × 75/100 = 417/2.\nSubtotal = 5 × 417/2 = 2085/2.\nCheck: recomputing the formula yields 2085/2.\nFinal answer: $\\frac{2085}{2}$", "final_answer": "\\frac{2085}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "424ec4cc4aff6afe", "problem_id": "be45d58217b76352", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_6=20$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_6=14.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=14 and k=6.\nFinal answer: $11628$", "final_answer": "11628", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d3add167efea2f5a", "problem_id": "be45d58217b76352", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_6=20$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_6=14.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=14 and k=6.\n4) Compute C(19,5) = 11628.\nFinal answer: $11628$", "final_answer": "11628", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e26db69a67e42272", "problem_id": "3a4036b8bcf4be27", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the multiplicative inverse of 7 modulo 23.", "output": "We want x such that 7x ≡ 1 (mod 23).\nUse extended Euclid to find s,t with 7s + 23t = 1.\nOne Bezout identity is: 7·(10) + 23·(-3) = 1.\nReduce s modulo 23: x ≡ 10 (mod 23).\nCheck: 7·10 mod 23 = 1.\nFinal answer: $x\\equiv 10\\pmod{23}$", "final_answer": "x\\equiv 10\\pmod{23}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7a05a26cb2c67543", "problem_id": "3a4036b8bcf4be27", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the multiplicative inverse of 7 modulo 23.", "output": "1) We want x such that 7x ≡ 1 (mod 23).\n2) Use extended Euclid to find s,t with 7s + 23t = 1.\n3) One Bezout identity is: 7·(10) + 23·(-3) = 1.\n4) Reduce s modulo 23: x ≡ 10 (mod 23).\nFinal answer: $x\\equiv 10\\pmod{23}$", "final_answer": "x\\equiv 10\\pmod{23}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "67813c3a8ada2f67", "problem_id": "fa06e5968b2870a6", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} 3 x^{3} + 2 x^{2} + 4 x + 4$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} + 2 x^{2} + 3 x + 4.\n3) Then the limit is 12.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: 13.\nFinal answer: $13$", "final_answer": "13", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "24797096be66b8a3", "problem_id": "f73c3e4425fea5ae", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 8 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=8, k=6, p=$\\frac{3}{8}$ and simplify.\n4) Here C(8,6) = 28.\nFinal answer: $\\frac{127575}{4194304}$", "final_answer": "\\frac{127575}{4194304}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "70b0876285979135", "problem_id": "77b0470762c3dcdd", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 12 times, what is the probability of exactly 6 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=12, k=6, p=$\\frac{1}{4}$ and simplify.\nHere C(12,6) = 924.\nCheck: expression simplifies to $\\frac{168399}{4194304}$.\nFinal answer: $\\frac{168399}{4194304}$", "final_answer": "\\frac{168399}{4194304}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cd88b54407c9ba96", "problem_id": "77b0470762c3dcdd", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 12 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=6, p=$\\frac{1}{4}$ and simplify.\n4) Here C(12,6) = 924.\nFinal answer: $\\frac{168399}{4194304}$", "final_answer": "\\frac{168399}{4194304}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2198df62ac4cca36", "problem_id": "3da69ede0c77730d", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} 4 x^{3} + x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $4 x^{3} + x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\n2) Evaluate at the bounds: F(2 \\pi) and F(0).\n3) Subtract: F(2 \\pi) − F(0) = $\\frac{2 \\pi \\left(3 + 9 \\pi + 4 \\pi^{2} + 24 \\pi^{3}\\right)}{3}$.\n4) Simplify the result.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(3 + 9\\cdot \\pi + 4\\cdot \\pi^{2} + 24\\cdot \\pi^{3}\\right)}{3}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(3 + 9\\cdot \\pi + 4\\cdot \\pi^{2} + 24\\cdot \\pi^{3}\\right)}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ee2aeb9df91858b5", "problem_id": "3da69ede0c77730d", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} 4 x^{3} + x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $4 x^{3} + x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\nEvaluate at the bounds: F(2 \\pi) and F(0).\nSubtract: F(2 \\pi) − F(0) = $\\frac{2 \\pi \\left(3 + 9 \\pi + 4 \\pi^{2} + 24 \\pi^{3}\\right)}{3}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(3 + 9\\cdot \\pi + 4\\cdot \\pi^{2} + 24\\cdot \\pi^{3}\\right)}{3}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(3 + 9\\cdot \\pi + 4\\cdot \\pi^{2} + 24\\cdot \\pi^{3}\\right)}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7dfa4d79f76c7512", "problem_id": "1f2071683d1842d5", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Two friends split $1635 in the ratio 9:6. How much does each receive?", "output": "Total ratio parts = 9+6 = 15.\nEach part = 1635 ÷ 15 = 109.\nFirst share = 9 × 109 = 981.\nSecond share = 6 × 109 = 654.\nCheck: 981+654=1635.\nFinal answer: $654$", "final_answer": "654", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3b08038e1eccb73d", "problem_id": "2f460b9885f27eca", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x^{2} - 4 x - 5$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} - 5 x - 5.\n3) Then the limit is -19.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -17.\nFinal answer: $-17$", "final_answer": "-17", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "cefdff1d2f995ede", "problem_id": "3027ee948e087fa6", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Find the multiplicative inverse of 11 modulo 29.", "output": "We want x such that 11x ≡ 1 (mod 29).\nUse extended Euclid to find s,t with 11s + 29t = 1.\nOne Bezout identity is: 11·(8) + 29·(-3) = 1.\nFinal answer: $x\\equiv 8\\pmod{29}$", "final_answer": "x\\equiv 8\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db2f8c8cfc541cb9", "problem_id": "3027ee948e087fa6", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the multiplicative inverse of 11 modulo 29.", "output": "1) We want x such that 11x ≡ 1 (mod 29).\n2) Use extended Euclid to find s,t with 11s + 29t = 1.\n3) One Bezout identity is: 11·(8) + 29·(-3) = 1.\n4) Reduce s modulo 29: x ≡ 8 (mod 29).\nFinal answer: $x\\equiv 8\\pmod{29}$", "final_answer": "x\\equiv 8\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "29f5add771ce6eec", "problem_id": "58975892db061809", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A store sells a backpack for $475. It is discounted by 15%. You buy 6 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 475 × (1 − 15/100) = 475 × 85/100 = 25*2**(314/427)*3**(213/427)*5**(142/427)*7**(59/61)/2.\n2) Subtotal = 6 × 25*2**(314/427)*3**(213/427)*5**(142/427)*7**(59/61)/2 = 75*2**(314/427)*3**(213/427)*5**(142/427)*7**(59/61).\n3) Total with tax = 75*2**(314/427)*3**(213/427)*5**(142/427)*7**(59/61) × 107/100 = 321*2**(314/427)*3**(213/427)*5**(142/427)*7**(59/61)/4.\nFinal answer: $\\frac{321\\cdot 2^{\\frac{314}{427}}\\cdot 3^{\\frac{213}{427}}\\cdot 5^{\\frac{142}{427}}\\cdot 7^{\\frac{59}{61}}}{4}$", "final_answer": "\\frac{321\\cdot 2^{\\frac{314}{427}}\\cdot 3^{\\frac{213}{427}}\\cdot 5^{\\frac{142}{427}}\\cdot 7^{\\frac{59}{61}}}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "518ff050ed3b91d5", "problem_id": "58975892db061809", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $475. It is discounted by 15%. You buy 6 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 475 × (1 − 15/100) = 475 × 85/100 = 25*2**(314/427)*3**(213/427)*5**(142/427)*7**(59/61)/2.\nSubtotal = 6 × 25*2**(314/427)*3**(213/427)*5**(142/427)*7**(59/61)/2 = 75*2**(314/427)*3**(213/427)*5**(142/427)*7**(59/61).\nTotal with tax = 75*2**(314/427)*3**(213/427)*5**(142/427)*7**(59/61) × 107/100 = 321*2**(314/427)*3**(213/427)*5**(142/427)*7**(59/61)/4.\nCheck: recomputing the formula yields 321*2**(314/427)*3**(213/427)*5**(142/427)*7**(59/61)/4.\nFinal answer: $\\frac{321\\cdot 2^{\\frac{314}{427}}\\cdot 3^{\\frac{213}{427}}\\cdot 5^{\\frac{142}{427}}\\cdot 7^{\\frac{59}{61}}}{4}$", "final_answer": "\\frac{321\\cdot 2^{\\frac{314}{427}}\\cdot 3^{\\frac{213}{427}}\\cdot 5^{\\frac{142}{427}}\\cdot 7^{\\frac{59}{61}}}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "27c3cea1e817df1a", "problem_id": "b32eb69975497d53", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A cyclist rides at 79 miles per hour for 1/2 hours, then at 24 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 79 × 1/2 = 79/2 miles.\n2) Second leg distance = 24 × 2 = 48 miles.\n3) Total distance = 79/2 + 48 = 175/2 miles.\nFinal answer: $\\frac{175}{2}$", "final_answer": "\\frac{175}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7c2a201cc7b67f93", "problem_id": "b32eb69975497d53", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A cyclist rides at 79 miles per hour for 1/2 hours, then at 24 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 79 × 1/2 = 79/2 miles.\nSecond leg distance = 24 × 2 = 48 miles.\nTotal distance = 79/2 + 48 = 175/2 miles.\nFinal answer: $\\frac{175}{2}$", "final_answer": "\\frac{175}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db7bd554b58d0580", "problem_id": "b124e2268a2f0ea2", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$5 x^{2} - 5 \\sqrt{6} x + 8 x - 8 \\sqrt{6}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$5$, b=$8 - 5 \\sqrt{6}$, c=$- 8 \\sqrt{6}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $80 \\sqrt{6} + 214$, so √Δ = $\\sqrt{80 \\sqrt{6} + 214}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{4}{5} + \\frac{\\sqrt{6}}{2} + \\frac{\\sqrt{80 \\sqrt{6} + 214}}{10}$.\n7) x2 = $- \\frac{\\sqrt{80 \\sqrt{6} + 214}}{10} - \\frac{4}{5} + \\frac{\\sqrt{6}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{4}{5} + \\frac{\\sqrt{6}}{2} + \\frac{\\sqrt{80\\cdot \\sqrt{6} + 214}}{10}, - \\frac{\\sqrt{80\\cdot \\sqrt{6} + 214}}{10} - \\frac{4}{5} + \\frac{\\sqrt{6}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{4}{5} + \\frac{\\sqrt{6}}{2} + \\frac{\\sqrt{80\\cdot \\sqrt{6} + 214}}{10}, - \\frac{\\sqrt{80\\cdot \\sqrt{6} + 214}}{10} - \\frac{4}{5} + \\frac{\\sqrt{6}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7df47fc4418e4cde", "problem_id": "b124e2268a2f0ea2", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$5 x^{2} - 5 \\sqrt{6} x + 8 x - 8 \\sqrt{6}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$5$, b=$8 - 5 \\sqrt{6}$, c=$- 8 \\sqrt{6}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $80 \\sqrt{6} + 214$, so √Δ = $\\sqrt{80 \\sqrt{6} + 214}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{4}{5} + \\frac{\\sqrt{6}}{2} + \\frac{\\sqrt{80 \\sqrt{6} + 214}}{10}$.\nx2 = $- \\frac{\\sqrt{80 \\sqrt{6} + 214}}{10} - \\frac{4}{5} + \\frac{\\sqrt{6}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{4}{5} + \\frac{\\sqrt{6}}{2} + \\frac{\\sqrt{80\\cdot \\sqrt{6} + 214}}{10}, - \\frac{\\sqrt{80\\cdot \\sqrt{6} + 214}}{10} - \\frac{4}{5} + \\frac{\\sqrt{6}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{4}{5} + \\frac{\\sqrt{6}}{2} + \\frac{\\sqrt{80\\cdot \\sqrt{6} + 214}}{10}, - \\frac{\\sqrt{80\\cdot \\sqrt{6} + 214}}{10} - \\frac{4}{5} + \\frac{\\sqrt{6}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ee1a37708ddf4426", "problem_id": "edfa031a837f901d", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} - 3 x^{2} + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 3$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $6 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)}$.\n2) Slope at x=-1: m = f'(-1) = $\\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)} + 12$.\n3) Point on curve: (-1, f(-1)) = (-1, $-8 + \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -8 + \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} = \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)} + 12(x − -1).\n6) Optionally expand: y = $\\frac{x}{2 e^{\\frac{1}{2}}} + 2 x \\sin{\\left(2 \\right)} + 12 x + \\cos{\\left(2 \\right)} + \\frac{3}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)} + 4$.\nFinal answer: $y = \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + 12\\cdot x + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)} + 4$", "final_answer": "y = \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + 12\\cdot x + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)} + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ce459ab9ed17d311", "problem_id": "edfa031a837f901d", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} - 3 x^{2} + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 3$ at $x=-1$.", "output": "Differentiate: f'(x) = $6 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)}$.\nSlope at x=-1: m = f'(-1) = $\\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)} + 12$.\nPoint on curve: (-1, f(-1)) = (-1, $-8 + \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -8 + \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} = \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)} + 12(x − -1).\nOptionally expand: y = $\\frac{x}{2 e^{\\frac{1}{2}}} + 2 x \\sin{\\left(2 \\right)} + 12 x + \\cos{\\left(2 \\right)} + \\frac{3}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)} + 4$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + 12\\cdot x + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)} + 4$", "final_answer": "y = \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + 12\\cdot x + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)} + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "daee4952f4f77faf", "problem_id": "a9964e54b88eefae", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "An urn has 15 red balls and 8 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(23,7).\n3) Favorable samples: choose k reds and n−k blues: C(15,2)·C(8,5).\n4) Compute totals: C(23,7) = 245157, and favorable = 5880.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1960}{81719}$", "final_answer": "\\frac{1960}{81719}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bb18cf92a6d47aa0", "problem_id": "a9964e54b88eefae", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 15 red balls and 8 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(23,7).\nFavorable samples: choose k reds and n−k blues: C(15,2)·C(8,5).\nCompute totals: C(23,7) = 245157, and favorable = 5880.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1960}{81719}$.\nFinal answer: $\\frac{1960}{81719}$", "final_answer": "\\frac{1960}{81719}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5bb78de9ee1fca57", "problem_id": "efc39c33de7d413c", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡3 (mod 11), x≡0 (mod 7). Give the solution modulo 77.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 77.\n3) Compute the CRT solution to get x ≡ 14 (mod 77).\nFinal answer: $x\\equiv 14\\pmod{77}$", "final_answer": "x\\equiv 14\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c9a7ec7bc888ad34", "problem_id": "efc39c33de7d413c", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡3 (mod 11), x≡0 (mod 7). Give the solution modulo 77.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 77.\nCompute the CRT solution to get x ≡ 14 (mod 77).\nCheck: 14 mod 11=3 and mod 7=0.\nFinal answer: $x\\equiv 14\\pmod{77}$", "final_answer": "x\\equiv 14\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fc1717d20ceb6a58", "problem_id": "830337ad7d8e10e7", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{39 x}{5} - 2 \\sqrt{11} x + \\frac{54}{5} + \\frac{18 \\sqrt{11}}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{39}{5} - 2 \\sqrt{11}$, c=$\\frac{54}{5} + \\frac{18 \\sqrt{11}}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{84 \\sqrt{11}}{5} + \\frac{1541}{25}$, so √Δ = $\\sqrt{\\frac{84 \\sqrt{11}}{5} + \\frac{1541}{25}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\sqrt{11} + \\frac{39}{10} + \\frac{\\sqrt{420 \\sqrt{11} + 1541}}{10}$.\nx2 = $- \\frac{\\sqrt{420 \\sqrt{11} + 1541}}{10} + \\sqrt{11} + \\frac{39}{10}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\sqrt{11} + \\frac{39}{10} + \\frac{\\sqrt{420\\cdot \\sqrt{11} + 1541}}{10}, - \\frac{\\sqrt{420\\cdot \\sqrt{11} + 1541}}{10} + \\sqrt{11} + \\frac{39}{10}\\}$", "final_answer": "x \\in \\{\\sqrt{11} + \\frac{39}{10} + \\frac{\\sqrt{420\\cdot \\sqrt{11} + 1541}}{10}, - \\frac{\\sqrt{420\\cdot \\sqrt{11} + 1541}}{10} + \\sqrt{11} + \\frac{39}{10}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5764458bad100962", "problem_id": "830337ad7d8e10e7", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{39 x}{5} - 2 \\sqrt{11} x + \\frac{54}{5} + \\frac{18 \\sqrt{11}}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{39}{5} - 2 \\sqrt{11}$, c=$\\frac{54}{5} + \\frac{18 \\sqrt{11}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{84 \\sqrt{11}}{5} + \\frac{1541}{25}$, so √Δ = $\\sqrt{\\frac{84 \\sqrt{11}}{5} + \\frac{1541}{25}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\sqrt{11} + \\frac{39}{10} + \\frac{\\sqrt{420 \\sqrt{11} + 1541}}{10}$.\n7) x2 = $- \\frac{\\sqrt{420 \\sqrt{11} + 1541}}{10} + \\sqrt{11} + \\frac{39}{10}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\sqrt{11} + \\frac{39}{10} + \\frac{\\sqrt{420\\cdot \\sqrt{11} + 1541}}{10}, - \\frac{\\sqrt{420\\cdot \\sqrt{11} + 1541}}{10} + \\sqrt{11} + \\frac{39}{10}\\}$", "final_answer": "x \\in \\{\\sqrt{11} + \\frac{39}{10} + \\frac{\\sqrt{420\\cdot \\sqrt{11} + 1541}}{10}, - \\frac{\\sqrt{420\\cdot \\sqrt{11} + 1541}}{10} + \\sqrt{11} + \\frac{39}{10}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "160cf1a111635c43", "problem_id": "874556957a732e33", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 11 times, what is the probability of exactly 8 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=8, p=$\\frac{2}{7}$ and simplify.\n4) Here C(11,8) = 165.\nFinal answer: $\\frac{5280000}{1977326743}$", "final_answer": "\\frac{5280000}{1977326743}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9dda61b84f98bd0a", "problem_id": "874556957a732e33", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 11 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=11, k=8, p=$\\frac{2}{7}$ and simplify.\nHere C(11,8) = 165.\nCheck: expression simplifies to $\\frac{5280000}{1977326743}$.\nFinal answer: $\\frac{5280000}{1977326743}$", "final_answer": "\\frac{5280000}{1977326743}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "65e9091e5380d273", "problem_id": "e5762cbcfe5c9410", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A store sells a backpack for $900. It is discounted by 10%. You buy 4 backpacks. What is the total cost?", "output": "1) Discounted price = 900 × (1 − 10/100) = 900 × 90/100 = 810.\n2) Subtotal = 4 × 810 = 3240.\nFinal answer: $3240$", "final_answer": "3240", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2b2b70bdbec6ed1c", "problem_id": "e5762cbcfe5c9410", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A store sells a backpack for $900. It is discounted by 10%. You buy 4 backpacks. What is the total cost?", "output": "Discounted price = 900 × (1 − 10/100) = 900 × 90/100 = 810.\nSubtotal = 4 × 810 = 3240.\nCheck: recomputing the formula yields 3240.\nFinal answer: $3240$", "final_answer": "3240", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c8897921aec92d56", "problem_id": "fc70afa8b307dbe2", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 22 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=22, k=3, p=$\\frac{1}{2}$ and simplify.\n4) Here C(22,3) = 1540.\nFinal answer: $\\frac{385}{1048576}$", "final_answer": "\\frac{385}{1048576}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "80868c0e6898db00", "problem_id": "fc70afa8b307dbe2", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 22 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=22, k=3, p=$\\frac{1}{2}$ and simplify.\nFinal answer: $\\frac{385}{1048576}$", "final_answer": "\\frac{385}{1048576}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dd6ae470cf32be4b", "problem_id": "c2077e609ca6b192", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$2 x - 6 y$ = $-22$\n$- 7 x - 4 y$ = $2$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-7$: $- 14 x + 42 y$ = $154$.\nMultiply the second equation by $2$: $- 14 x - 8 y$ = $4$.\nSubtract to eliminate x: $50 y$ = $150$.\nSolve for y: y = $3$.\nSubstitute back to find x: x = $-2$.\nCheck: plug (x,y)=(-2,3) into both equations.\nFinal answer: $(x,y)=\\left(-2, 3\\right)$", "final_answer": "(x,y)=\\left(-2, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "80f0abed4078e6b5", "problem_id": "c2077e609ca6b192", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$2 x - 6 y$ = $-22$\n$- 7 x - 4 y$ = $2$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-7$: $- 14 x + 42 y$ = $154$.\n3) Multiply the second equation by $2$: $- 14 x - 8 y$ = $4$.\n4) Subtract to eliminate x: $50 y$ = $150$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, 3\\right)$", "final_answer": "(x,y)=\\left(-2, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9d966d23eeeaf988", "problem_id": "f018355b7ee08b26", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 3 x^{2} - 1$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 3 x^{3} - 3 x^{2} - x$ and a = -1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 9 x^{2} - 6 x - 1$.\n5) Evaluate at a: p'(-1) = $-4$.\nFinal answer: $-4$", "final_answer": "-4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d3605ee2d8bdef99", "problem_id": "f018355b7ee08b26", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 3 x^{2} - 1$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 3 x^{3} - 3 x^{2} - x$ and a = -1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 9 x^{2} - 6 x - 1$.\nEvaluate at a: p'(-1) = $-4$.\nCheck: the identity gives $-4$.\nFinal answer: $-4$", "final_answer": "-4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "01f698ebc710de17", "problem_id": "071fad801159a5a8", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} x^{3} - x^{2} - 2 x - 5$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{4} - 2 x^{3} - x^{2} - 3 x + 1$ and a = 1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $4 x^{3} - 6 x^{2} - 2 x - 3$.\n5) Evaluate at a: p'(1) = $-7$.\nFinal answer: $-7$", "final_answer": "-7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "287ba159dad25a7b", "problem_id": "071fad801159a5a8", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} x^{3} - x^{2} - 2 x - 5$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $x^{4} - 2 x^{3} - x^{2} - 3 x + 1$ and a = 1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-7$", "final_answer": "-7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6051b681bcc02fbc", "problem_id": "021d7a605794ddd7", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "An urn has 6 red balls and 6 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(12,7).\n3) Favorable samples: choose k reds and n−k blues: C(6,1)·C(6,6).\n4) Compute totals: C(12,7) = 792, and favorable = 6.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1}{132}$", "final_answer": "\\frac{1}{132}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b68b32bea86c4a7a", "problem_id": "021d7a605794ddd7", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 6 red balls and 6 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(12,7).\nFavorable samples: choose k reds and n−k blues: C(6,1)·C(6,6).\nCompute totals: C(12,7) = 792, and favorable = 6.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1}{132}$.\nFinal answer: $\\frac{1}{132}$", "final_answer": "\\frac{1}{132}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "82ff0c786db884b6", "problem_id": "77bf007d2ad0bb7e", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} x - 3$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{2}$ and a = -3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $2 x$.\n5) Evaluate at a: p'(-3) = $-6$.\nFinal answer: $-6$", "final_answer": "-6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cd77aed91bda61b8", "problem_id": "77bf007d2ad0bb7e", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} x - 3$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $x^{2}$ and a = -3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $2 x$.\nEvaluate at a: p'(-3) = $-6$.\nCheck: the identity gives $-6$.\nFinal answer: $-6$", "final_answer": "-6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a02d19bfe4817a7", "problem_id": "46163e9d9c512159", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 2 x - 5$ = 0.", "output": "$x \\in \\{\\frac{5}{3}, -1\\}$", "final_answer": "x \\in \\{\\frac{5}{3}, -1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "52e38dfd38c62e28", "problem_id": "46163e9d9c512159", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 2 x - 5$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$-2$, c=$-5$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $64$, so √Δ = $8$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{5}{3}$.\nx2 = $-1$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{5}{3}, -1\\}$", "final_answer": "x \\in \\{\\frac{5}{3}, -1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "da54d7450a7e1c48", "problem_id": "5eb51c11e2d8f01f", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "A store sells a backpack for $220. It is discounted by 35%. You buy 7 backpacks. What is the total cost?", "output": "Discounted price = 220 × (1 − 35/100) = 220 × 65/100 = 143.\nSubtotal = 7 × 143 = 1001.\nFinal answer: $1001$", "final_answer": "1001", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "89264a761ec6ba98", "problem_id": "5eb51c11e2d8f01f", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A store sells a backpack for $220. It is discounted by 35%. You buy 7 backpacks. What is the total cost?", "output": "1) Discounted price = 220 × (1 − 35/100) = 220 × 65/100 = 143.\n2) Subtotal = 7 × 143 = 1001.\nFinal answer: $1001$", "final_answer": "1001", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "62a3080de1425ffa", "problem_id": "c482bc443e59cdea", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)}$ at $x=1$.", "output": "1) Differentiate: f'(x) = $- 9 x^{2} + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 1$.\n2) Slope at x=1: m = f'(1) = $-10 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}$.\n3) Point on curve: (1, f(1)) = (1, $-4 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -4 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} = -10 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}(x − 1).\n6) Optionally expand: y = $- 10 x - 2 x \\sin{\\left(2 \\right)} + \\frac{x e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2 \\sin{\\left(2 \\right)} + 6$.\nFinal answer: $y = - 10\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 6$", "final_answer": "y = - 10\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "76b4524977f5256a", "problem_id": "30d6cf9a97495d00", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{x^{2} + 3}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 3}) = \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} - \\frac{2 x^{4}}{2 x^{4} + 12 x^{2} + 18} + \\frac{6 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} + \\frac{12 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} - \\frac{12 x^{2}}{2 x^{4} + 12 x^{2} + 18} + \\frac{36 x}{2 x^{4} + 12 x^{2} + 18} + \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 6 x^{2} + 9} + \\frac{9 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} + \\frac{18 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} - \\frac{18}{2 x^{4} + 12 x^{2} + 18} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 6} - \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3} + \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 3}) = - \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3\\right)^{2}} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x^{2} + 3\\right)} + \\frac{3 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 4 x \\left(- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} + 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x^{2} + 3\\right)^{2}}$.\nFinal answer: $\\frac{- 4 x \\left(- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} + 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x^{2} + 3\\right)^{2}}$", "final_answer": "\\frac{- 4 x \\left(- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} + 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x^{2} + 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0b5ce5ce80937896", "problem_id": "23eafac488bc3cdd", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} x - 2$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to -2.\n3) Then the limit is -2.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -4.\nFinal answer: $-4$", "final_answer": "-4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e7a6135960852e1c", "problem_id": "adfe202ac964bfdc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{4} + 3 x^{3} + 3 x^{2} - 3 x + 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(3 x^{3}) = 9 x^{3}$.\n3) So $f'(x) = 25 x^{3} + 6 x - 3$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(3 x^{3}) = 9 x^{2}$.\nTherefore the correct derivative is $f'(x) = 16 x^{3} + 9 x^{2} + 6 x - 3$.\nFinal answer: $16 x^{3} + 9 x^{2} + 6 x - 3$", "final_answer": "16 x^{3} + 9 x^{2} + 6 x - 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7bd67ef53deca889", "problem_id": "a2e7696c6baa1e0a", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_9=24$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=24 and k=9.\n4) Compute C(32,8) = 10518300.\nFinal answer: $10518300$", "final_answer": "10518300", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "60778ab9e92737d7", "problem_id": "a2e7696c6baa1e0a", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_9=24$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=24 and k=9.\nCompute C(32,8) = 10518300.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $10518300$", "final_answer": "10518300", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "314813cfcc004a74", "problem_id": "dc02ee532430bfb0", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A cyclist rides at 71 miles per hour for 3/2 hours, then at 81 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 71 × 3/2 = 213/2 miles.\n2) Second leg distance = 81 × 1 = 81 miles.\n3) Total distance = 213/2 + 81 = 375/2 miles.\nFinal answer: $\\frac{375}{2}$", "final_answer": "\\frac{375}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "212ded3f3fd93f4f", "problem_id": "dc02ee532430bfb0", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "A cyclist rides at 71 miles per hour for 3/2 hours, then at 81 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 71 × 3/2 = 213/2 miles.\nSecond leg distance = 81 × 1 = 81 miles.\nTotal distance = 213/2 + 81 = 375/2 miles.\nFinal answer: $\\frac{375}{2}$", "final_answer": "\\frac{375}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "63fc2ebcae601ee1", "problem_id": "d3f8219d7c4ab7e0", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the linear system:\n$3 x - 6 y$ = $15$\n$- 4 x + 7 y$ = $-13$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $- 12 x + 24 y$ = $-60$.\n3) Multiply the second equation by $3$: $- 12 x + 21 y$ = $-39$.\n4) Subtract to eliminate x: $3 y$ = $-21$.\n5) Solve for y: y = $-7$.\n6) Substitute back to find x: x = $-9$.\nFinal answer: $(x,y)=\\left(-9, -7\\right)$", "final_answer": "(x,y)=\\left(-9, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d97153bbebe14d48", "problem_id": "d3f8219d7c4ab7e0", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$3 x - 6 y$ = $15$\n$- 4 x + 7 y$ = $-13$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $- 12 x + 24 y$ = $-60$.\nMultiply the second equation by $3$: $- 12 x + 21 y$ = $-39$.\nSubtract to eliminate x: $3 y$ = $-21$.\nSolve for y: y = $-7$.\nSubstitute back to find x: x = $-9$.\nCheck: plug (x,y)=(-9,-7) into both equations.\nFinal answer: $(x,y)=\\left(-9, -7\\right)$", "final_answer": "(x,y)=\\left(-9, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c11931cdae90cebf", "problem_id": "c2aa2ae946088360", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_7=28$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_7=14.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=14 and k=7.\n4) Compute C(20,6) = 38760.\nFinal answer: $38760$", "final_answer": "38760", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dc436c3f8e3c878e", "problem_id": "c2aa2ae946088360", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_7=28$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_7=14.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=14 and k=7.\nFinal answer: $38760$", "final_answer": "38760", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d84ff5226a228267", "problem_id": "c5cb5933d91cebfb", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_5=31$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_5=16.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=16 and k=5.\n4) Compute C(20,4) = 4845.\nFinal answer: $4845$", "final_answer": "4845", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "475cc78c267d0d72", "problem_id": "9d4cd9c05c99030c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x^{3} + 4 x^{2} + 7 x + 13$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to x^{3} + 4 x^{2} + 6 x + 13.\n3) Then the limit is 49.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 51.\nFinal answer: $51$", "final_answer": "51", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "1af351597e1ea4d1", "problem_id": "faf0c2f0ea36fe2f", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the congruences: x≡1 (mod 5), x≡9 (mod 13). Give the solution modulo 65.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 65.\nCompute the CRT solution to get x ≡ 61 (mod 65).\nFinal answer: $x\\equiv 61\\pmod{65}$", "final_answer": "x\\equiv 61\\pmod{65}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c1426578441f953c", "problem_id": "faf0c2f0ea36fe2f", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡1 (mod 5), x≡9 (mod 13). Give the solution modulo 65.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 65.\n3) Compute the CRT solution to get x ≡ 61 (mod 65).\nFinal answer: $x\\equiv 61\\pmod{65}$", "final_answer": "x\\equiv 61\\pmod{65}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1331f901aa03687f", "problem_id": "3825c6df7bc7f784", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_5=26$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_5=21.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=21 and k=5.\n4) Compute C(25,4) = 12650.\nFinal answer: $12650$", "final_answer": "12650", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "92267ee7298f28e8", "problem_id": "3825c6df7bc7f784", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_5=26$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_5=21.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=21 and k=5.\nCompute C(25,4) = 12650.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $12650$", "final_answer": "12650", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a79d4c4a7045f1b8", "problem_id": "a799b92543b82e7e", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 10 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=6, p=$\\frac{3}{8}$ and simplify.\n4) Here C(10,6) = 210.\nFinal answer: $\\frac{47840625}{536870912}$", "final_answer": "\\frac{47840625}{536870912}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ef0192bb681a076f", "problem_id": "0e94698edb22bdb2", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 2 x - 7$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to x - 7.\n3) Then the limit is -10.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -13.\nFinal answer: $-13$", "final_answer": "-13", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d4dbf494b68e16d4", "problem_id": "9b6ab941bbb2fbda", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Worker A finishes a job in 18 hours and Worker B finishes it in 11 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/18 job/hour. B's rate = 1/11 job/hour.\n2) Combined rate = 1/18 + 1/11 = 29/198 job/hour.\n3) Time = 1 ÷ (combined rate) = 198/29 hours.\nFinal answer: $\\frac{198}{29}$", "final_answer": "\\frac{198}{29}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f4360f7090202ce5", "problem_id": "7a1f0e318e19a3c0", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{4} + x^{3} + 2 x^{2} - 2 x + 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(2 x^{2}) = 4 x^{2}$.\n3) So $f'(x) = - 8 x^{3} + 7 x^{2} - 2$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(2 x^{2}) = 4 x$.\nTherefore the correct derivative is $f'(x) = - 8 x^{3} + 3 x^{2} + 4 x - 2$.\nFinal answer: $- 8 x^{3} + 3 x^{2} + 4 x - 2$", "final_answer": "- 8 x^{3} + 3 x^{2} + 4 x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e3cd4081fc0f87d0", "problem_id": "abfaffc3aaf0c0d8", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡6 (mod 8). Give the solution modulo 24.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 24.\n3) Compute the CRT solution to get x ≡ 22 (mod 24).\nFinal answer: $x\\equiv 22\\pmod{24}$", "final_answer": "x\\equiv 22\\pmod{24}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0570b5badd6cceda", "problem_id": "abfaffc3aaf0c0d8", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡6 (mod 8). Give the solution modulo 24.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 24.\nCompute the CRT solution to get x ≡ 22 (mod 24).\nCheck: 22 mod 3=1 and mod 8=6.\nFinal answer: $x\\equiv 22\\pmod{24}$", "final_answer": "x\\equiv 22\\pmod{24}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7b1120d2b51f0d0b", "problem_id": "bc42c927b289c24f", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡1 (mod 5), x≡7 (mod 11). Give the solution modulo 55.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 55.\n3) Compute the CRT solution to get x ≡ 51 (mod 55).\nFinal answer: $x\\equiv 51\\pmod{55}$", "final_answer": "x\\equiv 51\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7fa5d0b4695f93f8", "problem_id": "e4305ec474fba739", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\cos{\\left(2 x \\right)}) = - \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\cos{\\left(2 x \\right)}) = - 2 \\sin{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.\nFinal answer: $x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$", "final_answer": "x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "bb5250ba5da37a02", "problem_id": "422e48f31a3df5ba", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A cyclist rides at 44 miles per hour for 1/2 hours, then at 89 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 44 × 1/2 = 22 miles.\n2) Second leg distance = 89 × 3/2 = 267/2 miles.\n3) Total distance = 22 + 267/2 = 311/2 miles.\nFinal answer: $\\frac{311}{2}$", "final_answer": "\\frac{311}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "80e360f907ef0930", "problem_id": "d076fb0de7a8a9bc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(x^{2} + x - 3\\right) \\cos{\\left(3 x - 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x e^{x}) = x e^{x}$.\n3) So $f'(x) = - 3 x^{2} \\sin{\\left(3 x - 3 \\right)} + x e^{x} - 3 x \\sin{\\left(3 x - 3 \\right)} + 2 x \\cos{\\left(3 x - 3 \\right)} - 3 e^{x} + 9 \\sin{\\left(3 x - 3 \\right)} + \\cos{\\left(3 x - 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x e^{x}) = x e^{x} + e^{x}$.\nTherefore the correct derivative is $f'(x) = \\left(x - 3\\right) e^{x} + \\left(2 x + 1\\right) \\cos{\\left(3 x - 3 \\right)} - 3 \\left(x \\left(x + 1\\right) - 3\\right) \\sin{\\left(3 x - 3 \\right)} + e^{x}$.\nFinal answer: $\\left(x - 3\\right) e^{x} + \\left(2 x + 1\\right) \\cos{\\left(3 x - 3 \\right)} - 3 \\left(x \\left(x + 1\\right) - 3\\right) \\sin{\\left(3 x - 3 \\right)} + e^{x}$", "final_answer": "\\left(x - 3\\right) e^{x} + \\left(2 x + 1\\right) \\cos{\\left(3 x - 3 \\right)} - 3 \\left(x \\left(x + 1\\right) - 3\\right) \\sin{\\left(3 x - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b09bacd660b2ccca", "problem_id": "bd4f054ca1975717", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "You want 27 liters of a 30% salt solution. You have a 20% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 27−x liters is 50% solution.\n2) Salt equation: 20x + 50(27−x) = 30·27.\n3) (20-50)x = (30-50)·27 ⇒ x = 18.\n4) Then 27−x = 9.\nFinal answer: $18 L of 20% and 9 L of 50%$", "final_answer": "18 L of 20% and 9 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b28f934a5059ba90", "problem_id": "bd4f054ca1975717", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 27 liters of a 30% salt solution. You have a 20% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 27−x liters is 50% solution.\nSalt equation: 20x + 50(27−x) = 30·27.\n(20-50)x = (30-50)·27 ⇒ x = 18.\nThen 27−x = 9.\nCheck: 20·18 + 50·9 = 30·27.\nFinal answer: $18 L of 20% and 9 L of 50%$", "final_answer": "18 L of 20% and 9 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "00192f93662160d8", "problem_id": "e7873f473aded182", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "You want 51 liters of a 25% salt solution. You have a 10% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 51−x liters is 60% solution.\nSalt equation: 10x + 60(51−x) = 25·51.\n(10-60)x = (25-60)·51 ⇒ x = 357/10.\nThen 51−x = 153/10.\nCheck: 10·357/10 + 60·153/10 = 25·51.\nFinal answer: $357/10 L of 10% and 153/10 L of 60%$", "final_answer": "357/10 L of 10% and 153/10 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2173121accd19aa6", "problem_id": "e7873f473aded182", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "You want 51 liters of a 25% salt solution. You have a 10% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 51−x liters is 60% solution.\n2) Salt equation: 10x + 60(51−x) = 25·51.\n3) (10-60)x = (25-60)·51 ⇒ x = 357/10.\n4) Then 51−x = 153/10.\nFinal answer: $357/10 L of 10% and 153/10 L of 60%$", "final_answer": "357/10 L of 10% and 153/10 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c9834b152a64c8e0", "problem_id": "f3022ef618225512", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$- x - 4 y$ = $5$\n$5 x + 7 y$ = $14$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $5$: $- 5 x - 20 y$ = $25$.\n3) Multiply the second equation by $-1$: $- 5 x - 7 y$ = $-14$.\n4) Subtract to eliminate x: $- 13 y$ = $39$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $7$.\nFinal answer: $(x,y)=\\left(7, -3\\right)$", "final_answer": "(x,y)=\\left(7, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "69d2ddbe6eaa002a", "problem_id": "f3022ef618225512", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the linear system:\n$- x - 4 y$ = $5$\n$5 x + 7 y$ = $14$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $5$: $- 5 x - 20 y$ = $25$.\nMultiply the second equation by $-1$: $- 5 x - 7 y$ = $-14$.\nFinal answer: $(x,y)=\\left(7, -3\\right)$", "final_answer": "(x,y)=\\left(7, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4959e9681acb423b", "problem_id": "8abfd612f2e03612", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "You want 68 liters of a 40% salt solution. You have a 15% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 68−x liters is 60% solution.\nSalt equation: 15x + 60(68−x) = 40·68.\n(15-60)x = (40-60)·68 ⇒ x = 272/9.\nThen 68−x = 340/9.\nCheck: 15·272/9 + 60·340/9 = 40·68.\nFinal answer: $272/9 L of 15% and 340/9 L of 60%$", "final_answer": "272/9 L of 15% and 340/9 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0613dff507602daf", "problem_id": "8abfd612f2e03612", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "You want 68 liters of a 40% salt solution. You have a 15% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 68−x liters is 60% solution.\n2) Salt equation: 15x + 60(68−x) = 40·68.\n3) (15-60)x = (40-60)·68 ⇒ x = 272/9.\n4) Then 68−x = 340/9.\nFinal answer: $272/9 L of 15% and 340/9 L of 60%$", "final_answer": "272/9 L of 15% and 340/9 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a266dbbe616a8c2", "problem_id": "d845e49a5f85fac3", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡0 (mod 7), x≡1 (mod 13). Give the solution modulo 91.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 91.\n3) Compute the CRT solution to get x ≡ 14 (mod 91).\nFinal answer: $x\\equiv 14\\pmod{91}$", "final_answer": "x\\equiv 14\\pmod{91}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cf9a697bebf4926e", "problem_id": "d845e49a5f85fac3", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the congruences: x≡0 (mod 7), x≡1 (mod 13). Give the solution modulo 91.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 91.\nCompute the CRT solution to get x ≡ 14 (mod 91).\nFinal answer: $x\\equiv 14\\pmod{91}$", "final_answer": "x\\equiv 14\\pmod{91}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f325453b237ca70e", "problem_id": "58cd3efd96d88752", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 9 x + 20$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} + 9 x + 20$ = $\\left(x + 4\\right) \\left(x + 5\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $-4$.\nFrom the second factor: x = $-5$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-4, -5\\}$", "final_answer": "x \\in \\{-4, -5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b5c259d54f4bf910", "problem_id": "58cd3efd96d88752", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 9 x + 20$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} + 9 x + 20$ = $\\left(x + 4\\right) \\left(x + 5\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-4$.\n5) From the second factor: x = $-5$.\nFinal answer: $x \\in \\{-4, -5\\}$", "final_answer": "x \\in \\{-4, -5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c8b821ed23853695", "problem_id": "b7362e9f8b91bb7b", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(- 2 x^{3} - x^{2} + 3 x + 2\\right) \\cos{\\left(3 x^{2} - 2 x + 2 \\right)}$.", "output": "Let f(x) = $2 \\left(x - 3\\right) e^{x} + \\left(- 2 x^{3} - x^{2} + 3 x + 2\\right) \\cos{\\left(3 x^{2} - 2 x + 2 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $2 \\left(x - 3\\right) e^{x} - 2 \\left(3 x - 1\\right) \\left(- 2 x^{3} - x^{2} + 3 x + 2\\right) \\sin{\\left(3 x^{2} - 2 x + 2 \\right)} + \\left(- 6 x^{2} - 2 x + 3\\right) \\cos{\\left(3 x^{2} - 2 x + 2 \\right)} + 2 e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\left(2\\cdot x - 6\\right)\\cdot e^{x} - \\left(6\\cdot x - 2\\right)\\cdot \\left(- 2\\cdot x^{3} - x^{2} + 3\\cdot x + 2\\right)\\cdot \\sin{\\left(3\\cdot x^{2} - 2\\cdot x + 2 \\right)} + \\left(- 6\\cdot x^{2} - 2\\cdot x + 3\\right)\\cdot \\cos{\\left(3\\cdot x^{2} - 2\\cdot x + 2 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x - 6\\right)\\cdot e^{x} - \\left(6\\cdot x - 2\\right)\\cdot \\left(- 2\\cdot x^{3} - x^{2} + 3\\cdot x + 2\\right)\\cdot \\sin{\\left(3\\cdot x^{2} - 2\\cdot x + 2 \\right)} + \\left(- 6\\cdot x^{2} - 2\\cdot x + 3\\right)\\cdot \\cos{\\left(3\\cdot x^{2} - 2\\cdot x + 2 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f123a2b0330e3357", "problem_id": "b7362e9f8b91bb7b", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(- 2 x^{3} - x^{2} + 3 x + 2\\right) \\cos{\\left(3 x^{2} - 2 x + 2 \\right)}$.", "output": "1) Let f(x) = $2 \\left(x - 3\\right) e^{x} + \\left(- 2 x^{3} - x^{2} + 3 x + 2\\right) \\cos{\\left(3 x^{2} - 2 x + 2 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $2 \\left(x - 3\\right) e^{x} - 2 \\left(3 x - 1\\right) \\left(- 2 x^{3} - x^{2} + 3 x + 2\\right) \\sin{\\left(3 x^{2} - 2 x + 2 \\right)} + \\left(- 6 x^{2} - 2 x + 3\\right) \\cos{\\left(3 x^{2} - 2 x + 2 \\right)} + 2 e^{x}$.\nFinal answer: $\\left(2\\cdot x - 6\\right)\\cdot e^{x} - \\left(6\\cdot x - 2\\right)\\cdot \\left(- 2\\cdot x^{3} - x^{2} + 3\\cdot x + 2\\right)\\cdot \\sin{\\left(3\\cdot x^{2} - 2\\cdot x + 2 \\right)} + \\left(- 6\\cdot x^{2} - 2\\cdot x + 3\\right)\\cdot \\cos{\\left(3\\cdot x^{2} - 2\\cdot x + 2 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x - 6\\right)\\cdot e^{x} - \\left(6\\cdot x - 2\\right)\\cdot \\left(- 2\\cdot x^{3} - x^{2} + 3\\cdot x + 2\\right)\\cdot \\sin{\\left(3\\cdot x^{2} - 2\\cdot x + 2 \\right)} + \\left(- 6\\cdot x^{2} - 2\\cdot x + 3\\right)\\cdot \\cos{\\left(3\\cdot x^{2} - 2\\cdot x + 2 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "475cc78c267d0d72", "problem_id": "9d4cd9c05c99030c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x^{3} + 4 x^{2} + 7 x + 13$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to x^{3} + 4 x^{2} + 6 x + 13.\n3) Then the limit is 49.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 51.\nFinal answer: $51$", "final_answer": "51", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "857a31b3076ac3a9", "problem_id": "88f061b15e5ad780", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "You want 41 liters of a 45% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 41−x liters is 50% solution.\n2) Salt equation: 25x + 50(41−x) = 45·41.\n3) (25-50)x = (45-50)·41 ⇒ x = 41/5.\n4) Then 41−x = 164/5.\nFinal answer: $41/5 L of 25% and 164/5 L of 50%$", "final_answer": "41/5 L of 25% and 164/5 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "36cc93bd581ce03d", "problem_id": "0579e5ac8785cf33", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$x + 4 y$ = $7$\n$- x - y$ = $-1$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $- x - 4 y$ = $-7$.\n3) Multiply the second equation by $1$: $- x - y$ = $-1$.\n4) Subtract to eliminate x: $- 3 y$ = $-6$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, 2\\right)$", "final_answer": "(x,y)=\\left(-1, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a7d33026aa8c52b0", "problem_id": "0579e5ac8785cf33", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the linear system:\n$x + 4 y$ = $7$\n$- x - y$ = $-1$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $- x - 4 y$ = $-7$.\nMultiply the second equation by $1$: $- x - y$ = $-1$.\nFinal answer: $(x,y)=\\left(-1, 2\\right)$", "final_answer": "(x,y)=\\left(-1, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eabc95159469a72c", "problem_id": "746cebebf7c4d769", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Two friends split $436 in the ratio 1:3. How much does each receive?", "output": "1) Total ratio parts = 1+3 = 4.\n2) Each part = 436 ÷ 4 = 109.\n3) First share = 1 × 109 = 109.\n4) Second share = 3 × 109 = 327.\nFinal answer: $327$", "final_answer": "327", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6d405ecac9571f4c", "problem_id": "746cebebf7c4d769", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $436 in the ratio 1:3. How much does each receive?", "output": "Total ratio parts = 1+3 = 4.\nEach part = 436 ÷ 4 = 109.\nFirst share = 1 × 109 = 109.\nSecond share = 3 × 109 = 327.\nCheck: 109+327=436.\nFinal answer: $327$", "final_answer": "327", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3a57a6eb22d5fc30", "problem_id": "850cbb69fde8d4d5", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Two friends split $220 in the ratio 3:8. How much does each receive?", "output": "1) Total ratio parts = 3+8 = 11.\n2) Each part = 220 ÷ 11 = 20.\n3) First share = 3 × 20 = 60.\n4) Second share = 8 × 20 = 160.\nFinal answer: $160$", "final_answer": "160", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d9a99b92097002d", "problem_id": "850cbb69fde8d4d5", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Two friends split $220 in the ratio 3:8. How much does each receive?", "output": "Total ratio parts = 3+8 = 11.\nEach part = 220 ÷ 11 = 20.\nFirst share = 3 × 20 = 60.\nSecond share = 8 × 20 = 160.\nCheck: 60+160=220.\nFinal answer: $160$", "final_answer": "160", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ce5ce0ccc129f672", "problem_id": "b3ad8c1e464c33c5", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 16 red balls and 12 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(28,7).\nFavorable samples: choose k reds and n−k blues: C(16,3)·C(12,4).\nCompute totals: C(28,7) = 1184040, and favorable = 277200.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{70}{299}$.\nFinal answer: $\\frac{70}{299}$", "final_answer": "\\frac{70}{299}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b8942a3fc7f089e0", "problem_id": "b3ad8c1e464c33c5", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "An urn has 16 red balls and 12 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(28,7).\n3) Favorable samples: choose k reds and n−k blues: C(16,3)·C(12,4).\n4) Compute totals: C(28,7) = 1184040, and favorable = 277200.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{70}{299}$", "final_answer": "\\frac{70}{299}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d58ac762c2252486", "problem_id": "54988ac83e9f757e", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the equation of the tangent line to $y=- 4 x^{3} - x^{2} - 3 x - 4$ at $x=1$.", "output": "1) Differentiate: f'(x) = $- 12 x^{2} - 2 x - 3$.\n2) Slope at x=1: m = f'(1) = $-17$.\n3) Point on curve: (1, f(1)) = (1, $-12$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -12 = -17(x − 1).\n6) Optionally expand: y = $5 - 17 x$.\nFinal answer: $y = 5 - 17\\cdot x$", "final_answer": "y = 5 - 17\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8497b47c792d9850", "problem_id": "54988ac83e9f757e", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 4 x^{3} - x^{2} - 3 x - 4$ at $x=1$.", "output": "Differentiate: f'(x) = $- 12 x^{2} - 2 x - 3$.\nSlope at x=1: m = f'(1) = $-17$.\nPoint on curve: (1, f(1)) = (1, $-12$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -12 = -17(x − 1).\nOptionally expand: y = $5 - 17 x$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 5 - 17\\cdot x$", "final_answer": "y = 5 - 17\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8f98f54075075a27", "problem_id": "e4305ec474fba739", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $x^{3}\\cdot \\cos{\\left(x \\right)} + 3\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot x\\cdot \\cos{\\left(x \\right)} - 2\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + \\cos{\\left(x \\right)}$", "final_answer": "x^{3}\\cdot \\cos{\\left(x \\right)} + 3\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot x\\cdot \\cos{\\left(x \\right)} - 2\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8225a4b14bdc4215", "problem_id": "e4305ec474fba739", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.\nFinal answer: $x^{3}\\cdot \\cos{\\left(x \\right)} + 3\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot x\\cdot \\cos{\\left(x \\right)} - 2\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + \\cos{\\left(x \\right)}$", "final_answer": "x^{3}\\cdot \\cos{\\left(x \\right)} + 3\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot x\\cdot \\cos{\\left(x \\right)} - 2\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "34d601ada1b9b5a0", "problem_id": "4c0ac46a754d3c1b", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 2 x^{4} - x^{2} + x + 4\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 2 x^{4} - x^{2} + x + 4$.\n3) Compute an antiderivative: F(x) = $- \\frac{2 x^{5}}{5} - \\frac{x^{3}}{3} + \\frac{x^{2}}{2} + 4 x$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{2\\cdot x^{5}}{5} - \\frac{x^{3}}{3} + \\frac{x^{2}}{2} + 4\\cdot x$", "final_answer": "C - \\frac{2\\cdot x^{5}}{5} - \\frac{x^{3}}{3} + \\frac{x^{2}}{2} + 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8f662d81b2a28cba", "problem_id": "4c0ac46a754d3c1b", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 2 x^{4} - x^{2} + x + 4\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 2 x^{4} - x^{2} + x + 4$.\nCompute an antiderivative: F(x) = $- \\frac{2 x^{5}}{5} - \\frac{x^{3}}{3} + \\frac{x^{2}}{2} + 4 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - \\frac{2\\cdot x^{5}}{5} - \\frac{x^{3}}{3} + \\frac{x^{2}}{2} + 4\\cdot x$", "final_answer": "C - \\frac{2\\cdot x^{5}}{5} - \\frac{x^{3}}{3} + \\frac{x^{2}}{2} + 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7937b1640c54bf1d", "problem_id": "e9ee557d5751cab2", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 10 red balls and 16 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(26,11).\n3) Favorable samples: choose k reds and n−k blues: C(10,4)·C(16,7).\n4) Compute totals: C(26,11) = 7726160, and favorable = 2402400.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{2310}{7429}$", "final_answer": "\\frac{2310}{7429}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6bf80893ebd94a0b", "problem_id": "e9ee557d5751cab2", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 10 red balls and 16 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(26,11).\nFavorable samples: choose k reds and n−k blues: C(10,4)·C(16,7).\nCompute totals: C(26,11) = 7726160, and favorable = 2402400.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{2310}{7429}$.\nFinal answer: $\\frac{2310}{7429}$", "final_answer": "\\frac{2310}{7429}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4d3231df028b6f6a", "problem_id": "cdeefd635e7b63dc", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{3} + 2 x^{2} + 4 x - 4\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $2 x^{3} + 2 x^{2} + 4 x - 4$.\nCompute an antiderivative: F(x) = $\\frac{x^{4}}{2} + \\frac{2 x^{3}}{3} + 2 x^{2} - 4 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{x^{4}}{2} + \\frac{2\\cdot x^{3}}{3} + 2\\cdot x^{2} - 4\\cdot x$", "final_answer": "C + \\frac{x^{4}}{2} + \\frac{2\\cdot x^{3}}{3} + 2\\cdot x^{2} - 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "90dab8679c5db1f8", "problem_id": "cdeefd635e7b63dc", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{3} + 2 x^{2} + 4 x - 4\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $2 x^{3} + 2 x^{2} + 4 x - 4$.\nCompute an antiderivative: F(x) = $\\frac{x^{4}}{2} + \\frac{2 x^{3}}{3} + 2 x^{2} - 4 x$.\nFinal answer: $C + \\frac{x^{4}}{2} + \\frac{2\\cdot x^{3}}{3} + 2\\cdot x^{2} - 4\\cdot x$", "final_answer": "C + \\frac{x^{4}}{2} + \\frac{2\\cdot x^{3}}{3} + 2\\cdot x^{2} - 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "21c65c2771b617f3", "problem_id": "7f1d95a73b8cdab3", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "How many binary strings of length 18 contain exactly 9 ones with no two ones adjacent?", "output": "Reserve 8 mandatory zeros between the 9 ones.\nThis leaves n−k+1 = 10 valid slots for the ones.\nChoose the slots: C(10,9) = 10.\nFinal answer: $10$", "final_answer": "10", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e67ef034b505f909", "problem_id": "7f1d95a73b8cdab3", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many binary strings of length 18 contain exactly 9 ones with no two ones adjacent?", "output": "1) Reserve 8 mandatory zeros between the 9 ones.\n2) This leaves n−k+1 = 10 valid slots for the ones.\n3) Choose the slots: C(10,9) = 10.\nFinal answer: $10$", "final_answer": "10", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d2179dcc0f3ba3e0", "problem_id": "8a72a4a6c1c1defc", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x^{3} - 3 x^{2} - 3 x - 8$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 2 x^{4} + x^{3} + 3 x^{2} - 2 x - 3$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 8 x^{3} + 3 x^{2} + 6 x - 2$.\n5) Evaluate at a: p'(2) = $-42$.\nFinal answer: $-42$", "final_answer": "-42", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "913d6c2189cd77da", "problem_id": "8a72a4a6c1c1defc", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x^{3} - 3 x^{2} - 3 x - 8$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 2 x^{4} + x^{3} + 3 x^{2} - 2 x - 3$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 8 x^{3} + 3 x^{2} + 6 x - 2$.\nEvaluate at a: p'(2) = $-42$.\nCheck: the identity gives $-42$.\nFinal answer: $-42$", "final_answer": "-42", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d2ac8d5aba968222", "problem_id": "44da1682f1ccbbea", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 3 x^{3} - 4 x^{2} + 2 x + 1$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{4} - x^{3} - 2 x^{2} + 3 x - 1$ and a = -1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $12 x^{3} - 3 x^{2} - 4 x + 3$.\n5) Evaluate at a: p'(-1) = $-8$.\nFinal answer: $-8$", "final_answer": "-8", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f8e79b4ba1594234", "problem_id": "44da1682f1ccbbea", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 3 x^{3} - 4 x^{2} + 2 x + 1$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $3 x^{4} - x^{3} - 2 x^{2} + 3 x - 1$ and a = -1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $12 x^{3} - 3 x^{2} - 4 x + 3$.\nEvaluate at a: p'(-1) = $-8$.\nCheck: the identity gives $-8$.\nFinal answer: $-8$", "final_answer": "-8", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0b9300dc8752a6f6", "problem_id": "e72af8248c058793", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int x^{3} + 2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $x^{3} + 2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}$.\n3) Compute an antiderivative: F(x) = $\\frac{x^{4}}{4} + \\frac{2 x^{3}}{3} + x^{2} - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{x^{4}}{4} + \\frac{2\\cdot x^{3}}{3} + x^{2} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{4}}{4} + \\frac{2\\cdot x^{3}}{3} + x^{2} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9548c2949a96187e", "problem_id": "e72af8248c058793", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int x^{3} + 2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $x^{3} + 2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}$.\nCompute an antiderivative: F(x) = $\\frac{x^{4}}{4} + \\frac{2 x^{3}}{3} + x^{2} - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{x^{4}}{4} + \\frac{2\\cdot x^{3}}{3} + x^{2} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{4}}{4} + \\frac{2\\cdot x^{3}}{3} + x^{2} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b3dc3ce2d5b4890", "problem_id": "3ccb476d8383fbb1", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $962 in the ratio 6:7. How much does each receive?", "output": "Total ratio parts = 6+7 = 13.\nEach part = 962 ÷ 13 = 74.\nFirst share = 6 × 74 = 444.\nSecond share = 7 × 74 = 518.\nCheck: 444+518=962.\nFinal answer: $518$", "final_answer": "518", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8574f101650d4d3e", "problem_id": "3ccb476d8383fbb1", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Two friends split $962 in the ratio 6:7. How much does each receive?", "output": "Total ratio parts = 6+7 = 13.\nEach part = 962 ÷ 13 = 74.\nFirst share = 6 × 74 = 444.\nFinal answer: $518$", "final_answer": "518", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9bffd961401f4694", "problem_id": "a3ceea92f03c64f4", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A disease has prevalence $\\frac{3}{200}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{423}{10000}$.\n3) Compute P(D|+) = $\\frac{85}{282}$.\nFinal answer: $\\frac{85}{282}$", "final_answer": "\\frac{85}{282}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f77ca37cd8a33ba3", "problem_id": "a3ceea92f03c64f4", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A disease has prevalence $\\frac{3}{200}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{423}{10000}$.\nCompute P(D|+) = $\\frac{85}{282}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{85}{282}$", "final_answer": "\\frac{85}{282}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5db323f91e9868fa", "problem_id": "f4b22741548c30e2", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Two friends split $670 in the ratio 9:1. How much does each receive?", "output": "1) Total ratio parts = 9+1 = 10.\n2) Each part = 670 ÷ 10 = 67.\n3) First share = 9 × 67 = 603.\n4) Second share = 1 × 67 = 67.\nFinal answer: $67$", "final_answer": "67", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f2e05ae3edc1a2db", "problem_id": "f4b22741548c30e2", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $670 in the ratio 9:1. How much does each receive?", "output": "Total ratio parts = 9+1 = 10.\nEach part = 670 ÷ 10 = 67.\nFirst share = 9 × 67 = 603.\nSecond share = 1 × 67 = 67.\nCheck: 603+67=670.\nFinal answer: $67$", "final_answer": "67", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7a68ef1d4adfd9cc", "problem_id": "0639cd931b08127e", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x - 1$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 2 x^{2} + 3 x + 2$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $3 - 4 x$.\nEvaluate at a: p'(2) = $-5$.\nCheck: the identity gives $-5$.\nFinal answer: $-5$", "final_answer": "-5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3cebcfc8a192980a", "problem_id": "8becc4ad099fc928", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 3 x^{2} + 12 x - 35$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} + 11 x - 35.\n3) Then the limit is -95.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -98.\nFinal answer: $-98$", "final_answer": "-98", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b612c7786240b007", "problem_id": "50093220ea225e8e", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$- 2 x + 6 y$ = $-8$\n$3 x + y$ = $12$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $3$: $- 6 x + 18 y$ = $-24$.\n3) Multiply the second equation by $-2$: $- 6 x - 2 y$ = $-24$.\n4) Subtract to eliminate x: $20 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, 0\\right)$", "final_answer": "(x,y)=\\left(4, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f64f9b88a144798d", "problem_id": "f0edbf272a0e2609", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} + x^{2} - 2 x + 4$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x^{2}) = 2 x^{2}$.\n3) So $f'(x) = - 4 x^{2} - 2$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x^{2}) = 2 x$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- 3 x^{2} + x - 1\\right)$.\nFinal answer: $2 \\left(- 3 x^{2} + x - 1\\right)$", "final_answer": "2 \\left(- 3 x^{2} + x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "9aaf1f8d12b428cd", "problem_id": "052bfb2438851e2f", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 24 x + 70$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 24 x + 70$ = $2 \\left(x - 7\\right) \\left(x - 5\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $7$.\n5) From the second factor: x = $5$.\nFinal answer: $x \\in \\{7, 5\\}$", "final_answer": "x \\in \\{7, 5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "992a6389f1be6633", "problem_id": "052bfb2438851e2f", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 24 x + 70$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} - 24 x + 70$ = $2 \\left(x - 7\\right) \\left(x - 5\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $7$.\nFrom the second factor: x = $5$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{7, 5\\}$", "final_answer": "x \\in \\{7, 5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "41e252526f3d2c76", "problem_id": "c67b3085f2df2eba", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "How many binary strings of length 19 contain exactly 7 ones with no two ones adjacent?", "output": "1) Reserve 6 mandatory zeros between the 7 ones.\n2) This leaves n−k+1 = 13 valid slots for the ones.\n3) Choose the slots: C(13,7) = 1716.\nFinal answer: $1716$", "final_answer": "1716", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "223c3fffcf6d1e5e", "problem_id": "c67b3085f2df2eba", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many binary strings of length 19 contain exactly 7 ones with no two ones adjacent?", "output": "Reserve 6 mandatory zeros between the 7 ones.\nThis leaves n−k+1 = 13 valid slots for the ones.\nChoose the slots: C(13,7) = 1716.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $1716$", "final_answer": "1716", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9c7b236714df14ce", "problem_id": "eee0f338fc74cba4", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 15 x + 54$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} + 15 x + 54$ = $\\left(x + 6\\right) \\left(x + 9\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $-6$.\nFrom the second factor: x = $-9$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-6, -9\\}$", "final_answer": "x \\in \\{-6, -9\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1af87810311fb42b", "problem_id": "eee0f338fc74cba4", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 15 x + 54$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} + 15 x + 54$ = $\\left(x + 6\\right) \\left(x + 9\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-6$.\n5) From the second factor: x = $-9$.\nFinal answer: $x \\in \\{-6, -9\\}$", "final_answer": "x \\in \\{-6, -9\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "46bd8d6dac8c48a9", "problem_id": "2e78ef615df4b70c", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$x - 6 y$ = $-26$\n$7 x + 2 y$ = $-50$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $7$: $7 x - 42 y$ = $-182$.\n3) Multiply the second equation by $1$: $7 x + 2 y$ = $-50$.\n4) Subtract to eliminate x: $- 44 y$ = $-132$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $-8$.\nFinal answer: $(x,y)=\\left(-8, 3\\right)$", "final_answer": "(x,y)=\\left(-8, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "43beda04557acc6e", "problem_id": "2e78ef615df4b70c", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$x - 6 y$ = $-26$\n$7 x + 2 y$ = $-50$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $7$: $7 x - 42 y$ = $-182$.\nMultiply the second equation by $1$: $7 x + 2 y$ = $-50$.\nSubtract to eliminate x: $- 44 y$ = $-132$.\nSolve for y: y = $3$.\nSubstitute back to find x: x = $-8$.\nCheck: plug (x,y)=(-8,3) into both equations.\nFinal answer: $(x,y)=\\left(-8, 3\\right)$", "final_answer": "(x,y)=\\left(-8, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bbfb6cacec12d95e", "problem_id": "cbe4884831f96ad8", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_3=24$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=24 and k=3.\n4) Compute C(26,2) = 325.\nFinal answer: $325$", "final_answer": "325", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b7a2c4997dd7fc8e", "problem_id": "76ce63b064bca52e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{3 x^{2} + \\left(x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{x^{2} - x + 5}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 5}) = \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 5}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} - \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} - \\frac{4 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} + \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 11 x^{2} - 10 x + 25} + \\frac{11 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} + \\frac{22 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} - \\frac{6 x^{2}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 11 x^{2} - 10 x + 25} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 2 x + 10} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 5} - \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 5} - \\frac{10 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} - \\frac{20 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} + \\frac{72 x}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} - \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 5} + \\frac{25 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} + \\frac{50 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} - \\frac{6}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 5}) = \\frac{x^{2} \\left(1 - 2 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(x^{2} - x\\right) + 5\\right)^{2}} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(x^{2} - x\\right) + 5\\right)} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 5} + \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 5}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(1 - 2 x\\right) \\left(3 x^{2} + \\left(x \\left(x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x \\left(x - 1\\right) + 5\\right) \\left(12 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)}\\right)}{2 \\left(x \\left(x - 1\\right) + 5\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(1 - 2 x\\right) \\left(3 x^{2} + \\left(x \\left(x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x \\left(x - 1\\right) + 5\\right) \\left(12 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)}\\right)}{2 \\left(x \\left(x - 1\\right) + 5\\right)^{2}}$", "final_answer": "\\frac{2 \\left(1 - 2 x\\right) \\left(3 x^{2} + \\left(x \\left(x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x \\left(x - 1\\right) + 5\\right) \\left(12 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)}\\right)}{2 \\left(x \\left(x - 1\\right) + 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0e561a74735f1b6c", "problem_id": "4b33e38f8a7e16b4", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 11 times, what is the probability of exactly 7 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=7, p=$\\frac{2}{5}$ and simplify.\n4) Here C(11,7) = 330.\nFinal answer: $\\frac{684288}{9765625}$", "final_answer": "\\frac{684288}{9765625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bc3841b16b1a2a7a", "problem_id": "4b33e38f8a7e16b4", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 11 times, what is the probability of exactly 7 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=11, k=7, p=$\\frac{2}{5}$ and simplify.\nHere C(11,7) = 330.\nCheck: expression simplifies to $\\frac{684288}{9765625}$.\nFinal answer: $\\frac{684288}{9765625}$", "final_answer": "\\frac{684288}{9765625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b40da478e2b2b49", "problem_id": "69ea2375cb710953", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$- 7 x + y$ = $-29$\n$- 3 x + 5 y$ = $15$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $21 x - 3 y$ = $87$.\n3) Multiply the second equation by $-7$: $21 x - 35 y$ = $-105$.\n4) Subtract to eliminate x: $32 y$ = $192$.\n5) Solve for y: y = $6$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, 6\\right)$", "final_answer": "(x,y)=\\left(5, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "da256077e99bdd67", "problem_id": "69ea2375cb710953", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- 7 x + y$ = $-29$\n$- 3 x + 5 y$ = $15$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $21 x - 3 y$ = $87$.\nMultiply the second equation by $-7$: $21 x - 35 y$ = $-105$.\nSubtract to eliminate x: $32 y$ = $192$.\nSolve for y: y = $6$.\nSubstitute back to find x: x = $5$.\nCheck: plug (x,y)=(5,6) into both equations.\nFinal answer: $(x,y)=\\left(5, 6\\right)$", "final_answer": "(x,y)=\\left(5, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1b7acd7be98f7665", "problem_id": "a16a7eaf4ea87525", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A store sells a backpack for $31. It is discounted by 5%. You buy 6 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 31 × (1 − 5/100) = 31 × 95/100 = 589/20.\n2) Subtotal = 6 × 589/20 = 1767/10.\n3) Total with tax = 1767/10 × 107/100 = 189069/1000.\nFinal answer: $\\frac{189069}{1000}$", "final_answer": "\\frac{189069}{1000}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b20f076747b87f16", "problem_id": "a16a7eaf4ea87525", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $31. It is discounted by 5%. You buy 6 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 31 × (1 − 5/100) = 31 × 95/100 = 589/20.\nSubtotal = 6 × 589/20 = 1767/10.\nTotal with tax = 1767/10 × 107/100 = 189069/1000.\nCheck: recomputing the formula yields 189069/1000.\nFinal answer: $\\frac{189069}{1000}$", "final_answer": "\\frac{189069}{1000}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d55232948cf6424e", "problem_id": "4c2405f43e56810f", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 3 x - 4$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{2} - x - 2$ and a = -1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $6 x - 1$.\n5) Evaluate at a: p'(-1) = $-7$.\nFinal answer: $-7$", "final_answer": "-7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "431b6ea3467a3cd8", "problem_id": "4c2405f43e56810f", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 3 x - 4$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $3 x^{2} - x - 2$ and a = -1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $6 x - 1$.\nEvaluate at a: p'(-1) = $-7$.\nCheck: the identity gives $-7$.\nFinal answer: $-7$", "final_answer": "-7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9c2388d8785532a5", "problem_id": "9148025ce8f36299", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - x^{2} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=1$.", "output": "Differentiate: f'(x) = $- 9 x^{2} - 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\nSlope at x=1: m = f'(1) = $-13 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}$.\nPoint on curve: (1, f(1)) = (1, $-7 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -7 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} = -13 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}(x − 1).\nOptionally expand: y = $- 13 x - 2 x \\sin{\\left(2 \\right)} + \\frac{x e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2 \\sin{\\left(2 \\right)} + 6$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 13\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 6$", "final_answer": "y = - 13\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6131051b1942fb2e", "problem_id": "9148025ce8f36299", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - x^{2} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=1$.", "output": "1) Differentiate: f'(x) = $- 9 x^{2} - 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\n2) Slope at x=1: m = f'(1) = $-13 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}$.\n3) Point on curve: (1, f(1)) = (1, $-7 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -7 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} = -13 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}(x − 1).\n6) Optionally expand: y = $- 13 x - 2 x \\sin{\\left(2 \\right)} + \\frac{x e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2 \\sin{\\left(2 \\right)} + 6$.\nFinal answer: $y = - 13\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 6$", "final_answer": "y = - 13\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b602bea944240e36", "problem_id": "010b5b5f584aa21a", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Two friends split $702 in the ratio 9:4. How much does each receive?", "output": "Total ratio parts = 9+4 = 13.\nEach part = 702 ÷ 13 = 54.\nFirst share = 9 × 54 = 486.\nFinal answer: $216$", "final_answer": "216", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e7b8e9c34e8827e8", "problem_id": "010b5b5f584aa21a", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Two friends split $702 in the ratio 9:4. How much does each receive?", "output": "1) Total ratio parts = 9+4 = 13.\n2) Each part = 702 ÷ 13 = 54.\n3) First share = 9 × 54 = 486.\n4) Second share = 4 × 54 = 216.\nFinal answer: $216$", "final_answer": "216", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9e147935a9f88565", "problem_id": "17ae42742ad2c4f0", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} - 3 x^{2} + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=0$.", "output": "1) Differentiate: f'(x) = $- 3 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)}$.\n2) Slope at x=0: m = f'(0) = $\\frac{1}{2}$.\n3) Point on curve: (0, f(0)) = (0, $0$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 0 = \\frac{1}{2}(x − 0).\n6) Optionally expand: y = $\\frac{x}{2}$.\nFinal answer: $y = \\frac{x}{2}$", "final_answer": "y = \\frac{x}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "22c44e0ec1869452", "problem_id": "eb3dc200753b87c5", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 2 x - 2$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to x - 2.\n3) Then the limit is -3.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: -4.\nFinal answer: $-4$", "final_answer": "-4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "af5b95892db3b1c0", "problem_id": "3bba15df2f174dd4", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} - 2 x^{2} + 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 3$ at $x=1$.", "output": "Differentiate: f'(x) = $6 x^{2} - 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 2$.\nSlope at x=1: m = f'(1) = $- 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 4$.\nPoint on curve: (1, f(1)) = (1, $-1 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -1 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} = - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 4(x − 1).\nOptionally expand: y = $- 2 x \\sin{\\left(2 \\right)} + \\frac{x e^{\\frac{1}{2}}}{2} + 4 x - 5 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2 \\sin{\\left(2 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 4\\cdot x - 5 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 4\\cdot x - 5 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "121435aff46fbb73", "problem_id": "3bba15df2f174dd4", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} - 2 x^{2} + 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 3$ at $x=1$.", "output": "1) Differentiate: f'(x) = $6 x^{2} - 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 2$.\n2) Slope at x=1: m = f'(1) = $- 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 4$.\n3) Point on curve: (1, f(1)) = (1, $-1 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -1 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} = - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 4(x − 1).\n6) Optionally expand: y = $- 2 x \\sin{\\left(2 \\right)} + \\frac{x e^{\\frac{1}{2}}}{2} + 4 x - 5 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 4\\cdot x - 5 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 4\\cdot x - 5 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a5157cf36763119f", "problem_id": "bb14d54abf1c6cf7", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} 3 x^{3} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{3} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1$.\nEvaluate at the bounds: F(\\pi) and F(- \\pi).\nSubtract: F(\\pi) − F(- \\pi) = $2 \\pi$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $2\\cdot \\pi$", "final_answer": "2\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bbdbcc7ed9d353db", "problem_id": "bb14d54abf1c6cf7", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} 3 x^{3} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{3} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\pi).\n3) Subtract: F(\\pi) − F(- \\pi) = $2 \\pi$.\n4) Simplify the result.\nFinal answer: $2\\cdot \\pi$", "final_answer": "2\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "946d2a203585255b", "problem_id": "cfea77f4fb12f54b", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "You want 25 liters of a 25% salt solution. You have a 10% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 25−x liters is 40% solution.\nSalt equation: 10x + 40(25−x) = 25·25.\n(10-40)x = (25-40)·25 ⇒ x = 25/2.\nFinal answer: $25/2 L of 10% and 25/2 L of 40%$", "final_answer": "25/2 L of 10% and 25/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d455a6a894db8566", "problem_id": "cfea77f4fb12f54b", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "You want 25 liters of a 25% salt solution. You have a 10% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 25−x liters is 40% solution.\n2) Salt equation: 10x + 40(25−x) = 25·25.\n3) (10-40)x = (25-40)·25 ⇒ x = 25/2.\n4) Then 25−x = 25/2.\nFinal answer: $25/2 L of 10% and 25/2 L of 40%$", "final_answer": "25/2 L of 10% and 25/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a715da71a18e716c", "problem_id": "bb06b0d1689c71cd", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=41$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=41 and k=7.\nCompute C(47,6) = 10737573.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $10737573$", "final_answer": "10737573", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6314a17bb0eacaa9", "problem_id": "bb06b0d1689c71cd", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=41$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=41 and k=7.\n4) Compute C(47,6) = 10737573.\nFinal answer: $10737573$", "final_answer": "10737573", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "79246e6ac2421096", "problem_id": "f0b83e1da2b7e849", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "You want 58 liters of a 30% salt solution. You have a 10% solution and a 50% solution. How many liters of each should you mix?", "output": "29 L of 10% and 29 L of 50%", "final_answer": "29 L of 10% and 29 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fc8bd85b8aa03d18", "problem_id": "f0b83e1da2b7e849", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "You want 58 liters of a 30% salt solution. You have a 10% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 58−x liters is 50% solution.\n2) Salt equation: 10x + 50(58−x) = 30·58.\n3) (10-50)x = (30-50)·58 ⇒ x = 29.\n4) Then 58−x = 29.\nFinal answer: $29 L of 10% and 29 L of 50%$", "final_answer": "29 L of 10% and 29 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7cb1b2dbcf11dd55", "problem_id": "46b86b43360c6ddd", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{3} + 2 x^{2} - x + 1$.", "output": "1) Differentiate term-by-term:\n2) d/dx(1) = 0.\n3) d/dx(- x) = -1.\n4) d/dx(2 x^{2}) = 4 x.\n5) d/dx(4 x^{3}) = 12 x^{2}.\n6) Combine: f'(x) = 12 x^{2} + 4 x - 1.\nFinal answer: $12\\cdot x^{2} + 4\\cdot x - 1$", "final_answer": "12\\cdot x^{2} + 4\\cdot x - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ae98081e1a5eeb8d", "problem_id": "46b86b43360c6ddd", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{3} + 2 x^{2} - x + 1$.", "output": "Differentiate term-by-term:\nd/dx(1) = 0.\nd/dx(- x) = -1.\nFinal answer: $12\\cdot x^{2} + 4\\cdot x - 1$", "final_answer": "12\\cdot x^{2} + 4\\cdot x - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "682445640eb8c923", "problem_id": "11857de4ab1d55cc", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 30 x + 75$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} + 30 x + 75$ = $3 \\left(x + 5\\right)^{2}$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-5$.\n5) From the second factor: x = $-5$.\nFinal answer: $x \\in \\{-5, -5\\}$", "final_answer": "x \\in \\{-5, -5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "95c80e49cb3d61b9", "problem_id": "11857de4ab1d55cc", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 30 x + 75$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} + 30 x + 75$ = $3 \\left(x + 5\\right)^{2}$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $-5$.\nFrom the second factor: x = $-5$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-5, -5\\}$", "final_answer": "x \\in \\{-5, -5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b3a04c297c74f3bb", "problem_id": "cb83614046ed8d5a", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "An urn has 10 red balls and 16 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(26,10).\n3) Favorable samples: choose k reds and n−k blues: C(10,1)·C(16,9).\n4) Compute totals: C(26,10) = 5311735, and favorable = 114400.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{160}{7429}$", "final_answer": "\\frac{160}{7429}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3c42ce1a98e45672", "problem_id": "d26f10f35f204cfa", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} + 3 x - 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x^{2}) = 2 x^{2}$.\n3) So $f'(x) = 2 x^{2} + 3$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x^{2}) = 2 x$.\nTherefore the correct derivative is $f'(x) = 2 x + 3$.\nFinal answer: $2 x + 3$", "final_answer": "2 x + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0dbd93839be02cf0", "problem_id": "a268db6053c95afb", "style": "answer_only", "instruction": "Final answer only.", "input": "An urn has 18 red balls and 24 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "$\\frac{5313}{19721}$", "final_answer": "\\frac{5313}{19721}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ea240e99e7eecaac", "problem_id": "a268db6053c95afb", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "An urn has 18 red balls and 24 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(42,11).\n3) Favorable samples: choose k reds and n−k blues: C(18,5)·C(24,6).\n4) Compute totals: C(42,11) = 4280561376, and favorable = 1153218528.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{5313}{19721}$", "final_answer": "\\frac{5313}{19721}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "48cefbe6a95afce8", "problem_id": "bf3f90eb26274d99", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$2 x + 11 y$ = $-29$\n$- 8 x + 2 y$ = $-22$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-8$: $- 16 x - 88 y$ = $232$.\nMultiply the second equation by $2$: $- 16 x + 4 y$ = $-44$.\nSubtract to eliminate x: $- 92 y$ = $276$.\nSolve for y: y = $-3$.\nSubstitute back to find x: x = $2$.\nCheck: plug (x,y)=(2,-3) into both equations.\nFinal answer: $(x,y)=\\left(2, -3\\right)$", "final_answer": "(x,y)=\\left(2, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2c6e26461cf4fc80", "problem_id": "bf3f90eb26274d99", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the linear system:\n$2 x + 11 y$ = $-29$\n$- 8 x + 2 y$ = $-22$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-8$: $- 16 x - 88 y$ = $232$.\n3) Multiply the second equation by $2$: $- 16 x + 4 y$ = $-44$.\n4) Subtract to eliminate x: $- 92 y$ = $276$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $2$.\nFinal answer: $(x,y)=\\left(2, -3\\right)$", "final_answer": "(x,y)=\\left(2, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "42fab3f9ac42a8b0", "problem_id": "c08a168865e2a91c", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Two friends split $504 in the ratio 3:4. How much does each receive?", "output": "1) Total ratio parts = 3+4 = 7.\n2) Each part = 504 ÷ 7 = 72.\n3) First share = 3 × 72 = 216.\n4) Second share = 4 × 72 = 288.\nFinal answer: $288$", "final_answer": "288", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7596768297214003", "problem_id": "c08a168865e2a91c", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $504 in the ratio 3:4. How much does each receive?", "output": "Total ratio parts = 3+4 = 7.\nEach part = 504 ÷ 7 = 72.\nFirst share = 3 × 72 = 216.\nSecond share = 4 × 72 = 288.\nCheck: 216+288=504.\nFinal answer: $288$", "final_answer": "288", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4a4b286f2de709b7", "problem_id": "ca266f3e29b452e9", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡1 (mod 9), x≡10 (mod 11). Give the solution modulo 99.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 99.\n3) Compute the CRT solution to get x ≡ 10 (mod 99).\nFinal answer: $x\\equiv 10\\pmod{99}$", "final_answer": "x\\equiv 10\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "da727b22eb8bbc74", "problem_id": "ca266f3e29b452e9", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the congruences: x≡1 (mod 9), x≡10 (mod 11). Give the solution modulo 99.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 99.\nCompute the CRT solution to get x ≡ 10 (mod 99).\nFinal answer: $x\\equiv 10\\pmod{99}$", "final_answer": "x\\equiv 10\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9612d97aae22944f", "problem_id": "15257d80807afb4f", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$- 11 x + 3 y$ = $-59$\n$x - 9 y$ = $-47$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $- 11 x + 3 y$ = $-59$.\nMultiply the second equation by $-11$: $- 11 x + 99 y$ = $517$.\nSubtract to eliminate x: $- 96 y$ = $-576$.\nSolve for y: y = $6$.\nSubstitute back to find x: x = $7$.\nCheck: plug (x,y)=(7,6) into both equations.\nFinal answer: $(x,y)=\\left(7, 6\\right)$", "final_answer": "(x,y)=\\left(7, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cefade12c110ecdb", "problem_id": "15257d80807afb4f", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$- 11 x + 3 y$ = $-59$\n$x - 9 y$ = $-47$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $- 11 x + 3 y$ = $-59$.\n3) Multiply the second equation by $-11$: $- 11 x + 99 y$ = $517$.\n4) Subtract to eliminate x: $- 96 y$ = $-576$.\n5) Solve for y: y = $6$.\n6) Substitute back to find x: x = $7$.\nFinal answer: $(x,y)=\\left(7, 6\\right)$", "final_answer": "(x,y)=\\left(7, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "30f9d31838703601", "problem_id": "595143601fbc84f7", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡4 (mod 11), x≡8 (mod 10). Give the solution modulo 110.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 110.\nCompute the CRT solution to get x ≡ 48 (mod 110).\nCheck: 48 mod 11=4 and mod 10=8.\nFinal answer: $x\\equiv 48\\pmod{110}$", "final_answer": "x\\equiv 48\\pmod{110}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "44dd4e95be2473d8", "problem_id": "595143601fbc84f7", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡4 (mod 11), x≡8 (mod 10). Give the solution modulo 110.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 110.\n3) Compute the CRT solution to get x ≡ 48 (mod 110).\nFinal answer: $x\\equiv 48\\pmod{110}$", "final_answer": "x\\equiv 48\\pmod{110}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db74ff3c24231262", "problem_id": "d2cf990414d1a94b", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡0 (mod 3), x≡9 (mod 11). Give the solution modulo 33.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 33.\n3) Compute the CRT solution to get x ≡ 9 (mod 33).\nFinal answer: $x\\equiv 9\\pmod{33}$", "final_answer": "x\\equiv 9\\pmod{33}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fe01016a65728bf0", "problem_id": "ca1db2df9672f21b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 \\cos{\\left(2 x \\right)}) = - 2 \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 \\cos{\\left(2 x \\right)}) = - 4 \\sin{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$.\nFinal answer: $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$", "final_answer": "- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d4eeeedb0a294051", "problem_id": "883a3c838a58e50d", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 17 red balls and 17 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(34,10).\nFavorable samples: choose k reds and n−k blues: C(17,4)·C(17,6).\nCompute totals: C(34,10) = 131128140, and favorable = 29454880.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{6664}{29667}$.\nFinal answer: $\\frac{6664}{29667}$", "final_answer": "\\frac{6664}{29667}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7a023204ac97e64c", "problem_id": "883a3c838a58e50d", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 17 red balls and 17 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(34,10).\n3) Favorable samples: choose k reds and n−k blues: C(17,4)·C(17,6).\n4) Compute totals: C(34,10) = 131128140, and favorable = 29454880.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{6664}{29667}$", "final_answer": "\\frac{6664}{29667}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "66cdc6dba7517751", "problem_id": "d687b2960e4a67a1", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A store sells a backpack for $165. It is discounted by 5%. You buy 12 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 165 × (1 − 5/100) = 165 × 95/100 = 627/4.\n2) Subtotal = 12 × 627/4 = 1881.\n3) Total with tax = 1881 × 107/100 = 201267/100.\nFinal answer: $\\frac{201267}{100}$", "final_answer": "\\frac{201267}{100}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b3f6e5c689604ab", "problem_id": "d687b2960e4a67a1", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A store sells a backpack for $165. It is discounted by 5%. You buy 12 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 165 × (1 − 5/100) = 165 × 95/100 = 627/4.\nSubtotal = 12 × 627/4 = 1881.\nTotal with tax = 1881 × 107/100 = 201267/100.\nCheck: recomputing the formula yields 201267/100.\nFinal answer: $\\frac{201267}{100}$", "final_answer": "\\frac{201267}{100}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f59447b5c6ebf5ee", "problem_id": "e677d5d28f176f7a", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Worker A finishes a job in 4 hours and Worker B finishes it in 17 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/4 job/hour. B's rate = 1/17 job/hour.\nCombined rate = 1/4 + 1/17 = 21/68 job/hour.\nTime = 1 ÷ (combined rate) = 68/21 hours.\nCheck: (combined rate)×(time) = 21/68×68/21 = 1.\nFinal answer: $\\frac{68}{21}$", "final_answer": "\\frac{68}{21}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff12dd2d60bd4008", "problem_id": "e677d5d28f176f7a", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Worker A finishes a job in 4 hours and Worker B finishes it in 17 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/4 job/hour. B's rate = 1/17 job/hour.\n2) Combined rate = 1/4 + 1/17 = 21/68 job/hour.\n3) Time = 1 ÷ (combined rate) = 68/21 hours.\nFinal answer: $\\frac{68}{21}$", "final_answer": "\\frac{68}{21}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4fd3629bc59283f3", "problem_id": "7ab31a3a2183c34d", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- 3 x - 2 y$ = $10$\n$2 x + 2 y$ = $-8$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $2$: $- 6 x - 4 y$ = $20$.\nMultiply the second equation by $-3$: $- 6 x - 6 y$ = $24$.\nSubtract to eliminate x: $2 y$ = $-4$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $-2$.\nCheck: plug (x,y)=(-2,-2) into both equations.\nFinal answer: $(x,y)=\\left(-2, -2\\right)$", "final_answer": "(x,y)=\\left(-2, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cf2de2eb26f9e9e1", "problem_id": "7ab31a3a2183c34d", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$- 3 x - 2 y$ = $10$\n$2 x + 2 y$ = $-8$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $- 6 x - 4 y$ = $20$.\n3) Multiply the second equation by $-3$: $- 6 x - 6 y$ = $24$.\n4) Subtract to eliminate x: $2 y$ = $-4$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, -2\\right)$", "final_answer": "(x,y)=\\left(-2, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6990325049e7d8db", "problem_id": "c3018def236b3f67", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{5} x}{2} - \\frac{9 x}{5} + \\frac{4}{5} + \\frac{6 \\sqrt{5}}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{3 \\sqrt{5}}{2} - \\frac{9}{5}$, c=$\\frac{4}{5} + \\frac{6 \\sqrt{5}}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{3 \\sqrt{5}}{5} + \\frac{1129}{100}$, so √Δ = $\\sqrt{\\frac{3 \\sqrt{5}}{5} + \\frac{1129}{100}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{9}{10} + \\frac{3 \\sqrt{5}}{4} + \\frac{\\sqrt{60 \\sqrt{5} + 1129}}{20}$.\nx2 = $- \\frac{\\sqrt{60 \\sqrt{5} + 1129}}{20} + \\frac{9}{10} + \\frac{3 \\sqrt{5}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{9}{10} + \\frac{3\\cdot \\sqrt{5}}{4} + \\frac{\\sqrt{60\\cdot \\sqrt{5} + 1129}}{20}, - \\frac{\\sqrt{60\\cdot \\sqrt{5} + 1129}}{20} + \\frac{9}{10} + \\frac{3\\cdot \\sqrt{5}}{4}\\}$", "final_answer": "x \\in \\{\\frac{9}{10} + \\frac{3\\cdot \\sqrt{5}}{4} + \\frac{\\sqrt{60\\cdot \\sqrt{5} + 1129}}{20}, - \\frac{\\sqrt{60\\cdot \\sqrt{5} + 1129}}{20} + \\frac{9}{10} + \\frac{3\\cdot \\sqrt{5}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "711614cc271f9db8", "problem_id": "c3018def236b3f67", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{5} x}{2} - \\frac{9 x}{5} + \\frac{4}{5} + \\frac{6 \\sqrt{5}}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{3 \\sqrt{5}}{2} - \\frac{9}{5}$, c=$\\frac{4}{5} + \\frac{6 \\sqrt{5}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{3 \\sqrt{5}}{5} + \\frac{1129}{100}$, so √Δ = $\\sqrt{\\frac{3 \\sqrt{5}}{5} + \\frac{1129}{100}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{9}{10} + \\frac{3 \\sqrt{5}}{4} + \\frac{\\sqrt{60 \\sqrt{5} + 1129}}{20}$.\n7) x2 = $- \\frac{\\sqrt{60 \\sqrt{5} + 1129}}{20} + \\frac{9}{10} + \\frac{3 \\sqrt{5}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{9}{10} + \\frac{3\\cdot \\sqrt{5}}{4} + \\frac{\\sqrt{60\\cdot \\sqrt{5} + 1129}}{20}, - \\frac{\\sqrt{60\\cdot \\sqrt{5} + 1129}}{20} + \\frac{9}{10} + \\frac{3\\cdot \\sqrt{5}}{4}\\}$", "final_answer": "x \\in \\{\\frac{9}{10} + \\frac{3\\cdot \\sqrt{5}}{4} + \\frac{\\sqrt{60\\cdot \\sqrt{5} + 1129}}{20}, - \\frac{\\sqrt{60\\cdot \\sqrt{5} + 1129}}{20} + \\frac{9}{10} + \\frac{3\\cdot \\sqrt{5}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "835a0ab483f2e983", "problem_id": "4124296afa76dc5f", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Two friends split $665 in the ratio 3:4. How much does each receive?", "output": "Total ratio parts = 3+4 = 7.\nEach part = 665 ÷ 7 = 95.\nFirst share = 3 × 95 = 285.\nSecond share = 4 × 95 = 380.\nCheck: 285+380=665.\nFinal answer: $380$", "final_answer": "380", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e3175b4e9b16a86b", "problem_id": "4124296afa76dc5f", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Two friends split $665 in the ratio 3:4. How much does each receive?", "output": "1) Total ratio parts = 3+4 = 7.\n2) Each part = 665 ÷ 7 = 95.\n3) First share = 3 × 95 = 285.\n4) Second share = 4 × 95 = 380.\nFinal answer: $380$", "final_answer": "380", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0bdd2355799e9503", "problem_id": "d3544367d6419c3a", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$3 x + 6 y$ = $51$\n$- 7 x - 2 y$ = $-35$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-7$: $- 21 x - 42 y$ = $-357$.\n3) Multiply the second equation by $3$: $- 21 x - 6 y$ = $-105$.\n4) Subtract to eliminate x: $- 36 y$ = $-252$.\n5) Solve for y: y = $7$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, 7\\right)$", "final_answer": "(x,y)=\\left(3, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b338c560c65cd64f", "problem_id": "d3544367d6419c3a", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$3 x + 6 y$ = $51$\n$- 7 x - 2 y$ = $-35$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-7$: $- 21 x - 42 y$ = $-357$.\nMultiply the second equation by $3$: $- 21 x - 6 y$ = $-105$.\nSubtract to eliminate x: $- 36 y$ = $-252$.\nSolve for y: y = $7$.\nSubstitute back to find x: x = $3$.\nCheck: plug (x,y)=(3,7) into both equations.\nFinal answer: $(x,y)=\\left(3, 7\\right)$", "final_answer": "(x,y)=\\left(3, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "06473105ba856bd1", "problem_id": "68ab59a7803f9249", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{11} x + 5 x - 3 \\sqrt{11} + 4$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$5 - 3 \\sqrt{11}$, c=$4 - 3 \\sqrt{11}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $108 - 18 \\sqrt{11}$, so √Δ = $\\sqrt{108 - 18 \\sqrt{11}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{5}{2} + \\frac{3 \\sqrt{12 - 2 \\sqrt{11}}}{2} + \\frac{3 \\sqrt{11}}{2}$.\n7) x2 = $- \\frac{3 \\sqrt{12 - 2 \\sqrt{11}}}{2} - \\frac{5}{2} + \\frac{3 \\sqrt{11}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{5}{2} + \\frac{3\\cdot \\sqrt{12 - 2\\cdot \\sqrt{11}}}{2} + \\frac{3\\cdot \\sqrt{11}}{2}, - \\frac{3\\cdot \\sqrt{12 - 2\\cdot \\sqrt{11}}}{2} - \\frac{5}{2} + \\frac{3\\cdot \\sqrt{11}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{5}{2} + \\frac{3\\cdot \\sqrt{12 - 2\\cdot \\sqrt{11}}}{2} + \\frac{3\\cdot \\sqrt{11}}{2}, - \\frac{3\\cdot \\sqrt{12 - 2\\cdot \\sqrt{11}}}{2} - \\frac{5}{2} + \\frac{3\\cdot \\sqrt{11}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "14328323054f0a05", "problem_id": "68ab59a7803f9249", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{11} x + 5 x - 3 \\sqrt{11} + 4$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$5 - 3 \\sqrt{11}$, c=$4 - 3 \\sqrt{11}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{- \\frac{5}{2} + \\frac{3\\cdot \\sqrt{12 - 2\\cdot \\sqrt{11}}}{2} + \\frac{3\\cdot \\sqrt{11}}{2}, - \\frac{3\\cdot \\sqrt{12 - 2\\cdot \\sqrt{11}}}{2} - \\frac{5}{2} + \\frac{3\\cdot \\sqrt{11}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{5}{2} + \\frac{3\\cdot \\sqrt{12 - 2\\cdot \\sqrt{11}}}{2} + \\frac{3\\cdot \\sqrt{11}}{2}, - \\frac{3\\cdot \\sqrt{12 - 2\\cdot \\sqrt{11}}}{2} - \\frac{5}{2} + \\frac{3\\cdot \\sqrt{11}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8a697b316a1bbf51", "problem_id": "714fd15f1b0e5538", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Two friends split $1183 in the ratio 6:1. How much does each receive?", "output": "1) Total ratio parts = 6+1 = 7.\n2) Each part = 1183 ÷ 7 = 169.\n3) First share = 6 × 169 = 1014.\n4) Second share = 1 × 169 = 169.\nFinal answer: $169$", "final_answer": "169", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "042490be29b9dbfd", "problem_id": "714fd15f1b0e5538", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $1183 in the ratio 6:1. How much does each receive?", "output": "Total ratio parts = 6+1 = 7.\nEach part = 1183 ÷ 7 = 169.\nFirst share = 6 × 169 = 1014.\nSecond share = 1 × 169 = 169.\nCheck: 1014+169=1183.\nFinal answer: $169$", "final_answer": "169", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "879fccc1fc92e483", "problem_id": "be610ced86155035", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "An urn has 6 red balls and 8 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(14,6).\nFavorable samples: choose k reds and n−k blues: C(6,3)·C(8,3).\nFinal answer: $\\frac{160}{429}$", "final_answer": "\\frac{160}{429}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "404327e286e1cfcb", "problem_id": "be610ced86155035", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 6 red balls and 8 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(14,6).\n3) Favorable samples: choose k reds and n−k blues: C(6,3)·C(8,3).\n4) Compute totals: C(14,6) = 3003, and favorable = 1120.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{160}{429}$", "final_answer": "\\frac{160}{429}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "54b4814ff3e03987", "problem_id": "ae6c9c215e1038ce", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 18 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=18, k=8, p=$\\frac{3}{7}$ and simplify.\nHere C(18,8) = 43758.\nCheck: expression simplifies to $\\frac{301042224857088}{1628413597910449}$.\nFinal answer: $\\frac{301042224857088}{1628413597910449}$", "final_answer": "\\frac{301042224857088}{1628413597910449}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "402c72582b1271f0", "problem_id": "ae6c9c215e1038ce", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 18 times, what is the probability of exactly 8 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=18, k=8, p=$\\frac{3}{7}$ and simplify.\n4) Here C(18,8) = 43758.\nFinal answer: $\\frac{301042224857088}{1628413597910449}$", "final_answer": "\\frac{301042224857088}{1628413597910449}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db36b3a6a3834297", "problem_id": "bb3d27b8a7868843", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A store sells a backpack for $74. It is discounted by 30%. You buy 10 backpacks. What is the total cost?", "output": "1) Discounted price = 74 × (1 − 30/100) = 74 × 70/100 = 259/5.\n2) Subtotal = 10 × 259/5 = 518.\nFinal answer: $518$", "final_answer": "518", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6d512ac088d8bb04", "problem_id": "bb3d27b8a7868843", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A store sells a backpack for $74. It is discounted by 30%. You buy 10 backpacks. What is the total cost?", "output": "Discounted price = 74 × (1 − 30/100) = 74 × 70/100 = 259/5.\nSubtotal = 10 × 259/5 = 518.\nCheck: recomputing the formula yields 518.\nFinal answer: $518$", "final_answer": "518", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "89612dc744636cc1", "problem_id": "93a71b0588599438", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$7 x - 8 y$ = $-12$\n$7 x + 7 y$ = $63$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $7$: $49 x - 56 y$ = $-84$.\n3) Multiply the second equation by $7$: $49 x + 49 y$ = $441$.\n4) Subtract to eliminate x: $- 105 y$ = $-525$.\n5) Solve for y: y = $5$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, 5\\right)$", "final_answer": "(x,y)=\\left(4, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9d08d57b0f09cfdc", "problem_id": "a6d73985fdff9722", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(2 x^{3} - x^{2} - 3 x - 1\\right) \\cos{\\left(2 x - 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\cos{\\left(2 x - 2 \\right)}) = \\sin{\\left(2 x - 2 \\right)}$.\n3) So $f'(x) = - 4 x^{3} \\sin{\\left(2 x - 2 \\right)} + 2 x^{2} \\sin{\\left(2 x - 2 \\right)} + 6 x^{2} \\cos{\\left(2 x - 2 \\right)} + x e^{x} + 6 x \\sin{\\left(2 x - 2 \\right)} - 2 x \\cos{\\left(2 x - 2 \\right)} + 2 e^{x} + \\sin{\\left(2 x - 2 \\right)} - 3 \\cos{\\left(2 x - 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\cos{\\left(2 x - 2 \\right)}) = 2 \\sin{\\left(2 x - 2 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 1\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) - 3\\right) - 1\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(6 x^{2} - 2 x - 3\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 1\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) - 3\\right) - 1\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(6 x^{2} - 2 x - 3\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}$", "final_answer": "\\left(x + 1\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) - 3\\right) - 1\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(6 x^{2} - 2 x - 3\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "daef7f9e2f0df939", "problem_id": "4790be724c3fa1d1", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡4 (mod 11), x≡2 (mod 7). Give the solution modulo 77.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 77.\n3) Compute the CRT solution to get x ≡ 37 (mod 77).\nFinal answer: $x\\equiv 37\\pmod{77}$", "final_answer": "x\\equiv 37\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0848ebd408b770a9", "problem_id": "4790be724c3fa1d1", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡4 (mod 11), x≡2 (mod 7). Give the solution modulo 77.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 77.\nCompute the CRT solution to get x ≡ 37 (mod 77).\nCheck: 37 mod 11=4 and mod 7=2.\nFinal answer: $x\\equiv 37\\pmod{77}$", "final_answer": "x\\equiv 37\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "407e761dfc184201", "problem_id": "8ace5bf22ab7eae3", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{3} - x^{2} - x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1}{x^{2} + 2}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 2}) = \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 2}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 8 x^{2} + 8} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 8 x^{2} + 8} - \\frac{6 x^{4}}{2 x^{4} + 8 x^{2} + 8} + \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 4 x^{2} + 4} + \\frac{4 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 8 x^{2} + 8} + \\frac{8 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 8 x^{2} + 8} - \\frac{34 x^{2}}{2 x^{4} + 8 x^{2} + 8} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 4} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 2} - \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 2} - \\frac{4 x}{2 x^{4} + 8 x^{2} + 8} - \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 2} + \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 8 x^{2} + 8} + \\frac{8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 8 x^{2} + 8} - \\frac{4}{2 x^{4} + 8 x^{2} + 8}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 2}) = - \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 2\\right)^{2}} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x^{2} + 2\\right)} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 2} + \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 2}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 4 x \\left(- 3 x^{3} - x^{2} - x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) + \\left(x^{2} + 2\\right) \\left(- 18 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4 x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x^{2} + 2\\right)^{2}}$.\nFinal answer: $\\frac{- 4 x \\left(- 3 x^{3} - x^{2} - x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) + \\left(x^{2} + 2\\right) \\left(- 18 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4 x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x^{2} + 2\\right)^{2}}$", "final_answer": "\\frac{- 4 x \\left(- 3 x^{3} - x^{2} - x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) + \\left(x^{2} + 2\\right) \\left(- 18 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4 x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x^{2} + 2\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b80a67843b8e58a0", "problem_id": "d78080f149bde802", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 14 x + 12$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} - 14 x + 12$ = $2 \\left(x - 6\\right) \\left(x - 1\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $6$.\nFrom the second factor: x = $1$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{6, 1\\}$", "final_answer": "x \\in \\{6, 1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "67071f31e8463780", "problem_id": "e7d336d1f8d42aa6", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{2} + x + 2 \\sin{\\left(x \\right)} + 2$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $- 6 x + 2 \\cos{\\left(x \\right)} + 1$.\n2) Slope at x=-1: m = f'(-1) = $2 \\cos{\\left(1 \\right)} + 7$.\n3) Point on curve: (-1, f(-1)) = (-1, $-2 - 2 \\sin{\\left(1 \\right)}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -2 - 2 \\sin{\\left(1 \\right)} = 2 \\cos{\\left(1 \\right)} + 7(x − -1).\n6) Optionally expand: y = $2 x \\cos{\\left(1 \\right)} + 7 x - 2 \\sin{\\left(1 \\right)} + 2 \\cos{\\left(1 \\right)} + 5$.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 7\\cdot x - 2\\cdot \\sin{\\left(1 \\right)} + 2\\cdot \\cos{\\left(1 \\right)} + 5$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 7\\cdot x - 2\\cdot \\sin{\\left(1 \\right)} + 2\\cdot \\cos{\\left(1 \\right)} + 5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ae100e343465c30e", "problem_id": "3451b6d8ff262c62", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x^{3} \\sin{\\left(x \\right)}) = - x^{3} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - x^{3} \\cos{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x^{3} \\sin{\\left(x \\right)}) = - x^{3} \\cos{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - x^{3} \\cos{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.\nFinal answer: $- x^{3} \\cos{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$", "final_answer": "- x^{3} \\cos{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f901526cca7cf802", "problem_id": "6629b1228e0b98bf", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 12 times, what is the probability of exactly 7 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=7, p=$\\frac{2}{5}$ and simplify.\n4) Here C(12,7) = 792.\nFinal answer: $\\frac{24634368}{244140625}$", "final_answer": "\\frac{24634368}{244140625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ad374057c13763e5", "problem_id": "6629b1228e0b98bf", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 12 times, what is the probability of exactly 7 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=12, k=7, p=$\\frac{2}{5}$ and simplify.\nFinal answer: $\\frac{24634368}{244140625}$", "final_answer": "\\frac{24634368}{244140625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e8bff7b698532bed", "problem_id": "3ee881084e9c6459", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 7 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=7, k=5, p=$\\frac{1}{5}$ and simplify.\n4) Here C(7,5) = 21.\nFinal answer: $\\frac{336}{78125}$", "final_answer": "\\frac{336}{78125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ff092b0bb34fc34", "problem_id": "3ee881084e9c6459", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 7 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=7, k=5, p=$\\frac{1}{5}$ and simplify.\nFinal answer: $\\frac{336}{78125}$", "final_answer": "\\frac{336}{78125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a9270fe58a4c8555", "problem_id": "8401b803ad1745c7", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $- 2 x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - x\\cdot \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - x\\cdot \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1ade47e3a097af2e", "problem_id": "8401b803ad1745c7", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $- 2 x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - x\\cdot \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - x\\cdot \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d0c75c39c747044b", "problem_id": "da33d08499b0f399", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Worker A finishes a job in 22 hours and Worker B finishes it in 17 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/22 job/hour. B's rate = 1/17 job/hour.\n2) Combined rate = 1/22 + 1/17 = 39/374 job/hour.\n3) Time = 1 ÷ (combined rate) = 374/39 hours.\nFinal answer: $\\frac{374}{39}$", "final_answer": "\\frac{374}{39}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2aaf56c2c47f0d46", "problem_id": "da33d08499b0f399", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Worker A finishes a job in 22 hours and Worker B finishes it in 17 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/22 job/hour. B's rate = 1/17 job/hour.\nCombined rate = 1/22 + 1/17 = 39/374 job/hour.\nTime = 1 ÷ (combined rate) = 374/39 hours.\nCheck: (combined rate)×(time) = 39/374×374/39 = 1.\nFinal answer: $\\frac{374}{39}$", "final_answer": "\\frac{374}{39}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7045e33a0571088a", "problem_id": "1d3c3eec9c87cb0c", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $86. It is discounted by 35%. You buy 6 backpacks. What is the total cost?", "output": "Discounted price = 86 × (1 − 35/100) = 86 × 65/100 = 559/10.\nSubtotal = 6 × 559/10 = 1677/5.\nCheck: recomputing the formula yields 1677/5.\nFinal answer: $\\frac{1677}{5}$", "final_answer": "\\frac{1677}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3c69d67bc09afa3f", "problem_id": "1d3c3eec9c87cb0c", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A store sells a backpack for $86. It is discounted by 35%. You buy 6 backpacks. What is the total cost?", "output": "1) Discounted price = 86 × (1 − 35/100) = 86 × 65/100 = 559/10.\n2) Subtotal = 6 × 559/10 = 1677/5.\nFinal answer: $\\frac{1677}{5}$", "final_answer": "\\frac{1677}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4b773b15a7823937", "problem_id": "2b88eb96e7476d3d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{3} \\sin{\\left(x \\right)} - x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 \\cos{\\left(2 x \\right)}) = - 3 \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = 2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 3 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 \\cos{\\left(2 x \\right)}) = - 6 \\sin{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$", "final_answer": "2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b53c1d392370296a", "problem_id": "7743e790bfebc72f", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the multiplicative inverse of 28 modulo 31.", "output": "1) We want x such that 28x ≡ 1 (mod 31).\n2) Use extended Euclid to find s,t with 28s + 31t = 1.\n3) One Bezout identity is: 28·(10) + 31·(-9) = 1.\n4) Reduce s modulo 31: x ≡ 10 (mod 31).\nFinal answer: $x\\equiv 10\\pmod{31}$", "final_answer": "x\\equiv 10\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d46403d92538bad", "problem_id": "39a036e8e4b3525b", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 3 x - 12$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{2} - 3 x - 3$ and a = -3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $6 x - 3$.\n5) Evaluate at a: p'(-3) = $-21$.\nFinal answer: $-21$", "final_answer": "-21", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fc71e0667a8111df", "problem_id": "39a036e8e4b3525b", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 3 x - 12$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $3 x^{2} - 3 x - 3$ and a = -3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-21$", "final_answer": "-21", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "88a885021ae23e1b", "problem_id": "474f433e07e53049", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A cyclist rides at 61 miles per hour for 1/2 hours, then at 82 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 61 × 1/2 = 61/2 miles.\n2) Second leg distance = 82 × 1 = 82 miles.\n3) Total distance = 61/2 + 82 = 225/2 miles.\nFinal answer: $\\frac{225}{2}$", "final_answer": "\\frac{225}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ea44b0b4ad0b024a", "problem_id": "474f433e07e53049", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A cyclist rides at 61 miles per hour for 1/2 hours, then at 82 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 61 × 1/2 = 61/2 miles.\nSecond leg distance = 82 × 1 = 82 miles.\nTotal distance = 61/2 + 82 = 225/2 miles.\nCheck: adding both legs gives 225/2.\nFinal answer: $\\frac{225}{2}$", "final_answer": "\\frac{225}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d30a83e2c16c871", "problem_id": "7e5ae569fef0b133", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "You want 18 liters of a 35% salt solution. You have a 30% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 18−x liters is 50% solution.\nSalt equation: 30x + 50(18−x) = 35·18.\n(30-50)x = (35-50)·18 ⇒ x = 27/2.\nThen 18−x = 9/2.\nCheck: 30·27/2 + 50·9/2 = 35·18.\nFinal answer: $27/2 L of 30% and 9/2 L of 50%$", "final_answer": "27/2 L of 30% and 9/2 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "abc12147768ab952", "problem_id": "7e5ae569fef0b133", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "You want 18 liters of a 35% salt solution. You have a 30% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 18−x liters is 50% solution.\n2) Salt equation: 30x + 50(18−x) = 35·18.\n3) (30-50)x = (35-50)·18 ⇒ x = 27/2.\n4) Then 18−x = 9/2.\nFinal answer: $27/2 L of 30% and 9/2 L of 50%$", "final_answer": "27/2 L of 30% and 9/2 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3e879e55c729f4be", "problem_id": "5a1bcee1a7b9f155", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Worker A finishes a job in 4 hours and Worker B finishes it in 19 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/4 job/hour. B's rate = 1/19 job/hour.\n2) Combined rate = 1/4 + 1/19 = 23/76 job/hour.\n3) Time = 1 ÷ (combined rate) = 76/23 hours.\nFinal answer: $\\frac{76}{23}$", "final_answer": "\\frac{76}{23}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d1cf2b166410869d", "problem_id": "5a1bcee1a7b9f155", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Worker A finishes a job in 4 hours and Worker B finishes it in 19 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/4 job/hour. B's rate = 1/19 job/hour.\nCombined rate = 1/4 + 1/19 = 23/76 job/hour.\nTime = 1 ÷ (combined rate) = 76/23 hours.\nCheck: (combined rate)×(time) = 23/76×76/23 = 1.\nFinal answer: $\\frac{76}{23}$", "final_answer": "\\frac{76}{23}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d2b31423ff7d6d5b", "problem_id": "42ffaf92e63db81e", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - 3 x - 11$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 3 x^{2} - 2 x - 3$ and a = 3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 6 x - 2$.\n5) Evaluate at a: p'(3) = $-20$.\nFinal answer: $-20$", "final_answer": "-20", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "992a010642e77a00", "problem_id": "6f3a478956ce613a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} \\sin{\\left(x \\right)} + x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x^{3} \\sin{\\left(x \\right)}) = x^{3} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = x^{3} \\cos{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x^{3} \\sin{\\left(x \\right)}) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$.\nFinal answer: $x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$", "final_answer": "x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f8ef7b3f436148c9", "problem_id": "d219b2995f6de2d8", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "You want 28 liters of a 37% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 28−x liters is 50% solution.\nSalt equation: 25x + 50(28−x) = 37·28.\n(25-50)x = (37-50)·28 ⇒ x = 364/25.\nThen 28−x = 336/25.\nCheck: 25·364/25 + 50·336/25 = 37·28.\nFinal answer: $364/25 L of 25% and 336/25 L of 50%$", "final_answer": "364/25 L of 25% and 336/25 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eaf4ccc987631773", "problem_id": "d219b2995f6de2d8", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "You want 28 liters of a 37% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 28−x liters is 50% solution.\n2) Salt equation: 25x + 50(28−x) = 37·28.\n3) (25-50)x = (37-50)·28 ⇒ x = 364/25.\n4) Then 28−x = 336/25.\nFinal answer: $364/25 L of 25% and 336/25 L of 50%$", "final_answer": "364/25 L of 25% and 336/25 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ab33ada768d0ec88", "problem_id": "7d395e5261ab9753", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A cyclist rides at 36 miles per hour for 2 hours, then at 37 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 36 × 2 = 72 miles.\n2) Second leg distance = 37 × 1 = 37 miles.\n3) Total distance = 72 + 37 = 109 miles.\nFinal answer: $109$", "final_answer": "109", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b7de67bdaf965d34", "problem_id": "7d395e5261ab9753", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A cyclist rides at 36 miles per hour for 2 hours, then at 37 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 36 × 2 = 72 miles.\nSecond leg distance = 37 × 1 = 37 miles.\nTotal distance = 72 + 37 = 109 miles.\nCheck: adding both legs gives 109.\nFinal answer: $109$", "final_answer": "109", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f798151c0b802094", "problem_id": "abbc71750aa82b49", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x^{2} - 5 x - 11$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{3} - 3 x^{2} - x$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 3 x^{2} - 6 x - 1$.\n5) Evaluate at a: p'(2) = $-25$.\nFinal answer: $-25$", "final_answer": "-25", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6c866eb493a158f3", "problem_id": "abbc71750aa82b49", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x^{2} - 5 x - 11$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{3} - 3 x^{2} - x$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-25$", "final_answer": "-25", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a3b3e3245e80c307", "problem_id": "b236338c1b4e1204", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + \\frac{37 x}{6} + \\frac{28}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{37}{6}$, c=$\\frac{28}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{25}{36}$, so √Δ = $\\frac{5}{6}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{8}{3}$.\n7) x2 = $- \\frac{7}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{8}{3}, - \\frac{7}{2}\\}$", "final_answer": "x \\in \\{- \\frac{8}{3}, - \\frac{7}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f91f5cb54280a7e0", "problem_id": "98e765fee2d42e08", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} - 4 x + 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x^{2}) = - 4 x^{2}$.\n3) So $f'(x) = - 4 x^{2} - 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x^{2}) = - 4 x$.\nTherefore the correct derivative is $f'(x) = 4 \\left(- x - 1\\right)$.\nFinal answer: $4 \\left(- x - 1\\right)$", "final_answer": "4 \\left(- x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "92313fbd2a238dda", "problem_id": "1cfa224c0a0a1ac4", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$3 x - 4 y$ = $-7$\n$4 x - y$ = $-18$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $12 x - 16 y$ = $-28$.\n3) Multiply the second equation by $3$: $12 x - 3 y$ = $-54$.\n4) Subtract to eliminate x: $- 13 y$ = $26$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, -2\\right)$", "final_answer": "(x,y)=\\left(-5, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec7a0a04bece7e0d", "problem_id": "727bec7ffd87c9ba", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} + 2 x + \\left(2 x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{2 x^{2} - x + 5}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x^{2} - x\\right) + 5}) = \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - x + 5}$.\n3) So $f'(x) = \\frac{4 x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} - 8 x^{3} + 42 x^{2} - 20 x + 50} + \\frac{8 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} - 8 x^{3} + 42 x^{2} - 20 x + 50} - \\frac{4 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} - 8 x^{3} + 42 x^{2} - 20 x + 50} - \\frac{8 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} - 8 x^{3} + 42 x^{2} - 20 x + 50} + \\frac{21 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} - 8 x^{3} + 42 x^{2} - 20 x + 50} + \\frac{42 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} - 8 x^{3} + 42 x^{2} - 20 x + 50} - \\frac{6 x^{2}}{8 x^{4} - 8 x^{3} + 42 x^{2} - 20 x + 50} - \\frac{10 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} - 8 x^{3} + 42 x^{2} - 20 x + 50} - \\frac{20 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} - 8 x^{3} + 42 x^{2} - 20 x + 50} - \\frac{4 x}{8 x^{4} - 8 x^{3} + 42 x^{2} - 20 x + 50} + \\frac{20 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{4 x^{4} - 4 x^{3} + 21 x^{2} - 10 x + 25} + \\frac{25 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} - 8 x^{3} + 42 x^{2} - 20 x + 50} + \\frac{50 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} - 8 x^{3} + 42 x^{2} - 20 x + 50} + \\frac{16}{8 x^{4} - 8 x^{3} + 42 x^{2} - 20 x + 50} - \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{4 x^{4} - 4 x^{3} + 21 x^{2} - 10 x + 25} - \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{4 x^{2} - 2 x + 10} + \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - x + 5} - \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(2 x^{2} - x\\right) + 5}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x^{2} - x\\right) + 5}) = \\frac{5 \\left(1 - 4 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(2 x^{2} - x\\right) + 5\\right)^{2}} + \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(2 x^{2} - x\\right) + 5\\right)} + \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(2 x^{2} - x\\right) + 5}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(1 - 4 x\\right) \\left(- x^{2} + 2 x + \\left(x \\left(2 x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(x \\left(2 x - 1\\right) + 5\\right) \\left(- 4 x + 2 \\left(4 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(2 x - 1\\right) + 5\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(1 - 4 x\\right) \\left(- x^{2} + 2 x + \\left(x \\left(2 x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(x \\left(2 x - 1\\right) + 5\\right) \\left(- 4 x + 2 \\left(4 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(2 x - 1\\right) + 5\\right)^{2}}$", "final_answer": "\\frac{2 \\left(1 - 4 x\\right) \\left(- x^{2} + 2 x + \\left(x \\left(2 x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(x \\left(2 x - 1\\right) + 5\\right) \\left(- 4 x + 2 \\left(4 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(2 x - 1\\right) + 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b55b7fbc95b7da62", "problem_id": "553d469c254249f7", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Find the multiplicative inverse of 7 modulo 31.", "output": "We want x such that 7x ≡ 1 (mod 31).\nUse extended Euclid to find s,t with 7s + 31t = 1.\nOne Bezout identity is: 7·(9) + 31·(-2) = 1.\nFinal answer: $x\\equiv 9\\pmod{31}$", "final_answer": "x\\equiv 9\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3051798c245827e3", "problem_id": "553d469c254249f7", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the multiplicative inverse of 7 modulo 31.", "output": "1) We want x such that 7x ≡ 1 (mod 31).\n2) Use extended Euclid to find s,t with 7s + 31t = 1.\n3) One Bezout identity is: 7·(9) + 31·(-2) = 1.\n4) Reduce s modulo 31: x ≡ 9 (mod 31).\nFinal answer: $x\\equiv 9\\pmod{31}$", "final_answer": "x\\equiv 9\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "037847c6fc8d5afb", "problem_id": "3cbd332328aa3a94", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$- 2 x - 9 y$ = $17$\n$2 y$ = $-6$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $-2$: $- 4 y$ = $12$.\n4) Subtract to eliminate x: $4 y$ = $-12$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, -3\\right)$", "final_answer": "(x,y)=\\left(5, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ae05c6ef71bb2f8a", "problem_id": "3cbd332328aa3a94", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$- 2 x - 9 y$ = $17$\n$2 y$ = $-6$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $-2$: $- 4 y$ = $12$.\nSubtract to eliminate x: $4 y$ = $-12$.\nSolve for y: y = $-3$.\nSubstitute back to find x: x = $5$.\nCheck: plug (x,y)=(5,-3) into both equations.\nFinal answer: $(x,y)=\\left(5, -3\\right)$", "final_answer": "(x,y)=\\left(5, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "943163264ef697f5", "problem_id": "87a54dffd270e947", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Solve the congruences: x≡3 (mod 11), x≡3 (mod 8). Give the solution modulo 88.", "output": "$x\\equiv 3\\pmod{88}$", "final_answer": "x\\equiv 3\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "20c6f0d44dc3d290", "problem_id": "87a54dffd270e947", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡3 (mod 11), x≡3 (mod 8). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 3 (mod 88).\nFinal answer: $x\\equiv 3\\pmod{88}$", "final_answer": "x\\equiv 3\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "59953ae4b6114696", "problem_id": "11024044ec7ec250", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{3} + 4 x^{2} + 9 x + 20$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} + 4 x^{2} + 8 x + 20.\n3) Then the limit is 68.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 70.\nFinal answer: $70$", "final_answer": "70", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6ef44cd0e6587675", "problem_id": "2d18568d8cf35758", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$7 x + 5 y$ = $2$\n$8 x - 4 y$ = $-56$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $8$: $56 x + 40 y$ = $16$.\n3) Multiply the second equation by $7$: $56 x - 28 y$ = $-392$.\n4) Subtract to eliminate x: $68 y$ = $408$.\n5) Solve for y: y = $6$.\n6) Substitute back to find x: x = $-4$.\nFinal answer: $(x,y)=\\left(-4, 6\\right)$", "final_answer": "(x,y)=\\left(-4, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "97072b067a63ef68", "problem_id": "98c10fbad336aa3e", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_10=39$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_10=29.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=29 and k=10.\nCompute C(38,9) = 163011640.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $163011640$", "final_answer": "163011640", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b31d7ab4acf69729", "problem_id": "98c10fbad336aa3e", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_10=39$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_10=29.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=29 and k=10.\n4) Compute C(38,9) = 163011640.\nFinal answer: $163011640$", "final_answer": "163011640", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "30adfc8f114a0ca3", "problem_id": "322126f1744ebe3f", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{10} x}{2} + \\frac{9 x}{4} - \\frac{3 \\sqrt{10}}{8} + \\frac{1}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{9}{4} - \\frac{3 \\sqrt{10}}{2}$, c=$\\frac{1}{2} - \\frac{3 \\sqrt{10}}{8}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{409}{16} - \\frac{21 \\sqrt{10}}{4}$, so √Δ = $\\sqrt{\\frac{409}{16} - \\frac{21 \\sqrt{10}}{4}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{9}{8} + \\frac{\\sqrt{409 - 84 \\sqrt{10}}}{8} + \\frac{3 \\sqrt{10}}{4}$.\n7) x2 = $- \\frac{\\sqrt{409 - 84 \\sqrt{10}}}{8} - \\frac{9}{8} + \\frac{3 \\sqrt{10}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{9}{8} + \\frac{\\sqrt{409 - 84\\cdot \\sqrt{10}}}{8} + \\frac{3\\cdot \\sqrt{10}}{4}, - \\frac{\\sqrt{409 - 84\\cdot \\sqrt{10}}}{8} - \\frac{9}{8} + \\frac{3\\cdot \\sqrt{10}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{9}{8} + \\frac{\\sqrt{409 - 84\\cdot \\sqrt{10}}}{8} + \\frac{3\\cdot \\sqrt{10}}{4}, - \\frac{\\sqrt{409 - 84\\cdot \\sqrt{10}}}{8} - \\frac{9}{8} + \\frac{3\\cdot \\sqrt{10}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aa882351e13d38dd", "problem_id": "322126f1744ebe3f", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{10} x}{2} + \\frac{9 x}{4} - \\frac{3 \\sqrt{10}}{8} + \\frac{1}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{9}{4} - \\frac{3 \\sqrt{10}}{2}$, c=$\\frac{1}{2} - \\frac{3 \\sqrt{10}}{8}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{409}{16} - \\frac{21 \\sqrt{10}}{4}$, so √Δ = $\\sqrt{\\frac{409}{16} - \\frac{21 \\sqrt{10}}{4}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{9}{8} + \\frac{\\sqrt{409 - 84 \\sqrt{10}}}{8} + \\frac{3 \\sqrt{10}}{4}$.\nx2 = $- \\frac{\\sqrt{409 - 84 \\sqrt{10}}}{8} - \\frac{9}{8} + \\frac{3 \\sqrt{10}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{9}{8} + \\frac{\\sqrt{409 - 84\\cdot \\sqrt{10}}}{8} + \\frac{3\\cdot \\sqrt{10}}{4}, - \\frac{\\sqrt{409 - 84\\cdot \\sqrt{10}}}{8} - \\frac{9}{8} + \\frac{3\\cdot \\sqrt{10}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{9}{8} + \\frac{\\sqrt{409 - 84\\cdot \\sqrt{10}}}{8} + \\frac{3\\cdot \\sqrt{10}}{4}, - \\frac{\\sqrt{409 - 84\\cdot \\sqrt{10}}}{8} - \\frac{9}{8} + \\frac{3\\cdot \\sqrt{10}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e1c12e93cf3576c7", "problem_id": "2944680175593c9e", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{7 x}{3} - \\sqrt{5} x + \\frac{2}{3} + \\frac{\\sqrt{5}}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{7}{3} - \\sqrt{5}$, c=$\\frac{2}{3} + \\frac{\\sqrt{5}}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{10 \\sqrt{5}}{3} + \\frac{70}{9}$, so √Δ = $\\sqrt{\\frac{10 \\sqrt{5}}{3} + \\frac{70}{9}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{5}}{2} + \\frac{7}{6} + \\frac{\\sqrt{30 \\sqrt{5} + 70}}{6}$.\n7) x2 = $- \\frac{\\sqrt{30 \\sqrt{5} + 70}}{6} + \\frac{\\sqrt{5}}{2} + \\frac{7}{6}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{5}}{2} + \\frac{7}{6} + \\frac{\\sqrt{30\\cdot \\sqrt{5} + 70}}{6}, - \\frac{\\sqrt{30\\cdot \\sqrt{5} + 70}}{6} + \\frac{\\sqrt{5}}{2} + \\frac{7}{6}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{5}}{2} + \\frac{7}{6} + \\frac{\\sqrt{30\\cdot \\sqrt{5} + 70}}{6}, - \\frac{\\sqrt{30\\cdot \\sqrt{5} + 70}}{6} + \\frac{\\sqrt{5}}{2} + \\frac{7}{6}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b1dcfc7d1809861b", "problem_id": "2944680175593c9e", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{7 x}{3} - \\sqrt{5} x + \\frac{2}{3} + \\frac{\\sqrt{5}}{3}$ = 0.", "output": "$x \\in \\{\\frac{\\sqrt{5}}{2} + \\frac{7}{6} + \\frac{\\sqrt{30\\cdot \\sqrt{5} + 70}}{6}, - \\frac{\\sqrt{30\\cdot \\sqrt{5} + 70}}{6} + \\frac{\\sqrt{5}}{2} + \\frac{7}{6}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{5}}{2} + \\frac{7}{6} + \\frac{\\sqrt{30\\cdot \\sqrt{5} + 70}}{6}, - \\frac{\\sqrt{30\\cdot \\sqrt{5} + 70}}{6} + \\frac{\\sqrt{5}}{2} + \\frac{7}{6}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "474ef1a9cac18ebf", "problem_id": "1ef761d18dcb0f5c", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A store sells a backpack for $700. It is discounted by 20%. You buy 4 backpacks. What is the total cost?", "output": "1) Discounted price = 700 × (1 − 20/100) = 700 × 80/100 = 560.\n2) Subtotal = 4 × 560 = 2240.\nFinal answer: $2240$", "final_answer": "2240", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2fd309770e0a43f0", "problem_id": "1ef761d18dcb0f5c", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A store sells a backpack for $700. It is discounted by 20%. You buy 4 backpacks. What is the total cost?", "output": "Discounted price = 700 × (1 − 20/100) = 700 × 80/100 = 560.\nSubtotal = 4 × 560 = 2240.\nFinal answer: $2240$", "final_answer": "2240", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0348fcc87a1e12b2", "problem_id": "2a1b906d79c2a44e", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many binary strings of length 20 contain exactly 9 ones with no two ones adjacent?", "output": "Reserve 8 mandatory zeros between the 9 ones.\nThis leaves n−k+1 = 12 valid slots for the ones.\nChoose the slots: C(12,9) = 220.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $220$", "final_answer": "220", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "08c7d255d56854bd", "problem_id": "2a1b906d79c2a44e", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many binary strings of length 20 contain exactly 9 ones with no two ones adjacent?", "output": "1) Reserve 8 mandatory zeros between the 9 ones.\n2) This leaves n−k+1 = 12 valid slots for the ones.\n3) Choose the slots: C(12,9) = 220.\nFinal answer: $220$", "final_answer": "220", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cc229b44b4e19ebf", "problem_id": "92f60386c2ae2cbc", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A cyclist rides at 68 miles per hour for 1 hours, then at 33 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 68 × 1 = 68 miles.\n2) Second leg distance = 33 × 1 = 33 miles.\n3) Total distance = 68 + 33 = 101 miles.\nFinal answer: $101$", "final_answer": "101", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d5c430933f35cf2", "problem_id": "92f60386c2ae2cbc", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A cyclist rides at 68 miles per hour for 1 hours, then at 33 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 68 × 1 = 68 miles.\nSecond leg distance = 33 × 1 = 33 miles.\nTotal distance = 68 + 33 = 101 miles.\nCheck: adding both legs gives 101.\nFinal answer: $101$", "final_answer": "101", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "149bde7fed944aa8", "problem_id": "7e5a7c41652aa752", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "How many binary strings of length 19 contain exactly 6 ones with no two ones adjacent?", "output": "1) Reserve 5 mandatory zeros between the 6 ones.\n2) This leaves n−k+1 = 14 valid slots for the ones.\n3) Choose the slots: C(14,6) = 3003.\nFinal answer: $3003$", "final_answer": "3003", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8401c21132f5a4bd", "problem_id": "7e5a7c41652aa752", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many binary strings of length 19 contain exactly 6 ones with no two ones adjacent?", "output": "Reserve 5 mandatory zeros between the 6 ones.\nThis leaves n−k+1 = 14 valid slots for the ones.\nChoose the slots: C(14,6) = 3003.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $3003$", "final_answer": "3003", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a771ec08a39d16ec", "problem_id": "489a72dcdee6e07f", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Worker A finishes a job in 21 hours and Worker B finishes it in 17 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/21 job/hour. B's rate = 1/17 job/hour.\n2) Combined rate = 1/21 + 1/17 = 38/357 job/hour.\n3) Time = 1 ÷ (combined rate) = 357/38 hours.\nFinal answer: $\\frac{357}{38}$", "final_answer": "\\frac{357}{38}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "caffcf0a92e05ea0", "problem_id": "d37a8dd6d1d2c67d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{3} - 3 x^{2} - 2 x - 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 3 x^{2}) = - 6 x^{2}$.\n3) So $f'(x) = - 9 x^{2} - 2$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 3 x^{2}) = - 6 x$.\nTherefore the correct derivative is $f'(x) = - 3 x^{2} - 6 x - 2$.\nFinal answer: $- 3 x^{2} - 6 x - 2$", "final_answer": "- 3 x^{2} - 6 x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f7c35bdd06604f9b", "problem_id": "a7d2f6bfcd0d2ecd", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $396. It is discounted by 10%. You buy 14 backpacks. What is the total cost?", "output": "Discounted price = 396 × (1 − 10/100) = 396 × 90/100 = 1782/5.\nSubtotal = 14 × 1782/5 = 24948/5.\nCheck: recomputing the formula yields 24948/5.\nFinal answer: $\\frac{24948}{5}$", "final_answer": "\\frac{24948}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dfcd2976e09fde45", "problem_id": "a7d2f6bfcd0d2ecd", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A store sells a backpack for $396. It is discounted by 10%. You buy 14 backpacks. What is the total cost?", "output": "1) Discounted price = 396 × (1 − 10/100) = 396 × 90/100 = 1782/5.\n2) Subtotal = 14 × 1782/5 = 24948/5.\nFinal answer: $\\frac{24948}{5}$", "final_answer": "\\frac{24948}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4f9e06203f24ce4c", "problem_id": "d777936e17406dcb", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$- 2 y$ = $-10$\n$- 4 x + 2 y$ = $10$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $8 y$ = $40$.\n3) Multiply the second equation by $0$: $0$ = $0$.\n4) Subtract to eliminate x: $8 y$ = $40$.\n5) Solve for y: y = $5$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, 5\\right)$", "final_answer": "(x,y)=\\left(0, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8d8d528eea749af7", "problem_id": "d777936e17406dcb", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- 2 y$ = $-10$\n$- 4 x + 2 y$ = $10$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $8 y$ = $40$.\nMultiply the second equation by $0$: $0$ = $0$.\nSubtract to eliminate x: $8 y$ = $40$.\nSolve for y: y = $5$.\nSubstitute back to find x: x = $0$.\nCheck: plug (x,y)=(0,5) into both equations.\nFinal answer: $(x,y)=\\left(0, 5\\right)$", "final_answer": "(x,y)=\\left(0, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "53309474d6c379c3", "problem_id": "68fa5ec696958db4", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=19$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=19 and k=6.\n4) Compute C(24,5) = 42504.\nFinal answer: $42504$", "final_answer": "42504", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "735b983fd2d44ddb", "problem_id": "68fa5ec696958db4", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=19$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=19 and k=6.\nCompute C(24,5) = 42504.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $42504$", "final_answer": "42504", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aeb988cf4f4d1fdb", "problem_id": "eec14bcca5f66a20", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "You want 39 liters of a 40% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 39−x liters is 70% solution.\nSalt equation: 25x + 70(39−x) = 40·39.\n(25-70)x = (40-70)·39 ⇒ x = 26.\nFinal answer: $26 L of 25% and 13 L of 70%$", "final_answer": "26 L of 25% and 13 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e79a33e112b6b367", "problem_id": "eec14bcca5f66a20", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "You want 39 liters of a 40% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 39−x liters is 70% solution.\nSalt equation: 25x + 70(39−x) = 40·39.\n(25-70)x = (40-70)·39 ⇒ x = 26.\nThen 39−x = 13.\nCheck: 25·26 + 70·13 = 40·39.\nFinal answer: $26 L of 25% and 13 L of 70%$", "final_answer": "26 L of 25% and 13 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "582bab102d8fe79f", "problem_id": "ee2a845187ed539e", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x + 2$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{2}$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $2 x$.\n5) Evaluate at a: p'(2) = $4$.\nFinal answer: $4$", "final_answer": "4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db8da358982dd1b1", "problem_id": "f159eca792120ea4", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} 2 x^{3} + 11 x^{2} + 41 x + 162$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} + 11 x^{2} + 40 x + 162.\n3) Then the limit is 626.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: 630.\nFinal answer: $630$", "final_answer": "630", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "dc9111b4148868d4", "problem_id": "17ae47d9dad68f75", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{12 x}{5} - \\frac{3 \\sqrt{2} x}{2} + \\frac{7}{5} + \\frac{21 \\sqrt{2}}{10}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{12}{5} - \\frac{3 \\sqrt{2}}{2}$, c=$\\frac{7}{5} + \\frac{21 \\sqrt{2}}{10}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{233}{50} - \\frac{6 \\sqrt{2}}{5}$, so √Δ = $\\sqrt{\\frac{233}{50} - \\frac{6 \\sqrt{2}}{5}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{466 - 120 \\sqrt{2}}}{20} + \\frac{3 \\sqrt{2}}{4} + \\frac{6}{5}$.\nx2 = $- \\frac{\\sqrt{466 - 120 \\sqrt{2}}}{20} + \\frac{3 \\sqrt{2}}{4} + \\frac{6}{5}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{466 - 120\\cdot \\sqrt{2}}}{20} + \\frac{3\\cdot \\sqrt{2}}{4} + \\frac{6}{5}, - \\frac{\\sqrt{466 - 120\\cdot \\sqrt{2}}}{20} + \\frac{3\\cdot \\sqrt{2}}{4} + \\frac{6}{5}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{466 - 120\\cdot \\sqrt{2}}}{20} + \\frac{3\\cdot \\sqrt{2}}{4} + \\frac{6}{5}, - \\frac{\\sqrt{466 - 120\\cdot \\sqrt{2}}}{20} + \\frac{3\\cdot \\sqrt{2}}{4} + \\frac{6}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "59d9d84156ae2522", "problem_id": "17ae47d9dad68f75", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{12 x}{5} - \\frac{3 \\sqrt{2} x}{2} + \\frac{7}{5} + \\frac{21 \\sqrt{2}}{10}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{12}{5} - \\frac{3 \\sqrt{2}}{2}$, c=$\\frac{7}{5} + \\frac{21 \\sqrt{2}}{10}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{233}{50} - \\frac{6 \\sqrt{2}}{5}$, so √Δ = $\\sqrt{\\frac{233}{50} - \\frac{6 \\sqrt{2}}{5}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{466 - 120 \\sqrt{2}}}{20} + \\frac{3 \\sqrt{2}}{4} + \\frac{6}{5}$.\n7) x2 = $- \\frac{\\sqrt{466 - 120 \\sqrt{2}}}{20} + \\frac{3 \\sqrt{2}}{4} + \\frac{6}{5}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{466 - 120\\cdot \\sqrt{2}}}{20} + \\frac{3\\cdot \\sqrt{2}}{4} + \\frac{6}{5}, - \\frac{\\sqrt{466 - 120\\cdot \\sqrt{2}}}{20} + \\frac{3\\cdot \\sqrt{2}}{4} + \\frac{6}{5}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{466 - 120\\cdot \\sqrt{2}}}{20} + \\frac{3\\cdot \\sqrt{2}}{4} + \\frac{6}{5}, - \\frac{\\sqrt{466 - 120\\cdot \\sqrt{2}}}{20} + \\frac{3\\cdot \\sqrt{2}}{4} + \\frac{6}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dde08a9ecd819019", "problem_id": "62da60176287b7d5", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the congruences: x≡3 (mod 4), x≡3 (mod 7). Give the solution modulo 28.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 28.\nCompute the CRT solution to get x ≡ 3 (mod 28).\nFinal answer: $x\\equiv 3\\pmod{28}$", "final_answer": "x\\equiv 3\\pmod{28}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dc9ce8309dd7b8f5", "problem_id": "62da60176287b7d5", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡3 (mod 4), x≡3 (mod 7). Give the solution modulo 28.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 28.\n3) Compute the CRT solution to get x ≡ 3 (mod 28).\nFinal answer: $x\\equiv 3\\pmod{28}$", "final_answer": "x\\equiv 3\\pmod{28}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d591de0c57b3fed6", "problem_id": "f817667b684bf831", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$- 3 x - 7 y$ = $-48$\n$3 x - 7 y$ = $-78$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $3$: $- 9 x - 21 y$ = $-144$.\n3) Multiply the second equation by $-3$: $- 9 x + 21 y$ = $234$.\n4) Subtract to eliminate x: $- 42 y$ = $-378$.\n5) Solve for y: y = $9$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, 9\\right)$", "final_answer": "(x,y)=\\left(-5, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6e884dc35f9ce452", "problem_id": "f817667b684bf831", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$- 3 x - 7 y$ = $-48$\n$3 x - 7 y$ = $-78$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $- 9 x - 21 y$ = $-144$.\nMultiply the second equation by $-3$: $- 9 x + 21 y$ = $234$.\nSubtract to eliminate x: $- 42 y$ = $-378$.\nSolve for y: y = $9$.\nSubstitute back to find x: x = $-5$.\nCheck: plug (x,y)=(-5,9) into both equations.\nFinal answer: $(x,y)=\\left(-5, 9\\right)$", "final_answer": "(x,y)=\\left(-5, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3280ca74520b0e16", "problem_id": "02da5235a61ba49e", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Worker A finishes a job in 4 hours and Worker B finishes it in 5 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/4 job/hour. B's rate = 1/5 job/hour.\nCombined rate = 1/4 + 1/5 = 9/20 job/hour.\nTime = 1 ÷ (combined rate) = 20/9 hours.\nCheck: (combined rate)×(time) = 9/20×20/9 = 1.\nFinal answer: $\\frac{20}{9}$", "final_answer": "\\frac{20}{9}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bef2005e0b302138", "problem_id": "02da5235a61ba49e", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Worker A finishes a job in 4 hours and Worker B finishes it in 5 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/4 job/hour. B's rate = 1/5 job/hour.\n2) Combined rate = 1/4 + 1/5 = 9/20 job/hour.\n3) Time = 1 ÷ (combined rate) = 20/9 hours.\nFinal answer: $\\frac{20}{9}$", "final_answer": "\\frac{20}{9}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4e1eb5195caf4bf3", "problem_id": "075f3b878c420b38", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Find the multiplicative inverse of 19 modulo 43.", "output": "We want x such that 19x ≡ 1 (mod 43).\nUse extended Euclid to find s,t with 19s + 43t = 1.\nOne Bezout identity is: 19·(-9) + 43·(4) = 1.\nFinal answer: $x\\equiv 34\\pmod{43}$", "final_answer": "x\\equiv 34\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "422522f4ae8cd8c4", "problem_id": "075f3b878c420b38", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the multiplicative inverse of 19 modulo 43.", "output": "1) We want x such that 19x ≡ 1 (mod 43).\n2) Use extended Euclid to find s,t with 19s + 43t = 1.\n3) One Bezout identity is: 19·(-9) + 43·(4) = 1.\n4) Reduce s modulo 43: x ≡ 34 (mod 43).\nFinal answer: $x\\equiv 34\\pmod{43}$", "final_answer": "x\\equiv 34\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f98b9cc804d0736a", "problem_id": "26729ebaf0c2daf2", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Worker A finishes a job in 17 hours and Worker B finishes it in 18 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/17 job/hour. B's rate = 1/18 job/hour.\n2) Combined rate = 1/17 + 1/18 = 35/306 job/hour.\n3) Time = 1 ÷ (combined rate) = 306/35 hours.\nFinal answer: $\\frac{306}{35}$", "final_answer": "\\frac{306}{35}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "585b54f3b1aed6ec", "problem_id": "26729ebaf0c2daf2", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Worker A finishes a job in 17 hours and Worker B finishes it in 18 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/17 job/hour. B's rate = 1/18 job/hour.\nCombined rate = 1/17 + 1/18 = 35/306 job/hour.\nTime = 1 ÷ (combined rate) = 306/35 hours.\nFinal answer: $\\frac{306}{35}$", "final_answer": "\\frac{306}{35}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1d733f917f54c3b1", "problem_id": "084502bdb2df488d", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} + 3 x^{2} + 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=0$.", "output": "1) Differentiate: f'(x) = $- 3 x^{2} + 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 2$.\n2) Slope at x=0: m = f'(0) = $\\frac{5}{2}$.\n3) Point on curve: (0, f(0)) = (0, $0$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 0 = \\frac{5}{2}(x − 0).\n6) Optionally expand: y = $\\frac{5 x}{2}$.\nFinal answer: $y = \\frac{5\\cdot x}{2}$", "final_answer": "y = \\frac{5\\cdot x}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a252f3b0aa0b3cb", "problem_id": "d6334c18f690852d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} \\cos{\\left(x^{2} + 3 x + 1 \\right)} + 2 \\left(x + 1\\right) e^{x}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x e^{x}) = 2 x e^{x}$.\n3) So $f'(x) = - 6 x^{3} \\sin{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} - 9 x^{2} \\sin{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 2 x e^{x} + 6 x \\cos{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 2 e^{x}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x e^{x}) = 2 x e^{x} + 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = - 3 x^{2} \\left(2 x + 3\\right) \\sin{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 6 x \\cos{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 2 \\left(x + 1\\right) e^{x} + 2 e^{x}$.\nFinal answer: $- 3 x^{2} \\left(2 x + 3\\right) \\sin{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 6 x \\cos{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 2 \\left(x + 1\\right) e^{x} + 2 e^{x}$", "final_answer": "- 3 x^{2} \\left(2 x + 3\\right) \\sin{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 6 x \\cos{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 2 \\left(x + 1\\right) e^{x} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8bb3800e01af3f38", "problem_id": "df73769da04a24f4", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 9 red balls and 8 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(17,6).\n3) Favorable samples: choose k reds and n−k blues: C(9,1)·C(8,5).\n4) Compute totals: C(17,6) = 12376, and favorable = 504.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{9}{221}$", "final_answer": "\\frac{9}{221}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "79f5942f4e6b132e", "problem_id": "df73769da04a24f4", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 9 red balls and 8 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(17,6).\nFavorable samples: choose k reds and n−k blues: C(9,1)·C(8,5).\nCompute totals: C(17,6) = 12376, and favorable = 504.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{9}{221}$.\nFinal answer: $\\frac{9}{221}$", "final_answer": "\\frac{9}{221}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "38e75e944570998d", "problem_id": "f096d7019fec8096", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Two friends split $1350 in the ratio 8:1. How much does each receive?", "output": "1) Total ratio parts = 8+1 = 9.\n2) Each part = 1350 ÷ 9 = 150.\n3) First share = 8 × 150 = 1200.\n4) Second share = 1 × 150 = 150.\nFinal answer: $150$", "final_answer": "150", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e8d91e911b85e69", "problem_id": "d8ba5ecd6ddbe0bf", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 3 x^{3} - 6 x^{2} - 9 x - 15$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{3} - 6 x^{2} - 10 x - 15.\n3) Then the limit is -83.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -81.\nFinal answer: $-81$", "final_answer": "-81", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "139cb6fe77378549", "problem_id": "596cc6bf8630b02d", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $1441 in the ratio 6:5. How much does each receive?", "output": "Total ratio parts = 6+5 = 11.\nEach part = 1441 ÷ 11 = 131.\nFirst share = 6 × 131 = 786.\nSecond share = 5 × 131 = 655.\nCheck: 786+655=1441.\nFinal answer: $655$", "final_answer": "655", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "66dd24667351da68", "problem_id": "596cc6bf8630b02d", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $1441 in the ratio 6:5. How much does each receive?", "output": "1) Total ratio parts = 6+5 = 11.\n2) Each part = 1441 ÷ 11 = 131.\n3) First share = 6 × 131 = 786.\n4) Second share = 5 × 131 = 655.\nFinal answer: $655$", "final_answer": "655", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d65ca051daf73661", "problem_id": "e636bb92552ab755", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - x^{2} + x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=2$.", "output": "Differentiate: f'(x) = $- 9 x^{2} - 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 1$.\nSlope at x=2: m = f'(2) = $-39 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\nPoint on curve: (2, f(2)) = (2, $-27 + \\cos{\\left(4 \\right)} + e$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -27 + \\cos{\\left(4 \\right)} + e = -39 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\nOptionally expand: y = $- 39 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 51$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 39\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 51$", "final_answer": "y = - 39\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 51", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a07241951a9158e0", "problem_id": "e636bb92552ab755", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - x^{2} + x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 9 x^{2} - 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 1$.\n2) Slope at x=2: m = f'(2) = $-39 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\n3) Point on curve: (2, f(2)) = (2, $-27 + \\cos{\\left(4 \\right)} + e$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -27 + \\cos{\\left(4 \\right)} + e = -39 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\n6) Optionally expand: y = $- 39 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 51$.\nFinal answer: $y = - 39\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 51$", "final_answer": "y = - 39\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 51", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "86c00ca628866864", "problem_id": "01ea5dc6636699b2", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$5 x - y$ = $-20$\n$3 x + 2 y$ = $-25$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $3$: $15 x - 3 y$ = $-60$.\n3) Multiply the second equation by $5$: $15 x + 10 y$ = $-125$.\n4) Subtract to eliminate x: $- 13 y$ = $65$.\n5) Solve for y: y = $-5$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, -5\\right)$", "final_answer": "(x,y)=\\left(-5, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eaa3e71bb6f88f9c", "problem_id": "01ea5dc6636699b2", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$5 x - y$ = $-20$\n$3 x + 2 y$ = $-25$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $15 x - 3 y$ = $-60$.\nMultiply the second equation by $5$: $15 x + 10 y$ = $-125$.\nSubtract to eliminate x: $- 13 y$ = $65$.\nSolve for y: y = $-5$.\nSubstitute back to find x: x = $-5$.\nCheck: plug (x,y)=(-5,-5) into both equations.\nFinal answer: $(x,y)=\\left(-5, -5\\right)$", "final_answer": "(x,y)=\\left(-5, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2721acedf3aec646", "problem_id": "8c6c7805a65d73ba", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$x + 2 y$ = $-3$\n$4 x - 2 y$ = $-2$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $4 x + 8 y$ = $-12$.\n3) Multiply the second equation by $1$: $4 x - 2 y$ = $-2$.\n4) Subtract to eliminate x: $10 y$ = $-10$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, -1\\right)$", "final_answer": "(x,y)=\\left(-1, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a5b6d7cfce91495", "problem_id": "8c6c7805a65d73ba", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$x + 2 y$ = $-3$\n$4 x - 2 y$ = $-2$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $4 x + 8 y$ = $-12$.\nMultiply the second equation by $1$: $4 x - 2 y$ = $-2$.\nSubtract to eliminate x: $10 y$ = $-10$.\nSolve for y: y = $-1$.\nSubstitute back to find x: x = $-1$.\nCheck: plug (x,y)=(-1,-1) into both equations.\nFinal answer: $(x,y)=\\left(-1, -1\\right)$", "final_answer": "(x,y)=\\left(-1, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ab6e5608f83e65eb", "problem_id": "7781ebd081116ebb", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Worker A finishes a job in 21 hours and Worker B finishes it in 13 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/21 job/hour. B's rate = 1/13 job/hour.\n2) Combined rate = 1/21 + 1/13 = 34/273 job/hour.\n3) Time = 1 ÷ (combined rate) = 273/34 hours.\nFinal answer: $\\frac{273}{34}$", "final_answer": "\\frac{273}{34}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "36ba26efc666cd63", "problem_id": "7781ebd081116ebb", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Worker A finishes a job in 21 hours and Worker B finishes it in 13 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/21 job/hour. B's rate = 1/13 job/hour.\nCombined rate = 1/21 + 1/13 = 34/273 job/hour.\nTime = 1 ÷ (combined rate) = 273/34 hours.\nCheck: (combined rate)×(time) = 34/273×273/34 = 1.\nFinal answer: $\\frac{273}{34}$", "final_answer": "\\frac{273}{34}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "39c40153f8ccaac1", "problem_id": "28e4a0b32d43f2c9", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 15 red balls and 27 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(42,8).\nFavorable samples: choose k reds and n−k blues: C(15,2)·C(27,6).\nCompute totals: C(42,8) = 118030185, and favorable = 31081050.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{7590}{28823}$.\nFinal answer: $\\frac{7590}{28823}$", "final_answer": "\\frac{7590}{28823}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c4859ae64e980a65", "problem_id": "28e4a0b32d43f2c9", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "An urn has 15 red balls and 27 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(42,8).\n3) Favorable samples: choose k reds and n−k blues: C(15,2)·C(27,6).\n4) Compute totals: C(42,8) = 118030185, and favorable = 31081050.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{7590}{28823}$", "final_answer": "\\frac{7590}{28823}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fe42243dda5cfed0", "problem_id": "2ae7a5f235c59322", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A cyclist rides at 84 miles per hour for 2 hours, then at 36 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 84 × 2 = 168 miles.\n2) Second leg distance = 36 × 5/2 = 90 miles.\n3) Total distance = 168 + 90 = 258 miles.\nFinal answer: $258$", "final_answer": "258", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "478e34cb86030e67", "problem_id": "59c32a17a05252ed", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 3 x^{3} + 11 x^{2} - 31 x + 92$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{3} + 11 x^{2} - 32 x + 92.\n3) Then the limit is 368.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: 365.\nFinal answer: $365$", "final_answer": "365", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2b8e641abce9a510", "problem_id": "0eb6cb2e352a8e27", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} + 3 x^{2} + 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 3$ at $x=1$.", "output": "Differentiate: f'(x) = $9 x^{2} + 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 3$.\nSlope at x=1: m = f'(1) = $- 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 18$.\nPoint on curve: (1, f(1)) = (1, $\\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} + 6$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} + 6 = - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 18(x − 1).\nOptionally expand: y = $- 2 x \\sin{\\left(2 \\right)} + \\frac{x e^{\\frac{1}{2}}}{2} + 18 x - 12 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2 \\sin{\\left(2 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 18\\cdot x - 12 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 18\\cdot x - 12 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "07595d737b2b3e70", "problem_id": "0eb6cb2e352a8e27", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} + 3 x^{2} + 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 3$ at $x=1$.", "output": "1) Differentiate: f'(x) = $9 x^{2} + 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 3$.\n2) Slope at x=1: m = f'(1) = $- 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 18$.\n3) Point on curve: (1, f(1)) = (1, $\\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} + 6$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} + 6 = - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 18(x − 1).\n6) Optionally expand: y = $- 2 x \\sin{\\left(2 \\right)} + \\frac{x e^{\\frac{1}{2}}}{2} + 18 x - 12 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 18\\cdot x - 12 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 18\\cdot x - 12 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d106646d75f8270e", "problem_id": "25f93278bfbfee81", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A cyclist rides at 80 miles per hour for 3/2 hours, then at 66 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 80 × 3/2 = 120 miles.\nSecond leg distance = 66 × 3 = 198 miles.\nTotal distance = 120 + 198 = 318 miles.\nFinal answer: $318$", "final_answer": "318", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "31459c7f6502df3c", "problem_id": "25f93278bfbfee81", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A cyclist rides at 80 miles per hour for 3/2 hours, then at 66 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 80 × 3/2 = 120 miles.\nSecond leg distance = 66 × 3 = 198 miles.\nTotal distance = 120 + 198 = 318 miles.\nCheck: adding both legs gives 318.\nFinal answer: $318$", "final_answer": "318", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9c00ac500bc5874a", "problem_id": "cdfb3f1e13717526", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_5=27$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_5=17.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=17 and k=5.\nCompute C(21,4) = 5985.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $5985$", "final_answer": "5985", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a19493d824e12a28", "problem_id": "cdfb3f1e13717526", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_5=27$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_5=17.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=17 and k=5.\nFinal answer: $5985$", "final_answer": "5985", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "32f76c28b2b41fb8", "problem_id": "e4daba8c8a3696f3", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Two friends split $510 in the ratio 7:8. How much does each receive?", "output": "1) Total ratio parts = 7+8 = 15.\n2) Each part = 510 ÷ 15 = 34.\n3) First share = 7 × 34 = 238.\n4) Second share = 8 × 34 = 272.\nFinal answer: $272$", "final_answer": "272", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0f6ee3d9e5eb06f7", "problem_id": "e4daba8c8a3696f3", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Two friends split $510 in the ratio 7:8. How much does each receive?", "output": "Total ratio parts = 7+8 = 15.\nEach part = 510 ÷ 15 = 34.\nFirst share = 7 × 34 = 238.\nFinal answer: $272$", "final_answer": "272", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b43d9ab37fed6692", "problem_id": "3fabd6667125f0dc", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡5 (mod 9), x≡4 (mod 8). Give the solution modulo 72.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 72.\nCompute the CRT solution to get x ≡ 68 (mod 72).\nCheck: 68 mod 9=5 and mod 8=4.\nFinal answer: $x\\equiv 68\\pmod{72}$", "final_answer": "x\\equiv 68\\pmod{72}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "432502fec7b32822", "problem_id": "3fabd6667125f0dc", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡5 (mod 9), x≡4 (mod 8). Give the solution modulo 72.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 72.\n3) Compute the CRT solution to get x ≡ 68 (mod 72).\nFinal answer: $x\\equiv 68\\pmod{72}$", "final_answer": "x\\equiv 68\\pmod{72}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1dcc036de12239b2", "problem_id": "044e9b2474553f02", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the multiplicative inverse of 21 modulo 41.", "output": "1) We want x such that 21x ≡ 1 (mod 41).\n2) Use extended Euclid to find s,t with 21s + 41t = 1.\n3) One Bezout identity is: 21·(2) + 41·(-1) = 1.\n4) Reduce s modulo 41: x ≡ 2 (mod 41).\nFinal answer: $x\\equiv 2\\pmod{41}$", "final_answer": "x\\equiv 2\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "90b3972158a132c9", "problem_id": "044e9b2474553f02", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 21 modulo 41.", "output": "We want x such that 21x ≡ 1 (mod 41).\nUse extended Euclid to find s,t with 21s + 41t = 1.\nOne Bezout identity is: 21·(2) + 41·(-1) = 1.\nReduce s modulo 41: x ≡ 2 (mod 41).\nCheck: 21·2 mod 41 = 1.\nFinal answer: $x\\equiv 2\\pmod{41}$", "final_answer": "x\\equiv 2\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7a3a77c813408002", "problem_id": "901d42f9c7106fba", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 11 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=11, k=8, p=$\\frac{1}{5}$ and simplify.\nHere C(11,8) = 165.\nCheck: expression simplifies to $\\frac{2112}{9765625}$.\nFinal answer: $\\frac{2112}{9765625}$", "final_answer": "\\frac{2112}{9765625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "77696a25b406ce0b", "problem_id": "901d42f9c7106fba", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 11 times, what is the probability of exactly 8 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=8, p=$\\frac{1}{5}$ and simplify.\n4) Here C(11,8) = 165.\nFinal answer: $\\frac{2112}{9765625}$", "final_answer": "\\frac{2112}{9765625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "49ba00d5248552d0", "problem_id": "99645cc2e0769971", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$- 6 x - 7 y$ = $-78$\n$- 5 x - 5 y$ = $-60$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-5$: $30 x + 35 y$ = $390$.\n3) Multiply the second equation by $-6$: $30 x + 30 y$ = $360$.\n4) Subtract to eliminate x: $5 y$ = $30$.\n5) Solve for y: y = $6$.\n6) Substitute back to find x: x = $6$.\nFinal answer: $(x,y)=\\left(6, 6\\right)$", "final_answer": "(x,y)=\\left(6, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1b317b3e7e5b16a6", "problem_id": "99645cc2e0769971", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$- 6 x - 7 y$ = $-78$\n$- 5 x - 5 y$ = $-60$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-5$: $30 x + 35 y$ = $390$.\nMultiply the second equation by $-6$: $30 x + 30 y$ = $360$.\nSubtract to eliminate x: $5 y$ = $30$.\nSolve for y: y = $6$.\nSubstitute back to find x: x = $6$.\nCheck: plug (x,y)=(6,6) into both equations.\nFinal answer: $(x,y)=\\left(6, 6\\right)$", "final_answer": "(x,y)=\\left(6, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ae70cb32bcd68dbd", "problem_id": "876a97e316bf41fb", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡0 (mod 11), x≡5 (mod 9). Give the solution modulo 99.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 99.\n3) Compute the CRT solution to get x ≡ 77 (mod 99).\nFinal answer: $x\\equiv 77\\pmod{99}$", "final_answer": "x\\equiv 77\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "557b1e47c07fa429", "problem_id": "876a97e316bf41fb", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the congruences: x≡0 (mod 11), x≡5 (mod 9). Give the solution modulo 99.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 99.\nCompute the CRT solution to get x ≡ 77 (mod 99).\nFinal answer: $x\\equiv 77\\pmod{99}$", "final_answer": "x\\equiv 77\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "24c4148cd169046f", "problem_id": "4b3b2cf67a73a898", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "You want 32 liters of a 45% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 32−x liters is 50% solution.\n2) Salt equation: 25x + 50(32−x) = 45·32.\n3) (25-50)x = (45-50)·32 ⇒ x = 32/5.\n4) Then 32−x = 128/5.\nFinal answer: $32/5 L of 25% and 128/5 L of 50%$", "final_answer": "32/5 L of 25% and 128/5 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "50e6b337890f49ef", "problem_id": "4b3b2cf67a73a898", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "You want 32 liters of a 45% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 32−x liters is 50% solution.\nSalt equation: 25x + 50(32−x) = 45·32.\n(25-50)x = (45-50)·32 ⇒ x = 32/5.\nThen 32−x = 128/5.\nCheck: 25·32/5 + 50·128/5 = 45·32.\nFinal answer: $32/5 L of 25% and 128/5 L of 50%$", "final_answer": "32/5 L of 25% and 128/5 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5fb1d5dc2c90c0b4", "problem_id": "3de38b75669f2944", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A store sells a backpack for $346. It is discounted by 10%. You buy 9 backpacks. What is the total cost?", "output": "1) Discounted price = 346 × (1 − 10/100) = 346 × 90/100 = 1557/5.\n2) Subtotal = 9 × 1557/5 = 14013/5.\nFinal answer: $\\frac{14013}{5}$", "final_answer": "\\frac{14013}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4be64b4ea7319abd", "problem_id": "3de38b75669f2944", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A store sells a backpack for $346. It is discounted by 10%. You buy 9 backpacks. What is the total cost?", "output": "Discounted price = 346 × (1 − 10/100) = 346 × 90/100 = 1557/5.\nSubtotal = 9 × 1557/5 = 14013/5.\nCheck: recomputing the formula yields 14013/5.\nFinal answer: $\\frac{14013}{5}$", "final_answer": "\\frac{14013}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "669ef62309b82c6f", "problem_id": "df3a5de99aac9e76", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} + 3$ at $x=-1$.", "output": "Differentiate: f'(x) = $- 6 x + 2 \\cos{\\left(x \\right)} - 2$.\nSlope at x=-1: m = f'(-1) = $2 \\cos{\\left(1 \\right)} + 4$.\nPoint on curve: (-1, f(-1)) = (-1, $2 - 2 \\sin{\\left(1 \\right)}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 2 - 2 \\sin{\\left(1 \\right)} = 2 \\cos{\\left(1 \\right)} + 4(x − -1).\nOptionally expand: y = $2 x \\cos{\\left(1 \\right)} + 4 x - 2 \\sin{\\left(1 \\right)} + 2 \\cos{\\left(1 \\right)} + 6$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 4\\cdot x - 2\\cdot \\sin{\\left(1 \\right)} + 2\\cdot \\cos{\\left(1 \\right)} + 6$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 4\\cdot x - 2\\cdot \\sin{\\left(1 \\right)} + 2\\cdot \\cos{\\left(1 \\right)} + 6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9717f02fde4bba76", "problem_id": "df3a5de99aac9e76", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} + 3$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $- 6 x + 2 \\cos{\\left(x \\right)} - 2$.\n2) Slope at x=-1: m = f'(-1) = $2 \\cos{\\left(1 \\right)} + 4$.\n3) Point on curve: (-1, f(-1)) = (-1, $2 - 2 \\sin{\\left(1 \\right)}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 2 - 2 \\sin{\\left(1 \\right)} = 2 \\cos{\\left(1 \\right)} + 4(x − -1).\n6) Optionally expand: y = $2 x \\cos{\\left(1 \\right)} + 4 x - 2 \\sin{\\left(1 \\right)} + 2 \\cos{\\left(1 \\right)} + 6$.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 4\\cdot x - 2\\cdot \\sin{\\left(1 \\right)} + 2\\cdot \\cos{\\left(1 \\right)} + 6$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 4\\cdot x - 2\\cdot \\sin{\\left(1 \\right)} + 2\\cdot \\cos{\\left(1 \\right)} + 6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9b256efa6288c9f8", "problem_id": "1e679e243826c9e8", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Worker A finishes a job in 22 hours and Worker B finishes it in 8 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/22 job/hour. B's rate = 1/8 job/hour.\n2) Combined rate = 1/22 + 1/8 = 15/88 job/hour.\n3) Time = 1 ÷ (combined rate) = 88/15 hours.\nFinal answer: $\\frac{88}{15}$", "final_answer": "\\frac{88}{15}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d041ebe399f3197f", "problem_id": "1e679e243826c9e8", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Worker A finishes a job in 22 hours and Worker B finishes it in 8 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "$\\frac{88}{15}$", "final_answer": "\\frac{88}{15}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9662ba79ac40bfb9", "problem_id": "e0e4760e660d9ecf", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{5 x}{2} + \\frac{3}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$- \\frac{5}{2}$, c=$\\frac{3}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{1}{4}$, so √Δ = $\\frac{1}{2}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{3}{4}$.\n7) x2 = $\\frac{1}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{3}{4}, \\frac{1}{2}\\}$", "final_answer": "x \\in \\{\\frac{3}{4}, \\frac{1}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "66413d2ac3de6e32", "problem_id": "e0e4760e660d9ecf", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{5 x}{2} + \\frac{3}{4}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$- \\frac{5}{2}$, c=$\\frac{3}{4}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{1}{4}$, so √Δ = $\\frac{1}{2}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{3}{4}$.\nx2 = $\\frac{1}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{3}{4}, \\frac{1}{2}\\}$", "final_answer": "x \\in \\{\\frac{3}{4}, \\frac{1}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f0110ef150692666", "problem_id": "8eb508a8dbcdc0f8", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A cyclist rides at 21 miles per hour for 1/2 hours, then at 37 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 21 × 1/2 = 21/2 miles.\nSecond leg distance = 37 × 1 = 37 miles.\nTotal distance = 21/2 + 37 = 95/2 miles.\nCheck: adding both legs gives 95/2.\nFinal answer: $\\frac{95}{2}$", "final_answer": "\\frac{95}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "675d3bd3914dcd11", "problem_id": "8eb508a8dbcdc0f8", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A cyclist rides at 21 miles per hour for 1/2 hours, then at 37 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 21 × 1/2 = 21/2 miles.\n2) Second leg distance = 37 × 1 = 37 miles.\n3) Total distance = 21/2 + 37 = 95/2 miles.\nFinal answer: $\\frac{95}{2}$", "final_answer": "\\frac{95}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eafd664409157578", "problem_id": "0d3fb63f15e25210", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(2 x^{2} + 2 x - 3\\right) \\cos{\\left(3 x - 1 \\right)}$.", "output": "Let f(x) = $2 \\left(x + 2\\right) e^{x} + \\left(2 x^{2} + 2 x - 3\\right) \\cos{\\left(3 x - 1 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $2 \\left(x + 2\\right) e^{x} + 2 \\left(2 x + 1\\right) \\cos{\\left(3 x - 1 \\right)} - 3 \\left(2 x^{2} + 2 x - 3\\right) \\sin{\\left(3 x - 1 \\right)} + 2 e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(4\\cdot x + 2\\right)\\cdot \\cos{\\left(3\\cdot x - 1 \\right)} - \\left(6\\cdot x^{2} + 6\\cdot x - 9\\right)\\cdot \\sin{\\left(3\\cdot x - 1 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(4\\cdot x + 2\\right)\\cdot \\cos{\\left(3\\cdot x - 1 \\right)} - \\left(6\\cdot x^{2} + 6\\cdot x - 9\\right)\\cdot \\sin{\\left(3\\cdot x - 1 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d6715a3f9fc3dba", "problem_id": "0d3fb63f15e25210", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(2 x^{2} + 2 x - 3\\right) \\cos{\\left(3 x - 1 \\right)}$.", "output": "Let f(x) = $2 \\left(x + 2\\right) e^{x} + \\left(2 x^{2} + 2 x - 3\\right) \\cos{\\left(3 x - 1 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $2 \\left(x + 2\\right) e^{x} + 2 \\left(2 x + 1\\right) \\cos{\\left(3 x - 1 \\right)} - 3 \\left(2 x^{2} + 2 x - 3\\right) \\sin{\\left(3 x - 1 \\right)} + 2 e^{x}$.\nFinal answer: $\\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(4\\cdot x + 2\\right)\\cdot \\cos{\\left(3\\cdot x - 1 \\right)} - \\left(6\\cdot x^{2} + 6\\cdot x - 9\\right)\\cdot \\sin{\\left(3\\cdot x - 1 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(4\\cdot x + 2\\right)\\cdot \\cos{\\left(3\\cdot x - 1 \\right)} - \\left(6\\cdot x^{2} + 6\\cdot x - 9\\right)\\cdot \\sin{\\left(3\\cdot x - 1 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d2a69608f3dae421", "problem_id": "6ffbb966a04e4bba", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x + 3$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 3.\n3) Then the limit is 3.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 5.\nFinal answer: $5$", "final_answer": "5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7d5b247eac0a1b54", "problem_id": "a7fcd75d24aa26da", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 12 times, what is the probability of exactly 9 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=9, p=$\\frac{1}{8}$ and simplify.\n4) Here C(12,9) = 220.\nFinal answer: $\\frac{18865}{17179869184}$", "final_answer": "\\frac{18865}{17179869184}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9d3a977a7a7aacff", "problem_id": "7f2eee90b12eb3d8", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_7=34$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_7=13.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=13 and k=7.\n4) Compute C(19,6) = 27132.\nFinal answer: $27132$", "final_answer": "27132", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "841fbd3763458865", "problem_id": "f4bb53485da72686", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{4} - 3 x^{2} + x - 4$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x) = x$.\n3) So $f'(x) = 16 x^{3} - 5 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x) = 1$.\nTherefore the correct derivative is $f'(x) = 16 x^{3} - 6 x + 1$.\nFinal answer: $16 x^{3} - 6 x + 1$", "final_answer": "16 x^{3} - 6 x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "768b9343ebc8fa2f", "problem_id": "4386212b541ffa65", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A store sells a backpack for $320. It is discounted by 10%. You buy 4 backpacks. What is the total cost?", "output": "1) Discounted price = 320 × (1 − 10/100) = 320 × 90/100 = 288.\n2) Subtotal = 4 × 288 = 1152.\nFinal answer: $1152$", "final_answer": "1152", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "32299974a31216a0", "problem_id": "276c94ad624a95fe", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x^{3} - 4 x^{2} - 7 x - 12$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - x^{3} - 4 x^{2} - 8 x - 12.\n3) Then the limit is -52.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -50.\nFinal answer: $-50$", "final_answer": "-50", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5cc7c963d817291b", "problem_id": "a3c761c3af80b20d", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 8 x + 7$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} + 8 x + 7$ = $\\left(x + 1\\right) \\left(x + 7\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-1$.\n5) From the second factor: x = $-7$.\nFinal answer: $x \\in \\{-1, -7\\}$", "final_answer": "x \\in \\{-1, -7\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3dac2abc3a0a692c", "problem_id": "a3c761c3af80b20d", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 8 x + 7$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} + 8 x + 7$ = $\\left(x + 1\\right) \\left(x + 7\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $-1$.\nFrom the second factor: x = $-7$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-1, -7\\}$", "final_answer": "x \\in \\{-1, -7\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a4e67ee1a467a24", "problem_id": "a8f5b409f0508d7a", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡4 (mod 11), x≡7 (mod 8). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 15 (mod 88).\nFinal answer: $x\\equiv 15\\pmod{88}$", "final_answer": "x\\equiv 15\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b3d354133c07afde", "problem_id": "abbc71750aa82b49", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x^{2} - 5 x - 11$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} - 6 x - 11.\n3) Then the limit is -27.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -25.\nFinal answer: $-25$", "final_answer": "-25", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "dbc770fa748353d9", "problem_id": "81a39ca2a1c47c37", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} - 3 x^{3} + x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{3} + x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1$.\n2) Evaluate at the bounds: F(2 \\pi) and F(0).\n3) Subtract: F(2 \\pi) − F(0) = $\\frac{2 \\pi \\left(- 18 \\pi^{3} + 3 + 9 \\pi + 4 \\pi^{2}\\right)}{3}$.\n4) Simplify the result.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(- 18\\cdot \\pi^{3} + 3 + 9\\cdot \\pi + 4\\cdot \\pi^{2}\\right)}{3}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(- 18\\cdot \\pi^{3} + 3 + 9\\cdot \\pi + 4\\cdot \\pi^{2}\\right)}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8b7ea3a72b35eeb7", "problem_id": "81a39ca2a1c47c37", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} - 3 x^{3} + x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{3} + x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1$.\nEvaluate at the bounds: F(2 \\pi) and F(0).\nSubtract: F(2 \\pi) − F(0) = $\\frac{2 \\pi \\left(- 18 \\pi^{3} + 3 + 9 \\pi + 4 \\pi^{2}\\right)}{3}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(- 18\\cdot \\pi^{3} + 3 + 9\\cdot \\pi + 4\\cdot \\pi^{2}\\right)}{3}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(- 18\\cdot \\pi^{3} + 3 + 9\\cdot \\pi + 4\\cdot \\pi^{2}\\right)}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a70963bb61a4f902", "problem_id": "97f19d8f6c772a22", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=34$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=34 and k=10.\n4) Compute C(43,9) = 563921995.\nFinal answer: $563921995$", "final_answer": "563921995", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5ba50e0894cd7c81", "problem_id": "97f19d8f6c772a22", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=34$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=34 and k=10.\nCompute C(43,9) = 563921995.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $563921995$", "final_answer": "563921995", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a95e0a8ded24c573", "problem_id": "d9bb5932f303d88d", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the multiplicative inverse of 27 modulo 37.", "output": "1) We want x such that 27x ≡ 1 (mod 37).\n2) Use extended Euclid to find s,t with 27s + 37t = 1.\n3) One Bezout identity is: 27·(11) + 37·(-8) = 1.\n4) Reduce s modulo 37: x ≡ 11 (mod 37).\nFinal answer: $x\\equiv 11\\pmod{37}$", "final_answer": "x\\equiv 11\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8055073c399aee35", "problem_id": "d9bb5932f303d88d", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the multiplicative inverse of 27 modulo 37.", "output": "We want x such that 27x ≡ 1 (mod 37).\nUse extended Euclid to find s,t with 27s + 37t = 1.\nOne Bezout identity is: 27·(11) + 37·(-8) = 1.\nReduce s modulo 37: x ≡ 11 (mod 37).\nCheck: 27·11 mod 37 = 1.\nFinal answer: $x\\equiv 11\\pmod{37}$", "final_answer": "x\\equiv 11\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "59d9541cff4c58c2", "problem_id": "eb46a41234ce0fc6", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "An urn has 9 red balls and 8 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(17,6).\n3) Favorable samples: choose k reds and n−k blues: C(9,3)·C(8,3).\n4) Compute totals: C(17,6) = 12376, and favorable = 4704.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{84}{221}$", "final_answer": "\\frac{84}{221}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9359fc8af45118ec", "problem_id": "eb46a41234ce0fc6", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "An urn has 9 red balls and 8 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(17,6).\nFavorable samples: choose k reds and n−k blues: C(9,3)·C(8,3).\nFinal answer: $\\frac{84}{221}$", "final_answer": "\\frac{84}{221}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "31ada50fb36186c2", "problem_id": "9cd5cf978b9c2ce2", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{4} - x^{3} - 4 x^{2} - x - 1\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $3 x^{4} - x^{3} - 4 x^{2} - x - 1$.\n3) Compute an antiderivative: F(x) = $\\frac{3 x^{5}}{5} - \\frac{x^{4}}{4} - \\frac{4 x^{3}}{3} - \\frac{x^{2}}{2} - x$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{3\\cdot x^{5}}{5} - \\frac{x^{4}}{4} - \\frac{4\\cdot x^{3}}{3} - \\frac{x^{2}}{2} - x$", "final_answer": "C + \\frac{3\\cdot x^{5}}{5} - \\frac{x^{4}}{4} - \\frac{4\\cdot x^{3}}{3} - \\frac{x^{2}}{2} - x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6b2a12444e2d3ec1", "problem_id": "9cd5cf978b9c2ce2", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{4} - x^{3} - 4 x^{2} - x - 1\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $3 x^{4} - x^{3} - 4 x^{2} - x - 1$.\nCompute an antiderivative: F(x) = $\\frac{3 x^{5}}{5} - \\frac{x^{4}}{4} - \\frac{4 x^{3}}{3} - \\frac{x^{2}}{2} - x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{3\\cdot x^{5}}{5} - \\frac{x^{4}}{4} - \\frac{4\\cdot x^{3}}{3} - \\frac{x^{2}}{2} - x$", "final_answer": "C + \\frac{3\\cdot x^{5}}{5} - \\frac{x^{4}}{4} - \\frac{4\\cdot x^{3}}{3} - \\frac{x^{2}}{2} - x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0d8aaa338954014e", "problem_id": "eea7b3175756611f", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "You want 39 liters of a 30% salt solution. You have a 20% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 39−x liters is 60% solution.\n2) Salt equation: 20x + 60(39−x) = 30·39.\n3) (20-60)x = (30-60)·39 ⇒ x = 117/4.\n4) Then 39−x = 39/4.\nFinal answer: $117/4 L of 20% and 39/4 L of 60%$", "final_answer": "117/4 L of 20% and 39/4 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d929c22f7440532", "problem_id": "ac7aa19d28ca3e0c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 2 x^{3} + 8 x^{2} - 22 x + 68$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{3} + 8 x^{2} - 23 x + 68.\n3) Then the limit is 263.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: 260.\nFinal answer: $260$", "final_answer": "260", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "1de7fdbe6bd999e3", "problem_id": "515b81a07cef9576", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} - x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 6 x^{2} + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 1$.\n2) Slope at x=2: m = f'(2) = $-25 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\n3) Point on curve: (2, f(2)) = (2, $-17 + \\cos{\\left(4 \\right)} + e$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -17 + \\cos{\\left(4 \\right)} + e = -25 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\n6) Optionally expand: y = $- 25 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 33$.\nFinal answer: $y = - 25\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 33$", "final_answer": "y = - 25\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 33", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4ac75cf4ae67df3d", "problem_id": "515b81a07cef9576", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} - x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=2$.", "output": "Differentiate: f'(x) = $- 6 x^{2} + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 1$.\nSlope at x=2: m = f'(2) = $-25 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\nPoint on curve: (2, f(2)) = (2, $-17 + \\cos{\\left(4 \\right)} + e$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -17 + \\cos{\\left(4 \\right)} + e = -25 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\nOptionally expand: y = $- 25 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 33$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 25\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 33$", "final_answer": "y = - 25\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 33", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5336bcd494a9e58a", "problem_id": "12717b20a9259407", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x^{3} \\sin{\\left(x \\right)}) = 2 x^{3} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = 2 x^{3} \\cos{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x^{3} \\sin{\\left(x \\right)}) = 2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$", "final_answer": "2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "431dd64cf2a67514", "problem_id": "a0df7841a549c302", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=22$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=22 and k=6.\nCompute C(27,5) = 80730.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $80730$", "final_answer": "80730", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "678329a6ce1f306a", "problem_id": "505fd8ef0f26a20d", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A cyclist rides at 64 miles per hour for 1 hours, then at 74 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 64 × 1 = 64 miles.\n2) Second leg distance = 74 × 1 = 74 miles.\n3) Total distance = 64 + 74 = 138 miles.\nFinal answer: $138$", "final_answer": "138", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f684bc8986e08d9", "problem_id": "505fd8ef0f26a20d", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A cyclist rides at 64 miles per hour for 1 hours, then at 74 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 64 × 1 = 64 miles.\nSecond leg distance = 74 × 1 = 74 miles.\nTotal distance = 64 + 74 = 138 miles.\nCheck: adding both legs gives 138.\nFinal answer: $138$", "final_answer": "138", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c9cbcc509e82f2d3", "problem_id": "be51337964043403", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A cyclist rides at 53 miles per hour for 3/2 hours, then at 41 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 53 × 3/2 = 159/2 miles.\n2) Second leg distance = 41 × 5/2 = 205/2 miles.\n3) Total distance = 159/2 + 205/2 = 182 miles.\nFinal answer: $182$", "final_answer": "182", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6643bfeced0681b5", "problem_id": "be51337964043403", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 53 miles per hour for 3/2 hours, then at 41 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 53 × 3/2 = 159/2 miles.\nSecond leg distance = 41 × 5/2 = 205/2 miles.\nTotal distance = 159/2 + 205/2 = 182 miles.\nCheck: adding both legs gives 182.\nFinal answer: $182$", "final_answer": "182", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b63389a4e6e3d737", "problem_id": "366f1e8953d6dff7", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{3} + x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 3 x^{3} + x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3$.\n3) Compute an antiderivative: F(x) = $- \\frac{3 x^{4}}{4} + \\frac{x^{3}}{3} - x^{2} - 3 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{3\\cdot x^{4}}{4} + \\frac{x^{3}}{3} - x^{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{3\\cdot x^{4}}{4} + \\frac{x^{3}}{3} - x^{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7bbfdcc9396c004c", "problem_id": "dc42cd99772c0e8b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(x^{2} - x + 2\\right) \\cos{\\left(3 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 \\cos{\\left(3 x \\right)}) = - 2 \\sin{\\left(3 x \\right)}$.\n3) So $f'(x) = - 3 x^{2} \\sin{\\left(3 x \\right)} + x e^{x} + 3 x \\sin{\\left(3 x \\right)} + 2 x \\cos{\\left(3 x \\right)} - 2 e^{x} - 2 \\sin{\\left(3 x \\right)} - \\cos{\\left(3 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 \\cos{\\left(3 x \\right)}) = - 6 \\sin{\\left(3 x \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x - 3\\right) e^{x} + \\left(2 x - 1\\right) \\cos{\\left(3 x \\right)} - 3 \\left(x \\left(x - 1\\right) + 2\\right) \\sin{\\left(3 x \\right)} + e^{x}$.\nFinal answer: $\\left(x - 3\\right) e^{x} + \\left(2 x - 1\\right) \\cos{\\left(3 x \\right)} - 3 \\left(x \\left(x - 1\\right) + 2\\right) \\sin{\\left(3 x \\right)} + e^{x}$", "final_answer": "\\left(x - 3\\right) e^{x} + \\left(2 x - 1\\right) \\cos{\\left(3 x \\right)} - 3 \\left(x \\left(x - 1\\right) + 2\\right) \\sin{\\left(3 x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "bbe20f9e559f3f12", "problem_id": "d19ae3e66fbb87f7", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} + x^{2} + x - 1$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $9 x^{2} + 2 x + 1$.\n2) Slope at x=-2: m = f'(-2) = $33$.\n3) Point on curve: (-2, f(-2)) = (-2, $-23$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -23 = 33(x − -2).\n6) Optionally expand: y = $33 x + 43$.\nFinal answer: $y = 33\\cdot x + 43$", "final_answer": "y = 33\\cdot x + 43", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0857436a64f8926b", "problem_id": "7d441d4fa9f7e32e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 2 x^{3} + 2 x^{2} + x - 1$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{3} + 2 x^{2} - 1.\n3) Then the limit is 3.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: 2.\nFinal answer: $2$", "final_answer": "2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6d5058a088c137e6", "problem_id": "727817e90e192247", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Two friends split $742 in the ratio 4:3. How much does each receive?", "output": "$318$", "final_answer": "318", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "93899a3c86ea804f", "problem_id": "727817e90e192247", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Two friends split $742 in the ratio 4:3. How much does each receive?", "output": "Total ratio parts = 4+3 = 7.\nEach part = 742 ÷ 7 = 106.\nFirst share = 4 × 106 = 424.\nSecond share = 3 × 106 = 318.\nCheck: 424+318=742.\nFinal answer: $318$", "final_answer": "318", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c62433aac88c531b", "problem_id": "97bac04f2d2403a9", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} - x^{2} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $- 6 x^{2} - 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\n2) Slope at x=-2: m = f'(-2) = $-22 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}$.\n3) Point on curve: (-2, f(-2)) = (-2, $\\cos{\\left(4 \\right)} + e^{-1} + 14$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(4 \\right)} + e^{-1} + 14 = -22 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}(x − -2).\n6) Optionally expand: y = $- 22 x + 2 x \\sin{\\left(4 \\right)} + \\frac{x}{2 e} - 30 + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$.\nFinal answer: $y = - 22\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 30 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$", "final_answer": "y = - 22\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 30 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0beb80c81b5f1af5", "problem_id": "97bac04f2d2403a9", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} - x^{2} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=-2$.", "output": "$y = - 22\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 30 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$", "final_answer": "y = - 22\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 30 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "99781518ea600d13", "problem_id": "0c991e3e991284f9", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} + 3 x^{2} + 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=0$.", "output": "1) Differentiate: f'(x) = $- 9 x^{2} + 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 2$.\n2) Slope at x=0: m = f'(0) = $\\frac{5}{2}$.\n3) Point on curve: (0, f(0)) = (0, $1$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 1 = \\frac{5}{2}(x − 0).\n6) Optionally expand: y = $\\frac{5 x}{2} + 1$.\nFinal answer: $y = \\frac{5\\cdot x}{2} + 1$", "final_answer": "y = \\frac{5\\cdot x}{2} + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ffdcfebc1d9fed0d", "problem_id": "0c991e3e991284f9", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} + 3 x^{2} + 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=0$.", "output": "Differentiate: f'(x) = $- 9 x^{2} + 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 2$.\nSlope at x=0: m = f'(0) = $\\frac{5}{2}$.\nPoint on curve: (0, f(0)) = (0, $1$).\nFinal answer: $y = \\frac{5\\cdot x}{2} + 1$", "final_answer": "y = \\frac{5\\cdot x}{2} + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d6adc70575aa567", "problem_id": "2fe472136e0123c5", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the multiplicative inverse of 9 modulo 31.", "output": "1) We want x such that 9x ≡ 1 (mod 31).\n2) Use extended Euclid to find s,t with 9s + 31t = 1.\n3) One Bezout identity is: 9·(7) + 31·(-2) = 1.\n4) Reduce s modulo 31: x ≡ 7 (mod 31).\nFinal answer: $x\\equiv 7\\pmod{31}$", "final_answer": "x\\equiv 7\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ac58c4d0d8816768", "problem_id": "2fe472136e0123c5", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Find the multiplicative inverse of 9 modulo 31.", "output": "We want x such that 9x ≡ 1 (mod 31).\nUse extended Euclid to find s,t with 9s + 31t = 1.\nOne Bezout identity is: 9·(7) + 31·(-2) = 1.\nFinal answer: $x\\equiv 7\\pmod{31}$", "final_answer": "x\\equiv 7\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "acf3ae25075440ac", "problem_id": "136df5cfc429ff2c", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{13} x}{2} - 4 x + 4 + 3 \\sqrt{13}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{3 \\sqrt{13}}{2} - 4$, c=$4 + 3 \\sqrt{13}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{2 + \\frac{3\\cdot \\sqrt{13}}{2}, 2\\}$", "final_answer": "x \\in \\{2 + \\frac{3\\cdot \\sqrt{13}}{2}, 2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d78c0f3cc4eb5bda", "problem_id": "136df5cfc429ff2c", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{13} x}{2} - 4 x + 4 + 3 \\sqrt{13}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{3 \\sqrt{13}}{2} - 4$, c=$4 + 3 \\sqrt{13}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{117}{4}$, so √Δ = $\\frac{3 \\sqrt{13}}{2}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $2 + \\frac{3 \\sqrt{13}}{2}$.\nx2 = $2$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{2 + \\frac{3\\cdot \\sqrt{13}}{2}, 2\\}$", "final_answer": "x \\in \\{2 + \\frac{3\\cdot \\sqrt{13}}{2}, 2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bd6626861d31d8b0", "problem_id": "b8bd466d7887cb1c", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$x + y$ = $7$\n$3 x$ = $9$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $3$: $3 x + 3 y$ = $21$.\n3) Multiply the second equation by $1$: $3 x$ = $9$.\n4) Subtract to eliminate x: $3 y$ = $12$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, 4\\right)$", "final_answer": "(x,y)=\\left(3, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8a9b50199bf40975", "problem_id": "b8bd466d7887cb1c", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$x + y$ = $7$\n$3 x$ = $9$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $3 x + 3 y$ = $21$.\nMultiply the second equation by $1$: $3 x$ = $9$.\nSubtract to eliminate x: $3 y$ = $12$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $3$.\nCheck: plug (x,y)=(3,4) into both equations.\nFinal answer: $(x,y)=\\left(3, 4\\right)$", "final_answer": "(x,y)=\\left(3, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1bc72c2be14d79d0", "problem_id": "305ecb198891a27d", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the equation of the tangent line to $y=4 x^{3} - 2 x^{2} - 2$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $12 x^{2} - 4 x$.\n2) Slope at x=-1: m = f'(-1) = $16$.\n3) Point on curve: (-1, f(-1)) = (-1, $-8$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -8 = 16(x − -1).\n6) Optionally expand: y = $16 x + 8$.\nFinal answer: $y = 16\\cdot x + 8$", "final_answer": "y = 16\\cdot x + 8", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "de9410700e02de7a", "problem_id": "305ecb198891a27d", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Find the equation of the tangent line to $y=4 x^{3} - 2 x^{2} - 2$ at $x=-1$.", "output": "Differentiate: f'(x) = $12 x^{2} - 4 x$.\nSlope at x=-1: m = f'(-1) = $16$.\nPoint on curve: (-1, f(-1)) = (-1, $-8$).\nFinal answer: $y = 16\\cdot x + 8$", "final_answer": "y = 16\\cdot x + 8", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b8507f0c918ffd4a", "problem_id": "13b699b9f74ab877", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A store sells a backpack for $122. It is discounted by 5%. You buy 13 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 122 × (1 − 5/100) = 122 × 95/100 = 1159/10.\n2) Subtotal = 13 × 1159/10 = 15067/10.\n3) Total with tax = 15067/10 × 108/100 = 406809/250.\nFinal answer: $\\frac{406809}{250}$", "final_answer": "\\frac{406809}{250}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a12a0b5528b4e01b", "problem_id": "13b699b9f74ab877", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A store sells a backpack for $122. It is discounted by 5%. You buy 13 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 122 × (1 − 5/100) = 122 × 95/100 = 1159/10.\nSubtotal = 13 × 1159/10 = 15067/10.\nTotal with tax = 15067/10 × 108/100 = 406809/250.\nCheck: recomputing the formula yields 406809/250.\nFinal answer: $\\frac{406809}{250}$", "final_answer": "\\frac{406809}{250}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1db1bd767a2072b0", "problem_id": "fd2bcac6548591c0", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $x^{2}\\cdot \\cos{\\left(x \\right)} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}$", "final_answer": "x^{2}\\cdot \\cos{\\left(x \\right)} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5652cde983b4cf87", "problem_id": "fd2bcac6548591c0", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $x^{2}\\cdot \\cos{\\left(x \\right)} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}$", "final_answer": "x^{2}\\cdot \\cos{\\left(x \\right)} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6524bf379e624773", "problem_id": "7d288452adb227cc", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 13 times, what is the probability of exactly 10 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=13, k=10, p=$\\frac{1}{5}$ and simplify.\n4) Here C(13,10) = 286.\nFinal answer: $\\frac{18304}{1220703125}$", "final_answer": "\\frac{18304}{1220703125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2e27a9022b9f0c4f", "problem_id": "7d288452adb227cc", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 13 times, what is the probability of exactly 10 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=13, k=10, p=$\\frac{1}{5}$ and simplify.\nHere C(13,10) = 286.\nCheck: expression simplifies to $\\frac{18304}{1220703125}$.\nFinal answer: $\\frac{18304}{1220703125}$", "final_answer": "\\frac{18304}{1220703125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3fb2500392b43800", "problem_id": "95c7e275af4d8a66", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "You want 19 liters of a 20% salt solution. You have a 10% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 19−x liters is 40% solution.\n2) Salt equation: 10x + 40(19−x) = 20·19.\n3) (10-40)x = (20-40)·19 ⇒ x = 38/3.\n4) Then 19−x = 19/3.\nFinal answer: $38/3 L of 10% and 19/3 L of 40%$", "final_answer": "38/3 L of 10% and 19/3 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a465d044cb634863", "problem_id": "95c7e275af4d8a66", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "You want 19 liters of a 20% salt solution. You have a 10% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 19−x liters is 40% solution.\nSalt equation: 10x + 40(19−x) = 20·19.\n(10-40)x = (20-40)·19 ⇒ x = 38/3.\nFinal answer: $38/3 L of 10% and 19/3 L of 40%$", "final_answer": "38/3 L of 10% and 19/3 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "13c6ae599e672eb8", "problem_id": "ad0cb2fa6a9388dc", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 14 x - 36$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} - 14 x - 36$ = $2 \\left(x - 9\\right) \\left(x + 2\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $9$.\nFrom the second factor: x = $-2$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{9, -2\\}$", "final_answer": "x \\in \\{9, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8485806fef220151", "problem_id": "ad0cb2fa6a9388dc", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 14 x - 36$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 14 x - 36$ = $2 \\left(x - 9\\right) \\left(x + 2\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $9$.\n5) From the second factor: x = $-2$.\nFinal answer: $x \\in \\{9, -2\\}$", "final_answer": "x \\in \\{9, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6252185bc165d6e2", "problem_id": "109cefa4197417eb", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 10 x + 24$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} - 10 x + 24$ = $\\left(x - 6\\right) \\left(x - 4\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $6$.\n5) From the second factor: x = $4$.\nFinal answer: $x \\in \\{6, 4\\}$", "final_answer": "x \\in \\{6, 4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8ab9f1350dd72c9b", "problem_id": "109cefa4197417eb", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 10 x + 24$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} - 10 x + 24$ = $\\left(x - 6\\right) \\left(x - 4\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFinal answer: $x \\in \\{6, 4\\}$", "final_answer": "x \\in \\{6, 4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "34f506139fd64796", "problem_id": "c610b6dece7f6ab2", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Find the multiplicative inverse of 14 modulo 41.", "output": "We want x such that 14x ≡ 1 (mod 41).\nUse extended Euclid to find s,t with 14s + 41t = 1.\nOne Bezout identity is: 14·(3) + 41·(-1) = 1.\nFinal answer: $x\\equiv 3\\pmod{41}$", "final_answer": "x\\equiv 3\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f6598639ec23911e", "problem_id": "c610b6dece7f6ab2", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the multiplicative inverse of 14 modulo 41.", "output": "1) We want x such that 14x ≡ 1 (mod 41).\n2) Use extended Euclid to find s,t with 14s + 41t = 1.\n3) One Bezout identity is: 14·(3) + 41·(-1) = 1.\n4) Reduce s modulo 41: x ≡ 3 (mod 41).\nFinal answer: $x\\equiv 3\\pmod{41}$", "final_answer": "x\\equiv 3\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "469d9167e216ad7a", "problem_id": "707ef347742f79e4", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A disease has prevalence $\\frac{3}{200}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{251}{4000}$.\n3) Compute P(D|+) = $\\frac{54}{251}$.\nFinal answer: $\\frac{54}{251}$", "final_answer": "\\frac{54}{251}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "559d7f2bba4582a1", "problem_id": "707ef347742f79e4", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "A disease has prevalence $\\frac{3}{200}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{251}{4000}$.\nCompute P(D|+) = $\\frac{54}{251}$.\nFinal answer: $\\frac{54}{251}$", "final_answer": "\\frac{54}{251}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e2936527a399b14a", "problem_id": "25510e617dcdc250", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} - 4 x^{3} + 3 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 4 x^{3} + 3 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{3 \\pi^{2}}{8} + 5 + 3 \\pi + \\frac{9 \\pi^{3}}{8} + \\frac{15 \\pi^{4}}{16}$.\n4) Simplify the result.\nFinal answer: $- \\frac{3\\cdot \\pi^{2}}{8} + 5 + 3\\cdot \\pi + \\frac{9\\cdot \\pi^{3}}{8} + \\frac{15\\cdot \\pi^{4}}{16}$", "final_answer": "- \\frac{3\\cdot \\pi^{2}}{8} + 5 + 3\\cdot \\pi + \\frac{9\\cdot \\pi^{3}}{8} + \\frac{15\\cdot \\pi^{4}}{16}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "55f789fbe921a11d", "problem_id": "25510e617dcdc250", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} - 4 x^{3} + 3 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 4 x^{3} + 3 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{3 \\pi^{2}}{8} + 5 + 3 \\pi + \\frac{9 \\pi^{3}}{8} + \\frac{15 \\pi^{4}}{16}$.\nFinal answer: $- \\frac{3\\cdot \\pi^{2}}{8} + 5 + 3\\cdot \\pi + \\frac{9\\cdot \\pi^{3}}{8} + \\frac{15\\cdot \\pi^{4}}{16}$", "final_answer": "- \\frac{3\\cdot \\pi^{2}}{8} + 5 + 3\\cdot \\pi + \\frac{9\\cdot \\pi^{3}}{8} + \\frac{15\\cdot \\pi^{4}}{16}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "faead33f3ca7f977", "problem_id": "98a2e9a5196ba6d0", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡5 (mod 11), x≡2 (mod 7). Give the solution modulo 77.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 77.\n3) Compute the CRT solution to get x ≡ 16 (mod 77).\nFinal answer: $x\\equiv 16\\pmod{77}$", "final_answer": "x\\equiv 16\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "60a4b8d2b6efaeab", "problem_id": "98a2e9a5196ba6d0", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡5 (mod 11), x≡2 (mod 7). Give the solution modulo 77.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 77.\nCompute the CRT solution to get x ≡ 16 (mod 77).\nCheck: 16 mod 11=5 and mod 7=2.\nFinal answer: $x\\equiv 16\\pmod{77}$", "final_answer": "x\\equiv 16\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "490e8c64857e2257", "problem_id": "88abb3fdc41d3c72", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the congruences: x≡3 (mod 8), x≡1 (mod 13). Give the solution modulo 104.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 104.\n3) Compute the CRT solution to get x ≡ 27 (mod 104).\nFinal answer: $x\\equiv 27\\pmod{104}$", "final_answer": "x\\equiv 27\\pmod{104}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "caa2a63e0687c765", "problem_id": "88abb3fdc41d3c72", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡3 (mod 8), x≡1 (mod 13). Give the solution modulo 104.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 104.\nCompute the CRT solution to get x ≡ 27 (mod 104).\nCheck: 27 mod 8=3 and mod 13=1.\nFinal answer: $x\\equiv 27\\pmod{104}$", "final_answer": "x\\equiv 27\\pmod{104}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4ea654aea8d577e3", "problem_id": "25237ba4d603c8eb", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the congruences: x≡1 (mod 4), x≡0 (mod 11). Give the solution modulo 44.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 44.\n3) Compute the CRT solution to get x ≡ 33 (mod 44).\nFinal answer: $x\\equiv 33\\pmod{44}$", "final_answer": "x\\equiv 33\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "35c038e4a70c07ac", "problem_id": "25237ba4d603c8eb", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡1 (mod 4), x≡0 (mod 11). Give the solution modulo 44.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 44.\nCompute the CRT solution to get x ≡ 33 (mod 44).\nCheck: 33 mod 4=1 and mod 11=0.\nFinal answer: $x\\equiv 33\\pmod{44}$", "final_answer": "x\\equiv 33\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ab0f358e20bb03f6", "problem_id": "1b3e861b90629a33", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the multiplicative inverse of 17 modulo 29.", "output": "1) We want x such that 17x ≡ 1 (mod 29).\n2) Use extended Euclid to find s,t with 17s + 29t = 1.\n3) One Bezout identity is: 17·(12) + 29·(-7) = 1.\n4) Reduce s modulo 29: x ≡ 12 (mod 29).\nFinal answer: $x\\equiv 12\\pmod{29}$", "final_answer": "x\\equiv 12\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "841b9f384ea3dcd6", "problem_id": "1b3e861b90629a33", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the multiplicative inverse of 17 modulo 29.", "output": "We want x such that 17x ≡ 1 (mod 29).\nUse extended Euclid to find s,t with 17s + 29t = 1.\nOne Bezout identity is: 17·(12) + 29·(-7) = 1.\nReduce s modulo 29: x ≡ 12 (mod 29).\nCheck: 17·12 mod 29 = 1.\nFinal answer: $x\\equiv 12\\pmod{29}$", "final_answer": "x\\equiv 12\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "422407efb1690917", "problem_id": "240a28a71864f2b0", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Worker A finishes a job in 18 hours and Worker B finishes it in 5 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/18 job/hour. B's rate = 1/5 job/hour.\n2) Combined rate = 1/18 + 1/5 = 23/90 job/hour.\n3) Time = 1 ÷ (combined rate) = 90/23 hours.\nFinal answer: $\\frac{90}{23}$", "final_answer": "\\frac{90}{23}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4899df884d0c8cd7", "problem_id": "240a28a71864f2b0", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Worker A finishes a job in 18 hours and Worker B finishes it in 5 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/18 job/hour. B's rate = 1/5 job/hour.\nCombined rate = 1/18 + 1/5 = 23/90 job/hour.\nTime = 1 ÷ (combined rate) = 90/23 hours.\nCheck: (combined rate)×(time) = 23/90×90/23 = 1.\nFinal answer: $\\frac{90}{23}$", "final_answer": "\\frac{90}{23}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ab6199003a3efb3d", "problem_id": "854e6b638c0dcb8f", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{61 x}{15} + \\frac{56}{15}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{61}{15}$, c=$\\frac{56}{15}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{361}{225}$, so √Δ = $\\frac{19}{15}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{8}{3}$.\n7) x2 = $\\frac{7}{5}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{8}{3}, \\frac{7}{5}\\}$", "final_answer": "x \\in \\{\\frac{8}{3}, \\frac{7}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "00c6da854e4b5f6b", "problem_id": "854e6b638c0dcb8f", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{61 x}{15} + \\frac{56}{15}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{61}{15}$, c=$\\frac{56}{15}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{8}{3}, \\frac{7}{5}\\}$", "final_answer": "x \\in \\{\\frac{8}{3}, \\frac{7}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fc58da5108f80fd4", "problem_id": "689748c4ec545610", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$- 3 x + 4 y$ = $16$\n$- 4 x - 4 y$ = $12$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $12 x - 16 y$ = $-64$.\n3) Multiply the second equation by $-3$: $12 x + 12 y$ = $-36$.\n4) Subtract to eliminate x: $- 28 y$ = $-28$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $-4$.\nFinal answer: $(x,y)=\\left(-4, 1\\right)$", "final_answer": "(x,y)=\\left(-4, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4dd4b45b886978c9", "problem_id": "689748c4ec545610", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the linear system:\n$- 3 x + 4 y$ = $16$\n$- 4 x - 4 y$ = $12$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $12 x - 16 y$ = $-64$.\nMultiply the second equation by $-3$: $12 x + 12 y$ = $-36$.\nFinal answer: $(x,y)=\\left(-4, 1\\right)$", "final_answer": "(x,y)=\\left(-4, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f5d8a08ef8486a4b", "problem_id": "6c5362f6a5e82c18", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the multiplicative inverse of 13 modulo 29.", "output": "1) We want x such that 13x ≡ 1 (mod 29).\n2) Use extended Euclid to find s,t with 13s + 29t = 1.\n3) One Bezout identity is: 13·(9) + 29·(-4) = 1.\n4) Reduce s modulo 29: x ≡ 9 (mod 29).\nFinal answer: $x\\equiv 9\\pmod{29}$", "final_answer": "x\\equiv 9\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e7535a11f92ca764", "problem_id": "6c5362f6a5e82c18", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the multiplicative inverse of 13 modulo 29.", "output": "We want x such that 13x ≡ 1 (mod 29).\nUse extended Euclid to find s,t with 13s + 29t = 1.\nOne Bezout identity is: 13·(9) + 29·(-4) = 1.\nReduce s modulo 29: x ≡ 9 (mod 29).\nCheck: 13·9 mod 29 = 1.\nFinal answer: $x\\equiv 9\\pmod{29}$", "final_answer": "x\\equiv 9\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c3d5937301dfbb3d", "problem_id": "231ff801079198e0", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 13 red balls and 10 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(23,6).\nFavorable samples: choose k reds and n−k blues: C(13,1)·C(10,5).\nCompute totals: C(23,6) = 100947, and favorable = 3276.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{156}{4807}$.\nFinal answer: $\\frac{156}{4807}$", "final_answer": "\\frac{156}{4807}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9819a652784890bb", "problem_id": "231ff801079198e0", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 13 red balls and 10 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(23,6).\n3) Favorable samples: choose k reds and n−k blues: C(13,1)·C(10,5).\n4) Compute totals: C(23,6) = 100947, and favorable = 3276.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{156}{4807}$", "final_answer": "\\frac{156}{4807}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "610bae12ffe1afd1", "problem_id": "929473356136e5d9", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡5 (mod 9), x≡1 (mod 5). Give the solution modulo 45.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 45.\n3) Compute the CRT solution to get x ≡ 41 (mod 45).\nFinal answer: $x\\equiv 41\\pmod{45}$", "final_answer": "x\\equiv 41\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ce5153d324c62c5", "problem_id": "660f83bc358400a6", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{3} - 3 x^{2} - 2$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(4 x^{3}) = 12 x^{3}$.\n3) So $f'(x) = 12 x^{3} - 6 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(4 x^{3}) = 12 x^{2}$.\nTherefore the correct derivative is $f'(x) = 6 x \\left(2 x - 1\\right)$.\nFinal answer: $6 x \\left(2 x - 1\\right)$", "final_answer": "6 x \\left(2 x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a217f06dd18ca717", "problem_id": "786b6c1eb8676c9a", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_9=24$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_9=15.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=15 and k=9.\n4) Compute C(23,8) = 490314.\nFinal answer: $490314$", "final_answer": "490314", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f7e6c0582329bcbf", "problem_id": "37c1fa38fb04f96b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 2 x^{3} \\sin{\\left(x \\right)}) = - 2 x^{3} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 2 x^{3} \\cos{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 2 x^{3} \\sin{\\left(x \\right)}) = - 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- x^{3} \\cos{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 3 \\sin{\\left(2 x \\right)}\\right)$.\nFinal answer: $2 \\left(- x^{3} \\cos{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 3 \\sin{\\left(2 x \\right)}\\right)$", "final_answer": "2 \\left(- x^{3} \\cos{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 3 \\sin{\\left(2 x \\right)}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6f03f6fdfdc574c9", "problem_id": "24e25db991af6188", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} - 3 x^{2} + 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=1$.", "output": "Differentiate: f'(x) = $6 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 2$.\nSlope at x=1: m = f'(1) = $- 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2$.\nPoint on curve: (1, f(1)) = (1, $-1 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -1 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} = - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2(x − 1).\nOptionally expand: y = $- 2 x \\sin{\\left(2 \\right)} + \\frac{x e^{\\frac{1}{2}}}{2} + 2 x - 3 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2 \\sin{\\left(2 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 2\\cdot x - 3 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 2\\cdot x - 3 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "55d008c68ec70931", "problem_id": "24e25db991af6188", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Find the equation of the tangent line to $y=2 x^{3} - 3 x^{2} + 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=1$.", "output": "1) Differentiate: f'(x) = $6 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 2$.\n2) Slope at x=1: m = f'(1) = $- 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2$.\n3) Point on curve: (1, f(1)) = (1, $-1 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -1 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} = - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2(x − 1).\n6) Optionally expand: y = $- 2 x \\sin{\\left(2 \\right)} + \\frac{x e^{\\frac{1}{2}}}{2} + 2 x - 3 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 2\\cdot x - 3 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 2\\cdot x - 3 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5ad6380f92fa77d0", "problem_id": "39d8272f65ca84f1", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the linear system:\n$x - 8 y$ = $-8$\n$- 7 x + 4 y$ = $4$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-7$: $- 7 x + 56 y$ = $56$.\nMultiply the second equation by $1$: $- 7 x + 4 y$ = $4$.\nFinal answer: $(x,y)=\\left(0, 1\\right)$", "final_answer": "(x,y)=\\left(0, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5de3aefe94889ba5", "problem_id": "39d8272f65ca84f1", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$x - 8 y$ = $-8$\n$- 7 x + 4 y$ = $4$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-7$: $- 7 x + 56 y$ = $56$.\n3) Multiply the second equation by $1$: $- 7 x + 4 y$ = $4$.\n4) Subtract to eliminate x: $52 y$ = $52$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, 1\\right)$", "final_answer": "(x,y)=\\left(0, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a5de6cc9a5a23b2d", "problem_id": "2b698ddc2862d07d", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{58 x}{5} + \\frac{72}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$\\frac{58}{5}$, c=$\\frac{72}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{484}{25}$, so √Δ = $\\frac{22}{5}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{9}{5}$.\nx2 = $-4$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{9}{5}, -4\\}$", "final_answer": "x \\in \\{- \\frac{9}{5}, -4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6c7a0360da0abb77", "problem_id": "2b698ddc2862d07d", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{58 x}{5} + \\frac{72}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$\\frac{58}{5}$, c=$\\frac{72}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{484}{25}$, so √Δ = $\\frac{22}{5}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{9}{5}$.\n7) x2 = $-4$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{9}{5}, -4\\}$", "final_answer": "x \\in \\{- \\frac{9}{5}, -4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "401bf01d6e44bb84", "problem_id": "7440045c022782f9", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_5=35$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_5=25.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=25 and k=5.\n4) Compute C(29,4) = 23751.\nFinal answer: $23751$", "final_answer": "23751", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7e946566223328b1", "problem_id": "7440045c022782f9", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_5=35$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_5=25.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=25 and k=5.\nFinal answer: $23751$", "final_answer": "23751", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3fdb6771a1b74ebf", "problem_id": "a9191a04827a8ee3", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A cyclist rides at 82 miles per hour for 1 hours, then at 71 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 82 × 1 = 82 miles.\n2) Second leg distance = 71 × 3/2 = 213/2 miles.\n3) Total distance = 82 + 213/2 = 377/2 miles.\nFinal answer: $\\frac{377}{2}$", "final_answer": "\\frac{377}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "56df670d72329517", "problem_id": "a9191a04827a8ee3", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 82 miles per hour for 1 hours, then at 71 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 82 × 1 = 82 miles.\nSecond leg distance = 71 × 3/2 = 213/2 miles.\nTotal distance = 82 + 213/2 = 377/2 miles.\nCheck: adding both legs gives 377/2.\nFinal answer: $\\frac{377}{2}$", "final_answer": "\\frac{377}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d1f8c492ee1c26f6", "problem_id": "3d6682b7a8b79c0f", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡2 (mod 5), x≡0 (mod 9). Give the solution modulo 45.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 45.\n3) Compute the CRT solution to get x ≡ 27 (mod 45).\nFinal answer: $x\\equiv 27\\pmod{45}$", "final_answer": "x\\equiv 27\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f31cf1e7c3f08934", "problem_id": "3d6682b7a8b79c0f", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 5), x≡0 (mod 9). Give the solution modulo 45.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 45.\nCompute the CRT solution to get x ≡ 27 (mod 45).\nCheck: 27 mod 5=2 and mod 9=0.\nFinal answer: $x\\equiv 27\\pmod{45}$", "final_answer": "x\\equiv 27\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f5ba10d06c09047", "problem_id": "80cd2d9d70c1bc48", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{3} \\sin{\\left(x \\right)} - x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x \\sin{\\left(x \\right)}) = 2 x \\cos{\\left(x \\right)}$.\n3) So $f'(x) = 2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x \\sin{\\left(x \\right)}) = 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)}$.\nFinal answer: $2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)}$", "final_answer": "2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "01867999b3c7a5b9", "problem_id": "69b3bcbb89bf4194", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=12$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=12 and k=6.\nCompute C(17,5) = 6188.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $6188$", "final_answer": "6188", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "78b4dba08d8b36dc", "problem_id": "4d706b5fae16eeb0", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{3} - 3 x^{2} + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $3 x^{3} - 3 x^{2} + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 1$.\n3) Compute an antiderivative: F(x) = $\\frac{3 x^{4}}{4} - x^{3} + x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{3\\cdot x^{4}}{4} - x^{3} + x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{3\\cdot x^{4}}{4} - x^{3} + x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "805c2c2f459f68e0", "problem_id": "4d706b5fae16eeb0", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{3} - 3 x^{2} + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $3 x^{3} - 3 x^{2} + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 1$.\nCompute an antiderivative: F(x) = $\\frac{3 x^{4}}{4} - x^{3} + x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{3\\cdot x^{4}}{4} - x^{3} + x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{3\\cdot x^{4}}{4} - x^{3} + x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4d3e6e6f752dab45", "problem_id": "207c0b128c686018", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- \\frac{2 x^{3}}{3} - \\frac{x^{2}}{3} + x + \\left(x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1}{x - 2}$.", "output": "Let f(x) = $\\frac{- \\frac{2 x^{3}}{3} - \\frac{x^{2}}{3} + x + \\left(x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1}{x - 2}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{4 x^{3} + 2 x^{2} - 6 x + \\left(x - 2\\right) \\left(- 12 x^{2} - 4 x + 3 \\left(x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x - 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 6 \\left(x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6}{6 \\left(x - 2\\right)^{2}}$.\nFinal answer: $\\frac{4\\cdot x^{3} + 2\\cdot x^{2} - 6\\cdot x + \\left(x - 2\\right)\\cdot \\left(- 12\\cdot x^{2} - 4\\cdot x + \\left(3\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6\\right) - \\left(6\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6}{6\\cdot \\left(x - 2\\right)^{2}}$", "final_answer": "\\frac{4\\cdot x^{3} + 2\\cdot x^{2} - 6\\cdot x + \\left(x - 2\\right)\\cdot \\left(- 12\\cdot x^{2} - 4\\cdot x + \\left(3\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6\\right) - \\left(6\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6}{6\\cdot \\left(x - 2\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b52a9bbe543b974c", "problem_id": "207c0b128c686018", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- \\frac{2 x^{3}}{3} - \\frac{x^{2}}{3} + x + \\left(x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1}{x - 2}$.", "output": "1) Let f(x) = $\\frac{- \\frac{2 x^{3}}{3} - \\frac{x^{2}}{3} + x + \\left(x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1}{x - 2}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{4 x^{3} + 2 x^{2} - 6 x + \\left(x - 2\\right) \\left(- 12 x^{2} - 4 x + 3 \\left(x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x - 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 6 \\left(x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6}{6 \\left(x - 2\\right)^{2}}$.\nFinal answer: $\\frac{4\\cdot x^{3} + 2\\cdot x^{2} - 6\\cdot x + \\left(x - 2\\right)\\cdot \\left(- 12\\cdot x^{2} - 4\\cdot x + \\left(3\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6\\right) - \\left(6\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6}{6\\cdot \\left(x - 2\\right)^{2}}$", "final_answer": "\\frac{4\\cdot x^{3} + 2\\cdot x^{2} - 6\\cdot x + \\left(x - 2\\right)\\cdot \\left(- 12\\cdot x^{2} - 4\\cdot x + \\left(3\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6\\right) - \\left(6\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6}{6\\cdot \\left(x - 2\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f81033f05617e38e", "problem_id": "9ef4bedea90b0247", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(2 x^{2} - 3\\right) \\cos{\\left(x^{2} + x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 6 e^{x}) = - 6 e^{x}$.\n3) So $f'(x) = - 4 x^{3} \\sin{\\left(\\left(x^{2} + x\\right) + 3 \\right)} - 2 x^{2} \\sin{\\left(\\left(x^{2} + x\\right) + 3 \\right)} + 2 x e^{x} + 6 x \\sin{\\left(\\left(x^{2} + x\\right) + 3 \\right)} + 4 x \\cos{\\left(\\left(x^{2} + x\\right) + 3 \\right)} - 4 e^{x} + 3 \\sin{\\left(\\left(x^{2} + x\\right) + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 6 e^{x}) = - 6 e^{x}$.\nTherefore the correct derivative is $f'(x) = 4 x \\cos{\\left(\\left(x^{2} + x\\right) + 3 \\right)} + 2 \\left(x - 3\\right) e^{x} - \\left(2 x + 1\\right) \\left(2 x^{2} - 3\\right) \\sin{\\left(\\left(x^{2} + x\\right) + 3 \\right)} + 2 e^{x}$.\nFinal answer: $4 x \\cos{\\left(\\left(x^{2} + x\\right) + 3 \\right)} + 2 \\left(x - 3\\right) e^{x} - \\left(2 x + 1\\right) \\left(2 x^{2} - 3\\right) \\sin{\\left(\\left(x^{2} + x\\right) + 3 \\right)} + 2 e^{x}$", "final_answer": "4 x \\cos{\\left(\\left(x^{2} + x\\right) + 3 \\right)} + 2 \\left(x - 3\\right) e^{x} - \\left(2 x + 1\\right) \\left(2 x^{2} - 3\\right) \\sin{\\left(\\left(x^{2} + x\\right) + 3 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "14a6d76ab3f29dbc", "problem_id": "2427ac12c065ce58", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 20 red balls and 22 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(42,8).\n3) Favorable samples: choose k reds and n−k blues: C(20,2)·C(22,6).\n4) Compute totals: C(42,8) = 118030185, and favorable = 14176470.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{7106}{59163}$", "final_answer": "\\frac{7106}{59163}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "36f5a7edcc69e90d", "problem_id": "a932a22a8b83fe8b", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$11 x + 10 y$ = $-79$\n$6 x - y$ = $-56$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $6$: $66 x + 60 y$ = $-474$.\n3) Multiply the second equation by $11$: $66 x - 11 y$ = $-616$.\n4) Subtract to eliminate x: $71 y$ = $142$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $-9$.\nFinal answer: $(x,y)=\\left(-9, 2\\right)$", "final_answer": "(x,y)=\\left(-9, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "48f58e0136e6b9d6", "problem_id": "bc07347abcdf219c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 5 - x$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 5 - 2 x.\n3) Then the limit is 9.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 7.\nFinal answer: $7$", "final_answer": "7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "43a6a979cca7ed4e", "problem_id": "b667157e7d6f01e0", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$3 x - 2 y$ = $10$\n$- 3 x + 4 y$ = $-14$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $- 9 x + 6 y$ = $-30$.\nMultiply the second equation by $3$: $- 9 x + 12 y$ = $-42$.\nSubtract to eliminate x: $- 6 y$ = $12$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $2$.\nCheck: plug (x,y)=(2,-2) into both equations.\nFinal answer: $(x,y)=\\left(2, -2\\right)$", "final_answer": "(x,y)=\\left(2, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d7a060d0ac5a25c8", "problem_id": "b667157e7d6f01e0", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$3 x - 2 y$ = $10$\n$- 3 x + 4 y$ = $-14$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $- 9 x + 6 y$ = $-30$.\n3) Multiply the second equation by $3$: $- 9 x + 12 y$ = $-42$.\n4) Subtract to eliminate x: $- 6 y$ = $12$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $2$.\nFinal answer: $(x,y)=\\left(2, -2\\right)$", "final_answer": "(x,y)=\\left(2, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b16a7a18f9ed601f", "problem_id": "bbb92d67c0e97e7e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(3 x^{2} + 3\\right) \\cos{\\left(x^{2} - 3 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(e^{x}) = e^{x}$.\n3) So $f'(x) = - 6 x^{3} \\sin{\\left(x^{2} - 3 x \\right)} + 9 x^{2} \\sin{\\left(x^{2} - 3 x \\right)} + x e^{x} - 6 x \\sin{\\left(x^{2} - 3 x \\right)} + 6 x \\cos{\\left(x^{2} - 3 x \\right)} + 2 e^{x} + 9 \\sin{\\left(x^{2} - 3 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(e^{x}) = e^{x}$.\nTherefore the correct derivative is $f'(x) = 6 x \\cos{\\left(x^{2} - 3 x \\right)} + \\left(x + 1\\right) e^{x} - 3 \\left(2 x - 3\\right) \\left(x^{2} + 1\\right) \\sin{\\left(x^{2} - 3 x \\right)} + e^{x}$.\nFinal answer: $6 x \\cos{\\left(x^{2} - 3 x \\right)} + \\left(x + 1\\right) e^{x} - 3 \\left(2 x - 3\\right) \\left(x^{2} + 1\\right) \\sin{\\left(x^{2} - 3 x \\right)} + e^{x}$", "final_answer": "6 x \\cos{\\left(x^{2} - 3 x \\right)} + \\left(x + 1\\right) e^{x} - 3 \\left(2 x - 3\\right) \\left(x^{2} + 1\\right) \\sin{\\left(x^{2} - 3 x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "1c7314b3f233c6bf", "problem_id": "d076fb0de7a8a9bc", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(x^{2} + x - 3\\right) \\cos{\\left(3 x - 3 \\right)}$.", "output": "1) Let f(x) = $\\left(x - 3\\right) e^{x} + \\left(x^{2} + x - 3\\right) \\cos{\\left(3 x - 3 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\left(x - 3\\right) e^{x} + \\left(2 x + 1\\right) \\cos{\\left(3 x - 3 \\right)} - 3 \\left(x^{2} + x - 3\\right) \\sin{\\left(3 x - 3 \\right)} + e^{x}$.\nFinal answer: $\\left(x - 3\\right)\\cdot e^{x} + \\left(2\\cdot x + 1\\right)\\cdot \\cos{\\left(3\\cdot x - 3 \\right)} - \\left(3\\cdot x^{2} + 3\\cdot x - 9\\right)\\cdot \\sin{\\left(3\\cdot x - 3 \\right)} + e^{x}$", "final_answer": "\\left(x - 3\\right)\\cdot e^{x} + \\left(2\\cdot x + 1\\right)\\cdot \\cos{\\left(3\\cdot x - 3 \\right)} - \\left(3\\cdot x^{2} + 3\\cdot x - 9\\right)\\cdot \\sin{\\left(3\\cdot x - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e80a7235949222e7", "problem_id": "43ee3cf6a11c1077", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} 2 x^{3} - 3 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $2 x^{3} - 3 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $- \\frac{\\pi^{3}}{4} + 4 + 3 \\pi$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{\\pi^{3}}{4} + 4 + 3\\cdot \\pi$", "final_answer": "- \\frac{\\pi^{3}}{4} + 4 + 3\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "17e949fd506a2fa8", "problem_id": "43ee3cf6a11c1077", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} 2 x^{3} - 3 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{3} - 3 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $- \\frac{\\pi^{3}}{4} + 4 + 3 \\pi$.\n4) Simplify the result.\nFinal answer: $- \\frac{\\pi^{3}}{4} + 4 + 3\\cdot \\pi$", "final_answer": "- \\frac{\\pi^{3}}{4} + 4 + 3\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "63f15acaed4fdd79", "problem_id": "a0b5e033c7aa2ebe", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} 4 x^{3} - 3 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $4 x^{3} - 3 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{9 \\pi^{3}}{8} - 1 + \\frac{3 \\pi}{2} + \\frac{15 \\pi^{4}}{16}$.\n4) Simplify the result.\nFinal answer: $- \\frac{9\\cdot \\pi^{3}}{8} - 1 + \\frac{3\\cdot \\pi}{2} + \\frac{15\\cdot \\pi^{4}}{16}$", "final_answer": "- \\frac{9\\cdot \\pi^{3}}{8} - 1 + \\frac{3\\cdot \\pi}{2} + \\frac{15\\cdot \\pi^{4}}{16}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d2d9ee1b2da4e22f", "problem_id": "a0b5e033c7aa2ebe", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} 4 x^{3} - 3 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $4 x^{3} - 3 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\nEvaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{9 \\pi^{3}}{8} - 1 + \\frac{3 \\pi}{2} + \\frac{15 \\pi^{4}}{16}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{9\\cdot \\pi^{3}}{8} - 1 + \\frac{3\\cdot \\pi}{2} + \\frac{15\\cdot \\pi^{4}}{16}$", "final_answer": "- \\frac{9\\cdot \\pi^{3}}{8} - 1 + \\frac{3\\cdot \\pi}{2} + \\frac{15\\cdot \\pi^{4}}{16}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "31250964bba1c1b6", "problem_id": "83dbec25aef4b58a", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Worker A finishes a job in 7 hours and Worker B finishes it in 9 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/7 job/hour. B's rate = 1/9 job/hour.\n2) Combined rate = 1/7 + 1/9 = 16/63 job/hour.\n3) Time = 1 ÷ (combined rate) = 63/16 hours.\nFinal answer: $\\frac{63}{16}$", "final_answer": "\\frac{63}{16}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0439a716a84a5c28", "problem_id": "192e44a45e2bafd4", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} - x + 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x^{3}) = 3 x^{3}$.\n3) So $f'(x) = 3 x^{3} - 1$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x^{3}) = 3 x^{2}$.\nTherefore the correct derivative is $f'(x) = 3 x^{2} - 1$.\nFinal answer: $3 x^{2} - 1$", "final_answer": "3 x^{2} - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "783533001d93cd89", "problem_id": "755d5c0e083b972d", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "An urn has 24 red balls and 12 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(36,11).\nFavorable samples: choose k reds and n−k blues: C(24,5)·C(12,6).\nFinal answer: $\\frac{38962}{596037}$", "final_answer": "\\frac{38962}{596037}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "94131fb3ade459b5", "problem_id": "755d5c0e083b972d", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "An urn has 24 red balls and 12 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(36,11).\n3) Favorable samples: choose k reds and n−k blues: C(24,5)·C(12,6).\n4) Compute totals: C(36,11) = 600805296, and favorable = 39273696.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{38962}{596037}$", "final_answer": "\\frac{38962}{596037}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "34916b7928913dff", "problem_id": "2212aef6f786417e", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡0 (mod 4), x≡3 (mod 7). Give the solution modulo 28.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 28.\nCompute the CRT solution to get x ≡ 24 (mod 28).\nCheck: 24 mod 4=0 and mod 7=3.\nFinal answer: $x\\equiv 24\\pmod{28}$", "final_answer": "x\\equiv 24\\pmod{28}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b432eb30aefefac3", "problem_id": "2212aef6f786417e", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the congruences: x≡0 (mod 4), x≡3 (mod 7). Give the solution modulo 28.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 28.\nCompute the CRT solution to get x ≡ 24 (mod 28).\nFinal answer: $x\\equiv 24\\pmod{28}$", "final_answer": "x\\equiv 24\\pmod{28}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bdff1e0712961fc0", "problem_id": "edc590a0a6a6dc78", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡1 (mod 8), x≡6 (mod 13). Give the solution modulo 104.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 104.\nCompute the CRT solution to get x ≡ 97 (mod 104).\nCheck: 97 mod 8=1 and mod 13=6.\nFinal answer: $x\\equiv 97\\pmod{104}$", "final_answer": "x\\equiv 97\\pmod{104}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "84e4a3bce39825e1", "problem_id": "edc590a0a6a6dc78", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the congruences: x≡1 (mod 8), x≡6 (mod 13). Give the solution modulo 104.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 104.\n3) Compute the CRT solution to get x ≡ 97 (mod 104).\nFinal answer: $x\\equiv 97\\pmod{104}$", "final_answer": "x\\equiv 97\\pmod{104}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a881fc6074da9d1e", "problem_id": "7567574796b146a6", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Two friends split $1014 in the ratio 1:5. How much does each receive?", "output": "1) Total ratio parts = 1+5 = 6.\n2) Each part = 1014 ÷ 6 = 169.\n3) First share = 1 × 169 = 169.\n4) Second share = 5 × 169 = 845.\nFinal answer: $845$", "final_answer": "845", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2bd4ecf05e2faaf4", "problem_id": "7567574796b146a6", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $1014 in the ratio 1:5. How much does each receive?", "output": "Total ratio parts = 1+5 = 6.\nEach part = 1014 ÷ 6 = 169.\nFirst share = 1 × 169 = 169.\nSecond share = 5 × 169 = 845.\nCheck: 169+845=1014.\nFinal answer: $845$", "final_answer": "845", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "03735235915bdec3", "problem_id": "baae96f351cbea43", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A cyclist rides at 71 miles per hour for 1 hours, then at 51 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 71 × 1 = 71 miles.\n2) Second leg distance = 51 × 1 = 51 miles.\n3) Total distance = 71 + 51 = 122 miles.\nFinal answer: $122$", "final_answer": "122", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "696cce11096bece0", "problem_id": "baae96f351cbea43", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A cyclist rides at 71 miles per hour for 1 hours, then at 51 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 71 × 1 = 71 miles.\nSecond leg distance = 51 × 1 = 51 miles.\nTotal distance = 71 + 51 = 122 miles.\nFinal answer: $122$", "final_answer": "122", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d07a1001e602f90", "problem_id": "e48acf7751cbfad6", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the equation of the tangent line to $y=x^{3} - 2 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} + 3$ at $x=2$.", "output": "1) Differentiate: f'(x) = $3 x^{2} - 4 x + 2 \\cos{\\left(x \\right)} - 2$.\n2) Slope at x=2: m = f'(2) = $2 \\cos{\\left(2 \\right)} + 2$.\n3) Point on curve: (2, f(2)) = (2, $-1 + 2 \\sin{\\left(2 \\right)}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -1 + 2 \\sin{\\left(2 \\right)} = 2 \\cos{\\left(2 \\right)} + 2(x − 2).\n6) Optionally expand: y = $2 x \\cos{\\left(2 \\right)} + 2 x - 5 - 4 \\cos{\\left(2 \\right)} + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 2\\cdot x - 5 - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 2\\cdot x - 5 - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ba6bed1dcff40774", "problem_id": "e48acf7751cbfad6", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the equation of the tangent line to $y=x^{3} - 2 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} + 3$ at $x=2$.", "output": "Differentiate: f'(x) = $3 x^{2} - 4 x + 2 \\cos{\\left(x \\right)} - 2$.\nSlope at x=2: m = f'(2) = $2 \\cos{\\left(2 \\right)} + 2$.\nPoint on curve: (2, f(2)) = (2, $-1 + 2 \\sin{\\left(2 \\right)}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -1 + 2 \\sin{\\left(2 \\right)} = 2 \\cos{\\left(2 \\right)} + 2(x − 2).\nOptionally expand: y = $2 x \\cos{\\left(2 \\right)} + 2 x - 5 - 4 \\cos{\\left(2 \\right)} + 2 \\sin{\\left(2 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 2\\cdot x - 5 - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 2\\cdot x - 5 - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "47482553f792998c", "problem_id": "4f060fc8a978793b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(3 x^{2} + 2 x + 2\\right) \\cos{\\left(x - 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x e^{x}) = x e^{x}$.\n3) So $f'(x) = - 3 x^{2} \\sin{\\left(x - 3 \\right)} + x e^{x} - 2 x \\sin{\\left(x - 3 \\right)} + 6 x \\cos{\\left(x - 3 \\right)} + 2 e^{x} - 2 \\sin{\\left(x - 3 \\right)} + 2 \\cos{\\left(x - 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x e^{x}) = x e^{x} + e^{x}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 2\\right) e^{x} + 2 \\left(3 x + 1\\right) \\cos{\\left(x - 3 \\right)} - \\left(x \\left(3 x + 2\\right) + 2\\right) \\sin{\\left(x - 3 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 2\\right) e^{x} + 2 \\left(3 x + 1\\right) \\cos{\\left(x - 3 \\right)} - \\left(x \\left(3 x + 2\\right) + 2\\right) \\sin{\\left(x - 3 \\right)} + e^{x}$", "final_answer": "\\left(x + 2\\right) e^{x} + 2 \\left(3 x + 1\\right) \\cos{\\left(x - 3 \\right)} - \\left(x \\left(3 x + 2\\right) + 2\\right) \\sin{\\left(x - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "52ed7abec737dc53", "problem_id": "f0dcf08dee9e9d30", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - 3 x^{3} - 10 x^{2} - 39 x - 155$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 3 x^{4} + 2 x^{3} + x^{2} + x - 2$ and a = 4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 12 x^{3} + 6 x^{2} + 2 x + 1$.\n5) Evaluate at a: p'(4) = $-663$.\nFinal answer: $-663$", "final_answer": "-663", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1b63d77dc877c6f5", "problem_id": "87e1d1d1c6222c9e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(- 2 x^{2} - 2 x\\right) \\cos{\\left(- 3 x^{2} + 2 x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 2 x \\cos{\\left(\\left(- 3 x^{2} + 2 x\\right) + 3 \\right)}) = 2 x \\sin{\\left(- 3 x^{2} + 2 x + 3 \\right)}$.\n3) So $f'(x) = - 12 x^{3} \\sin{\\left(- 3 x^{2} + 2 x + 3 \\right)} + 4 x^{2} \\sin{\\left(- 3 x^{2} + 2 x + 3 \\right)} + 2 x e^{x} + 2 x \\sin{\\left(- 3 x^{2} + 2 x + 3 \\right)} - 4 x \\cos{\\left(\\left(- 3 x^{2} + 2 x\\right) + 3 \\right)} - 4 e^{x}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 2 x \\cos{\\left(\\left(- 3 x^{2} + 2 x\\right) + 3 \\right)}) = 2 x \\left(2 - 6 x\\right) \\sin{\\left(- 3 x^{2} + 2 x + 3 \\right)} - 2 \\cos{\\left(\\left(- 3 x^{2} + 2 x\\right) + 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- 2 x \\left(1 - 3 x\\right) \\left(- x - 1\\right) \\sin{\\left(- 3 x^{2} + 2 x + 3 \\right)} + \\left(- 2 x - 1\\right) \\cos{\\left(\\left(- 3 x^{2} + 2 x\\right) + 3 \\right)} + \\left(x - 3\\right) e^{x} + e^{x}\\right)$.\nFinal answer: $2 \\left(- 2 x \\left(1 - 3 x\\right) \\left(- x - 1\\right) \\sin{\\left(- 3 x^{2} + 2 x + 3 \\right)} + \\left(- 2 x - 1\\right) \\cos{\\left(\\left(- 3 x^{2} + 2 x\\right) + 3 \\right)} + \\left(x - 3\\right) e^{x} + e^{x}\\right)$", "final_answer": "2 \\left(- 2 x \\left(1 - 3 x\\right) \\left(- x - 1\\right) \\sin{\\left(- 3 x^{2} + 2 x + 3 \\right)} + \\left(- 2 x - 1\\right) \\cos{\\left(\\left(- 3 x^{2} + 2 x\\right) + 3 \\right)} + \\left(x - 3\\right) e^{x} + e^{x}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d59499db6bb33bfa", "problem_id": "970c0464fa1bef51", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{5}$. If it is flipped 15 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=15, k=2, p=$\\frac{3}{5}$ and simplify.\n4) Here C(15,2) = 105.\nFinal answer: $\\frac{1548288}{6103515625}$", "final_answer": "\\frac{1548288}{6103515625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9940b5edb5920768", "problem_id": "3343817ddf189073", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A store sells a backpack for $695. It is discounted by 25%. You buy 2 backpacks. What is the total cost?", "output": "1) Discounted price = 695 × (1 − 25/100) = 695 × 75/100 = 2085/4.\n2) Subtotal = 2 × 2085/4 = 2085/2.\nFinal answer: $\\frac{2085}{2}$", "final_answer": "\\frac{2085}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1505c40e6a2e8055", "problem_id": "3343817ddf189073", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $695. It is discounted by 25%. You buy 2 backpacks. What is the total cost?", "output": "Discounted price = 695 × (1 − 25/100) = 695 × 75/100 = 2085/4.\nSubtotal = 2 × 2085/4 = 2085/2.\nCheck: recomputing the formula yields 2085/2.\nFinal answer: $\\frac{2085}{2}$", "final_answer": "\\frac{2085}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "96380a36674d3609", "problem_id": "7dc35d24c40aa025", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $984 in the ratio 2:4. How much does each receive?", "output": "Total ratio parts = 2+4 = 6.\nEach part = 984 ÷ 6 = 164.\nFirst share = 2 × 164 = 328.\nSecond share = 4 × 164 = 656.\nCheck: 328+656=984.\nFinal answer: $656$", "final_answer": "656", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e8ae1e4801c8e811", "problem_id": "7dc35d24c40aa025", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Two friends split $984 in the ratio 2:4. How much does each receive?", "output": "Total ratio parts = 2+4 = 6.\nEach part = 984 ÷ 6 = 164.\nFirst share = 2 × 164 = 328.\nFinal answer: $656$", "final_answer": "656", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2dc6d1ebca5c9c00", "problem_id": "5b45ac11c1c21759", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$- 3 x + 2 y$ = $-12$\n$2 x - 3 y$ = $8$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $- 6 x + 4 y$ = $-24$.\n3) Multiply the second equation by $-3$: $- 6 x + 9 y$ = $-24$.\n4) Subtract to eliminate x: $- 5 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, 0\\right)$", "final_answer": "(x,y)=\\left(4, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "73c1b9c3489d7a94", "problem_id": "5b45ac11c1c21759", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the linear system:\n$- 3 x + 2 y$ = $-12$\n$2 x - 3 y$ = $8$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $2$: $- 6 x + 4 y$ = $-24$.\nMultiply the second equation by $-3$: $- 6 x + 9 y$ = $-24$.\nFinal answer: $(x,y)=\\left(4, 0\\right)$", "final_answer": "(x,y)=\\left(4, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "15214b57aa348380", "problem_id": "0ca2fe2256a902ae", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} - 2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\n3) Subtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{\\pi^{3}}{12} + 2 + \\frac{\\pi^{2}}{4} + \\pi$.\n4) Simplify the result.\nFinal answer: $- \\frac{\\pi^{3}}{12} + 2 + \\frac{\\pi^{2}}{4} + \\pi$", "final_answer": "- \\frac{\\pi^{3}}{12} + 2 + \\frac{\\pi^{2}}{4} + \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "257b01e62be5e537", "problem_id": "1c79bedd36efa53e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 3 x^{3} - 3 x^{2} + 2 x - 5$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} - 3 x^{2} + x - 5.\n3) Then the limit is -12.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: -13.\nFinal answer: $-13$", "final_answer": "-13", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a392fe847e9f6457", "problem_id": "dfb67074c6765c1b", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡3 (mod 8), x≡4 (mod 11). Give the solution modulo 88.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 88.\nCompute the CRT solution to get x ≡ 59 (mod 88).\nCheck: 59 mod 8=3 and mod 11=4.\nFinal answer: $x\\equiv 59\\pmod{88}$", "final_answer": "x\\equiv 59\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b77252d67539a894", "problem_id": "dfb67074c6765c1b", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the congruences: x≡3 (mod 8), x≡4 (mod 11). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 59 (mod 88).\nFinal answer: $x\\equiv 59\\pmod{88}$", "final_answer": "x\\equiv 59\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "42baee69924a2b06", "problem_id": "8000e0f99fbe8596", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{4} + 4 x^{3} - 2 x^{2} + 2 x - 2\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $2 x^{4} + 4 x^{3} - 2 x^{2} + 2 x - 2$.\nCompute an antiderivative: F(x) = $\\frac{2 x^{5}}{5} + x^{4} - \\frac{2 x^{3}}{3} + x^{2} - 2 x$.\nFinal answer: $C + \\frac{2\\cdot x^{5}}{5} + x^{4} - \\frac{2\\cdot x^{3}}{3} + x^{2} - 2\\cdot x$", "final_answer": "C + \\frac{2\\cdot x^{5}}{5} + x^{4} - \\frac{2\\cdot x^{3}}{3} + x^{2} - 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "29fbd6e5dd7af238", "problem_id": "8000e0f99fbe8596", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{4} + 4 x^{3} - 2 x^{2} + 2 x - 2\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $2 x^{4} + 4 x^{3} - 2 x^{2} + 2 x - 2$.\n3) Compute an antiderivative: F(x) = $\\frac{2 x^{5}}{5} + x^{4} - \\frac{2 x^{3}}{3} + x^{2} - 2 x$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{2\\cdot x^{5}}{5} + x^{4} - \\frac{2\\cdot x^{3}}{3} + x^{2} - 2\\cdot x$", "final_answer": "C + \\frac{2\\cdot x^{5}}{5} + x^{4} - \\frac{2\\cdot x^{3}}{3} + x^{2} - 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6cdb3340c1d7ab1f", "problem_id": "db26d52d5b2a23ff", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} - 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\nEvaluate at the bounds: F(\\pi) and F(0).\nSubtract: F(\\pi) − F(0) = $- \\pi^{3} - 3 \\pi + 4 + \\frac{3 \\pi^{2}}{2}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\pi^{3} - 3\\cdot \\pi + 4 + \\frac{3\\cdot \\pi^{2}}{2}$", "final_answer": "- \\pi^{3} - 3\\cdot \\pi + 4 + \\frac{3\\cdot \\pi^{2}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4b43b339cdeb26f1", "problem_id": "db26d52d5b2a23ff", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} - 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\n2) Evaluate at the bounds: F(\\pi) and F(0).\n3) Subtract: F(\\pi) − F(0) = $- \\pi^{3} - 3 \\pi + 4 + \\frac{3 \\pi^{2}}{2}$.\n4) Simplify the result.\nFinal answer: $- \\pi^{3} - 3\\cdot \\pi + 4 + \\frac{3\\cdot \\pi^{2}}{2}$", "final_answer": "- \\pi^{3} - 3\\cdot \\pi + 4 + \\frac{3\\cdot \\pi^{2}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "24eaaa78f7c2d78f", "problem_id": "ae4d5e74228cfa31", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A store sells a backpack for $830. It is discounted by 5%. You buy 13 backpacks. What is the total cost?", "output": "Discounted price = 830 × (1 − 5/100) = 830 × 95/100 = 1577/2.\nSubtotal = 13 × 1577/2 = 20501/2.\nCheck: recomputing the formula yields 20501/2.\nFinal answer: $\\frac{20501}{2}$", "final_answer": "\\frac{20501}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9f7973c84a4f9bf6", "problem_id": "ae4d5e74228cfa31", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A store sells a backpack for $830. It is discounted by 5%. You buy 13 backpacks. What is the total cost?", "output": "1) Discounted price = 830 × (1 − 5/100) = 830 × 95/100 = 1577/2.\n2) Subtotal = 13 × 1577/2 = 20501/2.\nFinal answer: $\\frac{20501}{2}$", "final_answer": "\\frac{20501}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "16616943f691d84a", "problem_id": "4beb45c214efd916", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A cyclist rides at 30 miles per hour for 1 hours, then at 64 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 30 × 1 = 30 miles.\n2) Second leg distance = 64 × 3/2 = 96 miles.\n3) Total distance = 30 + 96 = 126 miles.\nFinal answer: $126$", "final_answer": "126", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "07a1908046b52b5e", "problem_id": "4beb45c214efd916", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 30 miles per hour for 1 hours, then at 64 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 30 × 1 = 30 miles.\nSecond leg distance = 64 × 3/2 = 96 miles.\nTotal distance = 30 + 96 = 126 miles.\nCheck: adding both legs gives 126.\nFinal answer: $126$", "final_answer": "126", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "09567f0dd2317e5b", "problem_id": "e0b992559e0e860e", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "An urn has 19 red balls and 23 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(42,10).\n3) Favorable samples: choose k reds and n−k blues: C(19,1)·C(23,9).\n4) Compute totals: C(42,10) = 1471442973, and favorable = 15526610.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{4370}{414141}$", "final_answer": "\\frac{4370}{414141}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3dc71a93381d4310", "problem_id": "070b8d8e920c171e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- x^{2} + 3 x - 1\\right) \\cos{\\left(3 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 x \\cos{\\left(3 x \\right)}) = - 3 x \\sin{\\left(3 x \\right)}$.\n3) So $f'(x) = 3 x^{2} \\sin{\\left(3 x \\right)} + 2 x e^{x} - 3 x \\sin{\\left(3 x \\right)} - 2 x \\cos{\\left(3 x \\right)} + 4 e^{x} + 3 \\sin{\\left(3 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 x \\cos{\\left(3 x \\right)}) = - 9 x \\sin{\\left(3 x \\right)} + 3 \\cos{\\left(3 x \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(3 - 2 x\\right) \\cos{\\left(3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 3 \\left(x \\left(3 - x\\right) - 1\\right) \\sin{\\left(3 x \\right)} + 2 e^{x}$.\nFinal answer: $\\left(3 - 2 x\\right) \\cos{\\left(3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 3 \\left(x \\left(3 - x\\right) - 1\\right) \\sin{\\left(3 x \\right)} + 2 e^{x}$", "final_answer": "\\left(3 - 2 x\\right) \\cos{\\left(3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 3 \\left(x \\left(3 - x\\right) - 1\\right) \\sin{\\left(3 x \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "509c6f08e90b47b0", "problem_id": "b856749deafd33db", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$4 x + 6 y$ = $0$\n$2 x + 5 y$ = $0$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $2$: $8 x + 12 y$ = $0$.\nMultiply the second equation by $4$: $8 x + 20 y$ = $0$.\nSubtract to eliminate x: $- 8 y$ = $0$.\nSolve for y: y = $0$.\nSubstitute back to find x: x = $0$.\nCheck: plug (x,y)=(0,0) into both equations.\nFinal answer: $(x,y)=\\left(0, 0\\right)$", "final_answer": "(x,y)=\\left(0, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "33be56fcbdc49fb3", "problem_id": "b856749deafd33db", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$4 x + 6 y$ = $0$\n$2 x + 5 y$ = $0$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $8 x + 12 y$ = $0$.\n3) Multiply the second equation by $4$: $8 x + 20 y$ = $0$.\n4) Subtract to eliminate x: $- 8 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, 0\\right)$", "final_answer": "(x,y)=\\left(0, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "95db6ff9542c730f", "problem_id": "11f97f760a87fdad", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the multiplicative inverse of 9 modulo 37.", "output": "We want x such that 9x ≡ 1 (mod 37).\nUse extended Euclid to find s,t with 9s + 37t = 1.\nOne Bezout identity is: 9·(-4) + 37·(1) = 1.\nReduce s modulo 37: x ≡ 33 (mod 37).\nCheck: 9·33 mod 37 = 1.\nFinal answer: $x\\equiv 33\\pmod{37}$", "final_answer": "x\\equiv 33\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a77470b555257177", "problem_id": "11f97f760a87fdad", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the multiplicative inverse of 9 modulo 37.", "output": "1) We want x such that 9x ≡ 1 (mod 37).\n2) Use extended Euclid to find s,t with 9s + 37t = 1.\n3) One Bezout identity is: 9·(-4) + 37·(1) = 1.\n4) Reduce s modulo 37: x ≡ 33 (mod 37).\nFinal answer: $x\\equiv 33\\pmod{37}$", "final_answer": "x\\equiv 33\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6c161176c81db4eb", "problem_id": "6854d03417f22b81", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} + 4 x + 1$.", "output": "Differentiate term-by-term:\nd/dx(1) = 0.\nd/dx(x^{2}) = 2 x.\nd/dx(4 x) = 4.\nCombine: f'(x) = 2 \\left(x + 2\\right).\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $2\\cdot x + 4$", "final_answer": "2\\cdot x + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d467dd64d51802a", "problem_id": "77059cff0e0e9a9a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} x^{2} + 2$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to x^{2} - x + 2.\n3) Then the limit is 2.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: 3.\nFinal answer: $3$", "final_answer": "3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "58027bbdc2c8f03d", "problem_id": "cbc6ea08f638eab0", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 22 red balls and 10 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(32,8).\nFavorable samples: choose k reds and n−k blues: C(22,1)·C(10,7).\nCompute totals: C(32,8) = 10518300, and favorable = 2640.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{44}{175305}$.\nFinal answer: $\\frac{44}{175305}$", "final_answer": "\\frac{44}{175305}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "67f3a988a99d6882", "problem_id": "cbc6ea08f638eab0", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "An urn has 22 red balls and 10 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(32,8).\nFavorable samples: choose k reds and n−k blues: C(22,1)·C(10,7).\nFinal answer: $\\frac{44}{175305}$", "final_answer": "\\frac{44}{175305}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ba387c51182a10e4", "problem_id": "e2cf4665197f0ad9", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Two friends split $165 in the ratio 6:5. How much does each receive?", "output": "1) Total ratio parts = 6+5 = 11.\n2) Each part = 165 ÷ 11 = 15.\n3) First share = 6 × 15 = 90.\n4) Second share = 5 × 15 = 75.\nFinal answer: $75$", "final_answer": "75", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "60d80830014ed257", "problem_id": "e2cf4665197f0ad9", "style": "answer_only", "instruction": "Final answer only.", "input": "Two friends split $165 in the ratio 6:5. How much does each receive?", "output": "$75$", "final_answer": "75", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "27b716d8a62df9a5", "problem_id": "e1c55d2c1a3f1522", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A cyclist rides at 63 miles per hour for 1/2 hours, then at 45 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 63 × 1/2 = 63/2 miles.\n2) Second leg distance = 45 × 5/2 = 225/2 miles.\n3) Total distance = 63/2 + 225/2 = 144 miles.\nFinal answer: $144$", "final_answer": "144", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ac9204acfcf56e91", "problem_id": "e1c55d2c1a3f1522", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A cyclist rides at 63 miles per hour for 1/2 hours, then at 45 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 63 × 1/2 = 63/2 miles.\nSecond leg distance = 45 × 5/2 = 225/2 miles.\nTotal distance = 63/2 + 225/2 = 144 miles.\nCheck: adding both legs gives 144.\nFinal answer: $144$", "final_answer": "144", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db19e6aa48ddf2ae", "problem_id": "90700816846c085c", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Find the multiplicative inverse of 9 modulo 41.", "output": "We want x such that 9x ≡ 1 (mod 41).\nUse extended Euclid to find s,t with 9s + 41t = 1.\nOne Bezout identity is: 9·(-9) + 41·(2) = 1.\nFinal answer: $x\\equiv 32\\pmod{41}$", "final_answer": "x\\equiv 32\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c4892024a317cad5", "problem_id": "90700816846c085c", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Find the multiplicative inverse of 9 modulo 41.", "output": "1) We want x such that 9x ≡ 1 (mod 41).\n2) Use extended Euclid to find s,t with 9s + 41t = 1.\n3) One Bezout identity is: 9·(-9) + 41·(2) = 1.\n4) Reduce s modulo 41: x ≡ 32 (mod 41).\nFinal answer: $x\\equiv 32\\pmod{41}$", "final_answer": "x\\equiv 32\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7603a1868f2ebb32", "problem_id": "bc10dbfc4477761c", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{7}$. If it is flipped 6 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=6, k=3, p=$\\frac{4}{7}$ and simplify.\n4) Here C(6,3) = 20.\nFinal answer: $\\frac{34560}{117649}$", "final_answer": "\\frac{34560}{117649}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9b8cc653120c048d", "problem_id": "91858649a0e7970a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{3} + 2 x^{2} + 7 x + 15$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} + 2 x^{2} + 6 x + 15.\n3) Then the limit is 51.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 53.\nFinal answer: $53$", "final_answer": "53", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b9dd958326dcdc84", "problem_id": "b13909400809a0fc", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 68 miles per hour for 1 hours, then at 55 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 68 × 1 = 68 miles.\nSecond leg distance = 55 × 1 = 55 miles.\nTotal distance = 68 + 55 = 123 miles.\nCheck: adding both legs gives 123.\nFinal answer: $123$", "final_answer": "123", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8411ff310231cdd2", "problem_id": "798fa5b4e4c4f411", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} \\frac{- 2 x^{4} + 3 x^{2} + 20}{x + 2}$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to - \\frac{2 x^{4}}{x + 2} + \\frac{3 x^{2}}{x + 2} - x + \\frac{20}{x + 2}.\n3) Then the limit is \\text{NaN}.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 52.\nFinal answer: $52$", "final_answer": "52", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2a95685c56285563", "problem_id": "89a2bc9dd3788272", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡2 (mod 5), x≡0 (mod 11). Give the solution modulo 55.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 55.\n3) Compute the CRT solution to get x ≡ 22 (mod 55).\nFinal answer: $x\\equiv 22\\pmod{55}$", "final_answer": "x\\equiv 22\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "586a46fb440d7250", "problem_id": "89a2bc9dd3788272", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the congruences: x≡2 (mod 5), x≡0 (mod 11). Give the solution modulo 55.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 55.\nCompute the CRT solution to get x ≡ 22 (mod 55).\nFinal answer: $x\\equiv 22\\pmod{55}$", "final_answer": "x\\equiv 22\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d916e9e83a3fb10", "problem_id": "72a1763b38dfa521", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(- x^{3} - x^{2}\\right) \\cos{\\left(x^{2} + 2 x - 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x^{2} \\cos{\\left(\\left(x^{2} + 2 x\\right) - 2 \\right)}) = x^{2} \\sin{\\left(x^{2} + 2 x - 2 \\right)}$.\n3) So $f'(x) = 2 x^{4} \\sin{\\left(x^{2} + 2 x - 2 \\right)} + 2 x^{3} \\sin{\\left(x^{2} + 2 x - 2 \\right)} + x^{2} \\sin{\\left(x^{2} + 2 x - 2 \\right)} - 3 x^{2} \\cos{\\left(\\left(x^{2} + 2 x\\right) - 2 \\right)} + x e^{x} - 2 e^{x}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x^{2} \\cos{\\left(\\left(x^{2} + 2 x\\right) - 2 \\right)}) = x^{2} \\left(2 x + 2\\right) \\sin{\\left(x^{2} + 2 x - 2 \\right)} - 2 x \\cos{\\left(\\left(x^{2} + 2 x\\right) - 2 \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{2} \\left(- x - 1\\right) \\left(x + 1\\right) \\sin{\\left(x^{2} + 2 x - 2 \\right)} + x \\left(- 3 x - 2\\right) \\cos{\\left(\\left(x^{2} + 2 x\\right) - 2 \\right)} + \\left(x - 3\\right) e^{x} + e^{x}$.\nFinal answer: $- 2 x^{2} \\left(- x - 1\\right) \\left(x + 1\\right) \\sin{\\left(x^{2} + 2 x - 2 \\right)} + x \\left(- 3 x - 2\\right) \\cos{\\left(\\left(x^{2} + 2 x\\right) - 2 \\right)} + \\left(x - 3\\right) e^{x} + e^{x}$", "final_answer": "- 2 x^{2} \\left(- x - 1\\right) \\left(x + 1\\right) \\sin{\\left(x^{2} + 2 x - 2 \\right)} + x \\left(- 3 x - 2\\right) \\cos{\\left(\\left(x^{2} + 2 x\\right) - 2 \\right)} + \\left(x - 3\\right) e^{x} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0f0bd371447c26df", "problem_id": "65008ca2774abac3", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A store sells a backpack for $345. It is discounted by 20%. You buy 10 backpacks. What is the total cost?", "output": "Discounted price = 345 × (1 − 20/100) = 345 × 80/100 = 276.\nSubtotal = 10 × 276 = 2760.\nCheck: recomputing the formula yields 2760.\nFinal answer: $2760$", "final_answer": "2760", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "351ff007662fff7c", "problem_id": "051291cc9312935a", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$- 6 x - 5 y$ = $-6$\n$6 y$ = $0$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $-6$: $- 36 y$ = $0$.\nSubtract to eliminate x: $36 y$ = $0$.\nSolve for y: y = $0$.\nSubstitute back to find x: x = $1$.\nCheck: plug (x,y)=(1,0) into both equations.\nFinal answer: $(x,y)=\\left(1, 0\\right)$", "final_answer": "(x,y)=\\left(1, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1310c04de293d95d", "problem_id": "051291cc9312935a", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$- 6 x - 5 y$ = $-6$\n$6 y$ = $0$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $-6$: $- 36 y$ = $0$.\n4) Subtract to eliminate x: $36 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $1$.\nFinal answer: $(x,y)=\\left(1, 0\\right)$", "final_answer": "(x,y)=\\left(1, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db112cf84d333c8e", "problem_id": "1a9c1e882f4446d2", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many binary strings of length 13 contain exactly 2 ones with no two ones adjacent?", "output": "1) Reserve 1 mandatory zeros between the 2 ones.\n2) This leaves n−k+1 = 12 valid slots for the ones.\n3) Choose the slots: C(12,2) = 66.\nFinal answer: $66$", "final_answer": "66", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "433845cbe50b5720", "problem_id": "1a9c1e882f4446d2", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many binary strings of length 13 contain exactly 2 ones with no two ones adjacent?", "output": "Reserve 1 mandatory zeros between the 2 ones.\nThis leaves n−k+1 = 12 valid slots for the ones.\nChoose the slots: C(12,2) = 66.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $66$", "final_answer": "66", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e416d5bb3fd4d57f", "problem_id": "0a7cbf930d410060", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the multiplicative inverse of 20 modulo 29.", "output": "1) We want x such that 20x ≡ 1 (mod 29).\n2) Use extended Euclid to find s,t with 20s + 29t = 1.\n3) One Bezout identity is: 20·(-13) + 29·(9) = 1.\n4) Reduce s modulo 29: x ≡ 16 (mod 29).\nFinal answer: $x\\equiv 16\\pmod{29}$", "final_answer": "x\\equiv 16\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ea814e15f4605df4", "problem_id": "0a7cbf930d410060", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the multiplicative inverse of 20 modulo 29.", "output": "We want x such that 20x ≡ 1 (mod 29).\nUse extended Euclid to find s,t with 20s + 29t = 1.\nOne Bezout identity is: 20·(-13) + 29·(9) = 1.\nReduce s modulo 29: x ≡ 16 (mod 29).\nCheck: 20·16 mod 29 = 1.\nFinal answer: $x\\equiv 16\\pmod{29}$", "final_answer": "x\\equiv 16\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f57f5952d9fce4fa", "problem_id": "c0fbe97de884fdee", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} + 4 x + 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x^{3}) = 3 x^{3}$.\n3) So $f'(x) = 3 x^{3} + 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x^{3}) = 3 x^{2}$.\nTherefore the correct derivative is $f'(x) = 3 x^{2} + 4$.\nFinal answer: $3 x^{2} + 4$", "final_answer": "3 x^{2} + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d50ca79749f85847", "problem_id": "c8f2d5327aae1d14", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 3 x - 6$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} + 3 x - 6$ = $3 \\left(x - 1\\right) \\left(x + 2\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $1$.\n5) From the second factor: x = $-2$.\nFinal answer: $x \\in \\{1, -2\\}$", "final_answer": "x \\in \\{1, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b4172713e2eb6b66", "problem_id": "9047de539922cc99", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(- 3 x^{2} - x - 1\\right) \\cos{\\left(x - 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 3 x^{2} \\cos{\\left(x - 1 \\right)}) = 3 x^{2} \\sin{\\left(x - 1 \\right)}$.\n3) So $f'(x) = 3 x^{2} \\sin{\\left(x - 1 \\right)} + x e^{x} + x \\sin{\\left(x - 1 \\right)} - 2 e^{x} + \\sin{\\left(x - 1 \\right)} - \\cos{\\left(x - 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 3 x^{2} \\cos{\\left(x - 1 \\right)}) = 3 x^{2} \\sin{\\left(x - 1 \\right)} - 6 x \\cos{\\left(x - 1 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(- 6 x - 1\\right) \\cos{\\left(x - 1 \\right)} + \\left(x - 3\\right) e^{x} - \\left(x \\left(- 3 x - 1\\right) - 1\\right) \\sin{\\left(x - 1 \\right)} + e^{x}$.\nFinal answer: $\\left(- 6 x - 1\\right) \\cos{\\left(x - 1 \\right)} + \\left(x - 3\\right) e^{x} - \\left(x \\left(- 3 x - 1\\right) - 1\\right) \\sin{\\left(x - 1 \\right)} + e^{x}$", "final_answer": "\\left(- 6 x - 1\\right) \\cos{\\left(x - 1 \\right)} + \\left(x - 3\\right) e^{x} - \\left(x \\left(- 3 x - 1\\right) - 1\\right) \\sin{\\left(x - 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7d51a87d4bed1e4f", "problem_id": "32a7e3c5fd4238de", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the multiplicative inverse of 13 modulo 17.", "output": "1) We want x such that 13x ≡ 1 (mod 17).\n2) Use extended Euclid to find s,t with 13s + 17t = 1.\n3) One Bezout identity is: 13·(4) + 17·(-3) = 1.\n4) Reduce s modulo 17: x ≡ 4 (mod 17).\nFinal answer: $x\\equiv 4\\pmod{17}$", "final_answer": "x\\equiv 4\\pmod{17}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e636bda084566dd9", "problem_id": "48b177d48f951de9", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many binary strings of length 21 contain exactly 6 ones with no two ones adjacent?", "output": "1) Reserve 5 mandatory zeros between the 6 ones.\n2) This leaves n−k+1 = 16 valid slots for the ones.\n3) Choose the slots: C(16,6) = 8008.\nFinal answer: $8008$", "final_answer": "8008", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c816e14590863c89", "problem_id": "48b177d48f951de9", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many binary strings of length 21 contain exactly 6 ones with no two ones adjacent?", "output": "Reserve 5 mandatory zeros between the 6 ones.\nThis leaves n−k+1 = 16 valid slots for the ones.\nChoose the slots: C(16,6) = 8008.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $8008$", "final_answer": "8008", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8309586542b23a1a", "problem_id": "e89689863c0617d8", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Two friends split $2226 in the ratio 9:5. How much does each receive?", "output": "1) Total ratio parts = 9+5 = 14.\n2) Each part = 2226 ÷ 14 = 159.\n3) First share = 9 × 159 = 1431.\n4) Second share = 5 × 159 = 795.\nFinal answer: $795$", "final_answer": "795", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4b5c30e0117e9442", "problem_id": "e89689863c0617d8", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $2226 in the ratio 9:5. How much does each receive?", "output": "Total ratio parts = 9+5 = 14.\nEach part = 2226 ÷ 14 = 159.\nFirst share = 9 × 159 = 1431.\nSecond share = 5 × 159 = 795.\nCheck: 1431+795=2226.\nFinal answer: $795$", "final_answer": "795", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0235a5fca88b3d94", "problem_id": "146a8449e8c9727b", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡4 (mod 5), x≡2 (mod 7). Give the solution modulo 35.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 35.\nCompute the CRT solution to get x ≡ 9 (mod 35).\nCheck: 9 mod 5=4 and mod 7=2.\nFinal answer: $x\\equiv 9\\pmod{35}$", "final_answer": "x\\equiv 9\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a602a7b9be48734e", "problem_id": "146a8449e8c9727b", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the congruences: x≡4 (mod 5), x≡2 (mod 7). Give the solution modulo 35.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 35.\nCompute the CRT solution to get x ≡ 9 (mod 35).\nFinal answer: $x\\equiv 9\\pmod{35}$", "final_answer": "x\\equiv 9\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "319b09a5f61f8995", "problem_id": "5ad79604fefc56a0", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the congruences: x≡3 (mod 5), x≡3 (mod 8). Give the solution modulo 40.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 40.\n3) Compute the CRT solution to get x ≡ 3 (mod 40).\nFinal answer: $x\\equiv 3\\pmod{40}$", "final_answer": "x\\equiv 3\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f54f966122c202be", "problem_id": "5ad79604fefc56a0", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡3 (mod 5), x≡3 (mod 8). Give the solution modulo 40.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 40.\nCompute the CRT solution to get x ≡ 3 (mod 40).\nCheck: 3 mod 5=3 and mod 8=3.\nFinal answer: $x\\equiv 3\\pmod{40}$", "final_answer": "x\\equiv 3\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2870ee3835d73d41", "problem_id": "78a3c203f2f1a84a", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 14 red balls and 24 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(38,10).\n3) Favorable samples: choose k reds and n−k blues: C(14,4)·C(24,6).\n4) Compute totals: C(38,10) = 472733756, and favorable = 134730596.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{161161}{565471}$", "final_answer": "\\frac{161161}{565471}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b1a56663732dc09f", "problem_id": "78a3c203f2f1a84a", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 14 red balls and 24 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(38,10).\nFavorable samples: choose k reds and n−k blues: C(14,4)·C(24,6).\nCompute totals: C(38,10) = 472733756, and favorable = 134730596.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{161161}{565471}$.\nFinal answer: $\\frac{161161}{565471}$", "final_answer": "\\frac{161161}{565471}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f4bca3d4eac7b796", "problem_id": "060ef45041b19f8b", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{6} x}{3} - \\frac{x}{6} - \\frac{\\sqrt{6}}{6} - \\frac{1}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{\\sqrt{6}}{3} - \\frac{1}{6}$, c=$- \\frac{\\sqrt{6}}{6} - \\frac{1}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{7 \\sqrt{6}}{9} + \\frac{73}{36}$, so √Δ = $\\sqrt{\\frac{7 \\sqrt{6}}{9} + \\frac{73}{36}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{1}{12} + \\frac{\\sqrt{6}}{6} + \\frac{\\sqrt{28 \\sqrt{6} + 73}}{12}$.\n7) x2 = $- \\frac{\\sqrt{28 \\sqrt{6} + 73}}{12} + \\frac{1}{12} + \\frac{\\sqrt{6}}{6}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{1}{12} + \\frac{\\sqrt{6}}{6} + \\frac{\\sqrt{28\\cdot \\sqrt{6} + 73}}{12}, - \\frac{\\sqrt{28\\cdot \\sqrt{6} + 73}}{12} + \\frac{1}{12} + \\frac{\\sqrt{6}}{6}\\}$", "final_answer": "x \\in \\{\\frac{1}{12} + \\frac{\\sqrt{6}}{6} + \\frac{\\sqrt{28\\cdot \\sqrt{6} + 73}}{12}, - \\frac{\\sqrt{28\\cdot \\sqrt{6} + 73}}{12} + \\frac{1}{12} + \\frac{\\sqrt{6}}{6}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "299bf1782e7de8b6", "problem_id": "060ef45041b19f8b", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{6} x}{3} - \\frac{x}{6} - \\frac{\\sqrt{6}}{6} - \\frac{1}{3}$ = 0.", "output": "$x \\in \\{\\frac{1}{12} + \\frac{\\sqrt{6}}{6} + \\frac{\\sqrt{28\\cdot \\sqrt{6} + 73}}{12}, - \\frac{\\sqrt{28\\cdot \\sqrt{6} + 73}}{12} + \\frac{1}{12} + \\frac{\\sqrt{6}}{6}\\}$", "final_answer": "x \\in \\{\\frac{1}{12} + \\frac{\\sqrt{6}}{6} + \\frac{\\sqrt{28\\cdot \\sqrt{6} + 73}}{12}, - \\frac{\\sqrt{28\\cdot \\sqrt{6} + 73}}{12} + \\frac{1}{12} + \\frac{\\sqrt{6}}{6}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e734812d6c57014a", "problem_id": "a0cd7a21009eca00", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_9=43$?", "output": "Let y_i=x_i-3, then y_i≥0 and y_1+…+y_9=16.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=16 and k=9.\nCompute C(24,8) = 735471.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $735471$", "final_answer": "735471", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "74879831863bf3fb", "problem_id": "a0cd7a21009eca00", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_9=43$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_9=16.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=16 and k=9.\n4) Compute C(24,8) = 735471.\nFinal answer: $735471$", "final_answer": "735471", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9a54fb081e4c4369", "problem_id": "4dd4cc37b96485d5", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} + x - 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(2 x^{2}) = 4 x^{2}$.\n3) So $f'(x) = 4 x^{2} + 1$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(2 x^{2}) = 4 x$.\nTherefore the correct derivative is $f'(x) = 4 x + 1$.\nFinal answer: $4 x + 1$", "final_answer": "4 x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2d375d7336881d6e", "problem_id": "3bf9b66dd06668c7", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_8=18$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=18 and k=8.\nCompute C(25,7) = 480700.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $480700$", "final_answer": "480700", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7bc0cdda4f45a2a6", "problem_id": "d9a249add92fc5d1", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $2505 in the ratio 8:7. How much does each receive?", "output": "1) Total ratio parts = 8+7 = 15.\n2) Each part = 2505 ÷ 15 = 167.\n3) First share = 8 × 167 = 1336.\n4) Second share = 7 × 167 = 1169.\nFinal answer: $1169$", "final_answer": "1169", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "43ca96b469e75f3f", "problem_id": "d9a249add92fc5d1", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Two friends split $2505 in the ratio 8:7. How much does each receive?", "output": "Total ratio parts = 8+7 = 15.\nEach part = 2505 ÷ 15 = 167.\nFirst share = 8 × 167 = 1336.\nFinal answer: $1169$", "final_answer": "1169", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6ed20dbdf04a5761", "problem_id": "d8509bdb27162fcf", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Worker A finishes a job in 20 hours and Worker B finishes it in 13 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/20 job/hour. B's rate = 1/13 job/hour.\n2) Combined rate = 1/20 + 1/13 = 33/260 job/hour.\n3) Time = 1 ÷ (combined rate) = 260/33 hours.\nFinal answer: $\\frac{260}{33}$", "final_answer": "\\frac{260}{33}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4f246840ff58ee4f", "problem_id": "d8509bdb27162fcf", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Worker A finishes a job in 20 hours and Worker B finishes it in 13 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/20 job/hour. B's rate = 1/13 job/hour.\nCombined rate = 1/20 + 1/13 = 33/260 job/hour.\nTime = 1 ÷ (combined rate) = 260/33 hours.\nCheck: (combined rate)×(time) = 33/260×260/33 = 1.\nFinal answer: $\\frac{260}{33}$", "final_answer": "\\frac{260}{33}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c88bceaaf4f9a24e", "problem_id": "7975e440a88652f2", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$- 7 x - 5 y$ = $37$\n$x + 6 y$ = $-37$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $- 7 x - 5 y$ = $37$.\n3) Multiply the second equation by $-7$: $- 7 x - 42 y$ = $259$.\n4) Subtract to eliminate x: $37 y$ = $-222$.\n5) Solve for y: y = $-6$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, -6\\right)$", "final_answer": "(x,y)=\\left(-1, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a8121b413a17294f", "problem_id": "7975e440a88652f2", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the linear system:\n$- 7 x - 5 y$ = $37$\n$x + 6 y$ = $-37$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $- 7 x - 5 y$ = $37$.\nMultiply the second equation by $-7$: $- 7 x - 42 y$ = $259$.\nFinal answer: $(x,y)=\\left(-1, -6\\right)$", "final_answer": "(x,y)=\\left(-1, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4291d0098c300ab4", "problem_id": "af72c159de6e5f1b", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x + 2$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $2 x^{2} - 2 x + 2$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $4 x - 2$.\n5) Evaluate at a: p'(2) = $6$.\nFinal answer: $6$", "final_answer": "6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "878a851667da14d2", "problem_id": "af72c159de6e5f1b", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x + 2$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{2} - 2 x + 2$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $4 x - 2$.\nEvaluate at a: p'(2) = $6$.\nCheck: the identity gives $6$.\nFinal answer: $6$", "final_answer": "6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5cb24cdaad95ba9b", "problem_id": "88e13da041554d0c", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A store sells a backpack for $106. It is discounted by 10%. You buy 10 backpacks. What is the total cost?", "output": "1) Discounted price = 106 × (1 − 10/100) = 106 × 90/100 = 477/5.\n2) Subtotal = 10 × 477/5 = 954.\nFinal answer: $954$", "final_answer": "954", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0a47f6e32291858a", "problem_id": "77bf007d2ad0bb7e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} x - 3$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to -3.\n3) Then the limit is -3.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -6.\nFinal answer: $-6$", "final_answer": "-6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a93caa3b5630d262", "problem_id": "c1d34d67a269c9d7", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{2} - x + 3$.", "output": "1) Differentiate term-by-term:\n2) d/dx(3) = 0.\n3) d/dx(- x) = -1.\n4) d/dx(- 3 x^{2}) = - 6 x.\n5) Combine: f'(x) = - 6 x - 1.\nFinal answer: $- 6\\cdot x - 1$", "final_answer": "- 6\\cdot x - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "00194bf332298a9f", "problem_id": "c1d34d67a269c9d7", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{2} - x + 3$.", "output": "Differentiate term-by-term:\nd/dx(3) = 0.\nd/dx(- x) = -1.\nFinal answer: $- 6\\cdot x - 1$", "final_answer": "- 6\\cdot x - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0ff43007e9926b13", "problem_id": "9bfe61e156cce809", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A cyclist rides at 41 miles per hour for 1/2 hours, then at 68 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 41 × 1/2 = 41/2 miles.\n2) Second leg distance = 68 × 1 = 68 miles.\n3) Total distance = 41/2 + 68 = 177/2 miles.\nFinal answer: $\\frac{177}{2}$", "final_answer": "\\frac{177}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1ca5918f6b027b4d", "problem_id": "9bfe61e156cce809", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A cyclist rides at 41 miles per hour for 1/2 hours, then at 68 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 41 × 1/2 = 41/2 miles.\nSecond leg distance = 68 × 1 = 68 miles.\nTotal distance = 41/2 + 68 = 177/2 miles.\nFinal answer: $\\frac{177}{2}$", "final_answer": "\\frac{177}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "576d6c0083e4c717", "problem_id": "312bc297f9c2c3e1", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 6 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=6, k=4, p=$\\frac{1}{2}$ and simplify.\n4) Here C(6,4) = 15.\nFinal answer: $\\frac{15}{64}$", "final_answer": "\\frac{15}{64}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8f5c7d35a8ea227b", "problem_id": "312bc297f9c2c3e1", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 6 times, what is the probability of exactly 4 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=6, k=4, p=$\\frac{1}{2}$ and simplify.\nHere C(6,4) = 15.\nCheck: expression simplifies to $\\frac{15}{64}$.\nFinal answer: $\\frac{15}{64}$", "final_answer": "\\frac{15}{64}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e76ca69b887f3bc6", "problem_id": "ff8ee5f600f64d6f", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "You want 15 liters of a 35% salt solution. You have a 30% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 15−x liters is 40% solution.\n2) Salt equation: 30x + 40(15−x) = 35·15.\n3) (30-40)x = (35-40)·15 ⇒ x = 15/2.\n4) Then 15−x = 15/2.\nFinal answer: $15/2 L of 30% and 15/2 L of 40%$", "final_answer": "15/2 L of 30% and 15/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "202f2b807f21a754", "problem_id": "ff8ee5f600f64d6f", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "You want 15 liters of a 35% salt solution. You have a 30% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 15−x liters is 40% solution.\nSalt equation: 30x + 40(15−x) = 35·15.\n(30-40)x = (35-40)·15 ⇒ x = 15/2.\nFinal answer: $15/2 L of 30% and 15/2 L of 40%$", "final_answer": "15/2 L of 30% and 15/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "464f845b1635a635", "problem_id": "d9f8aa73ec777ba5", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} + 1$ at $x=1$.", "output": "Differentiate: f'(x) = $6 x + 2 \\cos{\\left(x \\right)} + 3$.\nSlope at x=1: m = f'(1) = $2 \\cos{\\left(1 \\right)} + 9$.\nPoint on curve: (1, f(1)) = (1, $2 \\sin{\\left(1 \\right)} + 7$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 2 \\sin{\\left(1 \\right)} + 7 = 2 \\cos{\\left(1 \\right)} + 9(x − 1).\nOptionally expand: y = $2 x \\cos{\\left(1 \\right)} + 9 x - 2 - 2 \\cos{\\left(1 \\right)} + 2 \\sin{\\left(1 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 9\\cdot x - 2 - 2\\cdot \\cos{\\left(1 \\right)} + 2\\cdot \\sin{\\left(1 \\right)}$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 9\\cdot x - 2 - 2\\cdot \\cos{\\left(1 \\right)} + 2\\cdot \\sin{\\left(1 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "41cc99bb54b752a5", "problem_id": "d9f8aa73ec777ba5", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Find the equation of the tangent line to $y=3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} + 1$ at $x=1$.", "output": "1) Differentiate: f'(x) = $6 x + 2 \\cos{\\left(x \\right)} + 3$.\n2) Slope at x=1: m = f'(1) = $2 \\cos{\\left(1 \\right)} + 9$.\n3) Point on curve: (1, f(1)) = (1, $2 \\sin{\\left(1 \\right)} + 7$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 2 \\sin{\\left(1 \\right)} + 7 = 2 \\cos{\\left(1 \\right)} + 9(x − 1).\n6) Optionally expand: y = $2 x \\cos{\\left(1 \\right)} + 9 x - 2 - 2 \\cos{\\left(1 \\right)} + 2 \\sin{\\left(1 \\right)}$.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 9\\cdot x - 2 - 2\\cdot \\cos{\\left(1 \\right)} + 2\\cdot \\sin{\\left(1 \\right)}$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 9\\cdot x - 2 - 2\\cdot \\cos{\\left(1 \\right)} + 2\\cdot \\sin{\\left(1 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "38fece2c7a76c194", "problem_id": "8cc196e22f91e90c", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "You want 10 liters of a 30% salt solution. You have a 15% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 10−x liters is 60% solution.\nSalt equation: 15x + 60(10−x) = 30·10.\n(15-60)x = (30-60)·10 ⇒ x = 20/3.\nThen 10−x = 10/3.\nCheck: 15·20/3 + 60·10/3 = 30·10.\nFinal answer: $20/3 L of 15% and 10/3 L of 60%$", "final_answer": "20/3 L of 15% and 10/3 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff2dcbddb73964cb", "problem_id": "8cc196e22f91e90c", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "You want 10 liters of a 30% salt solution. You have a 15% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 10−x liters is 60% solution.\n2) Salt equation: 15x + 60(10−x) = 30·10.\n3) (15-60)x = (30-60)·10 ⇒ x = 20/3.\n4) Then 10−x = 10/3.\nFinal answer: $20/3 L of 15% and 10/3 L of 60%$", "final_answer": "20/3 L of 15% and 10/3 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f2999f8ac3f5fa09", "problem_id": "54d9a8fa192fb813", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$x - y$ = $1$\n$- 3 x + y$ = $5$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $- 3 x + 3 y$ = $-3$.\n3) Multiply the second equation by $1$: $- 3 x + y$ = $5$.\n4) Subtract to eliminate x: $2 y$ = $-8$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, -4\\right)$", "final_answer": "(x,y)=\\left(-3, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d67e1de86e6d1989", "problem_id": "54d9a8fa192fb813", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$x - y$ = $1$\n$- 3 x + y$ = $5$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $- 3 x + 3 y$ = $-3$.\nMultiply the second equation by $1$: $- 3 x + y$ = $5$.\nSubtract to eliminate x: $2 y$ = $-8$.\nSolve for y: y = $-4$.\nSubstitute back to find x: x = $-3$.\nCheck: plug (x,y)=(-3,-4) into both equations.\nFinal answer: $(x,y)=\\left(-3, -4\\right)$", "final_answer": "(x,y)=\\left(-3, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e5b69faf8160fc1f", "problem_id": "220b1d991a18d9da", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(- 3 x^{2} - 3 x - 2\\right) \\cos{\\left(x - 3 \\right)}$.", "output": "Let f(x) = $\\left(x + 1\\right) e^{x} + \\left(- 3 x^{2} - 3 x - 2\\right) \\cos{\\left(x - 3 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $3 \\left(- 2 x - 1\\right) \\cos{\\left(x - 3 \\right)} + \\left(x + 1\\right) e^{x} - \\left(- 3 x^{2} - 3 x - 2\\right) \\sin{\\left(x - 3 \\right)} + e^{x}$.\nFinal answer: $\\left(- 6\\cdot x - 3\\right)\\cdot \\cos{\\left(x - 3 \\right)} + \\left(x + 1\\right)\\cdot e^{x} - \\left(- 3\\cdot x^{2} - 3\\cdot x - 2\\right)\\cdot \\sin{\\left(x - 3 \\right)} + e^{x}$", "final_answer": "\\left(- 6\\cdot x - 3\\right)\\cdot \\cos{\\left(x - 3 \\right)} + \\left(x + 1\\right)\\cdot e^{x} - \\left(- 3\\cdot x^{2} - 3\\cdot x - 2\\right)\\cdot \\sin{\\left(x - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "47ca718a404f1350", "problem_id": "220b1d991a18d9da", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(- 3 x^{2} - 3 x - 2\\right) \\cos{\\left(x - 3 \\right)}$.", "output": "1) Let f(x) = $\\left(x + 1\\right) e^{x} + \\left(- 3 x^{2} - 3 x - 2\\right) \\cos{\\left(x - 3 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $3 \\left(- 2 x - 1\\right) \\cos{\\left(x - 3 \\right)} + \\left(x + 1\\right) e^{x} - \\left(- 3 x^{2} - 3 x - 2\\right) \\sin{\\left(x - 3 \\right)} + e^{x}$.\nFinal answer: $\\left(- 6\\cdot x - 3\\right)\\cdot \\cos{\\left(x - 3 \\right)} + \\left(x + 1\\right)\\cdot e^{x} - \\left(- 3\\cdot x^{2} - 3\\cdot x - 2\\right)\\cdot \\sin{\\left(x - 3 \\right)} + e^{x}$", "final_answer": "\\left(- 6\\cdot x - 3\\right)\\cdot \\cos{\\left(x - 3 \\right)} + \\left(x + 1\\right)\\cdot e^{x} - \\left(- 3\\cdot x^{2} - 3\\cdot x - 2\\right)\\cdot \\sin{\\left(x - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a342a52047f1dde0", "problem_id": "cebf818cad64ef6f", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - x^{2} + 2 x - 1\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- x^{2} + 2 x - 1$.\n3) Compute an antiderivative: F(x) = $- \\frac{x^{3}}{3} + x^{2} - x$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{x^{3}}{3} + x^{2} - x$", "final_answer": "C - \\frac{x^{3}}{3} + x^{2} - x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "08f7b04f7d4f22f0", "problem_id": "cebf818cad64ef6f", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - x^{2} + 2 x - 1\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- x^{2} + 2 x - 1$.\nCompute an antiderivative: F(x) = $- \\frac{x^{3}}{3} + x^{2} - x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - \\frac{x^{3}}{3} + x^{2} - x$", "final_answer": "C - \\frac{x^{3}}{3} + x^{2} - x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0aa78102e477f1b2", "problem_id": "984e1aa42628536c", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 4 x + 2$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$4$, c=$2$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $0$, so √Δ = $0$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $-1$.\nx2 = $-1$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-1, -1\\}$", "final_answer": "x \\in \\{-1, -1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c2eea9805f80989e", "problem_id": "984e1aa42628536c", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 4 x + 2$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$4$, c=$2$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $0$, so √Δ = $0$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $-1$.\n7) x2 = $-1$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{-1, -1\\}$", "final_answer": "x \\in \\{-1, -1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0aa7095f9b1470e4", "problem_id": "1ae9634a07a93796", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 9 red balls and 12 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(21,8).\nFavorable samples: choose k reds and n−k blues: C(9,3)·C(12,5).\nCompute totals: C(21,8) = 203490, and favorable = 66528.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{528}{1615}$.\nFinal answer: $\\frac{528}{1615}$", "final_answer": "\\frac{528}{1615}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "880b4df2cc2454ed", "problem_id": "1ae9634a07a93796", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "An urn has 9 red balls and 12 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(21,8).\n3) Favorable samples: choose k reds and n−k blues: C(9,3)·C(12,5).\n4) Compute totals: C(21,8) = 203490, and favorable = 66528.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{528}{1615}$", "final_answer": "\\frac{528}{1615}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4c913e9e6d694996", "problem_id": "e634964889975fe1", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the linear system:\n$3 x + 2 y$ = $-2$\n$- x + 2 y$ = $14$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $- 3 x - 2 y$ = $2$.\nMultiply the second equation by $3$: $- 3 x + 6 y$ = $42$.\nFinal answer: $(x,y)=\\left(-4, 5\\right)$", "final_answer": "(x,y)=\\left(-4, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d66de9f58a3b2a84", "problem_id": "e634964889975fe1", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the linear system:\n$3 x + 2 y$ = $-2$\n$- x + 2 y$ = $14$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $- 3 x - 2 y$ = $2$.\n3) Multiply the second equation by $3$: $- 3 x + 6 y$ = $42$.\n4) Subtract to eliminate x: $- 8 y$ = $-40$.\n5) Solve for y: y = $5$.\n6) Substitute back to find x: x = $-4$.\nFinal answer: $(x,y)=\\left(-4, 5\\right)$", "final_answer": "(x,y)=\\left(-4, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e759cba0be51892a", "problem_id": "bd14cac53b312215", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 10 times, what is the probability of exactly 7 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=7, p=$\\frac{1}{8}$ and simplify.\nHere C(10,7) = 120.\nCheck: expression simplifies to $\\frac{5145}{134217728}$.\nFinal answer: $\\frac{5145}{134217728}$", "final_answer": "\\frac{5145}{134217728}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fbd93f585e0b9faf", "problem_id": "e54de4d1a7b1963c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 3 x^{2} - 9 x - 21$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} - 10 x - 21.\n3) Then the limit is -53.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -51.\nFinal answer: $-51$", "final_answer": "-51", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "45f9d2de369c7557", "problem_id": "0c058a2d5bed7f57", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{11} x + \\frac{19 x}{5} - \\frac{4 \\sqrt{11}}{5} + \\frac{12}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{19}{5} - \\sqrt{11}$, c=$\\frac{12}{5} - \\frac{4 \\sqrt{11}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{396}{25} - \\frac{22 \\sqrt{11}}{5}$, so √Δ = $\\sqrt{\\frac{396}{25} - \\frac{22 \\sqrt{11}}{5}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{19}{10} + \\frac{\\sqrt{396 - 110 \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}$.\n7) x2 = $- \\frac{19}{10} - \\frac{\\sqrt{396 - 110 \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{19}{10} + \\frac{\\sqrt{396 - 110\\cdot \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}, - \\frac{19}{10} - \\frac{\\sqrt{396 - 110\\cdot \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{19}{10} + \\frac{\\sqrt{396 - 110\\cdot \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}, - \\frac{19}{10} - \\frac{\\sqrt{396 - 110\\cdot \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "89108e8a6db729d3", "problem_id": "0c058a2d5bed7f57", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{11} x + \\frac{19 x}{5} - \\frac{4 \\sqrt{11}}{5} + \\frac{12}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{19}{5} - \\sqrt{11}$, c=$\\frac{12}{5} - \\frac{4 \\sqrt{11}}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{396}{25} - \\frac{22 \\sqrt{11}}{5}$, so √Δ = $\\sqrt{\\frac{396}{25} - \\frac{22 \\sqrt{11}}{5}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{19}{10} + \\frac{\\sqrt{396 - 110 \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}$.\nx2 = $- \\frac{19}{10} - \\frac{\\sqrt{396 - 110 \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{19}{10} + \\frac{\\sqrt{396 - 110\\cdot \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}, - \\frac{19}{10} - \\frac{\\sqrt{396 - 110\\cdot \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{19}{10} + \\frac{\\sqrt{396 - 110\\cdot \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}, - \\frac{19}{10} - \\frac{\\sqrt{396 - 110\\cdot \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ae06c8edc650afb1", "problem_id": "622f9230972b3858", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 3 x^{2} + 5 x - 4$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} + 4 x - 4.\n3) Then the limit is -11.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: -12.\nFinal answer: $-12$", "final_answer": "-12", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "1a234ff4abc2d929", "problem_id": "e21f03b6ed1414f2", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_9=42$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=42 and k=9.\n4) Compute C(50,8) = 536878650.\nFinal answer: $536878650$", "final_answer": "536878650", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d336746214093b1a", "problem_id": "b7362e9f8b91bb7b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(- 2 x^{3} - x^{2} + 3 x + 2\\right) \\cos{\\left(3 x^{2} - 2 x + 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 2 x^{3} \\cos{\\left(3 x^{2} - 2 x + 2 \\right)}) = 2 x^{3} \\sin{\\left(3 x^{2} - 2 x + 2 \\right)}$.\n3) So $f'(x) = 8 x^{3} \\sin{\\left(3 x^{2} - 2 x + 2 \\right)} - 20 x^{2} \\sin{\\left(3 x^{2} - 2 x + 2 \\right)} + 2 x e^{x} - 6 x \\sin{\\left(3 x^{2} - 2 x + 2 \\right)} - 2 x \\cos{\\left(3 x^{2} - 2 x + 2 \\right)} - 4 e^{x} + 4 \\sin{\\left(3 x^{2} - 2 x + 2 \\right)} + 3 \\cos{\\left(3 x^{2} - 2 x + 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 2 x^{3} \\cos{\\left(3 x^{2} - 2 x + 2 \\right)}) = 2 x^{3} \\left(6 x - 2\\right) \\sin{\\left(3 x^{2} - 2 x + 2 \\right)} - 6 x^{2} \\cos{\\left(3 x^{2} - 2 x + 2 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x - 3\\right) e^{x} - 2 \\left(3 x - 1\\right) \\left(x \\left(x \\left(- 2 x - 1\\right) + 3\\right) + 2\\right) \\sin{\\left(\\left(3 x^{2} - 2 x\\right) + 2 \\right)} + \\left(- 6 x^{2} - 2 x + 3\\right) \\cos{\\left(\\left(3 x^{2} - 2 x\\right) + 2 \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(x - 3\\right) e^{x} - 2 \\left(3 x - 1\\right) \\left(x \\left(x \\left(- 2 x - 1\\right) + 3\\right) + 2\\right) \\sin{\\left(\\left(3 x^{2} - 2 x\\right) + 2 \\right)} + \\left(- 6 x^{2} - 2 x + 3\\right) \\cos{\\left(\\left(3 x^{2} - 2 x\\right) + 2 \\right)} + 2 e^{x}$", "final_answer": "2 \\left(x - 3\\right) e^{x} - 2 \\left(3 x - 1\\right) \\left(x \\left(x \\left(- 2 x - 1\\right) + 3\\right) + 2\\right) \\sin{\\left(\\left(3 x^{2} - 2 x\\right) + 2 \\right)} + \\left(- 6 x^{2} - 2 x + 3\\right) \\cos{\\left(\\left(3 x^{2} - 2 x\\right) + 2 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "1fc45acc03ce152f", "problem_id": "6a0efee1761c333a", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 56 miles per hour for 2 hours, then at 31 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 56 × 2 = 112 miles.\nSecond leg distance = 31 × 2 = 62 miles.\nTotal distance = 112 + 62 = 174 miles.\nCheck: adding both legs gives 174.\nFinal answer: $174$", "final_answer": "174", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "348cb2993db255d1", "problem_id": "3357491f00dc3846", "style": "answer_only", "instruction": "Final answer only.", "input": "Solve the congruences: x≡9 (mod 11), x≡2 (mod 7). Give the solution modulo 77.", "output": "$x\\equiv 9\\pmod{77}$", "final_answer": "x\\equiv 9\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "501a45c70715023e", "problem_id": "3357491f00dc3846", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the congruences: x≡9 (mod 11), x≡2 (mod 7). Give the solution modulo 77.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 77.\nCompute the CRT solution to get x ≡ 9 (mod 77).\nFinal answer: $x\\equiv 9\\pmod{77}$", "final_answer": "x\\equiv 9\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "239f5cf856e03f5b", "problem_id": "4c067de813c10270", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} - 4 x^{4} + 2 x^{3} + 3 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 4 x^{4} + 2 x^{3} + 3 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1$.\nEvaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{33 \\pi^{5}}{40} - \\frac{9 \\pi^{2}}{8} - \\frac{3 \\pi}{2} - 1 + \\frac{9 \\pi^{3}}{8} + \\frac{15 \\pi^{4}}{32}$.\nFinal answer: $- \\frac{33\\cdot \\pi^{5}}{40} - \\frac{9\\cdot \\pi^{2}}{8} - \\frac{3\\cdot \\pi}{2} - 1 + \\frac{9\\cdot \\pi^{3}}{8} + \\frac{15\\cdot \\pi^{4}}{32}$", "final_answer": "- \\frac{33\\cdot \\pi^{5}}{40} - \\frac{9\\cdot \\pi^{2}}{8} - \\frac{3\\cdot \\pi}{2} - 1 + \\frac{9\\cdot \\pi^{3}}{8} + \\frac{15\\cdot \\pi^{4}}{32}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "71082ec38efe64ad", "problem_id": "4c067de813c10270", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} - 4 x^{4} + 2 x^{3} + 3 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 4 x^{4} + 2 x^{3} + 3 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{33 \\pi^{5}}{40} - \\frac{9 \\pi^{2}}{8} - \\frac{3 \\pi}{2} - 1 + \\frac{9 \\pi^{3}}{8} + \\frac{15 \\pi^{4}}{32}$.\n4) Simplify the result.\nFinal answer: $- \\frac{33\\cdot \\pi^{5}}{40} - \\frac{9\\cdot \\pi^{2}}{8} - \\frac{3\\cdot \\pi}{2} - 1 + \\frac{9\\cdot \\pi^{3}}{8} + \\frac{15\\cdot \\pi^{4}}{32}$", "final_answer": "- \\frac{33\\cdot \\pi^{5}}{40} - \\frac{9\\cdot \\pi^{2}}{8} - \\frac{3\\cdot \\pi}{2} - 1 + \\frac{9\\cdot \\pi^{3}}{8} + \\frac{15\\cdot \\pi^{4}}{32}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1154f443915e8670", "problem_id": "2c6b635f109d84c3", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} - 2 x^{3} + 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{3} + 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $\\frac{\\pi \\left(-12 + \\pi^{2}\\right)}{6}$.\nFinal answer: $\\frac{\\pi\\cdot \\left(-12 + \\pi^{2}\\right)}{6}$", "final_answer": "\\frac{\\pi\\cdot \\left(-12 + \\pi^{2}\\right)}{6}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f0062cb544c2410c", "problem_id": "2c6b635f109d84c3", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} - 2 x^{3} + 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{3} + 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $\\frac{\\pi \\left(-12 + \\pi^{2}\\right)}{6}$.\n4) Simplify the result.\nFinal answer: $\\frac{\\pi\\cdot \\left(-12 + \\pi^{2}\\right)}{6}$", "final_answer": "\\frac{\\pi\\cdot \\left(-12 + \\pi^{2}\\right)}{6}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b42916884596f65", "problem_id": "20e6e34ac6bb1843", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x - 1$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 2 x - 1.\n3) Then the limit is -5.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -3.\nFinal answer: $-3$", "final_answer": "-3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "74be2f946d9b8262", "problem_id": "2a1310ce045349d5", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A store sells a backpack for $200. It is discounted by 5%. You buy 5 backpacks. What is the total cost?", "output": "1) Discounted price = 200 × (1 − 5/100) = 200 × 95/100 = 190.\n2) Subtotal = 5 × 190 = 950.\nFinal answer: $950$", "final_answer": "950", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "47fcb1015403d6f9", "problem_id": "8ca33e415b9c53cf", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} + 3 x + 4$.", "output": "Differentiate term-by-term:\nd/dx(4) = 0.\nd/dx(- 2 x^{2}) = - 4 x.\nd/dx(3 x) = 3.\nCombine: f'(x) = 3 - 4 x.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $3 - 4\\cdot x$", "final_answer": "3 - 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "984757640273750b", "problem_id": "8ca33e415b9c53cf", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} + 3 x + 4$.", "output": "1) Differentiate term-by-term:\n2) d/dx(4) = 0.\n3) d/dx(- 2 x^{2}) = - 4 x.\n4) d/dx(3 x) = 3.\n5) Combine: f'(x) = 3 - 4 x.\nFinal answer: $3 - 4\\cdot x$", "final_answer": "3 - 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6b53526ca1e71f90", "problem_id": "ce8bb85ef01e6bfe", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the multiplicative inverse of 14 modulo 17.", "output": "1) We want x such that 14x ≡ 1 (mod 17).\n2) Use extended Euclid to find s,t with 14s + 17t = 1.\n3) One Bezout identity is: 14·(-6) + 17·(5) = 1.\n4) Reduce s modulo 17: x ≡ 11 (mod 17).\nFinal answer: $x\\equiv 11\\pmod{17}$", "final_answer": "x\\equiv 11\\pmod{17}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "31391182c1ad5a4e", "problem_id": "ce8bb85ef01e6bfe", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 14 modulo 17.", "output": "We want x such that 14x ≡ 1 (mod 17).\nUse extended Euclid to find s,t with 14s + 17t = 1.\nOne Bezout identity is: 14·(-6) + 17·(5) = 1.\nReduce s modulo 17: x ≡ 11 (mod 17).\nCheck: 14·11 mod 17 = 1.\nFinal answer: $x\\equiv 11\\pmod{17}$", "final_answer": "x\\equiv 11\\pmod{17}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "830cfcdbc1b68e3e", "problem_id": "dd4a3f18dcc1efe4", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡5 (mod 8), x≡4 (mod 5). Give the solution modulo 40.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 40.\n3) Compute the CRT solution to get x ≡ 29 (mod 40).\nFinal answer: $x\\equiv 29\\pmod{40}$", "final_answer": "x\\equiv 29\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "193f4a3e7d8c463f", "problem_id": "345a6508e6fb732d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} x^{3} - x^{2} + 2$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to x^{3} - x^{2} - x + 2.\n3) Then the limit is 1.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: 0.\nFinal answer: $0$", "final_answer": "0", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2fc1ea270a1c2e90", "problem_id": "c58b992cd3ccbd2d", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "You want 28 liters of a 40% salt solution. You have a 20% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 28−x liters is 50% solution.\n2) Salt equation: 20x + 50(28−x) = 40·28.\n3) (20-50)x = (40-50)·28 ⇒ x = 28/3.\n4) Then 28−x = 56/3.\nFinal answer: $28/3 L of 20% and 56/3 L of 50%$", "final_answer": "28/3 L of 20% and 56/3 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ac699e6e7821505", "problem_id": "52119ff68cb17c7e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(- 3 x^{3} + x^{2} + 2 x - 2\\right) \\cos{\\left(x - 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\n3) So $f'(x) = 3 x^{3} \\sin{\\left(x - 3 \\right)} - x^{2} \\sin{\\left(x - 3 \\right)} - 9 x^{2} \\cos{\\left(x - 3 \\right)} + x e^{x} - 2 x \\sin{\\left(x - 3 \\right)} + 2 x \\cos{\\left(x - 3 \\right)} + 3 e^{x} + 2 \\sin{\\left(x - 3 \\right)} + 2 \\cos{\\left(x - 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 2\\right) e^{x} - \\left(x \\left(x \\left(1 - 3 x\\right) + 2\\right) - 2\\right) \\sin{\\left(x - 3 \\right)} + \\left(- 9 x^{2} + 2 x + 2\\right) \\cos{\\left(x - 3 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 2\\right) e^{x} - \\left(x \\left(x \\left(1 - 3 x\\right) + 2\\right) - 2\\right) \\sin{\\left(x - 3 \\right)} + \\left(- 9 x^{2} + 2 x + 2\\right) \\cos{\\left(x - 3 \\right)} + e^{x}$", "final_answer": "\\left(x + 2\\right) e^{x} - \\left(x \\left(x \\left(1 - 3 x\\right) + 2\\right) - 2\\right) \\sin{\\left(x - 3 \\right)} + \\left(- 9 x^{2} + 2 x + 2\\right) \\cos{\\left(x - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "4b7419aa43c9e33e", "problem_id": "dd483e1a24528840", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$- 2 x + 8 y$ = $8$\n$- 3 x + 5 y$ = $19$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $6 x - 24 y$ = $-24$.\n3) Multiply the second equation by $-2$: $6 x - 10 y$ = $-38$.\n4) Subtract to eliminate x: $- 14 y$ = $14$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $-8$.\nFinal answer: $(x,y)=\\left(-8, -1\\right)$", "final_answer": "(x,y)=\\left(-8, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "19a37f0608b3699a", "problem_id": "dd483e1a24528840", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the linear system:\n$- 2 x + 8 y$ = $8$\n$- 3 x + 5 y$ = $19$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $6 x - 24 y$ = $-24$.\nMultiply the second equation by $-2$: $6 x - 10 y$ = $-38$.\nFinal answer: $(x,y)=\\left(-8, -1\\right)$", "final_answer": "(x,y)=\\left(-8, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "397d324d027e6216", "problem_id": "c4e4fad64e0bb43c", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the multiplicative inverse of 24 modulo 47.", "output": "1) We want x such that 24x ≡ 1 (mod 47).\n2) Use extended Euclid to find s,t with 24s + 47t = 1.\n3) One Bezout identity is: 24·(2) + 47·(-1) = 1.\n4) Reduce s modulo 47: x ≡ 2 (mod 47).\nFinal answer: $x\\equiv 2\\pmod{47}$", "final_answer": "x\\equiv 2\\pmod{47}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "86f21819383975aa", "problem_id": "c4e4fad64e0bb43c", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Find the multiplicative inverse of 24 modulo 47.", "output": "We want x such that 24x ≡ 1 (mod 47).\nUse extended Euclid to find s,t with 24s + 47t = 1.\nOne Bezout identity is: 24·(2) + 47·(-1) = 1.\nFinal answer: $x\\equiv 2\\pmod{47}$", "final_answer": "x\\equiv 2\\pmod{47}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "251ec124917ce1c8", "problem_id": "28ea9c30328ae44c", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Two friends split $1534 in the ratio 8:5. How much does each receive?", "output": "1) Total ratio parts = 8+5 = 13.\n2) Each part = 1534 ÷ 13 = 118.\n3) First share = 8 × 118 = 944.\n4) Second share = 5 × 118 = 590.\nFinal answer: $590$", "final_answer": "590", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6731afe02e75eda5", "problem_id": "28ea9c30328ae44c", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Two friends split $1534 in the ratio 8:5. How much does each receive?", "output": "$590$", "final_answer": "590", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "57ea6dc30935ce1a", "problem_id": "88fc4886983ce903", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A store sells a backpack for $324. It is discounted by 40%. You buy 14 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 324 × (1 − 40/100) = 324 × 60/100 = 972/5.\nSubtotal = 14 × 972/5 = 13608/5.\nTotal with tax = 13608/5 × 110/100 = 74844/25.\nCheck: recomputing the formula yields 74844/25.\nFinal answer: $\\frac{74844}{25}$", "final_answer": "\\frac{74844}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1cb2d7407416be4d", "problem_id": "88fc4886983ce903", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A store sells a backpack for $324. It is discounted by 40%. You buy 14 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 324 × (1 − 40/100) = 324 × 60/100 = 972/5.\n2) Subtotal = 14 × 972/5 = 13608/5.\n3) Total with tax = 13608/5 × 110/100 = 74844/25.\nFinal answer: $\\frac{74844}{25}$", "final_answer": "\\frac{74844}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2c7a4176ceec3d03", "problem_id": "ebc11044592e026a", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "You want 37 liters of a 45% salt solution. You have a 30% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 37−x liters is 50% solution.\n2) Salt equation: 30x + 50(37−x) = 45·37.\n3) (30-50)x = (45-50)·37 ⇒ x = 37/4.\n4) Then 37−x = 111/4.\nFinal answer: $37/4 L of 30% and 111/4 L of 50%$", "final_answer": "37/4 L of 30% and 111/4 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "78fc287371167334", "problem_id": "ebc11044592e026a", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "You want 37 liters of a 45% salt solution. You have a 30% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 37−x liters is 50% solution.\nSalt equation: 30x + 50(37−x) = 45·37.\n(30-50)x = (45-50)·37 ⇒ x = 37/4.\nFinal answer: $37/4 L of 30% and 111/4 L of 50%$", "final_answer": "37/4 L of 30% and 111/4 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "94fb60e950a4091e", "problem_id": "e55ca99decc2c37c", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A store sells a backpack for $36. It is discounted by 15%. You buy 3 backpacks. What is the total cost?", "output": "1) Discounted price = 36 × (1 − 15/100) = 36 × 85/100 = 153/5.\n2) Subtotal = 3 × 153/5 = 459/5.\nFinal answer: $\\frac{459}{5}$", "final_answer": "\\frac{459}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3c42ce1a98e45672", "problem_id": "d26f10f35f204cfa", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} + 3 x - 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x^{2}) = 2 x^{2}$.\n3) So $f'(x) = 2 x^{2} + 3$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x^{2}) = 2 x$.\nTherefore the correct derivative is $f'(x) = 2 x + 3$.\nFinal answer: $2 x + 3$", "final_answer": "2 x + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "cb9daa3af64aa907", "problem_id": "44f9a641df4f98ab", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A cyclist rides at 55 miles per hour for 1/2 hours, then at 63 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 55 × 1/2 = 55/2 miles.\n2) Second leg distance = 63 × 5/2 = 315/2 miles.\n3) Total distance = 55/2 + 315/2 = 185 miles.\nFinal answer: $185$", "final_answer": "185", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fa68ca2e40761284", "problem_id": "44f9a641df4f98ab", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "A cyclist rides at 55 miles per hour for 1/2 hours, then at 63 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 55 × 1/2 = 55/2 miles.\nSecond leg distance = 63 × 5/2 = 315/2 miles.\nTotal distance = 55/2 + 315/2 = 185 miles.\nFinal answer: $185$", "final_answer": "185", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "850916a2257e4009", "problem_id": "e387fe196834d189", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "An urn has 22 red balls and 27 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(49,12).\n3) Favorable samples: choose k reds and n−k blues: C(22,3)·C(27,9).\n4) Compute totals: C(49,12) = 92263734836, and favorable = 7217710500.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{45375}{580027}$", "final_answer": "\\frac{45375}{580027}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5484a0b453f58493", "problem_id": "e387fe196834d189", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 22 red balls and 27 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(49,12).\nFavorable samples: choose k reds and n−k blues: C(22,3)·C(27,9).\nCompute totals: C(49,12) = 92263734836, and favorable = 7217710500.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{45375}{580027}$.\nFinal answer: $\\frac{45375}{580027}$", "final_answer": "\\frac{45375}{580027}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c1e371ca3fe79380", "problem_id": "68774cf73aac8cbe", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Worker A finishes a job in 5 hours and Worker B finishes it in 18 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/5 job/hour. B's rate = 1/18 job/hour.\n2) Combined rate = 1/5 + 1/18 = 23/90 job/hour.\n3) Time = 1 ÷ (combined rate) = 90/23 hours.\nFinal answer: $\\frac{90}{23}$", "final_answer": "\\frac{90}{23}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0d288e7a006e2f10", "problem_id": "68774cf73aac8cbe", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Worker A finishes a job in 5 hours and Worker B finishes it in 18 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/5 job/hour. B's rate = 1/18 job/hour.\nCombined rate = 1/5 + 1/18 = 23/90 job/hour.\nTime = 1 ÷ (combined rate) = 90/23 hours.\nCheck: (combined rate)×(time) = 23/90×90/23 = 1.\nFinal answer: $\\frac{90}{23}$", "final_answer": "\\frac{90}{23}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e60a96e27023a694", "problem_id": "f09a43f31e9bf509", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A store sells a backpack for $308. It is discounted by 10%. You buy 4 backpacks. What is the total cost?", "output": "1) Discounted price = 308 × (1 − 10/100) = 308 × 90/100 = 1386/5.\n2) Subtotal = 4 × 1386/5 = 5544/5.\nFinal answer: $\\frac{5544}{5}$", "final_answer": "\\frac{5544}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "daaff289d3bff140", "problem_id": "f09a43f31e9bf509", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A store sells a backpack for $308. It is discounted by 10%. You buy 4 backpacks. What is the total cost?", "output": "Discounted price = 308 × (1 − 10/100) = 308 × 90/100 = 1386/5.\nSubtotal = 4 × 1386/5 = 5544/5.\nCheck: recomputing the formula yields 5544/5.\nFinal answer: $\\frac{5544}{5}$", "final_answer": "\\frac{5544}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6e6ed01ad17cda58", "problem_id": "b1c0222977612d3d", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=38$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=38 and k=5.\nFinal answer: $111930$", "final_answer": "111930", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0b878483b732fcb6", "problem_id": "b1c0222977612d3d", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=38$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=38 and k=5.\nCompute C(42,4) = 111930.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $111930$", "final_answer": "111930", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fa043b798e0fe282", "problem_id": "e709b76251ea7f02", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{4} - 4 x^{3} - 4 x - 3\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $2 x^{4} - 4 x^{3} - 4 x - 3$.\n3) Compute an antiderivative: F(x) = $\\frac{2 x^{5}}{5} - x^{4} - 2 x^{2} - 3 x$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{2\\cdot x^{5}}{5} - x^{4} - 2\\cdot x^{2} - 3\\cdot x$", "final_answer": "C + \\frac{2\\cdot x^{5}}{5} - x^{4} - 2\\cdot x^{2} - 3\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9a13da025526e13a", "problem_id": "e709b76251ea7f02", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{4} - 4 x^{3} - 4 x - 3\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $2 x^{4} - 4 x^{3} - 4 x - 3$.\nCompute an antiderivative: F(x) = $\\frac{2 x^{5}}{5} - x^{4} - 2 x^{2} - 3 x$.\nFinal answer: $C + \\frac{2\\cdot x^{5}}{5} - x^{4} - 2\\cdot x^{2} - 3\\cdot x$", "final_answer": "C + \\frac{2\\cdot x^{5}}{5} - x^{4} - 2\\cdot x^{2} - 3\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "efb5d940a15dbf44", "problem_id": "5815574d209b2101", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the congruences: x≡5 (mod 11), x≡4 (mod 7). Give the solution modulo 77.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 77.\nCompute the CRT solution to get x ≡ 60 (mod 77).\nFinal answer: $x\\equiv 60\\pmod{77}$", "final_answer": "x\\equiv 60\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8a029baa363519ba", "problem_id": "5815574d209b2101", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡5 (mod 11), x≡4 (mod 7). Give the solution modulo 77.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 77.\nCompute the CRT solution to get x ≡ 60 (mod 77).\nCheck: 60 mod 11=5 and mod 7=4.\nFinal answer: $x\\equiv 60\\pmod{77}$", "final_answer": "x\\equiv 60\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "64f6192a94d11882", "problem_id": "28af519710b354ed", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A store sells a backpack for $50. It is discounted by 5%. You buy 4 backpacks. What is the total cost?", "output": "1) Discounted price = 50 × (1 − 5/100) = 50 × 95/100 = 95/2.\n2) Subtotal = 4 × 95/2 = 190.\nFinal answer: $190$", "final_answer": "190", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ae06c8edc650afb1", "problem_id": "622f9230972b3858", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 3 x^{2} + 5 x - 4$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} + 4 x - 4.\n3) Then the limit is -11.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: -12.\nFinal answer: $-12$", "final_answer": "-12", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "23deaed3e9b3aa81", "problem_id": "140ac9627555efca", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- \\frac{x^{3}}{2} - x^{2} + x + \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - \\frac{1}{2}}{x + 2}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 2}) = \\frac{2 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 2}$.\n3) So $f'(x) = - \\frac{2 x^{3}}{2 x^{2} + 8 x + 8} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 8 x + 8} + \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + 8 x + 8} - \\frac{8 x^{2}}{2 x^{2} + 8 x + 8} + \\frac{4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 8 x + 8} + \\frac{8 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + 8 x + 8} - \\frac{8 x}{2 x^{2} + 8 x + 8} + \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 8 x + 8} + \\frac{8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + 8 x + 8} + \\frac{5}{2 x^{2} + 8 x + 8} + \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 4 x + 4} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 2} - \\frac{2 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 2} + \\frac{2 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 2}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 2}) = \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 2} + \\frac{2 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 2} - \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x + 2\\right)^{2}}$.\nTherefore the correct derivative is $f'(x) = \\frac{x^{3} + 2 x^{2} - 2 x + \\left(x + 2\\right) \\left(- 3 x^{2} - 4 x + \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2\\right) - 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{2 \\left(x + 2\\right)^{2}}$.\nFinal answer: $\\frac{x^{3} + 2 x^{2} - 2 x + \\left(x + 2\\right) \\left(- 3 x^{2} - 4 x + \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2\\right) - 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{2 \\left(x + 2\\right)^{2}}$", "final_answer": "\\frac{x^{3} + 2 x^{2} - 2 x + \\left(x + 2\\right) \\left(- 3 x^{2} - 4 x + \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2\\right) - 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{2 \\left(x + 2\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f72e8c91d83bae0d", "problem_id": "339f987e607f31d4", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "You want 30 liters of a 35% salt solution. You have a 30% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 30−x liters is 70% solution.\n2) Salt equation: 30x + 70(30−x) = 35·30.\n3) (30-70)x = (35-70)·30 ⇒ x = 105/4.\n4) Then 30−x = 15/4.\nFinal answer: $105/4 L of 30% and 15/4 L of 70%$", "final_answer": "105/4 L of 30% and 15/4 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ecab015c2be459b", "problem_id": "fd65afeae7de7e3f", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 3 x - 126$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} - 3 x - 126$ = $3 \\left(x - 7\\right) \\left(x + 6\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $7$.\n5) From the second factor: x = $-6$.\nFinal answer: $x \\in \\{7, -6\\}$", "final_answer": "x \\in \\{7, -6\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2bb7e5d103150a60", "problem_id": "fd65afeae7de7e3f", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 3 x - 126$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} - 3 x - 126$ = $3 \\left(x - 7\\right) \\left(x + 6\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFinal answer: $x \\in \\{7, -6\\}$", "final_answer": "x \\in \\{7, -6\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9161fbd5c305260c", "problem_id": "846deecb9de432dc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(x^{3} + 3 x^{2} - x - 1\\right) \\cos{\\left(x^{2} - 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x \\cos{\\left(x^{2} - 3 \\right)}) = x \\sin{\\left(x^{2} - 3 \\right)}$.\n3) So $f'(x) = - 2 x^{4} \\sin{\\left(x^{2} - 3 \\right)} - 6 x^{3} \\sin{\\left(x^{2} - 3 \\right)} + 3 x^{2} \\cos{\\left(x^{2} - 3 \\right)} + x e^{x} + 2 x \\sin{\\left(x^{2} - 3 \\right)} + x \\sin{\\left(x^{2} - 3 \\right)} + 6 x \\cos{\\left(x^{2} - 3 \\right)} + 2 e^{x}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x \\cos{\\left(x^{2} - 3 \\right)}) = 2 x^{2} \\sin{\\left(x^{2} - 3 \\right)} - \\cos{\\left(x^{2} - 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x \\left(x \\left(x \\left(x + 3\\right) - 1\\right) - 1\\right) \\sin{\\left(x^{2} - 3 \\right)} + \\left(x + 1\\right) e^{x} + \\left(3 x^{2} + 6 x - 1\\right) \\cos{\\left(x^{2} - 3 \\right)} + e^{x}$.\nFinal answer: $- 2 x \\left(x \\left(x \\left(x + 3\\right) - 1\\right) - 1\\right) \\sin{\\left(x^{2} - 3 \\right)} + \\left(x + 1\\right) e^{x} + \\left(3 x^{2} + 6 x - 1\\right) \\cos{\\left(x^{2} - 3 \\right)} + e^{x}$", "final_answer": "- 2 x \\left(x \\left(x \\left(x + 3\\right) - 1\\right) - 1\\right) \\sin{\\left(x^{2} - 3 \\right)} + \\left(x + 1\\right) e^{x} + \\left(3 x^{2} + 6 x - 1\\right) \\cos{\\left(x^{2} - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e31d337f3f7fe127", "problem_id": "2d99bab740831925", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A store sells a backpack for $126. It is discounted by 30%. You buy 3 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 126 × (1 − 30/100) = 126 × 70/100 = 441/5.\n2) Subtotal = 3 × 441/5 = 1323/5.\n3) Total with tax = 1323/5 × 109/100 = 144207/500.\nFinal answer: $\\frac{144207}{500}$", "final_answer": "\\frac{144207}{500}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "799efa4e020b5aa8", "problem_id": "6c19b9c79d2acab6", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "You want 14 liters of a 25% salt solution. You have a 15% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 14−x liters is 40% solution.\n2) Salt equation: 15x + 40(14−x) = 25·14.\n3) (15-40)x = (25-40)·14 ⇒ x = 42/5.\n4) Then 14−x = 28/5.\nFinal answer: $42/5 L of 15% and 28/5 L of 40%$", "final_answer": "42/5 L of 15% and 28/5 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff534312ac0915d7", "problem_id": "6c19b9c79d2acab6", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "You want 14 liters of a 25% salt solution. You have a 15% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 14−x liters is 40% solution.\nSalt equation: 15x + 40(14−x) = 25·14.\n(15-40)x = (25-40)·14 ⇒ x = 42/5.\nThen 14−x = 28/5.\nCheck: 15·42/5 + 40·28/5 = 25·14.\nFinal answer: $42/5 L of 15% and 28/5 L of 40%$", "final_answer": "42/5 L of 15% and 28/5 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8ed7d1c863064597", "problem_id": "156e242a5d1d3d55", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 8 red balls and 9 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(17,6).\nFavorable samples: choose k reds and n−k blues: C(8,2)·C(9,4).\nCompute totals: C(17,6) = 12376, and favorable = 3528.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{63}{221}$.\nFinal answer: $\\frac{63}{221}$", "final_answer": "\\frac{63}{221}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fb79d46156d50b8b", "problem_id": "156e242a5d1d3d55", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "An urn has 8 red balls and 9 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(17,6).\n3) Favorable samples: choose k reds and n−k blues: C(8,2)·C(9,4).\n4) Compute totals: C(17,6) = 12376, and favorable = 3528.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{63}{221}$", "final_answer": "\\frac{63}{221}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ae279923fdb5268e", "problem_id": "c0f9ed30797d389e", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} 4 x^{3} - 4 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $4 x^{3} - 4 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\n2) Evaluate at the bounds: F(\\pi) and F(0).\n3) Subtract: F(\\pi) − F(0) = $\\pi \\left(- \\frac{4 \\pi^{2}}{3} - 3 + \\pi^{3}\\right)$.\n4) Simplify the result.\nFinal answer: $\\pi\\cdot \\left(- \\frac{4\\cdot \\pi^{2}}{3} - 3 + \\pi^{3}\\right)$", "final_answer": "\\pi\\cdot \\left(- \\frac{4\\cdot \\pi^{2}}{3} - 3 + \\pi^{3}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d0a562df61cc5f28", "problem_id": "c0f9ed30797d389e", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} 4 x^{3} - 4 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $4 x^{3} - 4 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\nEvaluate at the bounds: F(\\pi) and F(0).\nSubtract: F(\\pi) − F(0) = $\\pi \\left(- \\frac{4 \\pi^{2}}{3} - 3 + \\pi^{3}\\right)$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\pi\\cdot \\left(- \\frac{4\\cdot \\pi^{2}}{3} - 3 + \\pi^{3}\\right)$", "final_answer": "\\pi\\cdot \\left(- \\frac{4\\cdot \\pi^{2}}{3} - 3 + \\pi^{3}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec9ff43afc123a0b", "problem_id": "3ef96dfabadc66d8", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 2 x^{3} + 5 x^{2} - 6 x + 8$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{3} + 5 x^{2} - 7 x + 8.\n3) Then the limit is 22.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: 21.\nFinal answer: $21$", "final_answer": "21", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c66c469ef4c4b5b2", "problem_id": "af6c296272f090fb", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$- 3 x - 2 y$ = $-19$\n$- 3 x + 2 y$ = $-11$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $9 x + 6 y$ = $57$.\n3) Multiply the second equation by $-3$: $9 x - 6 y$ = $33$.\n4) Subtract to eliminate x: $12 y$ = $24$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, 2\\right)$", "final_answer": "(x,y)=\\left(5, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "636996d909255a5f", "problem_id": "aef27228f3a18ecf", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the congruences: x≡1 (mod 4), x≡8 (mod 13). Give the solution modulo 52.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 52.\n3) Compute the CRT solution to get x ≡ 21 (mod 52).\nFinal answer: $x\\equiv 21\\pmod{52}$", "final_answer": "x\\equiv 21\\pmod{52}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "82f1e392a03f86e7", "problem_id": "aef27228f3a18ecf", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the congruences: x≡1 (mod 4), x≡8 (mod 13). Give the solution modulo 52.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 52.\nCompute the CRT solution to get x ≡ 21 (mod 52).\nFinal answer: $x\\equiv 21\\pmod{52}$", "final_answer": "x\\equiv 21\\pmod{52}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c9d0c56cd7195fdb", "problem_id": "b5753db2389d7636", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the equation of the tangent line to $y=x^{2} + 2 x + 2$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $2 x + 2$.\n2) Slope at x=-2: m = f'(-2) = $-2$.\n3) Point on curve: (-2, f(-2)) = (-2, $2$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 2 = -2(x − -2).\n6) Optionally expand: y = $- 2 x - 2$.\nFinal answer: $y = - 2\\cdot x - 2$", "final_answer": "y = - 2\\cdot x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a29fbd1e4d6f67a6", "problem_id": "b5753db2389d7636", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Find the equation of the tangent line to $y=x^{2} + 2 x + 2$ at $x=-2$.", "output": "Differentiate: f'(x) = $2 x + 2$.\nSlope at x=-2: m = f'(-2) = $-2$.\nPoint on curve: (-2, f(-2)) = (-2, $2$).\nFinal answer: $y = - 2\\cdot x - 2$", "final_answer": "y = - 2\\cdot x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "abdd688d76e21265", "problem_id": "b95f97726391fc8d", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A store sells a backpack for $86. It is discounted by 30%. You buy 12 backpacks. What is the total cost?", "output": "1) Discounted price = 86 × (1 − 30/100) = 86 × 70/100 = 301/5.\n2) Subtotal = 12 × 301/5 = 3612/5.\nFinal answer: $\\frac{3612}{5}$", "final_answer": "\\frac{3612}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "94f9c9b1c948d793", "problem_id": "b95f97726391fc8d", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A store sells a backpack for $86. It is discounted by 30%. You buy 12 backpacks. What is the total cost?", "output": "Discounted price = 86 × (1 − 30/100) = 86 × 70/100 = 301/5.\nSubtotal = 12 × 301/5 = 3612/5.\nCheck: recomputing the formula yields 3612/5.\nFinal answer: $\\frac{3612}{5}$", "final_answer": "\\frac{3612}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3622a56b2e672a6a", "problem_id": "d723cd3fbf671a71", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{3} + 2 x^{2} - x + \\left(2 x^{2} + x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{2 x^{2} + x - 6}$.", "output": "Let f(x) = $\\frac{- x^{3} + 2 x^{2} - x + \\left(2 x^{2} + x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{2 x^{2} + x - 6}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{2 \\left(- 4 x - 1\\right) \\left(- x^{3} + 2 x^{2} - x + \\left(2 x^{2} + x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(2 x^{2} + x - 6\\right) \\left(- 6 x^{2} + 8 x + 2 \\left(4 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(2 x^{2} + x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x^{2} + x - 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(2 x^{2} + x - 6\\right)^{2}}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\frac{\\left(- 8\\cdot x - 2\\right)\\cdot \\left(- x^{3} + 2\\cdot x^{2} - x + \\left(2\\cdot x^{2} + x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 3\\right) + \\left(2\\cdot x^{2} + x - 6\\right)\\cdot \\left(- 6\\cdot x^{2} + 8\\cdot x + \\left(8\\cdot x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x^{2} + 2\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 2\\right)}{2\\cdot \\left(2\\cdot x^{2} + x - 6\\right)^{2}}$", "final_answer": "\\frac{\\left(- 8\\cdot x - 2\\right)\\cdot \\left(- x^{3} + 2\\cdot x^{2} - x + \\left(2\\cdot x^{2} + x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 3\\right) + \\left(2\\cdot x^{2} + x - 6\\right)\\cdot \\left(- 6\\cdot x^{2} + 8\\cdot x + \\left(8\\cdot x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x^{2} + 2\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 2\\right)}{2\\cdot \\left(2\\cdot x^{2} + x - 6\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a40a774dacf110a6", "problem_id": "d723cd3fbf671a71", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{3} + 2 x^{2} - x + \\left(2 x^{2} + x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{2 x^{2} + x - 6}$.", "output": "1) Let f(x) = $\\frac{- x^{3} + 2 x^{2} - x + \\left(2 x^{2} + x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{2 x^{2} + x - 6}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{2 \\left(- 4 x - 1\\right) \\left(- x^{3} + 2 x^{2} - x + \\left(2 x^{2} + x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(2 x^{2} + x - 6\\right) \\left(- 6 x^{2} + 8 x + 2 \\left(4 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(2 x^{2} + x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x^{2} + x - 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(2 x^{2} + x - 6\\right)^{2}}$.\nFinal answer: $\\frac{\\left(- 8\\cdot x - 2\\right)\\cdot \\left(- x^{3} + 2\\cdot x^{2} - x + \\left(2\\cdot x^{2} + x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 3\\right) + \\left(2\\cdot x^{2} + x - 6\\right)\\cdot \\left(- 6\\cdot x^{2} + 8\\cdot x + \\left(8\\cdot x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x^{2} + 2\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 2\\right)}{2\\cdot \\left(2\\cdot x^{2} + x - 6\\right)^{2}}$", "final_answer": "\\frac{\\left(- 8\\cdot x - 2\\right)\\cdot \\left(- x^{3} + 2\\cdot x^{2} - x + \\left(2\\cdot x^{2} + x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 3\\right) + \\left(2\\cdot x^{2} + x - 6\\right)\\cdot \\left(- 6\\cdot x^{2} + 8\\cdot x + \\left(8\\cdot x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x^{2} + 2\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 2\\right)}{2\\cdot \\left(2\\cdot x^{2} + x - 6\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e16f169e5b21add3", "problem_id": "9d6d01414d68d7db", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A store sells a backpack for $335. It is discounted by 35%. You buy 5 backpacks. What is the total cost?", "output": "1) Discounted price = 335 × (1 − 35/100) = 335 × 65/100 = 871/4.\n2) Subtotal = 5 × 871/4 = 4355/4.\nFinal answer: $\\frac{4355}{4}$", "final_answer": "\\frac{4355}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "481ae5bf41a65fea", "problem_id": "9d6d01414d68d7db", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $335. It is discounted by 35%. You buy 5 backpacks. What is the total cost?", "output": "Discounted price = 335 × (1 − 35/100) = 335 × 65/100 = 871/4.\nSubtotal = 5 × 871/4 = 4355/4.\nCheck: recomputing the formula yields 4355/4.\nFinal answer: $\\frac{4355}{4}$", "final_answer": "\\frac{4355}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "84e23bb29d12de3a", "problem_id": "30ccc3f9815ec754", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "A store sells a backpack for $530. It is discounted by 20%. You buy 2 backpacks. What is the total cost?", "output": "$848$", "final_answer": "848", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f0aef0699cedc12e", "problem_id": "30ccc3f9815ec754", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $530. It is discounted by 20%. You buy 2 backpacks. What is the total cost?", "output": "Discounted price = 530 × (1 − 20/100) = 530 × 80/100 = 424.\nSubtotal = 2 × 424 = 848.\nCheck: recomputing the formula yields 848.\nFinal answer: $848$", "final_answer": "848", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b42f9fb6d3a62e45", "problem_id": "b8217404bce8c2af", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{2} - 2 x + 2\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 3 x^{2} - 2 x + 2$.\n3) Compute an antiderivative: F(x) = $- x^{3} - x^{2} + 2 x$.\n4) Include the constant of integration C.\nFinal answer: $C - x^{3} - x^{2} + 2\\cdot x$", "final_answer": "C - x^{3} - x^{2} + 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2f62b869dc9a75b8", "problem_id": "1098642163774e02", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x \\sin{\\left(x \\right)}) = 2 x \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x \\sin{\\left(x \\right)}) = 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}\\right)$.\nFinal answer: $2 \\left(- x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}\\right)$", "final_answer": "2 \\left(- x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3cfdcaa8e94d9fd8", "problem_id": "bb2b468f021a2552", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} - 3 x^{2} + x + 2 \\sin{\\left(x \\right)} - 2$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $- 3 x^{2} - 6 x + 2 \\cos{\\left(x \\right)} + 1$.\n2) Slope at x=-2: m = f'(-2) = $2 \\cos{\\left(2 \\right)} + 1$.\n3) Point on curve: (-2, f(-2)) = (-2, $-8 - 2 \\sin{\\left(2 \\right)}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -8 - 2 \\sin{\\left(2 \\right)} = 2 \\cos{\\left(2 \\right)} + 1(x − -2).\n6) Optionally expand: y = $2 x \\cos{\\left(2 \\right)} + x - 6 - 2 \\sin{\\left(2 \\right)} + 4 \\cos{\\left(2 \\right)}$.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + x - 6 - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)}$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + x - 6 - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "deb12c699b5e49fb", "problem_id": "bb2b468f021a2552", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} - 3 x^{2} + x + 2 \\sin{\\left(x \\right)} - 2$ at $x=-2$.", "output": "Differentiate: f'(x) = $- 3 x^{2} - 6 x + 2 \\cos{\\left(x \\right)} + 1$.\nSlope at x=-2: m = f'(-2) = $2 \\cos{\\left(2 \\right)} + 1$.\nPoint on curve: (-2, f(-2)) = (-2, $-8 - 2 \\sin{\\left(2 \\right)}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -8 - 2 \\sin{\\left(2 \\right)} = 2 \\cos{\\left(2 \\right)} + 1(x − -2).\nOptionally expand: y = $2 x \\cos{\\left(2 \\right)} + x - 6 - 2 \\sin{\\left(2 \\right)} + 4 \\cos{\\left(2 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + x - 6 - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)}$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + x - 6 - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6c45a638ea42bc46", "problem_id": "140ac9627555efca", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- \\frac{x^{3}}{2} - x^{2} + x + \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - \\frac{1}{2}}{x + 2}$.", "output": "1) Let f(x) = $\\frac{- \\frac{x^{3}}{2} - x^{2} + x + \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - \\frac{1}{2}}{x + 2}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{x^{3} + 2 x^{2} - 2 x + \\left(x + 2\\right) \\left(- 3 x^{2} - 4 x + \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2\\right) - 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{2 \\left(x + 2\\right)^{2}}$.\nFinal answer: $\\frac{x^{3} + 2\\cdot x^{2} - 2\\cdot x + \\left(x + 2\\right)\\cdot \\left(- 3\\cdot x^{2} - 4\\cdot x + \\left(x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 2\\right) - \\left(2\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 1}{2\\cdot \\left(x + 2\\right)^{2}}$", "final_answer": "\\frac{x^{3} + 2\\cdot x^{2} - 2\\cdot x + \\left(x + 2\\right)\\cdot \\left(- 3\\cdot x^{2} - 4\\cdot x + \\left(x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 2\\right) - \\left(2\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 1}{2\\cdot \\left(x + 2\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec5b011b1bd1221f", "problem_id": "140ac9627555efca", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- \\frac{x^{3}}{2} - x^{2} + x + \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - \\frac{1}{2}}{x + 2}$.", "output": "Let f(x) = $\\frac{- \\frac{x^{3}}{2} - x^{2} + x + \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - \\frac{1}{2}}{x + 2}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{x^{3} + 2 x^{2} - 2 x + \\left(x + 2\\right) \\left(- 3 x^{2} - 4 x + \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2\\right) - 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{2 \\left(x + 2\\right)^{2}}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\frac{x^{3} + 2\\cdot x^{2} - 2\\cdot x + \\left(x + 2\\right)\\cdot \\left(- 3\\cdot x^{2} - 4\\cdot x + \\left(x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 2\\right) - \\left(2\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 1}{2\\cdot \\left(x + 2\\right)^{2}}$", "final_answer": "\\frac{x^{3} + 2\\cdot x^{2} - 2\\cdot x + \\left(x + 2\\right)\\cdot \\left(- 3\\cdot x^{2} - 4\\cdot x + \\left(x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 2\\right) - \\left(2\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 1}{2\\cdot \\left(x + 2\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "91cad332ac27d697", "problem_id": "a00b0f574751024d", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "An urn has 7 red balls and 8 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(15,4).\n3) Favorable samples: choose k reds and n−k blues: C(7,1)·C(8,3).\n4) Compute totals: C(15,4) = 1365, and favorable = 392.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{56}{195}$", "final_answer": "\\frac{56}{195}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "42debff2aef8367c", "problem_id": "a00b0f574751024d", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 7 red balls and 8 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(15,4).\nFavorable samples: choose k reds and n−k blues: C(7,1)·C(8,3).\nCompute totals: C(15,4) = 1365, and favorable = 392.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{56}{195}$.\nFinal answer: $\\frac{56}{195}$", "final_answer": "\\frac{56}{195}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1f44053a504744cc", "problem_id": "b5b26c01fb38d5fc", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_9=45$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_9=18.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=18 and k=9.\n4) Compute C(26,8) = 1562275.\nFinal answer: $1562275$", "final_answer": "1562275", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "775613a1e34b9209", "problem_id": "b5b26c01fb38d5fc", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_9=45$?", "output": "Let y_i=x_i-3, then y_i≥0 and y_1+…+y_9=18.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=18 and k=9.\nCompute C(26,8) = 1562275.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $1562275$", "final_answer": "1562275", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f1a6d5f7da0e4c20", "problem_id": "993dd953f152f0a7", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A store sells a backpack for $33. It is discounted by 10%. You buy 14 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 33 × (1 − 10/100) = 33 × 90/100 = 297/10.\n2) Subtotal = 14 × 297/10 = 2079/5.\n3) Total with tax = 2079/5 × 109/100 = 226611/500.\nFinal answer: $\\frac{226611}{500}$", "final_answer": "\\frac{226611}{500}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2783c54051b9d5f3", "problem_id": "993dd953f152f0a7", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $33. It is discounted by 10%. You buy 14 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 33 × (1 − 10/100) = 33 × 90/100 = 297/10.\nSubtotal = 14 × 297/10 = 2079/5.\nTotal with tax = 2079/5 × 109/100 = 226611/500.\nCheck: recomputing the formula yields 226611/500.\nFinal answer: $\\frac{226611}{500}$", "final_answer": "\\frac{226611}{500}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "818992d77118757a", "problem_id": "28180b75b2542747", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Two friends split $870 in the ratio 6:9. How much does each receive?", "output": "1) Total ratio parts = 6+9 = 15.\n2) Each part = 870 ÷ 15 = 58.\n3) First share = 6 × 58 = 348.\n4) Second share = 9 × 58 = 522.\nFinal answer: $522$", "final_answer": "522", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dbdfb201ca05e350", "problem_id": "28180b75b2542747", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Two friends split $870 in the ratio 6:9. How much does each receive?", "output": "Total ratio parts = 6+9 = 15.\nEach part = 870 ÷ 15 = 58.\nFirst share = 6 × 58 = 348.\nFinal answer: $522$", "final_answer": "522", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ba08a219d4e88ba3", "problem_id": "f4ac4a51985af0c7", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 3 x^{3} - 4 x^{2} - 9 x - 17$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{3} - 4 x^{2} - 10 x - 17.\n3) Then the limit is -77.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -75.\nFinal answer: $-75$", "final_answer": "-75", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b7fc7471520e147b", "problem_id": "f80a0cd8ea6330f8", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$2 x + 2 y$ = $20$\n$- 2 x - 12 y$ = $-60$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $- 4 x - 4 y$ = $-40$.\n3) Multiply the second equation by $2$: $- 4 x - 24 y$ = $-120$.\n4) Subtract to eliminate x: $20 y$ = $80$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $6$.\nFinal answer: $(x,y)=\\left(6, 4\\right)$", "final_answer": "(x,y)=\\left(6, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "656647d2ed6ef063", "problem_id": "ca1db2df9672f21b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x \\sin{\\left(x \\right)}) = x \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x \\sin{\\left(x \\right)}) = x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$.\nFinal answer: $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$", "final_answer": "- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a0648a9a36d32a19", "problem_id": "b227543716344a1f", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡1 (mod 5), x≡2 (mod 9). Give the solution modulo 45.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 45.\n3) Compute the CRT solution to get x ≡ 11 (mod 45).\nFinal answer: $x\\equiv 11\\pmod{45}$", "final_answer": "x\\equiv 11\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d33cd89a5d3a3a4d", "problem_id": "d1ba675e040415f7", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the linear system:\n$- 2 x + 4 y$ = $-8$\n$3 x + 4 y$ = $2$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $3$: $- 6 x + 12 y$ = $-24$.\n3) Multiply the second equation by $-2$: $- 6 x - 8 y$ = $-4$.\n4) Subtract to eliminate x: $20 y$ = $-20$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $2$.\nFinal answer: $(x,y)=\\left(2, -1\\right)$", "final_answer": "(x,y)=\\left(2, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "03119e3f5f17a334", "problem_id": "d1ba675e040415f7", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the linear system:\n$- 2 x + 4 y$ = $-8$\n$3 x + 4 y$ = $2$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $- 6 x + 12 y$ = $-24$.\nMultiply the second equation by $-2$: $- 6 x - 8 y$ = $-4$.\nFinal answer: $(x,y)=\\left(2, -1\\right)$", "final_answer": "(x,y)=\\left(2, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b04350945b929c90", "problem_id": "6f9e912da189a7ca", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{10}$. If it is flipped 17 times, what is the probability of exactly 4 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=17, k=4, p=$\\frac{3}{10}$ and simplify.\nHere C(17,4) = 2380.\nCheck: expression simplifies to $\\frac{933913171313073}{5000000000000000}$.\nFinal answer: $\\frac{933913171313073}{5000000000000000}$", "final_answer": "\\frac{933913171313073}{5000000000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "095a7e728ad301fa", "problem_id": "6f9e912da189a7ca", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{10}$. If it is flipped 17 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=17, k=4, p=$\\frac{3}{10}$ and simplify.\n4) Here C(17,4) = 2380.\nFinal answer: $\\frac{933913171313073}{5000000000000000}$", "final_answer": "\\frac{933913171313073}{5000000000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "544f73fb42929caf", "problem_id": "1d882d9c1a8ad4b0", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{2 \\sqrt{3} x}{3} + \\frac{4 x}{3} - \\frac{32}{9} - \\frac{16 \\sqrt{3}}{9}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{4}{3} - \\frac{2 \\sqrt{3}}{3}$, c=$- \\frac{32}{9} - \\frac{16 \\sqrt{3}}{9}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{- \\frac{2}{3} + \\frac{\\sqrt{3}}{3} + \\frac{\\sqrt{12\\cdot \\sqrt{3} + 39}}{3}, - \\frac{\\sqrt{12\\cdot \\sqrt{3} + 39}}{3} - \\frac{2}{3} + \\frac{\\sqrt{3}}{3}\\}$", "final_answer": "x \\in \\{- \\frac{2}{3} + \\frac{\\sqrt{3}}{3} + \\frac{\\sqrt{12\\cdot \\sqrt{3} + 39}}{3}, - \\frac{\\sqrt{12\\cdot \\sqrt{3} + 39}}{3} - \\frac{2}{3} + \\frac{\\sqrt{3}}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "237f345bb1d5ad14", "problem_id": "1d882d9c1a8ad4b0", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{2 \\sqrt{3} x}{3} + \\frac{4 x}{3} - \\frac{32}{9} - \\frac{16 \\sqrt{3}}{9}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{4}{3} - \\frac{2 \\sqrt{3}}{3}$, c=$- \\frac{32}{9} - \\frac{16 \\sqrt{3}}{9}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{16 \\sqrt{3}}{3} + \\frac{52}{3}$, so √Δ = $\\sqrt{\\frac{16 \\sqrt{3}}{3} + \\frac{52}{3}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{2}{3} + \\frac{\\sqrt{3}}{3} + \\frac{\\sqrt{12 \\sqrt{3} + 39}}{3}$.\n7) x2 = $- \\frac{\\sqrt{12 \\sqrt{3} + 39}}{3} - \\frac{2}{3} + \\frac{\\sqrt{3}}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{2}{3} + \\frac{\\sqrt{3}}{3} + \\frac{\\sqrt{12\\cdot \\sqrt{3} + 39}}{3}, - \\frac{\\sqrt{12\\cdot \\sqrt{3} + 39}}{3} - \\frac{2}{3} + \\frac{\\sqrt{3}}{3}\\}$", "final_answer": "x \\in \\{- \\frac{2}{3} + \\frac{\\sqrt{3}}{3} + \\frac{\\sqrt{12\\cdot \\sqrt{3} + 39}}{3}, - \\frac{\\sqrt{12\\cdot \\sqrt{3} + 39}}{3} - \\frac{2}{3} + \\frac{\\sqrt{3}}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b53e161d775ead79", "problem_id": "1a7e28416d3bf1c7", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 89 miles per hour for 3/2 hours, then at 78 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 89 × 3/2 = 267/2 miles.\nSecond leg distance = 78 × 2 = 156 miles.\nTotal distance = 267/2 + 156 = 579/2 miles.\nCheck: adding both legs gives 579/2.\nFinal answer: $\\frac{579}{2}$", "final_answer": "\\frac{579}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aeb990e5b6a5d97c", "problem_id": "1a7e28416d3bf1c7", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A cyclist rides at 89 miles per hour for 3/2 hours, then at 78 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 89 × 3/2 = 267/2 miles.\n2) Second leg distance = 78 × 2 = 156 miles.\n3) Total distance = 267/2 + 156 = 579/2 miles.\nFinal answer: $\\frac{579}{2}$", "final_answer": "\\frac{579}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3dc1b708418a28ed", "problem_id": "c47914517390d476", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=14$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=14 and k=7.\nCompute C(20,6) = 38760.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $38760$", "final_answer": "38760", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b93956db50b9c8be", "problem_id": "c47914517390d476", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=14$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=14 and k=7.\n4) Compute C(20,6) = 38760.\nFinal answer: $38760$", "final_answer": "38760", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a0630dd4814cd967", "problem_id": "45794433fbba2c40", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} + 4 x^{2} + x + 1$ at $x=1$.", "output": "1) Differentiate: f'(x) = $- 6 x^{2} + 8 x + 1$.\n2) Slope at x=1: m = f'(1) = $3$.\n3) Point on curve: (1, f(1)) = (1, $4$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 4 = 3(x − 1).\n6) Optionally expand: y = $3 x + 1$.\nFinal answer: $y = 3\\cdot x + 1$", "final_answer": "y = 3\\cdot x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "99dbde0a7e594db9", "problem_id": "45794433fbba2c40", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} + 4 x^{2} + x + 1$ at $x=1$.", "output": "Differentiate: f'(x) = $- 6 x^{2} + 8 x + 1$.\nSlope at x=1: m = f'(1) = $3$.\nPoint on curve: (1, f(1)) = (1, $4$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 4 = 3(x − 1).\nOptionally expand: y = $3 x + 1$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 3\\cdot x + 1$", "final_answer": "y = 3\\cdot x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "79b6436398e9f726", "problem_id": "d235380932cdfa1b", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 20 red balls and 14 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(34,8).\nFavorable samples: choose k reds and n−k blues: C(20,1)·C(14,7).\nCompute totals: C(34,8) = 18156204, and favorable = 68640.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{520}{137547}$.\nFinal answer: $\\frac{520}{137547}$", "final_answer": "\\frac{520}{137547}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a335fbf38694d459", "problem_id": "8dafeffed0e67c36", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} 2 x - 9$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to x - 9.\n3) Then the limit is -13.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: -17.\nFinal answer: $-17$", "final_answer": "-17", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "972cbd0c4544d8b3", "problem_id": "01de01a000d83b7c", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 45 x + 168$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} - 45 x + 168$ = $3 \\left(x - 8\\right) \\left(x - 7\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $8$.\n5) From the second factor: x = $7$.\nFinal answer: $x \\in \\{8, 7\\}$", "final_answer": "x \\in \\{8, 7\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ab6af1d2b72f4754", "problem_id": "01de01a000d83b7c", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 45 x + 168$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} - 45 x + 168$ = $3 \\left(x - 8\\right) \\left(x - 7\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFinal answer: $x \\in \\{8, 7\\}$", "final_answer": "x \\in \\{8, 7\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff3f101aa9c0b575", "problem_id": "d14866f0ac52d8c3", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(3 x^{2} + x - 2\\right) \\cos{\\left(3 x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 2 \\cos{\\left(3 x + 3 \\right)}) = 2 \\sin{\\left(3 x + 3 \\right)}$.\n3) So $f'(x) = - 9 x^{2} \\sin{\\left(3 x + 3 \\right)} + 2 x e^{x} - 3 x \\sin{\\left(3 x + 3 \\right)} + 6 x \\cos{\\left(3 x + 3 \\right)} + 4 e^{x} + 2 \\sin{\\left(3 x + 3 \\right)} + \\cos{\\left(3 x + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 2 \\cos{\\left(3 x + 3 \\right)}) = 6 \\sin{\\left(3 x + 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x + 1\\right) e^{x} + \\left(6 x + 1\\right) \\cos{\\left(3 x + 3 \\right)} - 3 \\left(x \\left(3 x + 1\\right) - 2\\right) \\sin{\\left(3 x + 3 \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(x + 1\\right) e^{x} + \\left(6 x + 1\\right) \\cos{\\left(3 x + 3 \\right)} - 3 \\left(x \\left(3 x + 1\\right) - 2\\right) \\sin{\\left(3 x + 3 \\right)} + 2 e^{x}$", "final_answer": "2 \\left(x + 1\\right) e^{x} + \\left(6 x + 1\\right) \\cos{\\left(3 x + 3 \\right)} - 3 \\left(x \\left(3 x + 1\\right) - 2\\right) \\sin{\\left(3 x + 3 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f791ba2a2823cb15", "problem_id": "3a53ff7f3d90285a", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 5 x - 24$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} + 5 x - 24$ = $\\left(x - 3\\right) \\left(x + 8\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $3$.\n5) From the second factor: x = $-8$.\nFinal answer: $x \\in \\{3, -8\\}$", "final_answer": "x \\in \\{3, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e754bd8ce4d09217", "problem_id": "b3b8039ea8a706c0", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{3}$. If it is flipped 12 times, what is the probability of exactly 7 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=7, p=$\\frac{1}{3}$ and simplify.\n4) Here C(12,7) = 792.\nFinal answer: $\\frac{2816}{59049}$", "final_answer": "\\frac{2816}{59049}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "43e353e55b4c37ac", "problem_id": "b3b8039ea8a706c0", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{3}$. If it is flipped 12 times, what is the probability of exactly 7 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=12, k=7, p=$\\frac{1}{3}$ and simplify.\nHere C(12,7) = 792.\nCheck: expression simplifies to $\\frac{2816}{59049}$.\nFinal answer: $\\frac{2816}{59049}$", "final_answer": "\\frac{2816}{59049}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "22dcc4ec17db3b4e", "problem_id": "99afbb047b4d618a", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$4 x - 8 y$ = $60$\n$- 2 x - 8 y$ = $42$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $- 8 x + 16 y$ = $-120$.\n3) Multiply the second equation by $4$: $- 8 x - 32 y$ = $168$.\n4) Subtract to eliminate x: $48 y$ = $-288$.\n5) Solve for y: y = $-6$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, -6\\right)$", "final_answer": "(x,y)=\\left(3, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5cd768aa969a6441", "problem_id": "99afbb047b4d618a", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the linear system:\n$4 x - 8 y$ = $60$\n$- 2 x - 8 y$ = $42$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $- 8 x + 16 y$ = $-120$.\nMultiply the second equation by $4$: $- 8 x - 32 y$ = $168$.\nFinal answer: $(x,y)=\\left(3, -6\\right)$", "final_answer": "(x,y)=\\left(3, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d6a164c3d7a27b82", "problem_id": "e70274a2f9f6c7cf", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡2 (mod 7), x≡4 (mod 5). Give the solution modulo 35.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 35.\n3) Compute the CRT solution to get x ≡ 9 (mod 35).\nFinal answer: $x\\equiv 9\\pmod{35}$", "final_answer": "x\\equiv 9\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "daed6f27f12c348d", "problem_id": "e70274a2f9f6c7cf", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡2 (mod 7), x≡4 (mod 5). Give the solution modulo 35.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 35.\nCompute the CRT solution to get x ≡ 9 (mod 35).\nCheck: 9 mod 7=2 and mod 5=4.\nFinal answer: $x\\equiv 9\\pmod{35}$", "final_answer": "x\\equiv 9\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "13c1ca2ebf5ffa5f", "problem_id": "6623b8bb1449267b", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A store sells a backpack for $425. It is discounted by 30%. You buy 11 backpacks. What is the total cost?", "output": "1) Discounted price = 425 × (1 − 30/100) = 425 × 70/100 = 595/2.\n2) Subtotal = 11 × 595/2 = 6545/2.\nFinal answer: $\\frac{6545}{2}$", "final_answer": "\\frac{6545}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0908b83d0443d798", "problem_id": "6623b8bb1449267b", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A store sells a backpack for $425. It is discounted by 30%. You buy 11 backpacks. What is the total cost?", "output": "Discounted price = 425 × (1 − 30/100) = 425 × 70/100 = 595/2.\nSubtotal = 11 × 595/2 = 6545/2.\nFinal answer: $\\frac{6545}{2}$", "final_answer": "\\frac{6545}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0fb706cc4e5014b0", "problem_id": "76764a657d136529", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A store sells a backpack for $54. It is discounted by 40%. You buy 7 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 54 × (1 − 40/100) = 54 × 60/100 = 162/5.\n2) Subtotal = 7 × 162/5 = 1134/5.\n3) Total with tax = 1134/5 × 110/100 = 6237/25.\nFinal answer: $\\frac{6237}{25}$", "final_answer": "\\frac{6237}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9c4d535cb13934ae", "problem_id": "76764a657d136529", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $54. It is discounted by 40%. You buy 7 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 54 × (1 − 40/100) = 54 × 60/100 = 162/5.\nSubtotal = 7 × 162/5 = 1134/5.\nTotal with tax = 1134/5 × 110/100 = 6237/25.\nCheck: recomputing the formula yields 6237/25.\nFinal answer: $\\frac{6237}{25}$", "final_answer": "\\frac{6237}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1f136a90fd26f97c", "problem_id": "1520ca814888b83e", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $1100 in the ratio 4:7. How much does each receive?", "output": "Total ratio parts = 4+7 = 11.\nEach part = 1100 ÷ 11 = 100.\nFirst share = 4 × 100 = 400.\nSecond share = 7 × 100 = 700.\nCheck: 400+700=1100.\nFinal answer: $700$", "final_answer": "700", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ccf9249a6612a2f4", "problem_id": "1520ca814888b83e", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Two friends split $1100 in the ratio 4:7. How much does each receive?", "output": "1) Total ratio parts = 4+7 = 11.\n2) Each part = 1100 ÷ 11 = 100.\n3) First share = 4 × 100 = 400.\n4) Second share = 7 × 100 = 700.\nFinal answer: $700$", "final_answer": "700", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7521ad5c3e61c01f", "problem_id": "db249a016f6d7ae9", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "You want 15 liters of a 47% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 15−x liters is 70% solution.\n2) Salt equation: 25x + 70(15−x) = 47·15.\n3) (25-70)x = (47-70)·15 ⇒ x = 23/3.\n4) Then 15−x = 22/3.\nFinal answer: $23/3 L of 25% and 22/3 L of 70%$", "final_answer": "23/3 L of 25% and 22/3 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "43c01064c9202bce", "problem_id": "db249a016f6d7ae9", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "You want 15 liters of a 47% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 15−x liters is 70% solution.\nSalt equation: 25x + 70(15−x) = 47·15.\n(25-70)x = (47-70)·15 ⇒ x = 23/3.\nThen 15−x = 22/3.\nCheck: 25·23/3 + 70·22/3 = 47·15.\nFinal answer: $23/3 L of 25% and 22/3 L of 70%$", "final_answer": "23/3 L of 25% and 22/3 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b141479172c2bd7e", "problem_id": "5e5d41f7d9f0e5b9", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} - x^{3} - x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{3} - x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\n2) Evaluate at the bounds: F(\\pi) and F(0).\n3) Subtract: F(\\pi) − F(0) = $- \\frac{\\pi^{4}}{4} - \\frac{\\pi^{3}}{3} + 4 + \\frac{\\pi^{2}}{2} + 2 \\pi$.\n4) Simplify the result.\nFinal answer: $- \\frac{\\pi^{4}}{4} - \\frac{\\pi^{3}}{3} + 4 + \\frac{\\pi^{2}}{2} + 2\\cdot \\pi$", "final_answer": "- \\frac{\\pi^{4}}{4} - \\frac{\\pi^{3}}{3} + 4 + \\frac{\\pi^{2}}{2} + 2\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bb72837cb425b5f1", "problem_id": "5e5d41f7d9f0e5b9", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} - x^{3} - x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{3} - x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\nEvaluate at the bounds: F(\\pi) and F(0).\nSubtract: F(\\pi) − F(0) = $- \\frac{\\pi^{4}}{4} - \\frac{\\pi^{3}}{3} + 4 + \\frac{\\pi^{2}}{2} + 2 \\pi$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{\\pi^{4}}{4} - \\frac{\\pi^{3}}{3} + 4 + \\frac{\\pi^{2}}{2} + 2\\cdot \\pi$", "final_answer": "- \\frac{\\pi^{4}}{4} - \\frac{\\pi^{3}}{3} + 4 + \\frac{\\pi^{2}}{2} + 2\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "caab0058f8d85f9f", "problem_id": "44ae8186229bb9b0", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 6 x - 45$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} + 6 x - 45$ = $3 \\left(x - 3\\right) \\left(x + 5\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $3$.\nFrom the second factor: x = $-5$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{3, -5\\}$", "final_answer": "x \\in \\{3, -5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1cf9fa728274d828", "problem_id": "44ae8186229bb9b0", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 6 x - 45$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} + 6 x - 45$ = $3 \\left(x - 3\\right) \\left(x + 5\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $3$.\n5) From the second factor: x = $-5$.\nFinal answer: $x \\in \\{3, -5\\}$", "final_answer": "x \\in \\{3, -5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72138a9dd05b6e1c", "problem_id": "399ece830e72a8be", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - 2 x^{3} - 5 x^{2} - 21 x - 83$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{3} - 5 x^{2} - 22 x - 83.\n3) Then the limit is -379.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: -375.\nFinal answer: $-375$", "final_answer": "-375", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "fe5acba4710add32", "problem_id": "29afd3c3d157f0a6", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 10 times, what is the probability of exactly 4 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=4, p=$\\frac{1}{5}$ and simplify.\nHere C(10,4) = 210.\nCheck: expression simplifies to $\\frac{172032}{1953125}$.\nFinal answer: $\\frac{172032}{1953125}$", "final_answer": "\\frac{172032}{1953125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4091f3e26d2be5a9", "problem_id": "0294011514fe72df", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 14 red balls and 28 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(42,8).\nFavorable samples: choose k reds and n−k blues: C(14,4)·C(28,4).\nCompute totals: C(42,8) = 118030185, and favorable = 20495475.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{5005}{28823}$.\nFinal answer: $\\frac{5005}{28823}$", "final_answer": "\\frac{5005}{28823}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "69e3945635852faa", "problem_id": "0294011514fe72df", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "An urn has 14 red balls and 28 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(42,8).\n3) Favorable samples: choose k reds and n−k blues: C(14,4)·C(28,4).\n4) Compute totals: C(42,8) = 118030185, and favorable = 20495475.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{5005}{28823}$", "final_answer": "\\frac{5005}{28823}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c8e4a7c0365567df", "problem_id": "790a140c9e13e549", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "You want 53 liters of a 35% salt solution. You have a 30% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 53−x liters is 40% solution.\nSalt equation: 30x + 40(53−x) = 35·53.\n(30-40)x = (35-40)·53 ⇒ x = 53/2.\nFinal answer: $53/2 L of 30% and 53/2 L of 40%$", "final_answer": "53/2 L of 30% and 53/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2db0f285c960fa8e", "problem_id": "790a140c9e13e549", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "You want 53 liters of a 35% salt solution. You have a 30% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 53−x liters is 40% solution.\n2) Salt equation: 30x + 40(53−x) = 35·53.\n3) (30-40)x = (35-40)·53 ⇒ x = 53/2.\n4) Then 53−x = 53/2.\nFinal answer: $53/2 L of 30% and 53/2 L of 40%$", "final_answer": "53/2 L of 30% and 53/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "df2a09e0cb8a83af", "problem_id": "727d4a9dca73f60f", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the equation of the tangent line to $y=3 x^{3} - 3 x^{2} + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=0$.", "output": "1) Differentiate: f'(x) = $9 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)}$.\n2) Slope at x=0: m = f'(0) = $\\frac{1}{2}$.\n3) Point on curve: (0, f(0)) = (0, $1$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 1 = \\frac{1}{2}(x − 0).\n6) Optionally expand: y = $\\frac{x}{2} + 1$.\nFinal answer: $y = \\frac{x}{2} + 1$", "final_answer": "y = \\frac{x}{2} + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "61956448258affe1", "problem_id": "727d4a9dca73f60f", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} - 3 x^{2} + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=0$.", "output": "Differentiate: f'(x) = $9 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)}$.\nSlope at x=0: m = f'(0) = $\\frac{1}{2}$.\nPoint on curve: (0, f(0)) = (0, $1$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 1 = \\frac{1}{2}(x − 0).\nOptionally expand: y = $\\frac{x}{2} + 1$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = \\frac{x}{2} + 1$", "final_answer": "y = \\frac{x}{2} + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c1ca5fd12fbf657b", "problem_id": "b9c22c278144aeaf", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$3 x + y$ = $-4$\n$4 x + 4 y$ = $-8$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $12 x + 4 y$ = $-16$.\nMultiply the second equation by $3$: $12 x + 12 y$ = $-24$.\nSubtract to eliminate x: $- 8 y$ = $8$.\nSolve for y: y = $-1$.\nSubstitute back to find x: x = $-1$.\nCheck: plug (x,y)=(-1,-1) into both equations.\nFinal answer: $(x,y)=\\left(-1, -1\\right)$", "final_answer": "(x,y)=\\left(-1, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1f8c6deb22924205", "problem_id": "b9c22c278144aeaf", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$3 x + y$ = $-4$\n$4 x + 4 y$ = $-8$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $12 x + 4 y$ = $-16$.\n3) Multiply the second equation by $3$: $12 x + 12 y$ = $-24$.\n4) Subtract to eliminate x: $- 8 y$ = $8$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, -1\\right)$", "final_answer": "(x,y)=\\left(-1, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b90c674ae02d0ddd", "problem_id": "13e03390d8c0b1a2", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Two friends split $1450 in the ratio 4:6. How much does each receive?", "output": "Total ratio parts = 4+6 = 10.\nEach part = 1450 ÷ 10 = 145.\nFirst share = 4 × 145 = 580.\nFinal answer: $870$", "final_answer": "870", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cf207d9bba71ecfd", "problem_id": "13e03390d8c0b1a2", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $1450 in the ratio 4:6. How much does each receive?", "output": "Total ratio parts = 4+6 = 10.\nEach part = 1450 ÷ 10 = 145.\nFirst share = 4 × 145 = 580.\nSecond share = 6 × 145 = 870.\nCheck: 580+870=1450.\nFinal answer: $870$", "final_answer": "870", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fead95fb3d034cc1", "problem_id": "2458bd55b3c9d678", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 2 x^{2} - 5 x + 16$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $2 x^{3} + x^{2} + x - 3$ and a = -3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $6 x^{2} + 2 x + 1$.\n5) Evaluate at a: p'(-3) = $49$.\nFinal answer: $49$", "final_answer": "49", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "581d494836c2b660", "problem_id": "40056030801488e7", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{4} - 2 x^{3} + 4 x^{2} + 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x^{3}) = - 6 x^{3}$.\n3) So $f'(x) = - 2 x^{3} + 8 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x^{3}) = - 6 x^{2}$.\nTherefore the correct derivative is $f'(x) = 2 x \\left(2 x^{2} - 3 x + 4\\right)$.\nFinal answer: $2 x \\left(2 x^{2} - 3 x + 4\\right)$", "final_answer": "2 x \\left(2 x^{2} - 3 x + 4\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "be5145b6a6af2f04", "problem_id": "98d5173fe23db480", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=32$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=32 and k=10.\n4) Compute C(41,9) = 350343565.\nFinal answer: $350343565$", "final_answer": "350343565", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b92a04dbc1230bc9", "problem_id": "98d5173fe23db480", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=32$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=32 and k=10.\nCompute C(41,9) = 350343565.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $350343565$", "final_answer": "350343565", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "706283ab5563aa46", "problem_id": "cff7a2ac2739ad67", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡0 (mod 13). Give the solution modulo 39.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 39.\n3) Compute the CRT solution to get x ≡ 13 (mod 39).\nFinal answer: $x\\equiv 13\\pmod{39}$", "final_answer": "x\\equiv 13\\pmod{39}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "04bfa8285c991529", "problem_id": "cff7a2ac2739ad67", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡0 (mod 13). Give the solution modulo 39.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 39.\nCompute the CRT solution to get x ≡ 13 (mod 39).\nCheck: 13 mod 3=1 and mod 13=0.\nFinal answer: $x\\equiv 13\\pmod{39}$", "final_answer": "x\\equiv 13\\pmod{39}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7f77edb0004cf8ad", "problem_id": "266702b2874800c5", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} - 3 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\pi).\n3) Subtract: F(\\pi) − F(- \\pi) = $2 \\pi \\left(- \\pi^{2} - 2\\right)$.\n4) Simplify the result.\nFinal answer: $2\\cdot \\pi\\cdot \\left(- \\pi^{2} - 2\\right)$", "final_answer": "2\\cdot \\pi\\cdot \\left(- \\pi^{2} - 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8160d519962ecbf9", "problem_id": "266702b2874800c5", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} - 3 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\nEvaluate at the bounds: F(\\pi) and F(- \\pi).\nSubtract: F(\\pi) − F(- \\pi) = $2 \\pi \\left(- \\pi^{2} - 2\\right)$.\nFinal answer: $2\\cdot \\pi\\cdot \\left(- \\pi^{2} - 2\\right)$", "final_answer": "2\\cdot \\pi\\cdot \\left(- \\pi^{2} - 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1e918ffb9c50cbb1", "problem_id": "aed1a0eaae1d9cef", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{3} + x^{2} + 4 x + 6$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} + x^{2} + 3 x + 6.\n3) Then the limit is 32.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 34.\nFinal answer: $34$", "final_answer": "34", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "26285a4de3dc0b28", "problem_id": "8b9f2aac71a529c6", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{2 x^{3} + 3 x^{2} - 2 x + \\left(x^{2} + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{x^{2} + 5}$.", "output": "1) Let f(x) = $\\frac{2 x^{3} + 3 x^{2} - 2 x + \\left(x^{2} + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{x^{2} + 5}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{- 4 x \\left(2 x^{3} + 3 x^{2} - 2 x + \\left(x^{2} + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x^{2} + 5\\right) \\left(12 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 12 x + \\left(x^{2} + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 4\\right)}{2 \\left(x^{2} + 5\\right)^{2}}$.\nFinal answer: $\\frac{- 4\\cdot x\\cdot \\left(2\\cdot x^{3} + 3\\cdot x^{2} - 2\\cdot x + \\left(x^{2} + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 3\\right) + \\left(x^{2} + 5\\right)\\cdot \\left(12\\cdot x^{2} + 4\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 12\\cdot x + \\left(x^{2} + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 4\\right)}{2\\cdot \\left(x^{2} + 5\\right)^{2}}$", "final_answer": "\\frac{- 4\\cdot x\\cdot \\left(2\\cdot x^{3} + 3\\cdot x^{2} - 2\\cdot x + \\left(x^{2} + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 3\\right) + \\left(x^{2} + 5\\right)\\cdot \\left(12\\cdot x^{2} + 4\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 12\\cdot x + \\left(x^{2} + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 4\\right)}{2\\cdot \\left(x^{2} + 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5633012c180e83c0", "problem_id": "9b3b30ff03c74fc9", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_5=44$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_5=34.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=34 and k=5.\n4) Compute C(38,4) = 73815.\nFinal answer: $73815$", "final_answer": "73815", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "32cd067620624704", "problem_id": "acdd413b5fdca8a9", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x - 2$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 3 x - 2.\n3) Then the limit is -8.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -6.\nFinal answer: $-6$", "final_answer": "-6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ba9dad57d5be5efb", "problem_id": "7cc8b1d9a11d9659", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_4=20$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=20 and k=4.\nCompute C(23,3) = 1771.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $1771$", "final_answer": "1771", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "55fa4d79fbc17156", "problem_id": "7cc8b1d9a11d9659", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_4=20$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=20 and k=4.\n4) Compute C(23,3) = 1771.\nFinal answer: $1771$", "final_answer": "1771", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cd234a31a2f200b6", "problem_id": "6f7525808da7c68f", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A cyclist rides at 30 miles per hour for 2 hours, then at 60 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 30 × 2 = 60 miles.\nSecond leg distance = 60 × 3 = 180 miles.\nTotal distance = 60 + 180 = 240 miles.\nFinal answer: $240$", "final_answer": "240", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e7a97e3162d07a23", "problem_id": "6f7525808da7c68f", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A cyclist rides at 30 miles per hour for 2 hours, then at 60 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 30 × 2 = 60 miles.\n2) Second leg distance = 60 × 3 = 180 miles.\n3) Total distance = 60 + 180 = 240 miles.\nFinal answer: $240$", "final_answer": "240", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e5c2558b664d10a2", "problem_id": "070b8d8e920c171e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- x^{2} + 3 x - 1\\right) \\cos{\\left(3 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x^{2} \\cos{\\left(3 x \\right)}) = x^{2} \\sin{\\left(3 x \\right)}$.\n3) So $f'(x) = x^{2} \\sin{\\left(3 x \\right)} + 2 x e^{x} - 9 x \\sin{\\left(3 x \\right)} + 4 e^{x} + 3 \\sin{\\left(3 x \\right)} + 3 \\cos{\\left(3 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x^{2} \\cos{\\left(3 x \\right)}) = 3 x^{2} \\sin{\\left(3 x \\right)} - 2 x \\cos{\\left(3 x \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(3 - 2 x\\right) \\cos{\\left(3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 3 \\left(x \\left(3 - x\\right) - 1\\right) \\sin{\\left(3 x \\right)} + 2 e^{x}$.\nFinal answer: $\\left(3 - 2 x\\right) \\cos{\\left(3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 3 \\left(x \\left(3 - x\\right) - 1\\right) \\sin{\\left(3 x \\right)} + 2 e^{x}$", "final_answer": "\\left(3 - 2 x\\right) \\cos{\\left(3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 3 \\left(x \\left(3 - x\\right) - 1\\right) \\sin{\\left(3 x \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0c889ec338eaec12", "problem_id": "71f61a25d5cc0bc9", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{3} x + \\frac{5 x}{2} - 2 \\sqrt{3} + 1$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{5}{2} - \\sqrt{3}$, c=$1 - 2 \\sqrt{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $3 \\sqrt{3} + \\frac{21}{4}$, so √Δ = $\\sqrt{3 \\sqrt{3} + \\frac{21}{4}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{5}{4} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{12 \\sqrt{3} + 21}}{4}$.\nx2 = $- \\frac{\\sqrt{12 \\sqrt{3} + 21}}{4} - \\frac{5}{4} + \\frac{\\sqrt{3}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{5}{4} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{12\\cdot \\sqrt{3} + 21}}{4}, - \\frac{\\sqrt{12\\cdot \\sqrt{3} + 21}}{4} - \\frac{5}{4} + \\frac{\\sqrt{3}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{5}{4} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{12\\cdot \\sqrt{3} + 21}}{4}, - \\frac{\\sqrt{12\\cdot \\sqrt{3} + 21}}{4} - \\frac{5}{4} + \\frac{\\sqrt{3}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b2d5e8d37af62ce3", "problem_id": "854e0ce3f13153bd", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{11} x}{3} + \\frac{x}{15} - \\frac{2 \\sqrt{11}}{15} - \\frac{2}{15}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{1}{15} - \\frac{\\sqrt{11}}{3}$, c=$- \\frac{2 \\sqrt{11}}{15} - \\frac{2}{15}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{22 \\sqrt{11}}{45} + \\frac{44}{25}$, so √Δ = $\\sqrt{\\frac{22 \\sqrt{11}}{45} + \\frac{44}{25}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{1}{30} + \\frac{\\sqrt{11}}{6} + \\frac{\\sqrt{110 \\sqrt{11} + 396}}{30}$.\n7) x2 = $- \\frac{\\sqrt{110 \\sqrt{11} + 396}}{30} - \\frac{1}{30} + \\frac{\\sqrt{11}}{6}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{1}{30} + \\frac{\\sqrt{11}}{6} + \\frac{\\sqrt{110\\cdot \\sqrt{11} + 396}}{30}, - \\frac{\\sqrt{110\\cdot \\sqrt{11} + 396}}{30} - \\frac{1}{30} + \\frac{\\sqrt{11}}{6}\\}$", "final_answer": "x \\in \\{- \\frac{1}{30} + \\frac{\\sqrt{11}}{6} + \\frac{\\sqrt{110\\cdot \\sqrt{11} + 396}}{30}, - \\frac{\\sqrt{110\\cdot \\sqrt{11} + 396}}{30} - \\frac{1}{30} + \\frac{\\sqrt{11}}{6}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e5d3124ac78a3bea", "problem_id": "854e0ce3f13153bd", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{11} x}{3} + \\frac{x}{15} - \\frac{2 \\sqrt{11}}{15} - \\frac{2}{15}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{1}{15} - \\frac{\\sqrt{11}}{3}$, c=$- \\frac{2 \\sqrt{11}}{15} - \\frac{2}{15}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{22 \\sqrt{11}}{45} + \\frac{44}{25}$, so √Δ = $\\sqrt{\\frac{22 \\sqrt{11}}{45} + \\frac{44}{25}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{1}{30} + \\frac{\\sqrt{11}}{6} + \\frac{\\sqrt{110 \\sqrt{11} + 396}}{30}$.\nx2 = $- \\frac{\\sqrt{110 \\sqrt{11} + 396}}{30} - \\frac{1}{30} + \\frac{\\sqrt{11}}{6}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{1}{30} + \\frac{\\sqrt{11}}{6} + \\frac{\\sqrt{110\\cdot \\sqrt{11} + 396}}{30}, - \\frac{\\sqrt{110\\cdot \\sqrt{11} + 396}}{30} - \\frac{1}{30} + \\frac{\\sqrt{11}}{6}\\}$", "final_answer": "x \\in \\{- \\frac{1}{30} + \\frac{\\sqrt{11}}{6} + \\frac{\\sqrt{110\\cdot \\sqrt{11} + 396}}{30}, - \\frac{\\sqrt{110\\cdot \\sqrt{11} + 396}}{30} - \\frac{1}{30} + \\frac{\\sqrt{11}}{6}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "def24f5ec8cb2310", "problem_id": "5775a03e6b1659e5", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} - 3 x^{3} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{3} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\pi).\n3) Subtract: F(\\pi) − F(- \\pi) = $0$.\n4) Simplify the result.\nFinal answer: $0$", "final_answer": "0", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9c1381dcac5a1e19", "problem_id": "8a6217f9111d785d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 - 3 x$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 3 - 4 x.\n3) Then the limit is 11.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 9.\nFinal answer: $9$", "final_answer": "9", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c2b1e5bab1b75033", "problem_id": "6b4fb18756f3eb16", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 18 times, what is the probability of exactly 12 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=18, k=12, p=$\\frac{1}{10}$ and simplify.\n4) Here C(18,12) = 18564.\nFinal answer: $\\frac{2466417681}{250000000000000000}$", "final_answer": "\\frac{2466417681}{250000000000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "47df9eca55a52fd0", "problem_id": "6b4fb18756f3eb16", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 18 times, what is the probability of exactly 12 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=18, k=12, p=$\\frac{1}{10}$ and simplify.\nFinal answer: $\\frac{2466417681}{250000000000000000}$", "final_answer": "\\frac{2466417681}{250000000000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9813ebda049ed18c", "problem_id": "bf6a99df0acfd243", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} 2 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $2 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\nSubtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{\\pi^{2}}{4} + \\frac{\\pi}{2} + 2 + \\frac{\\pi^{3}}{12}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{\\pi^{2}}{4} + \\frac{\\pi}{2} + 2 + \\frac{\\pi^{3}}{12}$", "final_answer": "- \\frac{\\pi^{2}}{4} + \\frac{\\pi}{2} + 2 + \\frac{\\pi^{3}}{12}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "273367630d1ee786", "problem_id": "bf6a99df0acfd243", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} 2 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\n3) Subtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{\\pi^{2}}{4} + \\frac{\\pi}{2} + 2 + \\frac{\\pi^{3}}{12}$.\n4) Simplify the result.\nFinal answer: $- \\frac{\\pi^{2}}{4} + \\frac{\\pi}{2} + 2 + \\frac{\\pi^{3}}{12}$", "final_answer": "- \\frac{\\pi^{2}}{4} + \\frac{\\pi}{2} + 2 + \\frac{\\pi^{3}}{12}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8d4e3420a22be88b", "problem_id": "bd1a62826a338a55", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "You want 56 liters of a 35% salt solution. You have a 30% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 56−x liters is 40% solution.\n2) Salt equation: 30x + 40(56−x) = 35·56.\n3) (30-40)x = (35-40)·56 ⇒ x = 28.\n4) Then 56−x = 28.\nFinal answer: $28 L of 30% and 28 L of 40%$", "final_answer": "28 L of 30% and 28 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1221649538c58aab", "problem_id": "bd1a62826a338a55", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 56 liters of a 35% salt solution. You have a 30% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 56−x liters is 40% solution.\nSalt equation: 30x + 40(56−x) = 35·56.\n(30-40)x = (35-40)·56 ⇒ x = 28.\nThen 56−x = 28.\nCheck: 30·28 + 40·28 = 35·56.\nFinal answer: $28 L of 30% and 28 L of 40%$", "final_answer": "28 L of 30% and 28 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bdcaa3822a3f6e80", "problem_id": "fa06e5968b2870a6", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} 3 x^{3} + 2 x^{2} + 4 x + 4$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{4} - x^{3} + 2 x^{2} + 1$ and a = 1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $12 x^{3} - 3 x^{2} + 4 x$.\n5) Evaluate at a: p'(1) = $13$.\nFinal answer: $13$", "final_answer": "13", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0a8d44628a998940", "problem_id": "fa06e5968b2870a6", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} 3 x^{3} + 2 x^{2} + 4 x + 4$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $3 x^{4} - x^{3} + 2 x^{2} + 1$ and a = 1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $12 x^{3} - 3 x^{2} + 4 x$.\nEvaluate at a: p'(1) = $13$.\nCheck: the identity gives $13$.\nFinal answer: $13$", "final_answer": "13", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "14c74c9efefaf1df", "problem_id": "ded2825e9d1f2dcf", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 2 x + 1$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to x + 1.\n3) Then the limit is 0.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: -1.\nFinal answer: $-1$", "final_answer": "-1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b3d354133c07afde", "problem_id": "abbc71750aa82b49", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x^{2} - 5 x - 11$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} - 6 x - 11.\n3) Then the limit is -27.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -25.\nFinal answer: $-25$", "final_answer": "-25", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6a252f3b0aa0b3cb", "problem_id": "d6334c18f690852d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} \\cos{\\left(x^{2} + 3 x + 1 \\right)} + 2 \\left(x + 1\\right) e^{x}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x e^{x}) = 2 x e^{x}$.\n3) So $f'(x) = - 6 x^{3} \\sin{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} - 9 x^{2} \\sin{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 2 x e^{x} + 6 x \\cos{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 2 e^{x}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x e^{x}) = 2 x e^{x} + 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = - 3 x^{2} \\left(2 x + 3\\right) \\sin{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 6 x \\cos{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 2 \\left(x + 1\\right) e^{x} + 2 e^{x}$.\nFinal answer: $- 3 x^{2} \\left(2 x + 3\\right) \\sin{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 6 x \\cos{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 2 \\left(x + 1\\right) e^{x} + 2 e^{x}$", "final_answer": "- 3 x^{2} \\left(2 x + 3\\right) \\sin{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 6 x \\cos{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 2 \\left(x + 1\\right) e^{x} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "75d8be8a12f84d26", "problem_id": "30b493e250bba3f2", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the multiplicative inverse of 2 modulo 29.", "output": "1) We want x such that 2x ≡ 1 (mod 29).\n2) Use extended Euclid to find s,t with 2s + 29t = 1.\n3) One Bezout identity is: 2·(-14) + 29·(1) = 1.\n4) Reduce s modulo 29: x ≡ 15 (mod 29).\nFinal answer: $x\\equiv 15\\pmod{29}$", "final_answer": "x\\equiv 15\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4a3a85c785f0f948", "problem_id": "8a088a04b64ce077", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{2} + x - 1$ at $x=2$.", "output": "1) Differentiate: f'(x) = $1 - 6 x$.\n2) Slope at x=2: m = f'(2) = $-11$.\n3) Point on curve: (2, f(2)) = (2, $-11$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -11 = -11(x − 2).\n6) Optionally expand: y = $11 - 11 x$.\nFinal answer: $y = 11 - 11\\cdot x$", "final_answer": "y = 11 - 11\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9520ac050a0b80ad", "problem_id": "8a088a04b64ce077", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{2} + x - 1$ at $x=2$.", "output": "Differentiate: f'(x) = $1 - 6 x$.\nSlope at x=2: m = f'(2) = $-11$.\nPoint on curve: (2, f(2)) = (2, $-11$).\nFinal answer: $y = 11 - 11\\cdot x$", "final_answer": "y = 11 - 11\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d134d63e04c363d6", "problem_id": "37677161db5006b5", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} x \\left(3 x - 1\\right)$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{3} + 2 x^{2} - x - 3$ and a = -1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $9 x^{2} + 4 x - 1$.\n5) Evaluate at a: p'(-1) = $4$.\nFinal answer: $4$", "final_answer": "4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "62bb6d280cdeeefc", "problem_id": "37677161db5006b5", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} x \\left(3 x - 1\\right)$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $3 x^{3} + 2 x^{2} - x - 3$ and a = -1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $9 x^{2} + 4 x - 1$.\nEvaluate at a: p'(-1) = $4$.\nCheck: the identity gives $4$.\nFinal answer: $4$", "final_answer": "4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c127461dea9c3290", "problem_id": "e83cb3de2d8e44af", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A cyclist rides at 42 miles per hour for 3/2 hours, then at 47 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 42 × 3/2 = 63 miles.\n2) Second leg distance = 47 × 3 = 141 miles.\n3) Total distance = 63 + 141 = 204 miles.\nFinal answer: $204$", "final_answer": "204", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "37ee5a3146c5c917", "problem_id": "e83cb3de2d8e44af", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 42 miles per hour for 3/2 hours, then at 47 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 42 × 3/2 = 63 miles.\nSecond leg distance = 47 × 3 = 141 miles.\nTotal distance = 63 + 141 = 204 miles.\nCheck: adding both legs gives 204.\nFinal answer: $204$", "final_answer": "204", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5e9762aa6bfe6ff9", "problem_id": "c3b22a07797d36b0", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the equation of the tangent line to $y=4 x^{3} - 4 x^{2} - x - 1$ at $x=1$.", "output": "1) Differentiate: f'(x) = $12 x^{2} - 8 x - 1$.\n2) Slope at x=1: m = f'(1) = $3$.\n3) Point on curve: (1, f(1)) = (1, $-2$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -2 = 3(x − 1).\n6) Optionally expand: y = $3 x - 5$.\nFinal answer: $y = 3\\cdot x - 5$", "final_answer": "y = 3\\cdot x - 5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a757962b86f83a53", "problem_id": "4ca1963d6057596f", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} x^{3} + x^{2} - x + 1$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to x^{3} + x^{2} - 2 x + 1.\n3) Then the limit is 1.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -1.\nFinal answer: $-1$", "final_answer": "-1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ac29561062162668", "problem_id": "f9e31f868368b805", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{11} x}{3} + x - 2 + \\frac{\\sqrt{11}}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$1 - \\frac{\\sqrt{11}}{3}$, c=$-2 + \\frac{\\sqrt{11}}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{92}{9} - 2 \\sqrt{11}$, so √Δ = $\\sqrt{\\frac{92}{9} - 2 \\sqrt{11}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{1}{2} + \\frac{\\sqrt{11}}{6} + \\frac{\\sqrt{92 - 18 \\sqrt{11}}}{6}$.\n7) x2 = $- \\frac{\\sqrt{92 - 18 \\sqrt{11}}}{6} - \\frac{1}{2} + \\frac{\\sqrt{11}}{6}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{1}{2} + \\frac{\\sqrt{11}}{6} + \\frac{\\sqrt{92 - 18\\cdot \\sqrt{11}}}{6}, - \\frac{\\sqrt{92 - 18\\cdot \\sqrt{11}}}{6} - \\frac{1}{2} + \\frac{\\sqrt{11}}{6}\\}$", "final_answer": "x \\in \\{- \\frac{1}{2} + \\frac{\\sqrt{11}}{6} + \\frac{\\sqrt{92 - 18\\cdot \\sqrt{11}}}{6}, - \\frac{\\sqrt{92 - 18\\cdot \\sqrt{11}}}{6} - \\frac{1}{2} + \\frac{\\sqrt{11}}{6}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0d9e4b6c2358bc73", "problem_id": "f9e31f868368b805", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{11} x}{3} + x - 2 + \\frac{\\sqrt{11}}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$1 - \\frac{\\sqrt{11}}{3}$, c=$-2 + \\frac{\\sqrt{11}}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{92}{9} - 2 \\sqrt{11}$, so √Δ = $\\sqrt{\\frac{92}{9} - 2 \\sqrt{11}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{1}{2} + \\frac{\\sqrt{11}}{6} + \\frac{\\sqrt{92 - 18 \\sqrt{11}}}{6}$.\nx2 = $- \\frac{\\sqrt{92 - 18 \\sqrt{11}}}{6} - \\frac{1}{2} + \\frac{\\sqrt{11}}{6}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{1}{2} + \\frac{\\sqrt{11}}{6} + \\frac{\\sqrt{92 - 18\\cdot \\sqrt{11}}}{6}, - \\frac{\\sqrt{92 - 18\\cdot \\sqrt{11}}}{6} - \\frac{1}{2} + \\frac{\\sqrt{11}}{6}\\}$", "final_answer": "x \\in \\{- \\frac{1}{2} + \\frac{\\sqrt{11}}{6} + \\frac{\\sqrt{92 - 18\\cdot \\sqrt{11}}}{6}, - \\frac{\\sqrt{92 - 18\\cdot \\sqrt{11}}}{6} - \\frac{1}{2} + \\frac{\\sqrt{11}}{6}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2075f34f888e9964", "problem_id": "86b258ce2d1631ea", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(- x^{3} - 2 x^{2} + 3\\right) \\cos{\\left(x + 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 2 x^{2} \\cos{\\left(x + 1 \\right)}) = 2 x^{2} \\sin{\\left(x + 1 \\right)}$.\n3) So $f'(x) = x^{3} \\sin{\\left(x + 1 \\right)} + 2 x^{2} \\sin{\\left(x + 1 \\right)} - 3 x^{2} \\cos{\\left(x + 1 \\right)} + x e^{x} - 2 e^{x} - 3 \\sin{\\left(x + 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 2 x^{2} \\cos{\\left(x + 1 \\right)}) = 2 x^{2} \\sin{\\left(x + 1 \\right)} - 4 x \\cos{\\left(x + 1 \\right)}$.\nTherefore the correct derivative is $f'(x) = x \\left(- 3 x - 4\\right) \\cos{\\left(x + 1 \\right)} + \\left(x - 3\\right) e^{x} - \\left(x^{2} \\left(- x - 2\\right) + 3\\right) \\sin{\\left(x + 1 \\right)} + e^{x}$.\nFinal answer: $x \\left(- 3 x - 4\\right) \\cos{\\left(x + 1 \\right)} + \\left(x - 3\\right) e^{x} - \\left(x^{2} \\left(- x - 2\\right) + 3\\right) \\sin{\\left(x + 1 \\right)} + e^{x}$", "final_answer": "x \\left(- 3 x - 4\\right) \\cos{\\left(x + 1 \\right)} + \\left(x - 3\\right) e^{x} - \\left(x^{2} \\left(- x - 2\\right) + 3\\right) \\sin{\\left(x + 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8887345be46c0aa1", "problem_id": "dd8933b94667697b", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} - 3 x^{3} - x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{3} - x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{3 \\pi^{3}}{8} - 2 + \\frac{3 \\pi^{2}}{8} + 3 \\pi + \\frac{45 \\pi^{4}}{64}$.\n4) Simplify the result.\nFinal answer: $- \\frac{3\\cdot \\pi^{3}}{8} - 2 + \\frac{3\\cdot \\pi^{2}}{8} + 3\\cdot \\pi + \\frac{45\\cdot \\pi^{4}}{64}$", "final_answer": "- \\frac{3\\cdot \\pi^{3}}{8} - 2 + \\frac{3\\cdot \\pi^{2}}{8} + 3\\cdot \\pi + \\frac{45\\cdot \\pi^{4}}{64}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f788d9bd8edddb6", "problem_id": "271cbaf1c2dc99c9", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A cyclist rides at 76 miles per hour for 3/2 hours, then at 80 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 76 × 3/2 = 114 miles.\n2) Second leg distance = 80 × 5/2 = 200 miles.\n3) Total distance = 114 + 200 = 314 miles.\nFinal answer: $314$", "final_answer": "314", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "066df3bd8423728a", "problem_id": "271cbaf1c2dc99c9", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "A cyclist rides at 76 miles per hour for 3/2 hours, then at 80 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 76 × 3/2 = 114 miles.\nSecond leg distance = 80 × 5/2 = 200 miles.\nTotal distance = 114 + 200 = 314 miles.\nFinal answer: $314$", "final_answer": "314", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8b715b8b98a53bb1", "problem_id": "c50bf874f4a4ec8c", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡3 (mod 5), x≡10 (mod 11). Give the solution modulo 55.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 55.\n3) Compute the CRT solution to get x ≡ 43 (mod 55).\nFinal answer: $x\\equiv 43\\pmod{55}$", "final_answer": "x\\equiv 43\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7246c57b171c949f", "problem_id": "c50bf874f4a4ec8c", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡3 (mod 5), x≡10 (mod 11). Give the solution modulo 55.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 55.\nCompute the CRT solution to get x ≡ 43 (mod 55).\nCheck: 43 mod 5=3 and mod 11=10.\nFinal answer: $x\\equiv 43\\pmod{55}$", "final_answer": "x\\equiv 43\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "89b01e43787f2bdf", "problem_id": "8151bf25a6643895", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} + x^{2} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 3$ at $x=1$.", "output": "1) Differentiate: f'(x) = $- 6 x^{2} + 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\n2) Slope at x=1: m = f'(1) = $-6 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}$.\n3) Point on curve: (1, f(1)) = (1, $-6 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -6 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} = -6 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}(x − 1).\n6) Optionally expand: y = $- 6 x - 2 x \\sin{\\left(2 \\right)} + \\frac{x e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = - 6\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 6\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d944433de94814a2", "problem_id": "8151bf25a6643895", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} + x^{2} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 3$ at $x=1$.", "output": "Differentiate: f'(x) = $- 6 x^{2} + 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\nSlope at x=1: m = f'(1) = $-6 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}$.\nPoint on curve: (1, f(1)) = (1, $-6 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}}$).\nFinal answer: $y = - 6\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 6\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2fdc0ed62b6edd25", "problem_id": "c564970b5bc4e134", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the multiplicative inverse of 3 modulo 37.", "output": "1) We want x such that 3x ≡ 1 (mod 37).\n2) Use extended Euclid to find s,t with 3s + 37t = 1.\n3) One Bezout identity is: 3·(-12) + 37·(1) = 1.\n4) Reduce s modulo 37: x ≡ 25 (mod 37).\nFinal answer: $x\\equiv 25\\pmod{37}$", "final_answer": "x\\equiv 25\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7f7ac2a6414d1fc6", "problem_id": "e87817263420430d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(x^{3} + 2 x^{2} + 2 x + 3\\right) \\cos{\\left(3 x - 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 \\cos{\\left(3 x - 2 \\right)}) = - 3 \\sin{\\left(3 x - 2 \\right)}$.\n3) So $f'(x) = - 3 x^{3} \\sin{\\left(3 x - 2 \\right)} - 6 x^{2} \\sin{\\left(3 x - 2 \\right)} + 3 x^{2} \\cos{\\left(3 x - 2 \\right)} + x e^{x} - 6 x \\sin{\\left(3 x - 2 \\right)} + 4 x \\cos{\\left(3 x - 2 \\right)} + 2 e^{x} - 3 \\sin{\\left(3 x - 2 \\right)} + 2 \\cos{\\left(3 x - 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 \\cos{\\left(3 x - 2 \\right)}) = - 9 \\sin{\\left(3 x - 2 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 1\\right) e^{x} - 3 \\left(x \\left(x \\left(x + 2\\right) + 2\\right) + 3\\right) \\sin{\\left(3 x - 2 \\right)} + \\left(3 x^{2} + 4 x + 2\\right) \\cos{\\left(3 x - 2 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 1\\right) e^{x} - 3 \\left(x \\left(x \\left(x + 2\\right) + 2\\right) + 3\\right) \\sin{\\left(3 x - 2 \\right)} + \\left(3 x^{2} + 4 x + 2\\right) \\cos{\\left(3 x - 2 \\right)} + e^{x}$", "final_answer": "\\left(x + 1\\right) e^{x} - 3 \\left(x \\left(x \\left(x + 2\\right) + 2\\right) + 3\\right) \\sin{\\left(3 x - 2 \\right)} + \\left(3 x^{2} + 4 x + 2\\right) \\cos{\\left(3 x - 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8a2d2ecbcddd9e4f", "problem_id": "855ce794a1562ff9", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 9 x - 162$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} + 9 x - 162$ = $3 \\left(x - 6\\right) \\left(x + 9\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $6$.\nFrom the second factor: x = $-9$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{6, -9\\}$", "final_answer": "x \\in \\{6, -9\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "af3abb6524061925", "problem_id": "855ce794a1562ff9", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 9 x - 162$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} + 9 x - 162$ = $3 \\left(x - 6\\right) \\left(x + 9\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $6$.\n5) From the second factor: x = $-9$.\nFinal answer: $x \\in \\{6, -9\\}$", "final_answer": "x \\in \\{6, -9\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ad38b64493a67f45", "problem_id": "769160e9364c7dfd", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{14 x}{3} - \\frac{\\sqrt{3} x}{3} + \\frac{8 \\sqrt{3}}{9} + \\frac{16}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{14}{3} - \\frac{\\sqrt{3}}{3}$, c=$\\frac{8 \\sqrt{3}}{9} + \\frac{16}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{\\sqrt{7 - 4\\cdot \\sqrt{3}}}{6} + \\frac{\\sqrt{3}}{6} + \\frac{7}{3}, - \\frac{\\sqrt{7 - 4\\cdot \\sqrt{3}}}{6} + \\frac{\\sqrt{3}}{6} + \\frac{7}{3}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{7 - 4\\cdot \\sqrt{3}}}{6} + \\frac{\\sqrt{3}}{6} + \\frac{7}{3}, - \\frac{\\sqrt{7 - 4\\cdot \\sqrt{3}}}{6} + \\frac{\\sqrt{3}}{6} + \\frac{7}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9499f684d0dd57e9", "problem_id": "769160e9364c7dfd", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{14 x}{3} - \\frac{\\sqrt{3} x}{3} + \\frac{8 \\sqrt{3}}{9} + \\frac{16}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{14}{3} - \\frac{\\sqrt{3}}{3}$, c=$\\frac{8 \\sqrt{3}}{9} + \\frac{16}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{7}{9} - \\frac{4 \\sqrt{3}}{9}$, so √Δ = $\\sqrt{\\frac{7}{9} - \\frac{4 \\sqrt{3}}{9}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{7 - 4 \\sqrt{3}}}{6} + \\frac{\\sqrt{3}}{6} + \\frac{7}{3}$.\n7) x2 = $- \\frac{\\sqrt{7 - 4 \\sqrt{3}}}{6} + \\frac{\\sqrt{3}}{6} + \\frac{7}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{7 - 4\\cdot \\sqrt{3}}}{6} + \\frac{\\sqrt{3}}{6} + \\frac{7}{3}, - \\frac{\\sqrt{7 - 4\\cdot \\sqrt{3}}}{6} + \\frac{\\sqrt{3}}{6} + \\frac{7}{3}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{7 - 4\\cdot \\sqrt{3}}}{6} + \\frac{\\sqrt{3}}{6} + \\frac{7}{3}, - \\frac{\\sqrt{7 - 4\\cdot \\sqrt{3}}}{6} + \\frac{\\sqrt{3}}{6} + \\frac{7}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0d9663afe7e09b8c", "problem_id": "ce194fb3acc862cd", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 9 x - 10$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} + 9 x - 10$ = $\\left(x - 1\\right) \\left(x + 10\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $1$.\n5) From the second factor: x = $-10$.\nFinal answer: $x \\in \\{1, -10\\}$", "final_answer": "x \\in \\{1, -10\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e5a7c55c4ad1737c", "problem_id": "ce194fb3acc862cd", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 9 x - 10$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} + 9 x - 10$ = $\\left(x - 1\\right) \\left(x + 10\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $1$.\nFrom the second factor: x = $-10$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{1, -10\\}$", "final_answer": "x \\in \\{1, -10\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "834193a917ba4462", "problem_id": "c36d5cb6b54c49df", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 18 red balls and 18 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(36,11).\n3) Favorable samples: choose k reds and n−k blues: C(18,2)·C(18,9).\n4) Compute totals: C(36,11) = 600805296, and favorable = 7438860.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{935}{75516}$", "final_answer": "\\frac{935}{75516}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "909def8bfb9088ae", "problem_id": "c36d5cb6b54c49df", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 18 red balls and 18 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(36,11).\nFavorable samples: choose k reds and n−k blues: C(18,2)·C(18,9).\nCompute totals: C(36,11) = 600805296, and favorable = 7438860.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{935}{75516}$.\nFinal answer: $\\frac{935}{75516}$", "final_answer": "\\frac{935}{75516}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8abee24fb0f0e7d7", "problem_id": "53dc60affb920c92", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the linear system:\n$4 x - 2 y$ = $-18$\n$2 x$ = $-10$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $8 x - 4 y$ = $-36$.\n3) Multiply the second equation by $4$: $8 x$ = $-40$.\n4) Subtract to eliminate x: $- 4 y$ = $4$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, -1\\right)$", "final_answer": "(x,y)=\\left(-5, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "522d4c9d84d88a65", "problem_id": "53dc60affb920c92", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the linear system:\n$4 x - 2 y$ = $-18$\n$2 x$ = $-10$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $2$: $8 x - 4 y$ = $-36$.\nMultiply the second equation by $4$: $8 x$ = $-40$.\nFinal answer: $(x,y)=\\left(-5, -1\\right)$", "final_answer": "(x,y)=\\left(-5, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4b6d9fef0e845123", "problem_id": "d3b067b3b14a6674", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{3} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $- 3 x^{3} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.\nFinal answer: $- 3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} + \\cos{\\left(x \\right)}$", "final_answer": "- 3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} + \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a12d67a41eb63376", "problem_id": "d3b067b3b14a6674", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{3} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $- 3 x^{3} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} + \\cos{\\left(x \\right)}$", "final_answer": "- 3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} + \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6963088a83919bc8", "problem_id": "7229473dbffe18f6", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{10}$. If it is flipped 7 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=7, k=5, p=$\\frac{3}{10}$ and simplify.\n4) Here C(7,5) = 21.\nFinal answer: $\\frac{250047}{10000000}$", "final_answer": "\\frac{250047}{10000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "70704460dbb0fa08", "problem_id": "7229473dbffe18f6", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{10}$. If it is flipped 7 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=7, k=5, p=$\\frac{3}{10}$ and simplify.\nHere C(7,5) = 21.\nCheck: expression simplifies to $\\frac{250047}{10000000}$.\nFinal answer: $\\frac{250047}{10000000}$", "final_answer": "\\frac{250047}{10000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "080b9e85652f98c2", "problem_id": "83deeee066dd2a68", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{3} + x^{2} - 2 x + 1\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 3 x^{3} + x^{2} - 2 x + 1$.\nCompute an antiderivative: F(x) = $- \\frac{3 x^{4}}{4} + \\frac{x^{3}}{3} - x^{2} + x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - \\frac{3\\cdot x^{4}}{4} + \\frac{x^{3}}{3} - x^{2} + x$", "final_answer": "C - \\frac{3\\cdot x^{4}}{4} + \\frac{x^{3}}{3} - x^{2} + x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b0914d787df40706", "problem_id": "83deeee066dd2a68", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{3} + x^{2} - 2 x + 1\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 3 x^{3} + x^{2} - 2 x + 1$.\nCompute an antiderivative: F(x) = $- \\frac{3 x^{4}}{4} + \\frac{x^{3}}{3} - x^{2} + x$.\nFinal answer: $C - \\frac{3\\cdot x^{4}}{4} + \\frac{x^{3}}{3} - x^{2} + x$", "final_answer": "C - \\frac{3\\cdot x^{4}}{4} + \\frac{x^{3}}{3} - x^{2} + x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fcd8f077966e1885", "problem_id": "b25da1383a89c6c7", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{2} - 3 x + \\left(- x^{2} + 2 x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{- x^{2} + 2 x + 6}$.", "output": "Let f(x) = $\\frac{- 3 x^{2} - 3 x + \\left(- x^{2} + 2 x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{- x^{2} + 2 x + 6}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{4 \\left(x - 1\\right) \\left(- 3 x^{2} - 3 x + \\left(- x^{2} + 2 x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(- x^{2} + 2 x + 6\\right) \\left(- 12 x + 4 \\left(1 - x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(- x^{2} + 2 x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(- x^{2} + 2 x + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(- x^{2} + 2 x + 6\\right)^{2}}$.\nFinal answer: $\\frac{\\left(4\\cdot x - 4\\right)\\cdot \\left(- 3\\cdot x^{2} - 3\\cdot x + \\left(- x^{2} + 2\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 1\\right) + \\left(- x^{2} + 2\\cdot x + 6\\right)\\cdot \\left(- 12\\cdot x + \\left(4 - 4\\cdot x\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(- 2\\cdot x^{2} + 4\\cdot x + 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + \\left(- x^{2} + 2\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 6\\right)}{2\\cdot \\left(- x^{2} + 2\\cdot x + 6\\right)^{2}}$", "final_answer": "\\frac{\\left(4\\cdot x - 4\\right)\\cdot \\left(- 3\\cdot x^{2} - 3\\cdot x + \\left(- x^{2} + 2\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 1\\right) + \\left(- x^{2} + 2\\cdot x + 6\\right)\\cdot \\left(- 12\\cdot x + \\left(4 - 4\\cdot x\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(- 2\\cdot x^{2} + 4\\cdot x + 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + \\left(- x^{2} + 2\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 6\\right)}{2\\cdot \\left(- x^{2} + 2\\cdot x + 6\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f36d445eb69c916f", "problem_id": "b25da1383a89c6c7", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{2} - 3 x + \\left(- x^{2} + 2 x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{- x^{2} + 2 x + 6}$.", "output": "1) Let f(x) = $\\frac{- 3 x^{2} - 3 x + \\left(- x^{2} + 2 x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{- x^{2} + 2 x + 6}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{4 \\left(x - 1\\right) \\left(- 3 x^{2} - 3 x + \\left(- x^{2} + 2 x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(- x^{2} + 2 x + 6\\right) \\left(- 12 x + 4 \\left(1 - x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(- x^{2} + 2 x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(- x^{2} + 2 x + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(- x^{2} + 2 x + 6\\right)^{2}}$.\nFinal answer: $\\frac{\\left(4\\cdot x - 4\\right)\\cdot \\left(- 3\\cdot x^{2} - 3\\cdot x + \\left(- x^{2} + 2\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 1\\right) + \\left(- x^{2} + 2\\cdot x + 6\\right)\\cdot \\left(- 12\\cdot x + \\left(4 - 4\\cdot x\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(- 2\\cdot x^{2} + 4\\cdot x + 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + \\left(- x^{2} + 2\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 6\\right)}{2\\cdot \\left(- x^{2} + 2\\cdot x + 6\\right)^{2}}$", "final_answer": "\\frac{\\left(4\\cdot x - 4\\right)\\cdot \\left(- 3\\cdot x^{2} - 3\\cdot x + \\left(- x^{2} + 2\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 1\\right) + \\left(- x^{2} + 2\\cdot x + 6\\right)\\cdot \\left(- 12\\cdot x + \\left(4 - 4\\cdot x\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(- 2\\cdot x^{2} + 4\\cdot x + 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + \\left(- x^{2} + 2\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 6\\right)}{2\\cdot \\left(- x^{2} + 2\\cdot x + 6\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "58206635748de4bc", "problem_id": "e9f488a27db5aebd", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_9=27$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_9=9.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=9 and k=9.\n4) Compute C(17,8) = 24310.\nFinal answer: $24310$", "final_answer": "24310", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "caa5f0a6bb2a7d45", "problem_id": "64d4c2f2f046cdfc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x^{2} - 4 x - 7$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} - 5 x - 7.\n3) Then the limit is -21.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -19.\nFinal answer: $-19$", "final_answer": "-19", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f2fe2d8c097476b8", "problem_id": "232140c6d24e0324", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} x^{2} - 6 x + 22$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $x^{3} - 2 x^{2} - 2 x - 1$ and a = -4.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $3 x^{2} - 4 x - 2$.\nEvaluate at a: p'(-4) = $62$.\nCheck: the identity gives $62$.\nFinal answer: $62$", "final_answer": "62", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "44e2d16048b15050", "problem_id": "232140c6d24e0324", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} x^{2} - 6 x + 22$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{3} - 2 x^{2} - 2 x - 1$ and a = -4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $3 x^{2} - 4 x - 2$.\n5) Evaluate at a: p'(-4) = $62$.\nFinal answer: $62$", "final_answer": "62", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f6b61fdeeea2e850", "problem_id": "55042481eda39170", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A cyclist rides at 37 miles per hour for 1 hours, then at 77 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 37 × 1 = 37 miles.\n2) Second leg distance = 77 × 3 = 231 miles.\n3) Total distance = 37 + 231 = 268 miles.\nFinal answer: $268$", "final_answer": "268", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1ffb6d3b79fd8fe6", "problem_id": "55042481eda39170", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A cyclist rides at 37 miles per hour for 1 hours, then at 77 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 37 × 1 = 37 miles.\nSecond leg distance = 77 × 3 = 231 miles.\nTotal distance = 37 + 231 = 268 miles.\nCheck: adding both legs gives 268.\nFinal answer: $268$", "final_answer": "268", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8db8183219b093eb", "problem_id": "96c21ee930c79440", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 14 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=14, k=6, p=$\\frac{1}{6}$ and simplify.\n4) Here C(14,6) = 3003.\nFinal answer: $\\frac{391015625}{26121388032}$", "final_answer": "\\frac{391015625}{26121388032}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c6f7f981ef73fb25", "problem_id": "e4305ec474fba739", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\sin{\\left(x \\right)}) = \\cos{\\left(x \\right)}$.\n3) So $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\sin{\\left(x \\right)}) = \\cos{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.\nFinal answer: $x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$", "final_answer": "x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "73d31e0b46615d8b", "problem_id": "6823d11f274cecd0", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A cyclist rides at 32 miles per hour for 3/2 hours, then at 55 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 32 × 3/2 = 48 miles.\n2) Second leg distance = 55 × 3 = 165 miles.\n3) Total distance = 48 + 165 = 213 miles.\nFinal answer: $213$", "final_answer": "213", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "afa78f5f52c587ee", "problem_id": "6823d11f274cecd0", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A cyclist rides at 32 miles per hour for 3/2 hours, then at 55 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 32 × 3/2 = 48 miles.\nSecond leg distance = 55 × 3 = 165 miles.\nTotal distance = 48 + 165 = 213 miles.\nFinal answer: $213$", "final_answer": "213", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ce38b0f465b10636", "problem_id": "b4caba8669fa4e95", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} - 3 x^{3} + 2 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{3} + 2 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\nEvaluate at the bounds: F(\\pi) and F(- \\pi).\nSubtract: F(\\pi) − F(- \\pi) = $\\frac{4 \\pi^{3}}{3}$.\nFinal answer: $\\frac{4\\cdot \\pi^{3}}{3}$", "final_answer": "\\frac{4\\cdot \\pi^{3}}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "82f122bf8d7ce1c2", "problem_id": "b4caba8669fa4e95", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} - 3 x^{3} + 2 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{3} + 2 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\pi).\n3) Subtract: F(\\pi) − F(- \\pi) = $\\frac{4 \\pi^{3}}{3}$.\n4) Simplify the result.\nFinal answer: $\\frac{4\\cdot \\pi^{3}}{3}$", "final_answer": "\\frac{4\\cdot \\pi^{3}}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "63628b72a32e5092", "problem_id": "3d07f878cfe41ac7", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the multiplicative inverse of 19 modulo 41.", "output": "We want x such that 19x ≡ 1 (mod 41).\nUse extended Euclid to find s,t with 19s + 41t = 1.\nOne Bezout identity is: 19·(13) + 41·(-6) = 1.\nReduce s modulo 41: x ≡ 13 (mod 41).\nCheck: 19·13 mod 41 = 1.\nFinal answer: $x\\equiv 13\\pmod{41}$", "final_answer": "x\\equiv 13\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d6dbe6a6d661228", "problem_id": "ee2a845187ed539e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x + 2$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 2.\n3) Then the limit is 2.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 4.\nFinal answer: $4$", "final_answer": "4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2e8390af9e5b427c", "problem_id": "cae045131bacee8a", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Find the equation of the tangent line to $y=x^{3} + 2 x^{2} - 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 3$ at $x=1$.", "output": "1) Differentiate: f'(x) = $3 x^{2} + 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 3$.\n2) Slope at x=1: m = f'(1) = $- 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 4$.\n3) Point on curve: (1, f(1)) = (1, $-3 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -3 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} = - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 4(x − 1).\n6) Optionally expand: y = $- 2 x \\sin{\\left(2 \\right)} + \\frac{x e^{\\frac{1}{2}}}{2} + 4 x - 7 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 4\\cdot x - 7 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 4\\cdot x - 7 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d6dbe6a6d661228", "problem_id": "ee2a845187ed539e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x + 2$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 2.\n3) Then the limit is 2.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 4.\nFinal answer: $4$", "final_answer": "4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b6fc611edcf4d717", "problem_id": "e50ffcdbfc56dd5d", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A cyclist rides at 78 miles per hour for 1/2 hours, then at 82 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 78 × 1/2 = 39 miles.\n2) Second leg distance = 82 × 2 = 164 miles.\n3) Total distance = 39 + 164 = 203 miles.\nFinal answer: $203$", "final_answer": "203", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "131c813d54195550", "problem_id": "e50ffcdbfc56dd5d", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A cyclist rides at 78 miles per hour for 1/2 hours, then at 82 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 78 × 1/2 = 39 miles.\nSecond leg distance = 82 × 2 = 164 miles.\nTotal distance = 39 + 164 = 203 miles.\nCheck: adding both legs gives 203.\nFinal answer: $203$", "final_answer": "203", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1876ad7fbdcf0051", "problem_id": "171507cacba2478d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 2 x^{3} + x^{2} + 2 x + \\left(- 3 x^{2} + x + 8\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{- 3 x^{2} + x + 8}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{8 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- 3 x^{2} + x\\right) + 8}) = \\frac{8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{- 3 x^{2} + x + 8}$.\n3) So $f'(x) = \\frac{9 x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} + \\frac{18 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} + \\frac{12 x^{4}}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} - \\frac{6 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} - \\frac{12 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} - \\frac{8 x^{3}}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} - \\frac{47 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} - \\frac{94 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} - \\frac{82 x^{2}}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} + \\frac{16 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} + \\frac{32 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} - \\frac{4 x}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} - \\frac{48 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{4} - 6 x^{3} - 47 x^{2} + 16 x + 64} + \\frac{64 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} + \\frac{128 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} + \\frac{38}{18 x^{4} - 12 x^{3} - 94 x^{2} + 32 x + 128} + \\frac{8 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{4} - 6 x^{3} - 47 x^{2} + 16 x + 64} + \\frac{8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{- 3 x^{2} + x + 8} - \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- 3 x^{2} + x\\right) + 8} - \\frac{8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(- 3 x^{2} + x\\right) + 8}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{8 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- 3 x^{2} + x\\right) + 8}) = \\frac{8 \\left(6 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(- 3 x^{2} + x\\right) + 8\\right)^{2}} + \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- 3 x^{2} + x\\right) + 8} + \\frac{8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(- 3 x^{2} + x\\right) + 8}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(6 x - 1\\right) \\left(- 2 x^{3} + x^{2} + 2 x + \\left(x \\left(1 - 3 x\\right) + 8\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x \\left(1 - 3 x\\right) + 8\\right) \\left(- 12 x^{2} + 4 x + 2 \\left(1 - 6 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(1 - 3 x\\right) + 8\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(1 - 3 x\\right) + 8\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(1 - 3 x\\right) + 8\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(6 x - 1\\right) \\left(- 2 x^{3} + x^{2} + 2 x + \\left(x \\left(1 - 3 x\\right) + 8\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x \\left(1 - 3 x\\right) + 8\\right) \\left(- 12 x^{2} + 4 x + 2 \\left(1 - 6 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(1 - 3 x\\right) + 8\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(1 - 3 x\\right) + 8\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(1 - 3 x\\right) + 8\\right)^{2}}$", "final_answer": "\\frac{2 \\left(6 x - 1\\right) \\left(- 2 x^{3} + x^{2} + 2 x + \\left(x \\left(1 - 3 x\\right) + 8\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x \\left(1 - 3 x\\right) + 8\\right) \\left(- 12 x^{2} + 4 x + 2 \\left(1 - 6 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(1 - 3 x\\right) + 8\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(1 - 3 x\\right) + 8\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(1 - 3 x\\right) + 8\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "70896a4d7fed8e0b", "problem_id": "1f9de617945673eb", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$7 x - 2 y$ = $-58$\n$3 x + 6 y$ = $-18$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $3$: $21 x - 6 y$ = $-174$.\n3) Multiply the second equation by $7$: $21 x + 42 y$ = $-126$.\n4) Subtract to eliminate x: $- 48 y$ = $-48$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $-8$.\nFinal answer: $(x,y)=\\left(-8, 1\\right)$", "final_answer": "(x,y)=\\left(-8, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "281c091e88dd94f1", "problem_id": "4c7284f47c847b99", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$4 x - y$ = $28$\n$5 x - 7 y$ = $12$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $5$: $20 x - 5 y$ = $140$.\nMultiply the second equation by $4$: $20 x - 28 y$ = $48$.\nSubtract to eliminate x: $23 y$ = $92$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $8$.\nCheck: plug (x,y)=(8,4) into both equations.\nFinal answer: $(x,y)=\\left(8, 4\\right)$", "final_answer": "(x,y)=\\left(8, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "00080dc21435d91e", "problem_id": "4c7284f47c847b99", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$4 x - y$ = $28$\n$5 x - 7 y$ = $12$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $5$: $20 x - 5 y$ = $140$.\n3) Multiply the second equation by $4$: $20 x - 28 y$ = $48$.\n4) Subtract to eliminate x: $23 y$ = $92$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $8$.\nFinal answer: $(x,y)=\\left(8, 4\\right)$", "final_answer": "(x,y)=\\left(8, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1990330f1607ae20", "problem_id": "0e157e7bebf01dd6", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the multiplicative inverse of 24 modulo 29.", "output": "1) We want x such that 24x ≡ 1 (mod 29).\n2) Use extended Euclid to find s,t with 24s + 29t = 1.\n3) One Bezout identity is: 24·(-6) + 29·(5) = 1.\n4) Reduce s modulo 29: x ≡ 23 (mod 29).\nFinal answer: $x\\equiv 23\\pmod{29}$", "final_answer": "x\\equiv 23\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bbeb36ba84cbce74", "problem_id": "70f3d1d7a3038a3d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\left(x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)}\\right) - \\sin{\\left(x \\right)}) = \\left(x^{2} + x - 1\\right) \\cos{\\left(x \\right)}$.\n3) So $f'(x) = x^{2} \\cos{\\left(x \\right)} + x \\cos{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\left(x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)}\\right) - \\sin{\\left(x \\right)}) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - \\cos{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.\nFinal answer: $x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$", "final_answer": "x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "27e1a22844bfd352", "problem_id": "4f060fc8a978793b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(3 x^{2} + 2 x + 2\\right) \\cos{\\left(x - 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 x^{2} \\cos{\\left(x - 3 \\right)}) = - 3 x^{2} \\sin{\\left(x - 3 \\right)}$.\n3) So $f'(x) = - 3 x^{2} \\sin{\\left(x - 3 \\right)} + x e^{x} - 2 x \\sin{\\left(x - 3 \\right)} + 3 e^{x} - 2 \\sin{\\left(x - 3 \\right)} + 2 \\cos{\\left(x - 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 x^{2} \\cos{\\left(x - 3 \\right)}) = - 3 x^{2} \\sin{\\left(x - 3 \\right)} + 6 x \\cos{\\left(x - 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 2\\right) e^{x} + 2 \\left(3 x + 1\\right) \\cos{\\left(x - 3 \\right)} - \\left(x \\left(3 x + 2\\right) + 2\\right) \\sin{\\left(x - 3 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 2\\right) e^{x} + 2 \\left(3 x + 1\\right) \\cos{\\left(x - 3 \\right)} - \\left(x \\left(3 x + 2\\right) + 2\\right) \\sin{\\left(x - 3 \\right)} + e^{x}$", "final_answer": "\\left(x + 2\\right) e^{x} + 2 \\left(3 x + 1\\right) \\cos{\\left(x - 3 \\right)} - \\left(x \\left(3 x + 2\\right) + 2\\right) \\sin{\\left(x - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3ac274eaaa08c340", "problem_id": "366dd20ebfbc0dee", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡2 (mod 7), x≡6 (mod 13). Give the solution modulo 91.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 91.\n3) Compute the CRT solution to get x ≡ 58 (mod 91).\nFinal answer: $x\\equiv 58\\pmod{91}$", "final_answer": "x\\equiv 58\\pmod{91}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "17f24a8968bfa896", "problem_id": "f07440c39015021f", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Two friends split $428 in the ratio 1:3. How much does each receive?", "output": "1) Total ratio parts = 1+3 = 4.\n2) Each part = 428 ÷ 4 = 107.\n3) First share = 1 × 107 = 107.\n4) Second share = 3 × 107 = 321.\nFinal answer: $321$", "final_answer": "321", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b5d93e1c513f27be", "problem_id": "f07440c39015021f", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $428 in the ratio 1:3. How much does each receive?", "output": "Total ratio parts = 1+3 = 4.\nEach part = 428 ÷ 4 = 107.\nFirst share = 1 × 107 = 107.\nSecond share = 3 × 107 = 321.\nCheck: 107+321=428.\nFinal answer: $321$", "final_answer": "321", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f3102e44a70da8b9", "problem_id": "f50a77ce3a4f8971", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "You want 29 liters of a 25% salt solution. You have a 10% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 29−x liters is 40% solution.\n2) Salt equation: 10x + 40(29−x) = 25·29.\n3) (10-40)x = (25-40)·29 ⇒ x = 29/2.\n4) Then 29−x = 29/2.\nFinal answer: $29/2 L of 10% and 29/2 L of 40%$", "final_answer": "29/2 L of 10% and 29/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a28858a0b124ca37", "problem_id": "37677161db5006b5", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} x \\left(3 x - 1\\right)$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{2} - 2 x.\n3) Then the limit is 5.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: 4.\nFinal answer: $4$", "final_answer": "4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "04195c74afefad11", "problem_id": "89be46e5fef90b5f", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 3 x^{3} - x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{3} - x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $\\frac{3 \\pi \\left(- 4 \\pi^{2} - 8 \\pi + 8 + 15 \\pi^{3}\\right)}{4}$.\n4) Simplify the result.\nFinal answer: $\\frac{3\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} - 8\\cdot \\pi + 8 + 15\\cdot \\pi^{3}\\right)}{4}$", "final_answer": "\\frac{3\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} - 8\\cdot \\pi + 8 + 15\\cdot \\pi^{3}\\right)}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "be622ae8fef2eaf3", "problem_id": "89be46e5fef90b5f", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 3 x^{3} - x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{3} - x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $\\frac{3 \\pi \\left(- 4 \\pi^{2} - 8 \\pi + 8 + 15 \\pi^{3}\\right)}{4}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{3\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} - 8\\cdot \\pi + 8 + 15\\cdot \\pi^{3}\\right)}{4}$", "final_answer": "\\frac{3\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} - 8\\cdot \\pi + 8 + 15\\cdot \\pi^{3}\\right)}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6433e998353d95ef", "problem_id": "398f4b85637b18d7", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=17$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=17 and k=5.\n4) Compute C(21,4) = 5985.\nFinal answer: $5985$", "final_answer": "5985", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b5baa17f0af260e0", "problem_id": "398f4b85637b18d7", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=17$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=17 and k=5.\nCompute C(21,4) = 5985.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $5985$", "final_answer": "5985", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "118a35cb4f181af2", "problem_id": "2cf83b32af55ebb0", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$- x - 3 y$ = $-4$\n$7 x + 4 y$ = $-40$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $7$: $- 7 x - 21 y$ = $-28$.\n3) Multiply the second equation by $-1$: $- 7 x - 4 y$ = $40$.\n4) Subtract to eliminate x: $- 17 y$ = $-68$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $-8$.\nFinal answer: $(x,y)=\\left(-8, 4\\right)$", "final_answer": "(x,y)=\\left(-8, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "04d82a16b7be5360", "problem_id": "49882216d0f286e6", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - x^{3} - 7 x^{2} - 25 x - 99$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to - x^{3} - 7 x^{2} - 26 x - 99.\n3) Then the limit is -379.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: -375.\nFinal answer: $-375$", "final_answer": "-375", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a81840e2ad0676b2", "problem_id": "d95d2be1f7a7a43c", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Two friends split $270 in the ratio 1:4. How much does each receive?", "output": "Total ratio parts = 1+4 = 5.\nEach part = 270 ÷ 5 = 54.\nFirst share = 1 × 54 = 54.\nSecond share = 4 × 54 = 216.\nCheck: 54+216=270.\nFinal answer: $216$", "final_answer": "216", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1783ff8673cb3f50", "problem_id": "d95d2be1f7a7a43c", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Two friends split $270 in the ratio 1:4. How much does each receive?", "output": "Total ratio parts = 1+4 = 5.\nEach part = 270 ÷ 5 = 54.\nFirst share = 1 × 54 = 54.\nFinal answer: $216$", "final_answer": "216", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fcfd853743777a7e", "problem_id": "726de7c3634bffcd", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Two friends split $432 in the ratio 3:6. How much does each receive?", "output": "Total ratio parts = 3+6 = 9.\nEach part = 432 ÷ 9 = 48.\nFirst share = 3 × 48 = 144.\nSecond share = 6 × 48 = 288.\nCheck: 144+288=432.\nFinal answer: $288$", "final_answer": "288", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3c312b40d781417a", "problem_id": "726de7c3634bffcd", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Two friends split $432 in the ratio 3:6. How much does each receive?", "output": "1) Total ratio parts = 3+6 = 9.\n2) Each part = 432 ÷ 9 = 48.\n3) First share = 3 × 48 = 144.\n4) Second share = 6 × 48 = 288.\nFinal answer: $288$", "final_answer": "288", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "00f1fd54daaa1eed", "problem_id": "40375ef047c274f9", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 24 miles per hour for 2 hours, then at 85 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 24 × 2 = 48 miles.\nSecond leg distance = 85 × 3/2 = 255/2 miles.\nTotal distance = 48 + 255/2 = 351/2 miles.\nCheck: adding both legs gives 351/2.\nFinal answer: $\\frac{351}{2}$", "final_answer": "\\frac{351}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0939f0eea594f42a", "problem_id": "40375ef047c274f9", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A cyclist rides at 24 miles per hour for 2 hours, then at 85 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 24 × 2 = 48 miles.\n2) Second leg distance = 85 × 3/2 = 255/2 miles.\n3) Total distance = 48 + 255/2 = 351/2 miles.\nFinal answer: $\\frac{351}{2}$", "final_answer": "\\frac{351}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eb9d15de51cc5bf6", "problem_id": "233ef8312998889b", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the equation of the tangent line to $y=4 x^{3} - 2 x^{2} + 3 x - 3$ at $x=2$.", "output": "1) Differentiate: f'(x) = $12 x^{2} - 4 x + 3$.\n2) Slope at x=2: m = f'(2) = $43$.\n3) Point on curve: (2, f(2)) = (2, $27$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 27 = 43(x − 2).\n6) Optionally expand: y = $43 x - 59$.\nFinal answer: $y = 43\\cdot x - 59$", "final_answer": "y = 43\\cdot x - 59", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bac3a72ad188095f", "problem_id": "233ef8312998889b", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the equation of the tangent line to $y=4 x^{3} - 2 x^{2} + 3 x - 3$ at $x=2$.", "output": "Differentiate: f'(x) = $12 x^{2} - 4 x + 3$.\nSlope at x=2: m = f'(2) = $43$.\nPoint on curve: (2, f(2)) = (2, $27$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 27 = 43(x − 2).\nOptionally expand: y = $43 x - 59$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 43\\cdot x - 59$", "final_answer": "y = 43\\cdot x - 59", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "085d81cdfc4090df", "problem_id": "a181ae879b4bf103", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many binary strings of length 24 contain exactly 7 ones with no two ones adjacent?", "output": "Reserve 6 mandatory zeros between the 7 ones.\nThis leaves n−k+1 = 18 valid slots for the ones.\nChoose the slots: C(18,7) = 31824.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $31824$", "final_answer": "31824", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a09432fb7a155de", "problem_id": "a181ae879b4bf103", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "How many binary strings of length 24 contain exactly 7 ones with no two ones adjacent?", "output": "1) Reserve 6 mandatory zeros between the 7 ones.\n2) This leaves n−k+1 = 18 valid slots for the ones.\n3) Choose the slots: C(18,7) = 31824.\nFinal answer: $31824$", "final_answer": "31824", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "06054f5c563b5f91", "problem_id": "e42bbfe4f4b3b9f9", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $131. It is discounted by 30%. You buy 11 backpacks. What is the total cost?", "output": "Discounted price = 131 × (1 − 30/100) = 131 × 70/100 = 917/10.\nSubtotal = 11 × 917/10 = 10087/10.\nCheck: recomputing the formula yields 10087/10.\nFinal answer: $\\frac{10087}{10}$", "final_answer": "\\frac{10087}{10}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "973de038ac5b02d5", "problem_id": "e42bbfe4f4b3b9f9", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A store sells a backpack for $131. It is discounted by 30%. You buy 11 backpacks. What is the total cost?", "output": "1) Discounted price = 131 × (1 − 30/100) = 131 × 70/100 = 917/10.\n2) Subtotal = 11 × 917/10 = 10087/10.\nFinal answer: $\\frac{10087}{10}$", "final_answer": "\\frac{10087}{10}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1a3632e9f8da8fca", "problem_id": "d16111749b28c560", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$y$ = $4$\n$- x - y$ = $-1$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $- y$ = $-4$.\nMultiply the second equation by $0$: $0$ = $0$.\nSubtract to eliminate x: $- y$ = $-4$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $-3$.\nCheck: plug (x,y)=(-3,4) into both equations.\nFinal answer: $(x,y)=\\left(-3, 4\\right)$", "final_answer": "(x,y)=\\left(-3, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ad0d8f08e8f2fbf0", "problem_id": "d16111749b28c560", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$y$ = $4$\n$- x - y$ = $-1$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $- y$ = $-4$.\n3) Multiply the second equation by $0$: $0$ = $0$.\n4) Subtract to eliminate x: $- y$ = $-4$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, 4\\right)$", "final_answer": "(x,y)=\\left(-3, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a4a074d9f285f5c3", "problem_id": "50918079d7a6f4ca", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - x^{3} + 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- x^{3} + 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 1$.\n3) Compute an antiderivative: F(x) = $- \\frac{x^{4}}{4} + x^{3} + \\frac{3 x^{2}}{2} + x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{x^{4}}{4} + x^{3} + \\frac{3\\cdot x^{2}}{2} + x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{x^{4}}{4} + x^{3} + \\frac{3\\cdot x^{2}}{2} + x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4cfa1ffa85507cbe", "problem_id": "50918079d7a6f4ca", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - x^{3} + 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- x^{3} + 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 1$.\nCompute an antiderivative: F(x) = $- \\frac{x^{4}}{4} + x^{3} + \\frac{3 x^{2}}{2} + x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - \\frac{x^{4}}{4} + x^{3} + \\frac{3\\cdot x^{2}}{2} + x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{x^{4}}{4} + x^{3} + \\frac{3\\cdot x^{2}}{2} + x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0fda6ef7355e26b8", "problem_id": "a63e664af71373d9", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 14 red balls and 8 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(22,5).\nFavorable samples: choose k reds and n−k blues: C(14,2)·C(8,3).\nCompute totals: C(22,5) = 26334, and favorable = 5096.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{364}{1881}$.\nFinal answer: $\\frac{364}{1881}$", "final_answer": "\\frac{364}{1881}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fb07dd5e857bb82c", "problem_id": "a63e664af71373d9", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "An urn has 14 red balls and 8 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(22,5).\nFavorable samples: choose k reds and n−k blues: C(14,2)·C(8,3).\nFinal answer: $\\frac{364}{1881}$", "final_answer": "\\frac{364}{1881}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "06cbdf7ffd88a7ba", "problem_id": "5f7ce52afa2f4277", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 14 x + 40$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} - 14 x + 40$ = $\\left(x - 10\\right) \\left(x - 4\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $10$.\nFrom the second factor: x = $4$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{10, 4\\}$", "final_answer": "x \\in \\{10, 4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "953e01a3be2a506a", "problem_id": "5f7ce52afa2f4277", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 14 x + 40$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} - 14 x + 40$ = $\\left(x - 10\\right) \\left(x - 4\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $10$.\n5) From the second factor: x = $4$.\nFinal answer: $x \\in \\{10, 4\\}$", "final_answer": "x \\in \\{10, 4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b14ae61b5197384c", "problem_id": "fdbe7100b6380f31", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} - 3 x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $- \\frac{\\pi^{3}}{4} - \\pi$.\n4) Simplify the result.\nFinal answer: $- \\frac{\\pi^{3}}{4} - \\pi$", "final_answer": "- \\frac{\\pi^{3}}{4} - \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9c1381dcac5a1e19", "problem_id": "8a6217f9111d785d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 - 3 x$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 3 - 4 x.\n3) Then the limit is 11.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 9.\nFinal answer: $9$", "final_answer": "9", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "4a235028b611a752", "problem_id": "a29d67452e19b2e9", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "You want 38 liters of a 30% salt solution. You have a 15% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 38−x liters is 60% solution.\n2) Salt equation: 15x + 60(38−x) = 30·38.\n3) (15-60)x = (30-60)·38 ⇒ x = 76/3.\n4) Then 38−x = 38/3.\nFinal answer: $76/3 L of 15% and 38/3 L of 60%$", "final_answer": "76/3 L of 15% and 38/3 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "34eeddeda3148098", "problem_id": "a29d67452e19b2e9", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "You want 38 liters of a 30% salt solution. You have a 15% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 38−x liters is 60% solution.\nSalt equation: 15x + 60(38−x) = 30·38.\n(15-60)x = (30-60)·38 ⇒ x = 76/3.\nFinal answer: $76/3 L of 15% and 38/3 L of 60%$", "final_answer": "76/3 L of 15% and 38/3 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "539be20c54bba0be", "problem_id": "3055f9ef93c891b3", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A cyclist rides at 62 miles per hour for 3/2 hours, then at 80 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 62 × 3/2 = 93 miles.\nSecond leg distance = 80 × 3 = 240 miles.\nTotal distance = 93 + 240 = 333 miles.\nFinal answer: $333$", "final_answer": "333", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "21ae63481ff13333", "problem_id": "3055f9ef93c891b3", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A cyclist rides at 62 miles per hour for 3/2 hours, then at 80 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 62 × 3/2 = 93 miles.\n2) Second leg distance = 80 × 3 = 240 miles.\n3) Total distance = 93 + 240 = 333 miles.\nFinal answer: $333$", "final_answer": "333", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "696cda639985d4f4", "problem_id": "a83ceda577e0fae0", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A cyclist rides at 66 miles per hour for 3/2 hours, then at 90 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 66 × 3/2 = 99 miles.\nSecond leg distance = 90 × 2 = 180 miles.\nTotal distance = 99 + 180 = 279 miles.\nCheck: adding both legs gives 279.\nFinal answer: $279$", "final_answer": "279", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dabc3725bd6ddad4", "problem_id": "a83ceda577e0fae0", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A cyclist rides at 66 miles per hour for 3/2 hours, then at 90 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 66 × 3/2 = 99 miles.\n2) Second leg distance = 90 × 2 = 180 miles.\n3) Total distance = 99 + 180 = 279 miles.\nFinal answer: $279$", "final_answer": "279", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9bf15dbc8875912b", "problem_id": "d9817cea61b46ab0", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A store sells a backpack for $160. It is discounted by 15%. You buy 3 backpacks. What is the total cost?", "output": "Discounted price = 160 × (1 − 15/100) = 160 × 85/100 = 136.\nSubtotal = 3 × 136 = 408.\nFinal answer: $408$", "final_answer": "408", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "551f05ac4a0a1666", "problem_id": "d9817cea61b46ab0", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A store sells a backpack for $160. It is discounted by 15%. You buy 3 backpacks. What is the total cost?", "output": "1) Discounted price = 160 × (1 − 15/100) = 160 × 85/100 = 136.\n2) Subtotal = 3 × 136 = 408.\nFinal answer: $408$", "final_answer": "408", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "817ff0de2a1f088d", "problem_id": "605af96f1d9750ac", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{11} x + \\frac{2 x}{3} - \\frac{8}{9} + \\frac{2 \\sqrt{11}}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{2}{3} - \\sqrt{11}$, c=$- \\frac{8}{9} + \\frac{2 \\sqrt{11}}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $15 - 4 \\sqrt{11}$, so √Δ = $\\sqrt{15 - 4 \\sqrt{11}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{1}{3} + \\frac{\\sqrt{15 - 4 \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}$.\n7) x2 = $- \\frac{\\sqrt{15 - 4 \\sqrt{11}}}{2} - \\frac{1}{3} + \\frac{\\sqrt{11}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{1}{3} + \\frac{\\sqrt{15 - 4\\cdot \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}, - \\frac{\\sqrt{15 - 4\\cdot \\sqrt{11}}}{2} - \\frac{1}{3} + \\frac{\\sqrt{11}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{1}{3} + \\frac{\\sqrt{15 - 4\\cdot \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}, - \\frac{\\sqrt{15 - 4\\cdot \\sqrt{11}}}{2} - \\frac{1}{3} + \\frac{\\sqrt{11}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "11d389f31dbaeac2", "problem_id": "605af96f1d9750ac", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{11} x + \\frac{2 x}{3} - \\frac{8}{9} + \\frac{2 \\sqrt{11}}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{2}{3} - \\sqrt{11}$, c=$- \\frac{8}{9} + \\frac{2 \\sqrt{11}}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $15 - 4 \\sqrt{11}$, so √Δ = $\\sqrt{15 - 4 \\sqrt{11}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{1}{3} + \\frac{\\sqrt{15 - 4 \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}$.\nx2 = $- \\frac{\\sqrt{15 - 4 \\sqrt{11}}}{2} - \\frac{1}{3} + \\frac{\\sqrt{11}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{1}{3} + \\frac{\\sqrt{15 - 4\\cdot \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}, - \\frac{\\sqrt{15 - 4\\cdot \\sqrt{11}}}{2} - \\frac{1}{3} + \\frac{\\sqrt{11}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{1}{3} + \\frac{\\sqrt{15 - 4\\cdot \\sqrt{11}}}{2} + \\frac{\\sqrt{11}}{2}, - \\frac{\\sqrt{15 - 4\\cdot \\sqrt{11}}}{2} - \\frac{1}{3} + \\frac{\\sqrt{11}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a757962b86f83a53", "problem_id": "4ca1963d6057596f", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} x^{3} + x^{2} - x + 1$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to x^{3} + x^{2} - 2 x + 1.\n3) Then the limit is 1.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -1.\nFinal answer: $-1$", "final_answer": "-1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "09c11887e8c8f395", "problem_id": "7ec26898ccfa235f", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "An urn has 16 red balls and 20 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(36,11).\n3) Favorable samples: choose k reds and n−k blues: C(16,3)·C(20,8).\n4) Compute totals: C(36,11) = 600805296, and favorable = 70543200.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{950}{8091}$", "final_answer": "\\frac{950}{8091}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d3cdf3c31756c42c", "problem_id": "f63f7a44d91b8951", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 2 x^{2} + 3 x + \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 6}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 6}) = - \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 6}$.\n3) So $f'(x) = \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 24 x + 72} + \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 24 x + 72} - \\frac{4 x^{2}}{2 x^{2} - 24 x + 72} - \\frac{12 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 24 x + 72} - \\frac{24 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 24 x + 72} + \\frac{48 x}{2 x^{2} - 24 x + 72} + \\frac{36 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 24 x + 72} + \\frac{72 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 24 x + 72} - \\frac{36}{2 x^{2} - 24 x + 72} - \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 12 x + 36} - \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 6} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 6} + \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 6}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 6}) = - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 6} - \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 6} + \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x - 6\\right)^{2}}$.\nTherefore the correct derivative is $f'(x) = \\frac{4 x^{2} - 6 x + \\left(x - 6\\right) \\left(- 8 x + \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 2 \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x - 6\\right)^{2}}$.\nFinal answer: $\\frac{4 x^{2} - 6 x + \\left(x - 6\\right) \\left(- 8 x + \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 2 \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x - 6\\right)^{2}}$", "final_answer": "\\frac{4 x^{2} - 6 x + \\left(x - 6\\right) \\left(- 8 x + \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 2 \\left(x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x - 6\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "83de075dc5430645", "problem_id": "3143b25edd2dac50", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 7 x - 9$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$7$, c=$-9$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $121$, so √Δ = $11$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $1$.\n7) x2 = $- \\frac{9}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{1, - \\frac{9}{2}\\}$", "final_answer": "x \\in \\{1, - \\frac{9}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3a6d03a4507950c3", "problem_id": "8f0997d94f85b19f", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 26 red balls and 19 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(45,10).\nFavorable samples: choose k reds and n−k blues: C(26,3)·C(19,7).\nCompute totals: C(45,10) = 3190187286, and favorable = 131008800.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{88400}{2152623}$.\nFinal answer: $\\frac{88400}{2152623}$", "final_answer": "\\frac{88400}{2152623}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b1766e5c282766d4", "problem_id": "8f0997d94f85b19f", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "An urn has 26 red balls and 19 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(45,10).\n3) Favorable samples: choose k reds and n−k blues: C(26,3)·C(19,7).\n4) Compute totals: C(45,10) = 3190187286, and favorable = 131008800.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{88400}{2152623}$", "final_answer": "\\frac{88400}{2152623}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "233f4800c943720a", "problem_id": "a781cedaddc2e23a", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A store sells a backpack for $170. It is discounted by 15%. You buy 7 backpacks. What is the total cost?", "output": "1) Discounted price = 170 × (1 − 15/100) = 170 × 85/100 = 289/2.\n2) Subtotal = 7 × 289/2 = 2023/2.\nFinal answer: $\\frac{2023}{2}$", "final_answer": "\\frac{2023}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cb73f1aa80261d39", "problem_id": "b2ea521ab8090dcc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(2 x^{3} + x^{2} + 3 x - 3\\right) \\cos{\\left(2 x + 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x^{3} \\cos{\\left(2 x + 2 \\right)}) = - 2 x^{3} \\sin{\\left(2 x + 2 \\right)}$.\n3) So $f'(x) = - 2 x^{3} \\sin{\\left(2 x + 2 \\right)} - 2 x^{2} \\sin{\\left(2 x + 2 \\right)} + 2 x e^{x} - 6 x \\sin{\\left(2 x + 2 \\right)} + 2 x \\cos{\\left(2 x + 2 \\right)} - 4 e^{x} + 6 \\sin{\\left(2 x + 2 \\right)} + 3 \\cos{\\left(2 x + 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x^{3} \\cos{\\left(2 x + 2 \\right)}) = - 4 x^{3} \\sin{\\left(2 x + 2 \\right)} + 6 x^{2} \\cos{\\left(2 x + 2 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x - 3\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x + 1\\right) + 3\\right) - 3\\right) \\sin{\\left(2 x + 2 \\right)} + \\left(6 x^{2} + 2 x + 3\\right) \\cos{\\left(2 x + 2 \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(x - 3\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x + 1\\right) + 3\\right) - 3\\right) \\sin{\\left(2 x + 2 \\right)} + \\left(6 x^{2} + 2 x + 3\\right) \\cos{\\left(2 x + 2 \\right)} + 2 e^{x}$", "final_answer": "2 \\left(x - 3\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x + 1\\right) + 3\\right) - 3\\right) \\sin{\\left(2 x + 2 \\right)} + \\left(6 x^{2} + 2 x + 3\\right) \\cos{\\left(2 x + 2 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "fc95f69cf0778902", "problem_id": "d472b8dd2a954155", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡7 (mod 11), x≡12 (mod 13). Give the solution modulo 143.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 143.\n3) Compute the CRT solution to get x ≡ 51 (mod 143).\nFinal answer: $x\\equiv 51\\pmod{143}$", "final_answer": "x\\equiv 51\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "77c18198eab6f484", "problem_id": "d472b8dd2a954155", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡7 (mod 11), x≡12 (mod 13). Give the solution modulo 143.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 143.\nCompute the CRT solution to get x ≡ 51 (mod 143).\nCheck: 51 mod 11=7 and mod 13=12.\nFinal answer: $x\\equiv 51\\pmod{143}$", "final_answer": "x\\equiv 51\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c3cf7ed3e0c71a64", "problem_id": "741f4fb15e60eebd", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 20 times, what is the probability of exactly 7 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=20, k=7, p=$\\frac{1}{2}$ and simplify.\n4) Here C(20,7) = 77520.\nFinal answer: $\\frac{4845}{65536}$", "final_answer": "\\frac{4845}{65536}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7cd17f97c998a586", "problem_id": "741f4fb15e60eebd", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 20 times, what is the probability of exactly 7 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=20, k=7, p=$\\frac{1}{2}$ and simplify.\nFinal answer: $\\frac{4845}{65536}$", "final_answer": "\\frac{4845}{65536}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9558da6b24ab4482", "problem_id": "4781bfbd6a928e1d", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{19 x}{6} - \\sqrt{5} x + \\frac{5}{2} + \\frac{3 \\sqrt{5}}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{19}{6} - \\sqrt{5}$, c=$\\frac{5}{2} + \\frac{3 \\sqrt{5}}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{\\sqrt{5}}{3} + \\frac{181}{36}$, so √Δ = $\\sqrt{\\frac{\\sqrt{5}}{3} + \\frac{181}{36}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{5}}{2} + \\frac{\\sqrt{12 \\sqrt{5} + 181}}{12} + \\frac{19}{12}$.\n7) x2 = $- \\frac{\\sqrt{12 \\sqrt{5} + 181}}{12} + \\frac{\\sqrt{5}}{2} + \\frac{19}{12}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{5}}{2} + \\frac{\\sqrt{12\\cdot \\sqrt{5} + 181}}{12} + \\frac{19}{12}, - \\frac{\\sqrt{12\\cdot \\sqrt{5} + 181}}{12} + \\frac{\\sqrt{5}}{2} + \\frac{19}{12}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{5}}{2} + \\frac{\\sqrt{12\\cdot \\sqrt{5} + 181}}{12} + \\frac{19}{12}, - \\frac{\\sqrt{12\\cdot \\sqrt{5} + 181}}{12} + \\frac{\\sqrt{5}}{2} + \\frac{19}{12}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4b4855ad50aab994", "problem_id": "4781bfbd6a928e1d", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{19 x}{6} - \\sqrt{5} x + \\frac{5}{2} + \\frac{3 \\sqrt{5}}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{19}{6} - \\sqrt{5}$, c=$\\frac{5}{2} + \\frac{3 \\sqrt{5}}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{\\sqrt{5}}{2} + \\frac{\\sqrt{12\\cdot \\sqrt{5} + 181}}{12} + \\frac{19}{12}, - \\frac{\\sqrt{12\\cdot \\sqrt{5} + 181}}{12} + \\frac{\\sqrt{5}}{2} + \\frac{19}{12}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{5}}{2} + \\frac{\\sqrt{12\\cdot \\sqrt{5} + 181}}{12} + \\frac{19}{12}, - \\frac{\\sqrt{12\\cdot \\sqrt{5} + 181}}{12} + \\frac{\\sqrt{5}}{2} + \\frac{19}{12}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "27882d94e145fc90", "problem_id": "5927024d4b03cb07", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + \\frac{29 x}{10} + 1$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{29}{10}$, c=$1$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{441}{100}$, so √Δ = $\\frac{21}{10}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{2}{5}$.\n7) x2 = $- \\frac{5}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{2}{5}, - \\frac{5}{2}\\}$", "final_answer": "x \\in \\{- \\frac{2}{5}, - \\frac{5}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "475cc78c267d0d72", "problem_id": "9d4cd9c05c99030c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x^{3} + 4 x^{2} + 7 x + 13$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to x^{3} + 4 x^{2} + 6 x + 13.\n3) Then the limit is 49.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 51.\nFinal answer: $51$", "final_answer": "51", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "fc537e06e41f17be", "problem_id": "dad92afbe904f88a", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 10 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=8, p=$\\frac{1}{4}$ and simplify.\nHere C(10,8) = 45.\nCheck: expression simplifies to $\\frac{405}{1048576}$.\nFinal answer: $\\frac{405}{1048576}$", "final_answer": "\\frac{405}{1048576}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c8e544a01bcaa10e", "problem_id": "dad92afbe904f88a", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 10 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=8, p=$\\frac{1}{4}$ and simplify.\nFinal answer: $\\frac{405}{1048576}$", "final_answer": "\\frac{405}{1048576}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a4521bf328435264", "problem_id": "155ff48c4ad963d3", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$- 3 x$ = $15$\n$3 x - 4 y$ = $-3$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $3$: $- 9 x$ = $45$.\n3) Multiply the second equation by $-3$: $- 9 x + 12 y$ = $9$.\n4) Subtract to eliminate x: $- 12 y$ = $36$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, -3\\right)$", "final_answer": "(x,y)=\\left(-5, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cbd916a6e267642f", "problem_id": "155ff48c4ad963d3", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- 3 x$ = $15$\n$3 x - 4 y$ = $-3$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $- 9 x$ = $45$.\nMultiply the second equation by $-3$: $- 9 x + 12 y$ = $9$.\nSubtract to eliminate x: $- 12 y$ = $36$.\nSolve for y: y = $-3$.\nSubstitute back to find x: x = $-5$.\nCheck: plug (x,y)=(-5,-3) into both equations.\nFinal answer: $(x,y)=\\left(-5, -3\\right)$", "final_answer": "(x,y)=\\left(-5, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c1e062a73b0855e1", "problem_id": "baa0586f365f7413", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{5 x}{3} + \\frac{1}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$\\frac{5}{3}$, c=$\\frac{1}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{1}{9}$, so √Δ = $\\frac{1}{3}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{1}{3}$.\nx2 = $- \\frac{1}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{1}{3}, - \\frac{1}{2}\\}$", "final_answer": "x \\in \\{- \\frac{1}{3}, - \\frac{1}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d510c91e17de7d89", "problem_id": "baa0586f365f7413", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{5 x}{3} + \\frac{1}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$\\frac{5}{3}$, c=$\\frac{1}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{- \\frac{1}{3}, - \\frac{1}{2}\\}$", "final_answer": "x \\in \\{- \\frac{1}{3}, - \\frac{1}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c159f51c5d64d8ff", "problem_id": "43ee62114f310790", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A store sells a backpack for $163. It is discounted by 25%. You buy 5 backpacks. What is the total cost?", "output": "Discounted price = 163 × (1 − 25/100) = 163 × 75/100 = 489/4.\nSubtotal = 5 × 489/4 = 2445/4.\nCheck: recomputing the formula yields 2445/4.\nFinal answer: $\\frac{2445}{4}$", "final_answer": "\\frac{2445}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a4250154a8d7b4c0", "problem_id": "43ee62114f310790", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "A store sells a backpack for $163. It is discounted by 25%. You buy 5 backpacks. What is the total cost?", "output": "Discounted price = 163 × (1 − 25/100) = 163 × 75/100 = 489/4.\nSubtotal = 5 × 489/4 = 2445/4.\nFinal answer: $\\frac{2445}{4}$", "final_answer": "\\frac{2445}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7d798582e4cb6a06", "problem_id": "c9726155d58e10e3", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 22 times, what is the probability of exactly 11 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=22, k=11, p=$\\frac{1}{5}$ and simplify.\n4) Here C(22,11) = 705432.\nFinal answer: $\\frac{2958796259328}{2384185791015625}$", "final_answer": "\\frac{2958796259328}{2384185791015625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ba729ab231d91bca", "problem_id": "c9726155d58e10e3", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 22 times, what is the probability of exactly 11 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=22, k=11, p=$\\frac{1}{5}$ and simplify.\nHere C(22,11) = 705432.\nCheck: expression simplifies to $\\frac{2958796259328}{2384185791015625}$.\nFinal answer: $\\frac{2958796259328}{2384185791015625}$", "final_answer": "\\frac{2958796259328}{2384185791015625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "302f5f5d9c782921", "problem_id": "8a7dcb5c2f7eb76c", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "You want 29 liters of a 37% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 29−x liters is 50% solution.\n2) Salt equation: 25x + 50(29−x) = 37·29.\n3) (25-50)x = (37-50)·29 ⇒ x = 377/25.\n4) Then 29−x = 348/25.\nFinal answer: $377/25 L of 25% and 348/25 L of 50%$", "final_answer": "377/25 L of 25% and 348/25 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8559ba6e485631b3", "problem_id": "fb66368f2415621d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(- 3 x^{3} - x + 1\\right) \\cos{\\left(- x^{2} + x + 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\cos{\\left(\\left(- x^{2} + x\\right) + 2 \\right)}) = - \\sin{\\left(- x^{2} + x + 2 \\right)}$.\n3) So $f'(x) = - 6 x^{4} \\sin{\\left(- x^{2} + x + 2 \\right)} + 3 x^{3} \\sin{\\left(- x^{2} + x + 2 \\right)} - 2 x^{2} \\sin{\\left(- x^{2} + x + 2 \\right)} - 9 x^{2} \\cos{\\left(\\left(- x^{2} + x\\right) + 2 \\right)} + 2 x e^{x} + x \\sin{\\left(- x^{2} + x + 2 \\right)} + 6 e^{x} - \\sin{\\left(- x^{2} + x + 2 \\right)} - \\cos{\\left(\\left(- x^{2} + x\\right) + 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\cos{\\left(\\left(- x^{2} + x\\right) + 2 \\right)}) = - \\left(1 - 2 x\\right) \\sin{\\left(- x^{2} + x + 2 \\right)}$.\nTherefore the correct derivative is $f'(x) = - \\left(1 - 2 x\\right) \\left(x \\left(- 3 x^{2} - 1\\right) + 1\\right) \\sin{\\left(- x^{2} + x + 2 \\right)} + 2 \\left(x + 2\\right) e^{x} + \\left(- 9 x^{2} - 1\\right) \\cos{\\left(\\left(- x^{2} + x\\right) + 2 \\right)} + 2 e^{x}$.\nFinal answer: $- \\left(1 - 2 x\\right) \\left(x \\left(- 3 x^{2} - 1\\right) + 1\\right) \\sin{\\left(- x^{2} + x + 2 \\right)} + 2 \\left(x + 2\\right) e^{x} + \\left(- 9 x^{2} - 1\\right) \\cos{\\left(\\left(- x^{2} + x\\right) + 2 \\right)} + 2 e^{x}$", "final_answer": "- \\left(1 - 2 x\\right) \\left(x \\left(- 3 x^{2} - 1\\right) + 1\\right) \\sin{\\left(- x^{2} + x + 2 \\right)} + 2 \\left(x + 2\\right) e^{x} + \\left(- 9 x^{2} - 1\\right) \\cos{\\left(\\left(- x^{2} + x\\right) + 2 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f235f07b372aeab9", "problem_id": "43f395ffbeeca9e8", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$- 6 y$ = $0$\n$- x + 8 y$ = $-3$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $6 y$ = $0$.\n3) Multiply the second equation by $0$: $0$ = $0$.\n4) Subtract to eliminate x: $6 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, 0\\right)$", "final_answer": "(x,y)=\\left(3, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "51d06bf053dbb8ec", "problem_id": "5196868997c8b2a9", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{3} - x^{2} - 4$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 3 x^{3}) = - 9 x^{3}$.\n3) So $f'(x) = - 9 x^{3} - 2 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 3 x^{3}) = - 9 x^{2}$.\nTherefore the correct derivative is $f'(x) = x \\left(- 9 x - 2\\right)$.\nFinal answer: $x \\left(- 9 x - 2\\right)$", "final_answer": "x \\left(- 9 x - 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "72bc72440bb51b6d", "problem_id": "13e34a26e202e462", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "An urn has 15 red balls and 16 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 6 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(31,12).\n3) Favorable samples: choose k reds and n−k blues: C(15,6)·C(16,6).\n4) Compute totals: C(31,12) = 141120525, and favorable = 40080040.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{88088}{310155}$", "final_answer": "\\frac{88088}{310155}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3cb2f7bd508710a3", "problem_id": "ee8ca69cc5ede5ab", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} x + 1$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 1.\n3) Then the limit is 1.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -1.\nFinal answer: $-1$", "final_answer": "-1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "adf2edb3af989b73", "problem_id": "c715212560b3d3d1", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{5} x}{2} + 2 x - \\frac{21 \\sqrt{5}}{4} - \\frac{21}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$2 - \\frac{3 \\sqrt{5}}{2}$, c=$- \\frac{21 \\sqrt{5}}{4} - \\frac{21}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $15 \\sqrt{5} + \\frac{145}{4}$, so √Δ = $\\sqrt{15 \\sqrt{5} + \\frac{145}{4}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $-1 + \\frac{3 \\sqrt{5}}{4} + \\frac{\\sqrt{60 \\sqrt{5} + 145}}{4}$.\n7) x2 = $- \\frac{\\sqrt{60 \\sqrt{5} + 145}}{4} - 1 + \\frac{3 \\sqrt{5}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{-1 + \\frac{3\\cdot \\sqrt{5}}{4} + \\frac{\\sqrt{60\\cdot \\sqrt{5} + 145}}{4}, - \\frac{\\sqrt{60\\cdot \\sqrt{5} + 145}}{4} - 1 + \\frac{3\\cdot \\sqrt{5}}{4}\\}$", "final_answer": "x \\in \\{-1 + \\frac{3\\cdot \\sqrt{5}}{4} + \\frac{\\sqrt{60\\cdot \\sqrt{5} + 145}}{4}, - \\frac{\\sqrt{60\\cdot \\sqrt{5} + 145}}{4} - 1 + \\frac{3\\cdot \\sqrt{5}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "321b402293a3ce43", "problem_id": "c715212560b3d3d1", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{5} x}{2} + 2 x - \\frac{21 \\sqrt{5}}{4} - \\frac{21}{4}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$2 - \\frac{3 \\sqrt{5}}{2}$, c=$- \\frac{21 \\sqrt{5}}{4} - \\frac{21}{4}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $15 \\sqrt{5} + \\frac{145}{4}$, so √Δ = $\\sqrt{15 \\sqrt{5} + \\frac{145}{4}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $-1 + \\frac{3 \\sqrt{5}}{4} + \\frac{\\sqrt{60 \\sqrt{5} + 145}}{4}$.\nx2 = $- \\frac{\\sqrt{60 \\sqrt{5} + 145}}{4} - 1 + \\frac{3 \\sqrt{5}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-1 + \\frac{3\\cdot \\sqrt{5}}{4} + \\frac{\\sqrt{60\\cdot \\sqrt{5} + 145}}{4}, - \\frac{\\sqrt{60\\cdot \\sqrt{5} + 145}}{4} - 1 + \\frac{3\\cdot \\sqrt{5}}{4}\\}$", "final_answer": "x \\in \\{-1 + \\frac{3\\cdot \\sqrt{5}}{4} + \\frac{\\sqrt{60\\cdot \\sqrt{5} + 145}}{4}, - \\frac{\\sqrt{60\\cdot \\sqrt{5} + 145}}{4} - 1 + \\frac{3\\cdot \\sqrt{5}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e9c75438c3f7ab88", "problem_id": "59a370a90a25c1f1", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} + 2 x^{2} - 4 x + 3$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $9 x^{2} + 4 x - 4$.\n2) Slope at x=-2: m = f'(-2) = $24$.\n3) Point on curve: (-2, f(-2)) = (-2, $-5$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -5 = 24(x − -2).\n6) Optionally expand: y = $24 x + 43$.\nFinal answer: $y = 24\\cdot x + 43$", "final_answer": "y = 24\\cdot x + 43", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d23bcdcfc11a54eb", "problem_id": "59a370a90a25c1f1", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} + 2 x^{2} - 4 x + 3$ at $x=-2$.", "output": "Differentiate: f'(x) = $9 x^{2} + 4 x - 4$.\nSlope at x=-2: m = f'(-2) = $24$.\nPoint on curve: (-2, f(-2)) = (-2, $-5$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -5 = 24(x − -2).\nOptionally expand: y = $24 x + 43$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 24\\cdot x + 43$", "final_answer": "y = 24\\cdot x + 43", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "08a0e72b2b739bd4", "problem_id": "142b2b622291e22d", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$4 x - 6 y$ = $-46$\n$3 x + 7 y$ = $0$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $3$: $12 x - 18 y$ = $-138$.\n3) Multiply the second equation by $4$: $12 x + 28 y$ = $0$.\n4) Subtract to eliminate x: $- 46 y$ = $-138$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $-7$.\nFinal answer: $(x,y)=\\left(-7, 3\\right)$", "final_answer": "(x,y)=\\left(-7, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ac8823dc1e494ed7", "problem_id": "142b2b622291e22d", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the linear system:\n$4 x - 6 y$ = $-46$\n$3 x + 7 y$ = $0$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $12 x - 18 y$ = $-138$.\nMultiply the second equation by $4$: $12 x + 28 y$ = $0$.\nFinal answer: $(x,y)=\\left(-7, 3\\right)$", "final_answer": "(x,y)=\\left(-7, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7bc1ef9fb3e7568f", "problem_id": "deeac36f3e76b040", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A cyclist rides at 76 miles per hour for 3/2 hours, then at 64 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 76 × 3/2 = 114 miles.\n2) Second leg distance = 64 × 2 = 128 miles.\n3) Total distance = 114 + 128 = 242 miles.\nFinal answer: $242$", "final_answer": "242", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a02535c7e1529e2", "problem_id": "deeac36f3e76b040", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 76 miles per hour for 3/2 hours, then at 64 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 76 × 3/2 = 114 miles.\nSecond leg distance = 64 × 2 = 128 miles.\nTotal distance = 114 + 128 = 242 miles.\nCheck: adding both legs gives 242.\nFinal answer: $242$", "final_answer": "242", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0857436a64f8926b", "problem_id": "7d441d4fa9f7e32e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 2 x^{3} + 2 x^{2} + x - 1$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{3} + 2 x^{2} - 1.\n3) Then the limit is 3.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: 2.\nFinal answer: $2$", "final_answer": "2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "45fd95f288a521dd", "problem_id": "639620f801975644", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many binary strings of length 10 contain exactly 3 ones with no two ones adjacent?", "output": "Reserve 2 mandatory zeros between the 3 ones.\nThis leaves n−k+1 = 8 valid slots for the ones.\nChoose the slots: C(8,3) = 56.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $56$", "final_answer": "56", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2cd2829bfe820d49", "problem_id": "92ba9b564be0f9ee", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 45 x + 162$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} + 45 x + 162$ = $3 \\left(x + 6\\right) \\left(x + 9\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $-6$.\nFrom the second factor: x = $-9$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-6, -9\\}$", "final_answer": "x \\in \\{-6, -9\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b6a86539a410dfbc", "problem_id": "92ba9b564be0f9ee", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 45 x + 162$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} + 45 x + 162$ = $3 \\left(x + 6\\right) \\left(x + 9\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-6$.\n5) From the second factor: x = $-9$.\nFinal answer: $x \\in \\{-6, -9\\}$", "final_answer": "x \\in \\{-6, -9\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "70ba9a22b621ee74", "problem_id": "53e2d97f7f5d58dd", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{4} + x^{3} - 2 x^{2} + 4 x + 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x^{3}) = 3 x^{3}$.\n3) So $f'(x) = 15 x^{3} - 4 x + 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x^{3}) = 3 x^{2}$.\nTherefore the correct derivative is $f'(x) = 12 x^{3} + 3 x^{2} - 4 x + 4$.\nFinal answer: $12 x^{3} + 3 x^{2} - 4 x + 4$", "final_answer": "12 x^{3} + 3 x^{2} - 4 x + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "252b08df36a07cf9", "problem_id": "784e0ea0fdf61229", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Find the multiplicative inverse of 16 modulo 29.", "output": "We want x such that 16x ≡ 1 (mod 29).\nUse extended Euclid to find s,t with 16s + 29t = 1.\nOne Bezout identity is: 16·(-9) + 29·(5) = 1.\nFinal answer: $x\\equiv 20\\pmod{29}$", "final_answer": "x\\equiv 20\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f2e75e7a99ef2e0a", "problem_id": "f78eca9e51fae563", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{5} x + \\frac{24 x}{5} - \\frac{36}{5} + \\frac{18 \\sqrt{5}}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{24}{5} - 3 \\sqrt{5}$, c=$- \\frac{36}{5} + \\frac{18 \\sqrt{5}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{2421}{25} - \\frac{216 \\sqrt{5}}{5}$, so √Δ = $\\sqrt{\\frac{2421}{25} - \\frac{216 \\sqrt{5}}{5}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{12}{5} + \\frac{3 \\sqrt{269 - 120 \\sqrt{5}}}{10} + \\frac{3 \\sqrt{5}}{2}$.\n7) x2 = $- \\frac{12}{5} - \\frac{3 \\sqrt{269 - 120 \\sqrt{5}}}{10} + \\frac{3 \\sqrt{5}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{12}{5} + \\frac{3\\cdot \\sqrt{269 - 120\\cdot \\sqrt{5}}}{10} + \\frac{3\\cdot \\sqrt{5}}{2}, - \\frac{12}{5} - \\frac{3\\cdot \\sqrt{269 - 120\\cdot \\sqrt{5}}}{10} + \\frac{3\\cdot \\sqrt{5}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{12}{5} + \\frac{3\\cdot \\sqrt{269 - 120\\cdot \\sqrt{5}}}{10} + \\frac{3\\cdot \\sqrt{5}}{2}, - \\frac{12}{5} - \\frac{3\\cdot \\sqrt{269 - 120\\cdot \\sqrt{5}}}{10} + \\frac{3\\cdot \\sqrt{5}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e889628de1fc0f7a", "problem_id": "f78eca9e51fae563", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{5} x + \\frac{24 x}{5} - \\frac{36}{5} + \\frac{18 \\sqrt{5}}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{24}{5} - 3 \\sqrt{5}$, c=$- \\frac{36}{5} + \\frac{18 \\sqrt{5}}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{2421}{25} - \\frac{216 \\sqrt{5}}{5}$, so √Δ = $\\sqrt{\\frac{2421}{25} - \\frac{216 \\sqrt{5}}{5}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{12}{5} + \\frac{3 \\sqrt{269 - 120 \\sqrt{5}}}{10} + \\frac{3 \\sqrt{5}}{2}$.\nx2 = $- \\frac{12}{5} - \\frac{3 \\sqrt{269 - 120 \\sqrt{5}}}{10} + \\frac{3 \\sqrt{5}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{12}{5} + \\frac{3\\cdot \\sqrt{269 - 120\\cdot \\sqrt{5}}}{10} + \\frac{3\\cdot \\sqrt{5}}{2}, - \\frac{12}{5} - \\frac{3\\cdot \\sqrt{269 - 120\\cdot \\sqrt{5}}}{10} + \\frac{3\\cdot \\sqrt{5}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{12}{5} + \\frac{3\\cdot \\sqrt{269 - 120\\cdot \\sqrt{5}}}{10} + \\frac{3\\cdot \\sqrt{5}}{2}, - \\frac{12}{5} - \\frac{3\\cdot \\sqrt{269 - 120\\cdot \\sqrt{5}}}{10} + \\frac{3\\cdot \\sqrt{5}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "83fea311948b5369", "problem_id": "ab4d06f163ecb35d", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int x^{2} - 2 x - 4\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $x^{2} - 2 x - 4$.\n3) Compute an antiderivative: F(x) = $\\frac{x^{3}}{3} - x^{2} - 4 x$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{x^{3}}{3} - x^{2} - 4\\cdot x$", "final_answer": "C + \\frac{x^{3}}{3} - x^{2} - 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a6dc968ae8e51b1f", "problem_id": "e3905de450773855", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} - x - 4$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to - 2 x - 4.\n3) Then the limit is -6.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: -5.\nFinal answer: $-5$", "final_answer": "-5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "382069027e79acdf", "problem_id": "5ce64a88a47e0d79", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 16 red balls and 8 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(24,5).\n3) Favorable samples: choose k reds and n−k blues: C(16,1)·C(8,4).\n4) Compute totals: C(24,5) = 42504, and favorable = 1120.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{20}{759}$", "final_answer": "\\frac{20}{759}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a8296377bbed1ecc", "problem_id": "683fef02f3e46d41", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 4 x^{3} - 2 x^{2} - 4 x + 4$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x^{2}) = - 4 x^{2}$.\n3) So $f'(x) = - 16 x^{2} - 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x^{2}) = - 4 x$.\nTherefore the correct derivative is $f'(x) = 4 \\left(- 3 x^{2} - x - 1\\right)$.\nFinal answer: $4 \\left(- 3 x^{2} - x - 1\\right)$", "final_answer": "4 \\left(- 3 x^{2} - x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ef199a6151fd020a", "problem_id": "2d9bdfb89c843dc3", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} 3 x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\n2) Evaluate at the bounds: F(\\pi) and F(0).\n3) Subtract: F(\\pi) − F(0) = $4 + 2 \\pi + \\pi^{3}$.\n4) Simplify the result.\nFinal answer: $4 + 2\\cdot \\pi + \\pi^{3}$", "final_answer": "4 + 2\\cdot \\pi + \\pi^{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f9e8a02aba8c08ad", "problem_id": "2d9bdfb89c843dc3", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} 3 x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\nEvaluate at the bounds: F(\\pi) and F(0).\nSubtract: F(\\pi) − F(0) = $4 + 2 \\pi + \\pi^{3}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $4 + 2\\cdot \\pi + \\pi^{3}$", "final_answer": "4 + 2\\cdot \\pi + \\pi^{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "50bcd4a021950a4c", "problem_id": "acd89fd59be73df7", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=23$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=23 and k=6.\n4) Compute C(28,5) = 98280.\nFinal answer: $98280$", "final_answer": "98280", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3b08038e1eccb73d", "problem_id": "2f460b9885f27eca", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x^{2} - 4 x - 5$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} - 5 x - 5.\n3) Then the limit is -19.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -17.\nFinal answer: $-17$", "final_answer": "-17", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8ea55ae59408766e", "problem_id": "1ea224a21e5049d7", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} 2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $- \\pi + \\frac{\\pi^{3}}{6}$.\n4) Simplify the result.\nFinal answer: $- \\pi + \\frac{\\pi^{3}}{6}$", "final_answer": "- \\pi + \\frac{\\pi^{3}}{6}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8cd953abf093530d", "problem_id": "1ea224a21e5049d7", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} 2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $- \\pi + \\frac{\\pi^{3}}{6}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\pi + \\frac{\\pi^{3}}{6}$", "final_answer": "- \\pi + \\frac{\\pi^{3}}{6}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d04368f26bc55d4b", "problem_id": "24400c0b627b367f", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A store sells a backpack for $70. It is discounted by 5%. You buy 13 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 70 × (1 − 5/100) = 70 × 95/100 = 133/2.\n2) Subtotal = 13 × 133/2 = 1729/2.\n3) Total with tax = 1729/2 × 105/100 = 36309/40.\nFinal answer: $\\frac{36309}{40}$", "final_answer": "\\frac{36309}{40}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "79c3794b1e3b835a", "problem_id": "4d555e3f0bec517a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 2 x^{2} - 2 x - 1$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{2} - 3 x - 1.\n3) Then the limit is 4.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: 3.\nFinal answer: $3$", "final_answer": "3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b0f674f36c2ad753", "problem_id": "7d8f4768c706d81a", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 30 red balls and 10 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(40,12).\n3) Favorable samples: choose k reds and n−k blues: C(30,2)·C(10,10).\n4) Compute totals: C(40,12) = 5586853480, and favorable = 435.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{3}{38530024}$", "final_answer": "\\frac{3}{38530024}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4248795a0a0eb581", "problem_id": "7d8f4768c706d81a", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 30 red balls and 10 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(40,12).\nFavorable samples: choose k reds and n−k blues: C(30,2)·C(10,10).\nCompute totals: C(40,12) = 5586853480, and favorable = 435.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{3}{38530024}$.\nFinal answer: $\\frac{3}{38530024}$", "final_answer": "\\frac{3}{38530024}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "59e44b6a68bbbcd5", "problem_id": "18bdb9ec0b12e2f7", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A store sells a backpack for $330. It is discounted by 40%. You buy 4 backpacks. What is the total cost?", "output": "1) Discounted price = 330 × (1 − 40/100) = 330 × 60/100 = 198.\n2) Subtotal = 4 × 198 = 792.\nFinal answer: $792$", "final_answer": "792", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fe2bf11434ef754a", "problem_id": "18bdb9ec0b12e2f7", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A store sells a backpack for $330. It is discounted by 40%. You buy 4 backpacks. What is the total cost?", "output": "Discounted price = 330 × (1 − 40/100) = 330 × 60/100 = 198.\nSubtotal = 4 × 198 = 792.\nCheck: recomputing the formula yields 792.\nFinal answer: $792$", "final_answer": "792", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d2809d225ff555f9", "problem_id": "a686660a43b7a82d", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int x^{4} - 4 x^{3} + 3 x - 1\\,dx$.", "output": "$C + \\frac{x^{5}}{5} - x^{4} + \\frac{3\\cdot x^{2}}{2} - x$", "final_answer": "C + \\frac{x^{5}}{5} - x^{4} + \\frac{3\\cdot x^{2}}{2} - x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f42b38329e23e697", "problem_id": "a686660a43b7a82d", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int x^{4} - 4 x^{3} + 3 x - 1\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $x^{4} - 4 x^{3} + 3 x - 1$.\nCompute an antiderivative: F(x) = $\\frac{x^{5}}{5} - x^{4} + \\frac{3 x^{2}}{2} - x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{x^{5}}{5} - x^{4} + \\frac{3\\cdot x^{2}}{2} - x$", "final_answer": "C + \\frac{x^{5}}{5} - x^{4} + \\frac{3\\cdot x^{2}}{2} - x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "616bb7204ac617f7", "problem_id": "8dafeffed0e67c36", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} 2 x - 9$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $2 x^{2} - x + 3$ and a = -4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $4 x - 1$.\n5) Evaluate at a: p'(-4) = $-17$.\nFinal answer: $-17$", "final_answer": "-17", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8c531306f660cffb", "problem_id": "8dafeffed0e67c36", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} 2 x - 9$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{2} - x + 3$ and a = -4.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-17$", "final_answer": "-17", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "603651446d619437", "problem_id": "a4a589186b453cec", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{98}{9}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$0$, c=$- \\frac{98}{9}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{784}{9}$, so √Δ = $\\frac{28}{3}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{7}{3}$.\n7) x2 = $- \\frac{7}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{7}{3}, - \\frac{7}{3}\\}$", "final_answer": "x \\in \\{\\frac{7}{3}, - \\frac{7}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "82b1437400328d2b", "problem_id": "a4a589186b453cec", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{98}{9}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$0$, c=$- \\frac{98}{9}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{784}{9}$, so √Δ = $\\frac{28}{3}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{7}{3}$.\nx2 = $- \\frac{7}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{7}{3}, - \\frac{7}{3}\\}$", "final_answer": "x \\in \\{\\frac{7}{3}, - \\frac{7}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f3651bb76f17bc4a", "problem_id": "7de519621d8574c2", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Find the multiplicative inverse of 39 modulo 41.", "output": "1) We want x such that 39x ≡ 1 (mod 41).\n2) Use extended Euclid to find s,t with 39s + 41t = 1.\n3) One Bezout identity is: 39·(20) + 41·(-19) = 1.\n4) Reduce s modulo 41: x ≡ 20 (mod 41).\nFinal answer: $x\\equiv 20\\pmod{41}$", "final_answer": "x\\equiv 20\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7e19c892373cdba7", "problem_id": "af7b0f60f2a2007c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{3} - 3 x^{2} + x - 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x) = x$.\n3) So $f'(x) = 12 x^{2} - 5 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x) = 1$.\nTherefore the correct derivative is $f'(x) = 12 x^{2} - 6 x + 1$.\nFinal answer: $12 x^{2} - 6 x + 1$", "final_answer": "12 x^{2} - 6 x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c0100df27d837bca", "problem_id": "426b58a28bca5c5e", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{11} x}{2} - \\frac{5 x}{4} - \\frac{3}{2} - \\frac{3 \\sqrt{11}}{8}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{\\sqrt{11}}{2} - \\frac{5}{4}$, c=$- \\frac{3}{2} - \\frac{3 \\sqrt{11}}{8}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{11 \\sqrt{11}}{4} + \\frac{165}{16}$, so √Δ = $\\sqrt{\\frac{11 \\sqrt{11}}{4} + \\frac{165}{16}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{5}{8} + \\frac{\\sqrt{11}}{4} + \\frac{\\sqrt{44 \\sqrt{11} + 165}}{8}$.\n7) x2 = $- \\frac{\\sqrt{44 \\sqrt{11} + 165}}{8} + \\frac{5}{8} + \\frac{\\sqrt{11}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{5}{8} + \\frac{\\sqrt{11}}{4} + \\frac{\\sqrt{44\\cdot \\sqrt{11} + 165}}{8}, - \\frac{\\sqrt{44\\cdot \\sqrt{11} + 165}}{8} + \\frac{5}{8} + \\frac{\\sqrt{11}}{4}\\}$", "final_answer": "x \\in \\{\\frac{5}{8} + \\frac{\\sqrt{11}}{4} + \\frac{\\sqrt{44\\cdot \\sqrt{11} + 165}}{8}, - \\frac{\\sqrt{44\\cdot \\sqrt{11} + 165}}{8} + \\frac{5}{8} + \\frac{\\sqrt{11}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "90ad76e67fa70ea2", "problem_id": "426b58a28bca5c5e", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{11} x}{2} - \\frac{5 x}{4} - \\frac{3}{2} - \\frac{3 \\sqrt{11}}{8}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{\\sqrt{11}}{2} - \\frac{5}{4}$, c=$- \\frac{3}{2} - \\frac{3 \\sqrt{11}}{8}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{11 \\sqrt{11}}{4} + \\frac{165}{16}$, so √Δ = $\\sqrt{\\frac{11 \\sqrt{11}}{4} + \\frac{165}{16}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{5}{8} + \\frac{\\sqrt{11}}{4} + \\frac{\\sqrt{44 \\sqrt{11} + 165}}{8}$.\nx2 = $- \\frac{\\sqrt{44 \\sqrt{11} + 165}}{8} + \\frac{5}{8} + \\frac{\\sqrt{11}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{5}{8} + \\frac{\\sqrt{11}}{4} + \\frac{\\sqrt{44\\cdot \\sqrt{11} + 165}}{8}, - \\frac{\\sqrt{44\\cdot \\sqrt{11} + 165}}{8} + \\frac{5}{8} + \\frac{\\sqrt{11}}{4}\\}$", "final_answer": "x \\in \\{\\frac{5}{8} + \\frac{\\sqrt{11}}{4} + \\frac{\\sqrt{44\\cdot \\sqrt{11} + 165}}{8}, - \\frac{\\sqrt{44\\cdot \\sqrt{11} + 165}}{8} + \\frac{5}{8} + \\frac{\\sqrt{11}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d71de60040f4f567", "problem_id": "cb10ca8c0e2e1fb1", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{11} x - x - 6 + 9 \\sqrt{11}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- 3 \\sqrt{11} - 1$, c=$-6 + 9 \\sqrt{11}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $124 - 30 \\sqrt{11}$, so √Δ = $\\sqrt{124 - 30 \\sqrt{11}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{1}{2} + \\frac{\\sqrt{124 - 30 \\sqrt{11}}}{2} + \\frac{3 \\sqrt{11}}{2}$.\nx2 = $- \\frac{\\sqrt{124 - 30 \\sqrt{11}}}{2} + \\frac{1}{2} + \\frac{3 \\sqrt{11}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{1}{2} + \\frac{\\sqrt{124 - 30\\cdot \\sqrt{11}}}{2} + \\frac{3\\cdot \\sqrt{11}}{2}, - \\frac{\\sqrt{124 - 30\\cdot \\sqrt{11}}}{2} + \\frac{1}{2} + \\frac{3\\cdot \\sqrt{11}}{2}\\}$", "final_answer": "x \\in \\{\\frac{1}{2} + \\frac{\\sqrt{124 - 30\\cdot \\sqrt{11}}}{2} + \\frac{3\\cdot \\sqrt{11}}{2}, - \\frac{\\sqrt{124 - 30\\cdot \\sqrt{11}}}{2} + \\frac{1}{2} + \\frac{3\\cdot \\sqrt{11}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f39da047309afec8", "problem_id": "cb10ca8c0e2e1fb1", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{11} x - x - 6 + 9 \\sqrt{11}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- 3 \\sqrt{11} - 1$, c=$-6 + 9 \\sqrt{11}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $124 - 30 \\sqrt{11}$, so √Δ = $\\sqrt{124 - 30 \\sqrt{11}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{1}{2} + \\frac{\\sqrt{124 - 30 \\sqrt{11}}}{2} + \\frac{3 \\sqrt{11}}{2}$.\n7) x2 = $- \\frac{\\sqrt{124 - 30 \\sqrt{11}}}{2} + \\frac{1}{2} + \\frac{3 \\sqrt{11}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{1}{2} + \\frac{\\sqrt{124 - 30\\cdot \\sqrt{11}}}{2} + \\frac{3\\cdot \\sqrt{11}}{2}, - \\frac{\\sqrt{124 - 30\\cdot \\sqrt{11}}}{2} + \\frac{1}{2} + \\frac{3\\cdot \\sqrt{11}}{2}\\}$", "final_answer": "x \\in \\{\\frac{1}{2} + \\frac{\\sqrt{124 - 30\\cdot \\sqrt{11}}}{2} + \\frac{3\\cdot \\sqrt{11}}{2}, - \\frac{\\sqrt{124 - 30\\cdot \\sqrt{11}}}{2} + \\frac{1}{2} + \\frac{3\\cdot \\sqrt{11}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a09db453567a2e0e", "problem_id": "1346f397ed755d42", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A cyclist rides at 20 miles per hour for 2 hours, then at 81 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 20 × 2 = 40 miles.\n2) Second leg distance = 81 × 2 = 162 miles.\n3) Total distance = 40 + 162 = 202 miles.\nFinal answer: $202$", "final_answer": "202", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0c2aec2dc7fd4286", "problem_id": "f94bb98cea397d4b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{3} \\sin{\\left(x \\right)} + x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x \\sin{\\left(x \\right)}) = x \\cos{\\left(x \\right)}$.\n3) So $f'(x) = 3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x \\sin{\\left(x \\right)}) = x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$", "final_answer": "3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8a7f66fe396c63ae", "problem_id": "03c68353edf9245f", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 28 red balls and 25 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(53,11).\nFavorable samples: choose k reds and n−k blues: C(28,2)·C(25,9).\nCompute totals: C(53,11) = 76223753060, and favorable = 772244550.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{28215}{2784938}$.\nFinal answer: $\\frac{28215}{2784938}$", "final_answer": "\\frac{28215}{2784938}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "08b27a0bd681197c", "problem_id": "03c68353edf9245f", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 28 red balls and 25 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(53,11).\n3) Favorable samples: choose k reds and n−k blues: C(28,2)·C(25,9).\n4) Compute totals: C(53,11) = 76223753060, and favorable = 772244550.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{28215}{2784938}$", "final_answer": "\\frac{28215}{2784938}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c46b6485ebb603a9", "problem_id": "51e61aed591dc615", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A cyclist rides at 43 miles per hour for 2 hours, then at 22 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 43 × 2 = 86 miles.\nSecond leg distance = 22 × 3/2 = 33 miles.\nTotal distance = 86 + 33 = 119 miles.\nCheck: adding both legs gives 119.\nFinal answer: $119$", "final_answer": "119", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a57dcab703205bb", "problem_id": "51e61aed591dc615", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A cyclist rides at 43 miles per hour for 2 hours, then at 22 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 43 × 2 = 86 miles.\n2) Second leg distance = 22 × 3/2 = 33 miles.\n3) Total distance = 86 + 33 = 119 miles.\nFinal answer: $119$", "final_answer": "119", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a2ef8609fb3bdda6", "problem_id": "3f3d9473e75655f1", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Two friends split $150 in the ratio 1:1. How much does each receive?", "output": "Total ratio parts = 1+1 = 2.\nEach part = 150 ÷ 2 = 75.\nFirst share = 1 × 75 = 75.\nFinal answer: $75$", "final_answer": "75", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9201fde67402ff07", "problem_id": "3f3d9473e75655f1", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Two friends split $150 in the ratio 1:1. How much does each receive?", "output": "1) Total ratio parts = 1+1 = 2.\n2) Each part = 150 ÷ 2 = 75.\n3) First share = 1 × 75 = 75.\n4) Second share = 1 × 75 = 75.\nFinal answer: $75$", "final_answer": "75", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "86691de6ae7e2d50", "problem_id": "ba1b08180bbc1187", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Worker A finishes a job in 13 hours and Worker B finishes it in 22 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/13 job/hour. B's rate = 1/22 job/hour.\nCombined rate = 1/13 + 1/22 = 35/286 job/hour.\nTime = 1 ÷ (combined rate) = 286/35 hours.\nCheck: (combined rate)×(time) = 35/286×286/35 = 1.\nFinal answer: $\\frac{286}{35}$", "final_answer": "\\frac{286}{35}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ee27352f2f25fdb6", "problem_id": "ba1b08180bbc1187", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Worker A finishes a job in 13 hours and Worker B finishes it in 22 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/13 job/hour. B's rate = 1/22 job/hour.\nCombined rate = 1/13 + 1/22 = 35/286 job/hour.\nTime = 1 ÷ (combined rate) = 286/35 hours.\nFinal answer: $\\frac{286}{35}$", "final_answer": "\\frac{286}{35}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "827d739b36f2cf17", "problem_id": "f9dadd44ea2bd1dd", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 24 x + 54$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 24 x + 54$ = $2 \\left(x - 9\\right) \\left(x - 3\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $9$.\n5) From the second factor: x = $3$.\nFinal answer: $x \\in \\{9, 3\\}$", "final_answer": "x \\in \\{9, 3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "859e5185e085d13d", "problem_id": "f9dadd44ea2bd1dd", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 24 x + 54$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} - 24 x + 54$ = $2 \\left(x - 9\\right) \\left(x - 3\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFinal answer: $x \\in \\{9, 3\\}$", "final_answer": "x \\in \\{9, 3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "493b2fb68dd9bc97", "problem_id": "9776533017892f15", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "You want 62 liters of a 45% salt solution. You have a 10% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 62−x liters is 70% solution.\n2) Salt equation: 10x + 70(62−x) = 45·62.\n3) (10-70)x = (45-70)·62 ⇒ x = 155/6.\n4) Then 62−x = 217/6.\nFinal answer: $155/6 L of 10% and 217/6 L of 70%$", "final_answer": "155/6 L of 10% and 217/6 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fa9c7cf2de8ebc9a", "problem_id": "9776533017892f15", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "You want 62 liters of a 45% salt solution. You have a 10% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 62−x liters is 70% solution.\nSalt equation: 10x + 70(62−x) = 45·62.\n(10-70)x = (45-70)·62 ⇒ x = 155/6.\nThen 62−x = 217/6.\nCheck: 10·155/6 + 70·217/6 = 45·62.\nFinal answer: $155/6 L of 10% and 217/6 L of 70%$", "final_answer": "155/6 L of 10% and 217/6 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0a57d73f8992e641", "problem_id": "6acefecb917a6890", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 14 red balls and 11 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(25,6).\n3) Favorable samples: choose k reds and n−k blues: C(14,1)·C(11,5).\n4) Compute totals: C(25,6) = 177100, and favorable = 6468.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{21}{575}$", "final_answer": "\\frac{21}{575}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "84984131817b456a", "problem_id": "6acefecb917a6890", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 14 red balls and 11 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(25,6).\nFavorable samples: choose k reds and n−k blues: C(14,1)·C(11,5).\nCompute totals: C(25,6) = 177100, and favorable = 6468.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{21}{575}$.\nFinal answer: $\\frac{21}{575}$", "final_answer": "\\frac{21}{575}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "add0672a91b91418", "problem_id": "4dc7351d3207582c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} 2 x^{2} + 9 x + 34$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{2} + 8 x + 34.\n3) Then the limit is 98.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: 102.\nFinal answer: $102$", "final_answer": "102", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b50232e3151f8bb1", "problem_id": "7de1f884c986b08d", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 14 red balls and 28 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 6 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(42,12).\n3) Favorable samples: choose k reds and n−k blues: C(14,6)·C(28,6).\n4) Compute totals: C(42,12) = 11058116888, and favorable = 1131350220.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{3108105}{30379442}$", "final_answer": "\\frac{3108105}{30379442}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3cf0bf4f6f73b031", "problem_id": "349196f4956472ce", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$3 x - 2 y$ = $-5$\n$4 x - 4 y$ = $-4$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $12 x - 8 y$ = $-20$.\nMultiply the second equation by $3$: $12 x - 12 y$ = $-12$.\nSubtract to eliminate x: $4 y$ = $-8$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $-3$.\nCheck: plug (x,y)=(-3,-2) into both equations.\nFinal answer: $(x,y)=\\left(-3, -2\\right)$", "final_answer": "(x,y)=\\left(-3, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7d277a941d9dff9f", "problem_id": "349196f4956472ce", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$3 x - 2 y$ = $-5$\n$4 x - 4 y$ = $-4$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $12 x - 8 y$ = $-20$.\n3) Multiply the second equation by $3$: $12 x - 12 y$ = $-12$.\n4) Subtract to eliminate x: $4 y$ = $-8$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, -2\\right)$", "final_answer": "(x,y)=\\left(-3, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "91dc857065e1033e", "problem_id": "562df06bbfce7a8c", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{3} x - \\frac{5 x}{4} + \\frac{3}{8} + \\frac{3 \\sqrt{3}}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\sqrt{3} - \\frac{5}{4}$, c=$\\frac{3}{8} + \\frac{3 \\sqrt{3}}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{49}{16} - \\frac{\\sqrt{3}}{2}$, so √Δ = $\\sqrt{\\frac{49}{16} - \\frac{\\sqrt{3}}{2}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{5}{8} + \\frac{\\sqrt{49 - 8 \\sqrt{3}}}{8} + \\frac{\\sqrt{3}}{2}$.\n7) x2 = $- \\frac{\\sqrt{49 - 8 \\sqrt{3}}}{8} + \\frac{5}{8} + \\frac{\\sqrt{3}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{5}{8} + \\frac{\\sqrt{49 - 8\\cdot \\sqrt{3}}}{8} + \\frac{\\sqrt{3}}{2}, - \\frac{\\sqrt{49 - 8\\cdot \\sqrt{3}}}{8} + \\frac{5}{8} + \\frac{\\sqrt{3}}{2}\\}$", "final_answer": "x \\in \\{\\frac{5}{8} + \\frac{\\sqrt{49 - 8\\cdot \\sqrt{3}}}{8} + \\frac{\\sqrt{3}}{2}, - \\frac{\\sqrt{49 - 8\\cdot \\sqrt{3}}}{8} + \\frac{5}{8} + \\frac{\\sqrt{3}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9052f62bc9de46e5", "problem_id": "562df06bbfce7a8c", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{3} x - \\frac{5 x}{4} + \\frac{3}{8} + \\frac{3 \\sqrt{3}}{4}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\sqrt{3} - \\frac{5}{4}$, c=$\\frac{3}{8} + \\frac{3 \\sqrt{3}}{4}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{49}{16} - \\frac{\\sqrt{3}}{2}$, so √Δ = $\\sqrt{\\frac{49}{16} - \\frac{\\sqrt{3}}{2}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{5}{8} + \\frac{\\sqrt{49 - 8 \\sqrt{3}}}{8} + \\frac{\\sqrt{3}}{2}$.\nx2 = $- \\frac{\\sqrt{49 - 8 \\sqrt{3}}}{8} + \\frac{5}{8} + \\frac{\\sqrt{3}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{5}{8} + \\frac{\\sqrt{49 - 8\\cdot \\sqrt{3}}}{8} + \\frac{\\sqrt{3}}{2}, - \\frac{\\sqrt{49 - 8\\cdot \\sqrt{3}}}{8} + \\frac{5}{8} + \\frac{\\sqrt{3}}{2}\\}$", "final_answer": "x \\in \\{\\frac{5}{8} + \\frac{\\sqrt{49 - 8\\cdot \\sqrt{3}}}{8} + \\frac{\\sqrt{3}}{2}, - \\frac{\\sqrt{49 - 8\\cdot \\sqrt{3}}}{8} + \\frac{5}{8} + \\frac{\\sqrt{3}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0bcc4ee92c05f50c", "problem_id": "97e18d7b75fc58db", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - \\frac{47 x}{4} + \\frac{45}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$- \\frac{47}{4}$, c=$\\frac{45}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{49}{16}$, so √Δ = $\\frac{7}{4}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{9}{4}$.\n7) x2 = $\\frac{5}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{9}{4}, \\frac{5}{3}\\}$", "final_answer": "x \\in \\{\\frac{9}{4}, \\frac{5}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9e2b540055a8ae6c", "problem_id": "97e18d7b75fc58db", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - \\frac{47 x}{4} + \\frac{45}{4}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$- \\frac{47}{4}$, c=$\\frac{45}{4}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{49}{16}$, so √Δ = $\\frac{7}{4}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{9}{4}$.\nx2 = $\\frac{5}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{9}{4}, \\frac{5}{3}\\}$", "final_answer": "x \\in \\{\\frac{9}{4}, \\frac{5}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "02e6d66fb8ae644e", "problem_id": "21404154525ab2e0", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{6} x - x - 2 + 2 \\sqrt{6}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\sqrt{6} - 1$, c=$-2 + 2 \\sqrt{6}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $15 - 6 \\sqrt{6}$, so √Δ = $\\sqrt{15 - 6 \\sqrt{6}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{15 - 6 \\sqrt{6}}}{2} + \\frac{1}{2} + \\frac{\\sqrt{6}}{2}$.\n7) x2 = $- \\frac{\\sqrt{15 - 6 \\sqrt{6}}}{2} + \\frac{1}{2} + \\frac{\\sqrt{6}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{15 - 6\\cdot \\sqrt{6}}}{2} + \\frac{1}{2} + \\frac{\\sqrt{6}}{2}, - \\frac{\\sqrt{15 - 6\\cdot \\sqrt{6}}}{2} + \\frac{1}{2} + \\frac{\\sqrt{6}}{2}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{15 - 6\\cdot \\sqrt{6}}}{2} + \\frac{1}{2} + \\frac{\\sqrt{6}}{2}, - \\frac{\\sqrt{15 - 6\\cdot \\sqrt{6}}}{2} + \\frac{1}{2} + \\frac{\\sqrt{6}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "033ae80012ace99b", "problem_id": "21404154525ab2e0", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{6} x - x - 2 + 2 \\sqrt{6}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\sqrt{6} - 1$, c=$-2 + 2 \\sqrt{6}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $15 - 6 \\sqrt{6}$, so √Δ = $\\sqrt{15 - 6 \\sqrt{6}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{15 - 6 \\sqrt{6}}}{2} + \\frac{1}{2} + \\frac{\\sqrt{6}}{2}$.\nx2 = $- \\frac{\\sqrt{15 - 6 \\sqrt{6}}}{2} + \\frac{1}{2} + \\frac{\\sqrt{6}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{15 - 6\\cdot \\sqrt{6}}}{2} + \\frac{1}{2} + \\frac{\\sqrt{6}}{2}, - \\frac{\\sqrt{15 - 6\\cdot \\sqrt{6}}}{2} + \\frac{1}{2} + \\frac{\\sqrt{6}}{2}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{15 - 6\\cdot \\sqrt{6}}}{2} + \\frac{1}{2} + \\frac{\\sqrt{6}}{2}, - \\frac{\\sqrt{15 - 6\\cdot \\sqrt{6}}}{2} + \\frac{1}{2} + \\frac{\\sqrt{6}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c5e04d8b70bd2cc7", "problem_id": "1503bc9420e3e4e6", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 2 \\sqrt{3} x - x - 4 \\sqrt{3} - 6$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- 2 \\sqrt{3} - 1$, c=$- 4 \\sqrt{3} - 6$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $20 \\sqrt{3} + 37$, so √Δ = $\\sqrt{20 \\sqrt{3} + 37}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{1}{2} + \\sqrt{3} + \\frac{\\sqrt{20 \\sqrt{3} + 37}}{2}$.\nx2 = $- \\frac{\\sqrt{20 \\sqrt{3} + 37}}{2} + \\frac{1}{2} + \\sqrt{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{1}{2} + \\sqrt{3} + \\frac{\\sqrt{20\\cdot \\sqrt{3} + 37}}{2}, - \\frac{\\sqrt{20\\cdot \\sqrt{3} + 37}}{2} + \\frac{1}{2} + \\sqrt{3}\\}$", "final_answer": "x \\in \\{\\frac{1}{2} + \\sqrt{3} + \\frac{\\sqrt{20\\cdot \\sqrt{3} + 37}}{2}, - \\frac{\\sqrt{20\\cdot \\sqrt{3} + 37}}{2} + \\frac{1}{2} + \\sqrt{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "387a291cf16fdc9d", "problem_id": "1503bc9420e3e4e6", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 2 \\sqrt{3} x - x - 4 \\sqrt{3} - 6$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- 2 \\sqrt{3} - 1$, c=$- 4 \\sqrt{3} - 6$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $20 \\sqrt{3} + 37$, so √Δ = $\\sqrt{20 \\sqrt{3} + 37}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{1}{2} + \\sqrt{3} + \\frac{\\sqrt{20 \\sqrt{3} + 37}}{2}$.\n7) x2 = $- \\frac{\\sqrt{20 \\sqrt{3} + 37}}{2} + \\frac{1}{2} + \\sqrt{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{1}{2} + \\sqrt{3} + \\frac{\\sqrt{20\\cdot \\sqrt{3} + 37}}{2}, - \\frac{\\sqrt{20\\cdot \\sqrt{3} + 37}}{2} + \\frac{1}{2} + \\sqrt{3}\\}$", "final_answer": "x \\in \\{\\frac{1}{2} + \\sqrt{3} + \\frac{\\sqrt{20\\cdot \\sqrt{3} + 37}}{2}, - \\frac{\\sqrt{20\\cdot \\sqrt{3} + 37}}{2} + \\frac{1}{2} + \\sqrt{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0353303bca30c740", "problem_id": "a6d73985fdff9722", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(2 x^{3} - x^{2} - 3 x - 1\\right) \\cos{\\left(2 x - 2 \\right)}$.", "output": "1) Let f(x) = $\\left(x + 1\\right) e^{x} + \\left(2 x^{3} - x^{2} - 3 x - 1\\right) \\cos{\\left(2 x - 2 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\left(x + 1\\right) e^{x} + \\left(6 x^{2} - 2 x - 3\\right) \\cos{\\left(2 x - 2 \\right)} - 2 \\left(2 x^{3} - x^{2} - 3 x - 1\\right) \\sin{\\left(2 x - 2 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 1\\right)\\cdot e^{x} + \\left(6\\cdot x^{2} - 2\\cdot x - 3\\right)\\cdot \\cos{\\left(2\\cdot x - 2 \\right)} - \\left(4\\cdot x^{3} - 2\\cdot x^{2} - 6\\cdot x - 2\\right)\\cdot \\sin{\\left(2\\cdot x - 2 \\right)} + e^{x}$", "final_answer": "\\left(x + 1\\right)\\cdot e^{x} + \\left(6\\cdot x^{2} - 2\\cdot x - 3\\right)\\cdot \\cos{\\left(2\\cdot x - 2 \\right)} - \\left(4\\cdot x^{3} - 2\\cdot x^{2} - 6\\cdot x - 2\\right)\\cdot \\sin{\\left(2\\cdot x - 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "569cc3e59e74aed1", "problem_id": "a6d73985fdff9722", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(2 x^{3} - x^{2} - 3 x - 1\\right) \\cos{\\left(2 x - 2 \\right)}$.", "output": "Let f(x) = $\\left(x + 1\\right) e^{x} + \\left(2 x^{3} - x^{2} - 3 x - 1\\right) \\cos{\\left(2 x - 2 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\left(x + 1\\right) e^{x} + \\left(6 x^{2} - 2 x - 3\\right) \\cos{\\left(2 x - 2 \\right)} - 2 \\left(2 x^{3} - x^{2} - 3 x - 1\\right) \\sin{\\left(2 x - 2 \\right)} + e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\left(x + 1\\right)\\cdot e^{x} + \\left(6\\cdot x^{2} - 2\\cdot x - 3\\right)\\cdot \\cos{\\left(2\\cdot x - 2 \\right)} - \\left(4\\cdot x^{3} - 2\\cdot x^{2} - 6\\cdot x - 2\\right)\\cdot \\sin{\\left(2\\cdot x - 2 \\right)} + e^{x}$", "final_answer": "\\left(x + 1\\right)\\cdot e^{x} + \\left(6\\cdot x^{2} - 2\\cdot x - 3\\right)\\cdot \\cos{\\left(2\\cdot x - 2 \\right)} - \\left(4\\cdot x^{3} - 2\\cdot x^{2} - 6\\cdot x - 2\\right)\\cdot \\sin{\\left(2\\cdot x - 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d6ac873b6e602b72", "problem_id": "8d436169cd3f3653", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 12 red balls and 8 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(20,5).\nFavorable samples: choose k reds and n−k blues: C(12,1)·C(8,4).\nCompute totals: C(20,5) = 15504, and favorable = 840.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{35}{646}$.\nFinal answer: $\\frac{35}{646}$", "final_answer": "\\frac{35}{646}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5643e5c7b23fcff7", "problem_id": "8d436169cd3f3653", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "An urn has 12 red balls and 8 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(20,5).\nFavorable samples: choose k reds and n−k blues: C(12,1)·C(8,4).\nFinal answer: $\\frac{35}{646}$", "final_answer": "\\frac{35}{646}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2318cf005831e240", "problem_id": "168252852ce36a51", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Two friends split $2076 in the ratio 7:5. How much does each receive?", "output": "1) Total ratio parts = 7+5 = 12.\n2) Each part = 2076 ÷ 12 = 173.\n3) First share = 7 × 173 = 1211.\n4) Second share = 5 × 173 = 865.\nFinal answer: $865$", "final_answer": "865", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "090c64126f7a9564", "problem_id": "168252852ce36a51", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $2076 in the ratio 7:5. How much does each receive?", "output": "Total ratio parts = 7+5 = 12.\nEach part = 2076 ÷ 12 = 173.\nFirst share = 7 × 173 = 1211.\nSecond share = 5 × 173 = 865.\nCheck: 1211+865=2076.\nFinal answer: $865$", "final_answer": "865", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3af39ba87dfc511a", "problem_id": "0879c7cb736340f6", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} 3 x^{3} + 2 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{3} + 2 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{45 \\pi^{4}}{64} - \\frac{9 \\pi}{2} + 5 + \\frac{3 \\pi^{2}}{4} + \\frac{3 \\pi^{3}}{4}$.\n4) Simplify the result.\nFinal answer: $- \\frac{45\\cdot \\pi^{4}}{64} - \\frac{9\\cdot \\pi}{2} + 5 + \\frac{3\\cdot \\pi^{2}}{4} + \\frac{3\\cdot \\pi^{3}}{4}$", "final_answer": "- \\frac{45\\cdot \\pi^{4}}{64} - \\frac{9\\cdot \\pi}{2} + 5 + \\frac{3\\cdot \\pi^{2}}{4} + \\frac{3\\cdot \\pi^{3}}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2dfae5dcb331379f", "problem_id": "0879c7cb736340f6", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} 3 x^{3} + 2 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{3} + 2 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{45 \\pi^{4}}{64} - \\frac{9 \\pi}{2} + 5 + \\frac{3 \\pi^{2}}{4} + \\frac{3 \\pi^{3}}{4}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{45\\cdot \\pi^{4}}{64} - \\frac{9\\cdot \\pi}{2} + 5 + \\frac{3\\cdot \\pi^{2}}{4} + \\frac{3\\cdot \\pi^{3}}{4}$", "final_answer": "- \\frac{45\\cdot \\pi^{4}}{64} - \\frac{9\\cdot \\pi}{2} + 5 + \\frac{3\\cdot \\pi^{2}}{4} + \\frac{3\\cdot \\pi^{3}}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "04d82a16b7be5360", "problem_id": "49882216d0f286e6", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - x^{3} - 7 x^{2} - 25 x - 99$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to - x^{3} - 7 x^{2} - 26 x - 99.\n3) Then the limit is -379.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: -375.\nFinal answer: $-375$", "final_answer": "-375", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "40da708cd01398b1", "problem_id": "4be834d9346f579e", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A disease has prevalence $\\frac{3}{100}$. A test has sensitivity $\\frac{19}{20}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{251}{2000}$.\n3) Compute P(D|+) = $\\frac{57}{251}$.\nFinal answer: $\\frac{57}{251}$", "final_answer": "\\frac{57}{251}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "22a33ee9664cc40f", "problem_id": "3e24d211fe80c868", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "A store sells a backpack for $930. It is discounted by 10%. You buy 9 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 930 × (1 − 10/100) = 930 × 90/100 = 837.\nSubtotal = 9 × 837 = 7533.\nTotal with tax = 7533 × 105/100 = 158193/20.\nFinal answer: $\\frac{158193}{20}$", "final_answer": "\\frac{158193}{20}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2793881947c3dc3c", "problem_id": "3e24d211fe80c868", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A store sells a backpack for $930. It is discounted by 10%. You buy 9 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 930 × (1 − 10/100) = 930 × 90/100 = 837.\n2) Subtotal = 9 × 837 = 7533.\n3) Total with tax = 7533 × 105/100 = 158193/20.\nFinal answer: $\\frac{158193}{20}$", "final_answer": "\\frac{158193}{20}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1cce6bc36ace0ed6", "problem_id": "088bedd917213ca0", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 58 liters of a 35% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 58−x liters is 50% solution.\nSalt equation: 25x + 50(58−x) = 35·58.\n(25-50)x = (35-50)·58 ⇒ x = 174/5.\nThen 58−x = 116/5.\nCheck: 25·174/5 + 50·116/5 = 35·58.\nFinal answer: $174/5 L of 25% and 116/5 L of 50%$", "final_answer": "174/5 L of 25% and 116/5 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c96bf6427bb1a856", "problem_id": "088bedd917213ca0", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "You want 58 liters of a 35% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 58−x liters is 50% solution.\n2) Salt equation: 25x + 50(58−x) = 35·58.\n3) (25-50)x = (35-50)·58 ⇒ x = 174/5.\n4) Then 58−x = 116/5.\nFinal answer: $174/5 L of 25% and 116/5 L of 50%$", "final_answer": "174/5 L of 25% and 116/5 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1f7b7c38e87ff4d5", "problem_id": "f99bb73a21a9e783", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{5}$. If it is flipped 10 times, what is the probability of exactly 8 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=8, p=$\\frac{3}{5}$ and simplify.\n4) Here C(10,8) = 45.\nFinal answer: $\\frac{236196}{1953125}$", "final_answer": "\\frac{236196}{1953125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fe12432f0093cc1c", "problem_id": "f99bb73a21a9e783", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{5}$. If it is flipped 10 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=8, p=$\\frac{3}{5}$ and simplify.\nHere C(10,8) = 45.\nCheck: expression simplifies to $\\frac{236196}{1953125}$.\nFinal answer: $\\frac{236196}{1953125}$", "final_answer": "\\frac{236196}{1953125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "785d7b302a76fcf4", "problem_id": "abdc202e2c12ea5f", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Find the equation of the tangent line to $y=2 x^{3} + x^{2} + 2 x + 2 \\sin{\\left(x \\right)} + 3$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $6 x^{2} + 2 x + 2 \\cos{\\left(x \\right)} + 2$.\n2) Slope at x=-1: m = f'(-1) = $2 \\cos{\\left(1 \\right)} + 6$.\n3) Point on curve: (-1, f(-1)) = (-1, $- 2 \\sin{\\left(1 \\right)}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − - 2 \\sin{\\left(1 \\right)} = 2 \\cos{\\left(1 \\right)} + 6(x − -1).\n6) Optionally expand: y = $2 x \\cos{\\left(1 \\right)} + 6 x - 2 \\sin{\\left(1 \\right)} + 2 \\cos{\\left(1 \\right)} + 6$.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 6\\cdot x - 2\\cdot \\sin{\\left(1 \\right)} + 2\\cdot \\cos{\\left(1 \\right)} + 6$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 6\\cdot x - 2\\cdot \\sin{\\left(1 \\right)} + 2\\cdot \\cos{\\left(1 \\right)} + 6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1a0aef7aafc12a1c", "problem_id": "abdc202e2c12ea5f", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} + x^{2} + 2 x + 2 \\sin{\\left(x \\right)} + 3$ at $x=-1$.", "output": "Differentiate: f'(x) = $6 x^{2} + 2 x + 2 \\cos{\\left(x \\right)} + 2$.\nSlope at x=-1: m = f'(-1) = $2 \\cos{\\left(1 \\right)} + 6$.\nPoint on curve: (-1, f(-1)) = (-1, $- 2 \\sin{\\left(1 \\right)}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − - 2 \\sin{\\left(1 \\right)} = 2 \\cos{\\left(1 \\right)} + 6(x − -1).\nOptionally expand: y = $2 x \\cos{\\left(1 \\right)} + 6 x - 2 \\sin{\\left(1 \\right)} + 2 \\cos{\\left(1 \\right)} + 6$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 6\\cdot x - 2\\cdot \\sin{\\left(1 \\right)} + 2\\cdot \\cos{\\left(1 \\right)} + 6$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} + 6\\cdot x - 2\\cdot \\sin{\\left(1 \\right)} + 2\\cdot \\cos{\\left(1 \\right)} + 6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c54f484b1a3922b8", "problem_id": "8c4001df6f819bf6", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "An urn has 19 red balls and 20 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(39,7).\n3) Favorable samples: choose k reds and n−k blues: C(19,2)·C(20,5).\n4) Compute totals: C(39,7) = 15380937, and favorable = 2651184.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{912}{5291}$", "final_answer": "\\frac{912}{5291}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "23025bd4baca431c", "problem_id": "8c4001df6f819bf6", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 19 red balls and 20 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(39,7).\nFavorable samples: choose k reds and n−k blues: C(19,2)·C(20,5).\nCompute totals: C(39,7) = 15380937, and favorable = 2651184.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{912}{5291}$.\nFinal answer: $\\frac{912}{5291}$", "final_answer": "\\frac{912}{5291}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b2ca3f9e27bef591", "problem_id": "076d8238b392350c", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - 3 x^{3} - 3 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{3} - 3 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{765 \\pi^{4}}{64} - \\frac{65 \\pi^{3}}{8} - 2 + 5 \\pi + \\frac{15 \\pi^{2}}{8}$.\n4) Simplify the result.\nFinal answer: $- \\frac{765\\cdot \\pi^{4}}{64} - \\frac{65\\cdot \\pi^{3}}{8} - 2 + 5\\cdot \\pi + \\frac{15\\cdot \\pi^{2}}{8}$", "final_answer": "- \\frac{765\\cdot \\pi^{4}}{64} - \\frac{65\\cdot \\pi^{3}}{8} - 2 + 5\\cdot \\pi + \\frac{15\\cdot \\pi^{2}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e43c2d530fa127e8", "problem_id": "076d8238b392350c", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - 3 x^{3} - 3 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{3} - 3 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{765 \\pi^{4}}{64} - \\frac{65 \\pi^{3}}{8} - 2 + 5 \\pi + \\frac{15 \\pi^{2}}{8}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{765\\cdot \\pi^{4}}{64} - \\frac{65\\cdot \\pi^{3}}{8} - 2 + 5\\cdot \\pi + \\frac{15\\cdot \\pi^{2}}{8}$", "final_answer": "- \\frac{765\\cdot \\pi^{4}}{64} - \\frac{65\\cdot \\pi^{3}}{8} - 2 + 5\\cdot \\pi + \\frac{15\\cdot \\pi^{2}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "90833d24249cdd00", "problem_id": "fb2fcde9ac121f56", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 15 red balls and 9 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(24,6).\nFavorable samples: choose k reds and n−k blues: C(15,2)·C(9,4).\nCompute totals: C(24,6) = 134596, and favorable = 13230.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{945}{9614}$.\nFinal answer: $\\frac{945}{9614}$", "final_answer": "\\frac{945}{9614}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e674d8f4d271663a", "problem_id": "fb2fcde9ac121f56", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 15 red balls and 9 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(24,6).\n3) Favorable samples: choose k reds and n−k blues: C(15,2)·C(9,4).\n4) Compute totals: C(24,6) = 134596, and favorable = 13230.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{945}{9614}$", "final_answer": "\\frac{945}{9614}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7f805b9eec170993", "problem_id": "c839d512da3040b3", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 27 red balls and 22 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(49,8).\n3) Favorable samples: choose k reds and n−k blues: C(27,4)·C(22,4).\n4) Compute totals: C(49,8) = 450978066, and favorable = 128378250.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{92625}{325381}$", "final_answer": "\\frac{92625}{325381}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fc3e1626cb696bf2", "problem_id": "c839d512da3040b3", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "An urn has 27 red balls and 22 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(49,8).\nFavorable samples: choose k reds and n−k blues: C(27,4)·C(22,4).\nFinal answer: $\\frac{92625}{325381}$", "final_answer": "\\frac{92625}{325381}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5748229959457f7e", "problem_id": "896153e4ff6d0e72", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Two friends split $644 in the ratio 2:5. How much does each receive?", "output": "Total ratio parts = 2+5 = 7.\nEach part = 644 ÷ 7 = 92.\nFirst share = 2 × 92 = 184.\nSecond share = 5 × 92 = 460.\nCheck: 184+460=644.\nFinal answer: $460$", "final_answer": "460", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4bfe8e37cc773e6d", "problem_id": "6ba99aa955fab2e0", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} + 4 x$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(4 x) = 4 x$.\n3) So $f'(x) = - 6 x^{2} + 4 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(4 x) = 4$.\nTherefore the correct derivative is $f'(x) = 2 \\left(2 - 3 x^{2}\\right)$.\nFinal answer: $2 \\left(2 - 3 x^{2}\\right)$", "final_answer": "2 \\left(2 - 3 x^{2}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "fd78b18cdfa52237", "problem_id": "40bb79ea3fccd1dc", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 7 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=7, k=3, p=$\\frac{1}{5}$ and simplify.\n4) Here C(7,3) = 35.\nFinal answer: $\\frac{1792}{15625}$", "final_answer": "\\frac{1792}{15625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "83cb8da266daa7a8", "problem_id": "40bb79ea3fccd1dc", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 7 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=7, k=3, p=$\\frac{1}{5}$ and simplify.\nHere C(7,3) = 35.\nCheck: expression simplifies to $\\frac{1792}{15625}$.\nFinal answer: $\\frac{1792}{15625}$", "final_answer": "\\frac{1792}{15625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a8efaa11a958a6b2", "problem_id": "5b0dc193733791a4", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 21 red balls and 11 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(32,10).\nFavorable samples: choose k reds and n−k blues: C(21,2)·C(11,8).\nCompute totals: C(32,10) = 64512240, and favorable = 34650.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1155}{2150408}$.\nFinal answer: $\\frac{1155}{2150408}$", "final_answer": "\\frac{1155}{2150408}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2fcd0cddd9f2e591", "problem_id": "5b0dc193733791a4", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "An urn has 21 red balls and 11 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(32,10).\n3) Favorable samples: choose k reds and n−k blues: C(21,2)·C(11,8).\n4) Compute totals: C(32,10) = 64512240, and favorable = 34650.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1155}{2150408}$", "final_answer": "\\frac{1155}{2150408}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "554aef37942051ea", "problem_id": "366ffa34a081caad", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡0 (mod 3), x≡10 (mod 11). Give the solution modulo 33.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 33.\n3) Compute the CRT solution to get x ≡ 21 (mod 33).\nFinal answer: $x\\equiv 21\\pmod{33}$", "final_answer": "x\\equiv 21\\pmod{33}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b2a98414d5f23e62", "problem_id": "366ffa34a081caad", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡0 (mod 3), x≡10 (mod 11). Give the solution modulo 33.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 33.\nCompute the CRT solution to get x ≡ 21 (mod 33).\nCheck: 21 mod 3=0 and mod 11=10.\nFinal answer: $x\\equiv 21\\pmod{33}$", "final_answer": "x\\equiv 21\\pmod{33}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fac3c47359531596", "problem_id": "1639ced25ef9a536", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$- 5 x - 2 y$ = $41$\n$7 y$ = $-21$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $-5$: $- 35 y$ = $105$.\n4) Subtract to eliminate x: $35 y$ = $-105$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $-7$.\nFinal answer: $(x,y)=\\left(-7, -3\\right)$", "final_answer": "(x,y)=\\left(-7, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4103c2b28c6ce49d", "problem_id": "1639ced25ef9a536", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- 5 x - 2 y$ = $41$\n$7 y$ = $-21$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $-5$: $- 35 y$ = $105$.\nSubtract to eliminate x: $35 y$ = $-105$.\nSolve for y: y = $-3$.\nSubstitute back to find x: x = $-7$.\nCheck: plug (x,y)=(-7,-3) into both equations.\nFinal answer: $(x,y)=\\left(-7, -3\\right)$", "final_answer": "(x,y)=\\left(-7, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "859300b73be1092d", "problem_id": "1c1b26e5f76a574f", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "You want 43 liters of a 25% salt solution. You have a 20% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 43−x liters is 60% solution.\n2) Salt equation: 20x + 60(43−x) = 25·43.\n3) (20-60)x = (25-60)·43 ⇒ x = 301/8.\n4) Then 43−x = 43/8.\nFinal answer: $301/8 L of 20% and 43/8 L of 60%$", "final_answer": "301/8 L of 20% and 43/8 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ae8d048df5c24783", "problem_id": "1c1b26e5f76a574f", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 43 liters of a 25% salt solution. You have a 20% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 43−x liters is 60% solution.\nSalt equation: 20x + 60(43−x) = 25·43.\n(20-60)x = (25-60)·43 ⇒ x = 301/8.\nThen 43−x = 43/8.\nCheck: 20·301/8 + 60·43/8 = 25·43.\nFinal answer: $301/8 L of 20% and 43/8 L of 60%$", "final_answer": "301/8 L of 20% and 43/8 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "29ec3038ed8e2a62", "problem_id": "7e0def97577ffd06", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 11 times, what is the probability of exactly 9 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=9, p=$\\frac{1}{5}$ and simplify.\n4) Here C(11,9) = 55.\nFinal answer: $\\frac{176}{9765625}$", "final_answer": "\\frac{176}{9765625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7d10fcca9a343dec", "problem_id": "a296c28adb8b96c6", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 4 - 3 x$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 4 - 4 x.\n3) Then the limit is 12.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 10.\nFinal answer: $10$", "final_answer": "10", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "062f04c2bc1ab55b", "problem_id": "e1e2b21eb7a793e4", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the multiplicative inverse of 31 modulo 47.", "output": "1) We want x such that 31x ≡ 1 (mod 47).\n2) Use extended Euclid to find s,t with 31s + 47t = 1.\n3) One Bezout identity is: 31·(-3) + 47·(2) = 1.\n4) Reduce s modulo 47: x ≡ 44 (mod 47).\nFinal answer: $x\\equiv 44\\pmod{47}$", "final_answer": "x\\equiv 44\\pmod{47}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "701f2911e4d69181", "problem_id": "e1e2b21eb7a793e4", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Find the multiplicative inverse of 31 modulo 47.", "output": "We want x such that 31x ≡ 1 (mod 47).\nUse extended Euclid to find s,t with 31s + 47t = 1.\nOne Bezout identity is: 31·(-3) + 47·(2) = 1.\nFinal answer: $x\\equiv 44\\pmod{47}$", "final_answer": "x\\equiv 44\\pmod{47}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4c378ec0c77c62a1", "problem_id": "7fd2ca2a65a5c148", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Worker A finishes a job in 19 hours and Worker B finishes it in 8 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/19 job/hour. B's rate = 1/8 job/hour.\nCombined rate = 1/19 + 1/8 = 27/152 job/hour.\nTime = 1 ÷ (combined rate) = 152/27 hours.\nCheck: (combined rate)×(time) = 27/152×152/27 = 1.\nFinal answer: $\\frac{152}{27}$", "final_answer": "\\frac{152}{27}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ae35bad299a1800e", "problem_id": "7fd2ca2a65a5c148", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Worker A finishes a job in 19 hours and Worker B finishes it in 8 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/19 job/hour. B's rate = 1/8 job/hour.\nCombined rate = 1/19 + 1/8 = 27/152 job/hour.\nTime = 1 ÷ (combined rate) = 152/27 hours.\nFinal answer: $\\frac{152}{27}$", "final_answer": "\\frac{152}{27}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1910dc74ead53288", "problem_id": "9740c42157209668", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A store sells a backpack for $545. It is discounted by 35%. You buy 4 backpacks. What is the total cost?", "output": "1) Discounted price = 545 × (1 − 35/100) = 545 × 65/100 = 1417/4.\n2) Subtotal = 4 × 1417/4 = 1417.\nFinal answer: $1417$", "final_answer": "1417", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "00cefb07776f226c", "problem_id": "9740c42157209668", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $545. It is discounted by 35%. You buy 4 backpacks. What is the total cost?", "output": "Discounted price = 545 × (1 − 35/100) = 545 × 65/100 = 1417/4.\nSubtotal = 4 × 1417/4 = 1417.\nCheck: recomputing the formula yields 1417.\nFinal answer: $1417$", "final_answer": "1417", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "936de283a8cedee0", "problem_id": "beaf37cafc8cbb15", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "How many binary strings of length 22 contain exactly 6 ones with no two ones adjacent?", "output": "Reserve 5 mandatory zeros between the 6 ones.\nThis leaves n−k+1 = 17 valid slots for the ones.\nChoose the slots: C(17,6) = 12376.\nFinal answer: $12376$", "final_answer": "12376", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b55c063533648df3", "problem_id": "beaf37cafc8cbb15", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "How many binary strings of length 22 contain exactly 6 ones with no two ones adjacent?", "output": "$12376$", "final_answer": "12376", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1c37194587dc7295", "problem_id": "3cc7277d83655261", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "You want 46 liters of a 25% salt solution. You have a 20% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 46−x liters is 40% solution.\n2) Salt equation: 20x + 40(46−x) = 25·46.\n3) (20-40)x = (25-40)·46 ⇒ x = 69/2.\n4) Then 46−x = 23/2.\nFinal answer: $69/2 L of 20% and 23/2 L of 40%$", "final_answer": "69/2 L of 20% and 23/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2fe769035c283672", "problem_id": "3cc7277d83655261", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 46 liters of a 25% salt solution. You have a 20% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 46−x liters is 40% solution.\nSalt equation: 20x + 40(46−x) = 25·46.\n(20-40)x = (25-40)·46 ⇒ x = 69/2.\nThen 46−x = 23/2.\nCheck: 20·69/2 + 40·23/2 = 25·46.\nFinal answer: $69/2 L of 20% and 23/2 L of 40%$", "final_answer": "69/2 L of 20% and 23/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d65f2a4620e8f856", "problem_id": "c12001569a296350", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "An urn has 27 red balls and 27 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(54,10).\n3) Favorable samples: choose k reds and n−k blues: C(27,2)·C(27,8).\n4) Compute totals: C(54,10) = 23930713170, and favorable = 779246325.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{19305}{592858}$", "final_answer": "\\frac{19305}{592858}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d5f9438fd1f7afb7", "problem_id": "c12001569a296350", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 27 red balls and 27 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(54,10).\nFavorable samples: choose k reds and n−k blues: C(27,2)·C(27,8).\nCompute totals: C(54,10) = 23930713170, and favorable = 779246325.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{19305}{592858}$.\nFinal answer: $\\frac{19305}{592858}$", "final_answer": "\\frac{19305}{592858}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1f4c5d65b1acf0d1", "problem_id": "4dfdd8d335aaba13", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A store sells a backpack for $224. It is discounted by 5%. You buy 14 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 224 × (1 − 5/100) = 224 × 95/100 = 1064/5.\nSubtotal = 14 × 1064/5 = 14896/5.\nTotal with tax = 14896/5 × 105/100 = 78204/25.\nCheck: recomputing the formula yields 78204/25.\nFinal answer: $\\frac{78204}{25}$", "final_answer": "\\frac{78204}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d4b87210958fc9a1", "problem_id": "4dfdd8d335aaba13", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A store sells a backpack for $224. It is discounted by 5%. You buy 14 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 224 × (1 − 5/100) = 224 × 95/100 = 1064/5.\n2) Subtotal = 14 × 1064/5 = 14896/5.\n3) Total with tax = 14896/5 × 105/100 = 78204/25.\nFinal answer: $\\frac{78204}{25}$", "final_answer": "\\frac{78204}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "49c13acde02c6ce1", "problem_id": "509e5685bd8cb857", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A cyclist rides at 36 miles per hour for 3/2 hours, then at 65 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 36 × 3/2 = 54 miles.\n2) Second leg distance = 65 × 1 = 65 miles.\n3) Total distance = 54 + 65 = 119 miles.\nFinal answer: $119$", "final_answer": "119", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d2c7c241cdfc7ef2", "problem_id": "509e5685bd8cb857", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A cyclist rides at 36 miles per hour for 3/2 hours, then at 65 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 36 × 3/2 = 54 miles.\nSecond leg distance = 65 × 1 = 65 miles.\nTotal distance = 54 + 65 = 119 miles.\nCheck: adding both legs gives 119.\nFinal answer: $119$", "final_answer": "119", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d810ff197559aae3", "problem_id": "3532616070f49954", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A cyclist rides at 88 miles per hour for 2 hours, then at 49 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 88 × 2 = 176 miles.\n2) Second leg distance = 49 × 3/2 = 147/2 miles.\n3) Total distance = 176 + 147/2 = 499/2 miles.\nFinal answer: $\\frac{499}{2}$", "final_answer": "\\frac{499}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bab3d6b217df6ae7", "problem_id": "fa30c41e2f53e39c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(- 2 x^{3} + 2 x^{2} - 2 x - 3\\right) \\cos{\\left(x - 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 2 x^{3} \\cos{\\left(x - 3 \\right)}) = 2 x^{3} \\sin{\\left(x - 3 \\right)}$.\n3) So $f'(x) = 2 x^{3} \\sin{\\left(x - 3 \\right)} - 2 x^{2} \\sin{\\left(x - 3 \\right)} + 2 x e^{x} + 2 x \\sin{\\left(x - 3 \\right)} + 4 x \\cos{\\left(x - 3 \\right)} - 4 e^{x} + 3 \\sin{\\left(x - 3 \\right)} - 2 \\cos{\\left(x - 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 2 x^{3} \\cos{\\left(x - 3 \\right)}) = 2 x^{3} \\sin{\\left(x - 3 \\right)} - 6 x^{2} \\cos{\\left(x - 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x - 3\\right) e^{x} - \\left(2 x \\left(x \\left(1 - x\\right) - 1\\right) - 3\\right) \\sin{\\left(x - 3 \\right)} + 2 \\left(- 3 x^{2} + 2 x - 1\\right) \\cos{\\left(x - 3 \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(x - 3\\right) e^{x} - \\left(2 x \\left(x \\left(1 - x\\right) - 1\\right) - 3\\right) \\sin{\\left(x - 3 \\right)} + 2 \\left(- 3 x^{2} + 2 x - 1\\right) \\cos{\\left(x - 3 \\right)} + 2 e^{x}$", "final_answer": "2 \\left(x - 3\\right) e^{x} - \\left(2 x \\left(x \\left(1 - x\\right) - 1\\right) - 3\\right) \\sin{\\left(x - 3 \\right)} + 2 \\left(- 3 x^{2} + 2 x - 1\\right) \\cos{\\left(x - 3 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c248cb536d8b58f2", "problem_id": "2032381000011739", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡0 (mod 4), x≡4 (mod 7). Give the solution modulo 28.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 28.\n3) Compute the CRT solution to get x ≡ 4 (mod 28).\nFinal answer: $x\\equiv 4\\pmod{28}$", "final_answer": "x\\equiv 4\\pmod{28}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "af97502bc4ac9810", "problem_id": "2032381000011739", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡0 (mod 4), x≡4 (mod 7). Give the solution modulo 28.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 28.\nCompute the CRT solution to get x ≡ 4 (mod 28).\nCheck: 4 mod 4=0 and mod 7=4.\nFinal answer: $x\\equiv 4\\pmod{28}$", "final_answer": "x\\equiv 4\\pmod{28}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3eb248848a6fedb3", "problem_id": "141ad1a2ae347f7a", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $210. It is discounted by 15%. You buy 8 backpacks. What is the total cost?", "output": "Discounted price = 210 × (1 − 15/100) = 210 × 85/100 = 357/2.\nSubtotal = 8 × 357/2 = 1428.\nCheck: recomputing the formula yields 1428.\nFinal answer: $1428$", "final_answer": "1428", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1fd99d3b43d80e95", "problem_id": "141ad1a2ae347f7a", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A store sells a backpack for $210. It is discounted by 15%. You buy 8 backpacks. What is the total cost?", "output": "1) Discounted price = 210 × (1 − 15/100) = 210 × 85/100 = 357/2.\n2) Subtotal = 8 × 357/2 = 1428.\nFinal answer: $1428$", "final_answer": "1428", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6b9cad547c39c8dd", "problem_id": "dfa2d55586e8175f", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A cyclist rides at 20 miles per hour for 1/2 hours, then at 53 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 20 × 1/2 = 10 miles.\n2) Second leg distance = 53 × 2 = 106 miles.\n3) Total distance = 10 + 106 = 116 miles.\nFinal answer: $116$", "final_answer": "116", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4977def328da2f03", "problem_id": "dfa2d55586e8175f", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 20 miles per hour for 1/2 hours, then at 53 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 20 × 1/2 = 10 miles.\nSecond leg distance = 53 × 2 = 106 miles.\nTotal distance = 10 + 106 = 116 miles.\nCheck: adding both legs gives 116.\nFinal answer: $116$", "final_answer": "116", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d41cadb83fbfd377", "problem_id": "13cef4c003b4fd84", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡5 (mod 7), x≡3 (mod 8). Give the solution modulo 56.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 56.\n3) Compute the CRT solution to get x ≡ 19 (mod 56).\nFinal answer: $x\\equiv 19\\pmod{56}$", "final_answer": "x\\equiv 19\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b73a6acbe519c669", "problem_id": "93c560e9cdb2555e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{3} + 4 x^{2} + 10 x + 17$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} + 4 x^{2} + 9 x + 17.\n3) Then the limit is 67.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 69.\nFinal answer: $69$", "final_answer": "69", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "98ee626a2e688736", "problem_id": "945e671dba341443", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$7 x + y$ = $1$\n$3 y$ = $3$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $7$: $21 y$ = $21$.\n4) Subtract to eliminate x: $- 21 y$ = $-21$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, 1\\right)$", "final_answer": "(x,y)=\\left(0, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3ffa7c3762464144", "problem_id": "945e671dba341443", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the linear system:\n$7 x + y$ = $1$\n$3 y$ = $3$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $7$: $21 y$ = $21$.\nFinal answer: $(x,y)=\\left(0, 1\\right)$", "final_answer": "(x,y)=\\left(0, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e55b970310dc0653", "problem_id": "61f8e3681d7e0065", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 13 times, what is the probability of exactly 8 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=13, k=8, p=$\\frac{1}{5}$ and simplify.\n4) Here C(13,8) = 1287.\nFinal answer: $\\frac{1317888}{1220703125}$", "final_answer": "\\frac{1317888}{1220703125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "96a3565822399d08", "problem_id": "61f8e3681d7e0065", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 13 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=13, k=8, p=$\\frac{1}{5}$ and simplify.\nHere C(13,8) = 1287.\nCheck: expression simplifies to $\\frac{1317888}{1220703125}$.\nFinal answer: $\\frac{1317888}{1220703125}$", "final_answer": "\\frac{1317888}{1220703125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1440f057e5eba77d", "problem_id": "cb3d8976c90e974a", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$- 3 x + 2 y$ = $15$\n$4 x - 4 y$ = $-24$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $- 12 x + 8 y$ = $60$.\nMultiply the second equation by $-3$: $- 12 x + 12 y$ = $72$.\nSubtract to eliminate x: $- 4 y$ = $-12$.\nSolve for y: y = $3$.\nSubstitute back to find x: x = $-3$.\nCheck: plug (x,y)=(-3,3) into both equations.\nFinal answer: $(x,y)=\\left(-3, 3\\right)$", "final_answer": "(x,y)=\\left(-3, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "100a3373afd65982", "problem_id": "cb3d8976c90e974a", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the linear system:\n$- 3 x + 2 y$ = $15$\n$4 x - 4 y$ = $-24$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $- 12 x + 8 y$ = $60$.\n3) Multiply the second equation by $-3$: $- 12 x + 12 y$ = $72$.\n4) Subtract to eliminate x: $- 4 y$ = $-12$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, 3\\right)$", "final_answer": "(x,y)=\\left(-3, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "99aba26d4922146b", "problem_id": "f0d0320c453803d8", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Worker A finishes a job in 8 hours and Worker B finishes it in 13 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/8 job/hour. B's rate = 1/13 job/hour.\n2) Combined rate = 1/8 + 1/13 = 21/104 job/hour.\n3) Time = 1 ÷ (combined rate) = 104/21 hours.\nFinal answer: $\\frac{104}{21}$", "final_answer": "\\frac{104}{21}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a23723a7854df3ca", "problem_id": "f0d0320c453803d8", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Worker A finishes a job in 8 hours and Worker B finishes it in 13 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/8 job/hour. B's rate = 1/13 job/hour.\nCombined rate = 1/8 + 1/13 = 21/104 job/hour.\nTime = 1 ÷ (combined rate) = 104/21 hours.\nCheck: (combined rate)×(time) = 21/104×104/21 = 1.\nFinal answer: $\\frac{104}{21}$", "final_answer": "\\frac{104}{21}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "682427203de31263", "problem_id": "77ca41f980dc3dbe", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 16 red balls and 9 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(25,6).\n3) Favorable samples: choose k reds and n−k blues: C(16,3)·C(9,3).\n4) Compute totals: C(25,6) = 177100, and favorable = 47040.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{336}{1265}$", "final_answer": "\\frac{336}{1265}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "30fdecdee74e5abf", "problem_id": "77ca41f980dc3dbe", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 16 red balls and 9 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(25,6).\nFavorable samples: choose k reds and n−k blues: C(16,3)·C(9,3).\nCompute totals: C(25,6) = 177100, and favorable = 47040.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{336}{1265}$.\nFinal answer: $\\frac{336}{1265}$", "final_answer": "\\frac{336}{1265}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "403bfd5eb1a65edf", "problem_id": "0dbacd13adbc5aed", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- 4 x - 2 y$ = $22$\n$- 4 x + 8 y$ = $92$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $16 x + 8 y$ = $-88$.\nMultiply the second equation by $-4$: $16 x - 32 y$ = $-368$.\nSubtract to eliminate x: $40 y$ = $280$.\nSolve for y: y = $7$.\nSubstitute back to find x: x = $-9$.\nCheck: plug (x,y)=(-9,7) into both equations.\nFinal answer: $(x,y)=\\left(-9, 7\\right)$", "final_answer": "(x,y)=\\left(-9, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5f4333d71833ae51", "problem_id": "0dbacd13adbc5aed", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$- 4 x - 2 y$ = $22$\n$- 4 x + 8 y$ = $92$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $16 x + 8 y$ = $-88$.\n3) Multiply the second equation by $-4$: $16 x - 32 y$ = $-368$.\n4) Subtract to eliminate x: $40 y$ = $280$.\n5) Solve for y: y = $7$.\n6) Substitute back to find x: x = $-9$.\nFinal answer: $(x,y)=\\left(-9, 7\\right)$", "final_answer": "(x,y)=\\left(-9, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "845d09a5d2d6eace", "problem_id": "d440cd917ae79255", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the linear system:\n$4 x - 8 y$ = $44$\n$- 5 x - y$ = $0$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-5$: $- 20 x + 40 y$ = $-220$.\nMultiply the second equation by $4$: $- 20 x - 4 y$ = $0$.\nFinal answer: $(x,y)=\\left(1, -5\\right)$", "final_answer": "(x,y)=\\left(1, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3d4d62d9636700ac", "problem_id": "d440cd917ae79255", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$4 x - 8 y$ = $44$\n$- 5 x - y$ = $0$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-5$: $- 20 x + 40 y$ = $-220$.\n3) Multiply the second equation by $4$: $- 20 x - 4 y$ = $0$.\n4) Subtract to eliminate x: $44 y$ = $-220$.\n5) Solve for y: y = $-5$.\n6) Substitute back to find x: x = $1$.\nFinal answer: $(x,y)=\\left(1, -5\\right)$", "final_answer": "(x,y)=\\left(1, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f69096690833ff7d", "problem_id": "a901955fcf344cbf", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 3 x^{3} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{3} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $\\frac{3 \\pi \\left(2 \\pi + 12 + 15 \\pi^{3}\\right)}{4}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{3\\cdot \\pi\\cdot \\left(2\\cdot \\pi + 12 + 15\\cdot \\pi^{3}\\right)}{4}$", "final_answer": "\\frac{3\\cdot \\pi\\cdot \\left(2\\cdot \\pi + 12 + 15\\cdot \\pi^{3}\\right)}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fa9c83f9dfcdf388", "problem_id": "a901955fcf344cbf", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 3 x^{3} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{3} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $\\frac{3 \\pi \\left(2 \\pi + 12 + 15 \\pi^{3}\\right)}{4}$.\nFinal answer: $\\frac{3\\cdot \\pi\\cdot \\left(2\\cdot \\pi + 12 + 15\\cdot \\pi^{3}\\right)}{4}$", "final_answer": "\\frac{3\\cdot \\pi\\cdot \\left(2\\cdot \\pi + 12 + 15\\cdot \\pi^{3}\\right)}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bd5148403bcef9c4", "problem_id": "988132d1899c4697", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{2 \\sqrt{11} x}{3} - \\frac{11 x}{15} + \\frac{2}{15} + \\frac{4 \\sqrt{11}}{15}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{2 \\sqrt{11}}{3} - \\frac{11}{15}$, c=$\\frac{2}{15} + \\frac{4 \\sqrt{11}}{15}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{367}{75} - \\frac{4 \\sqrt{11}}{45}$, so √Δ = $\\sqrt{\\frac{367}{75} - \\frac{4 \\sqrt{11}}{45}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{11}{30} + \\frac{\\sqrt{1101 - 20 \\sqrt{11}}}{30} + \\frac{\\sqrt{11}}{3}$.\n7) x2 = $- \\frac{\\sqrt{1101 - 20 \\sqrt{11}}}{30} + \\frac{11}{30} + \\frac{\\sqrt{11}}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{11}{30} + \\frac{\\sqrt{1101 - 20\\cdot \\sqrt{11}}}{30} + \\frac{\\sqrt{11}}{3}, - \\frac{\\sqrt{1101 - 20\\cdot \\sqrt{11}}}{30} + \\frac{11}{30} + \\frac{\\sqrt{11}}{3}\\}$", "final_answer": "x \\in \\{\\frac{11}{30} + \\frac{\\sqrt{1101 - 20\\cdot \\sqrt{11}}}{30} + \\frac{\\sqrt{11}}{3}, - \\frac{\\sqrt{1101 - 20\\cdot \\sqrt{11}}}{30} + \\frac{11}{30} + \\frac{\\sqrt{11}}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4970084014a3473d", "problem_id": "988132d1899c4697", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{2 \\sqrt{11} x}{3} - \\frac{11 x}{15} + \\frac{2}{15} + \\frac{4 \\sqrt{11}}{15}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{2 \\sqrt{11}}{3} - \\frac{11}{15}$, c=$\\frac{2}{15} + \\frac{4 \\sqrt{11}}{15}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{367}{75} - \\frac{4 \\sqrt{11}}{45}$, so √Δ = $\\sqrt{\\frac{367}{75} - \\frac{4 \\sqrt{11}}{45}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{11}{30} + \\frac{\\sqrt{1101 - 20 \\sqrt{11}}}{30} + \\frac{\\sqrt{11}}{3}$.\nx2 = $- \\frac{\\sqrt{1101 - 20 \\sqrt{11}}}{30} + \\frac{11}{30} + \\frac{\\sqrt{11}}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{11}{30} + \\frac{\\sqrt{1101 - 20\\cdot \\sqrt{11}}}{30} + \\frac{\\sqrt{11}}{3}, - \\frac{\\sqrt{1101 - 20\\cdot \\sqrt{11}}}{30} + \\frac{11}{30} + \\frac{\\sqrt{11}}{3}\\}$", "final_answer": "x \\in \\{\\frac{11}{30} + \\frac{\\sqrt{1101 - 20\\cdot \\sqrt{11}}}{30} + \\frac{\\sqrt{11}}{3}, - \\frac{\\sqrt{1101 - 20\\cdot \\sqrt{11}}}{30} + \\frac{11}{30} + \\frac{\\sqrt{11}}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "174334242ec9564b", "problem_id": "ad0e1bd0fab2a83a", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - 3 x^{2} + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)}$ at $x=2$.", "output": "Differentiate: f'(x) = $- 9 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)}$.\nSlope at x=2: m = f'(2) = $-48 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\nPoint on curve: (2, f(2)) = (2, $-36 + \\cos{\\left(4 \\right)} + e$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -36 + \\cos{\\left(4 \\right)} + e = -48 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\nOptionally expand: y = $- 48 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 60$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 48\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 60$", "final_answer": "y = - 48\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 60", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2bb8a65cf9848123", "problem_id": "ad0e1bd0fab2a83a", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - 3 x^{2} + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)}$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 9 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)}$.\n2) Slope at x=2: m = f'(2) = $-48 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\n3) Point on curve: (2, f(2)) = (2, $-36 + \\cos{\\left(4 \\right)} + e$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -36 + \\cos{\\left(4 \\right)} + e = -48 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\n6) Optionally expand: y = $- 48 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 60$.\nFinal answer: $y = - 48\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 60$", "final_answer": "y = - 48\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 60", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "78964c1c16b4cacf", "problem_id": "91f4506bed4f1387", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "An urn has 13 red balls and 14 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(27,11).\nFavorable samples: choose k reds and n−k blues: C(13,2)·C(14,9).\nFinal answer: $\\frac{4004}{334305}$", "final_answer": "\\frac{4004}{334305}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5635f67d146f0bd5", "problem_id": "91f4506bed4f1387", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "An urn has 13 red balls and 14 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(27,11).\n3) Favorable samples: choose k reds and n−k blues: C(13,2)·C(14,9).\n4) Compute totals: C(27,11) = 13037895, and favorable = 156156.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{4004}{334305}$", "final_answer": "\\frac{4004}{334305}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a509d5037963a960", "problem_id": "91bcf2820c73f844", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡1 (mod 4), x≡1 (mod 7). Give the solution modulo 28.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 28.\n3) Compute the CRT solution to get x ≡ 1 (mod 28).\nFinal answer: $x\\equiv 1\\pmod{28}$", "final_answer": "x\\equiv 1\\pmod{28}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dfb781d1b7c3c8a6", "problem_id": "91bcf2820c73f844", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡1 (mod 4), x≡1 (mod 7). Give the solution modulo 28.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 28.\nCompute the CRT solution to get x ≡ 1 (mod 28).\nCheck: 1 mod 4=1 and mod 7=1.\nFinal answer: $x\\equiv 1\\pmod{28}$", "final_answer": "x\\equiv 1\\pmod{28}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a07b9a1449d85277", "problem_id": "815871738e4ccb3f", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} - 3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1$.\n2) Evaluate at the bounds: F(2 \\pi) and F(0).\n3) Subtract: F(2 \\pi) − F(0) = $2 \\pi \\left(- 4 \\pi^{2} - 3 \\pi + 1\\right)$.\n4) Simplify the result.\nFinal answer: $2\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} - 3\\cdot \\pi + 1\\right)$", "final_answer": "2\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} - 3\\cdot \\pi + 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6accb85ed293649e", "problem_id": "815871738e4ccb3f", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} - 3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1$.\nEvaluate at the bounds: F(2 \\pi) and F(0).\nSubtract: F(2 \\pi) − F(0) = $2 \\pi \\left(- 4 \\pi^{2} - 3 \\pi + 1\\right)$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $2\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} - 3\\cdot \\pi + 1\\right)$", "final_answer": "2\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} - 3\\cdot \\pi + 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9cb9140305ae91b6", "problem_id": "43495be8e9653dbf", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Worker A finishes a job in 4 hours and Worker B finishes it in 6 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/4 job/hour. B's rate = 1/6 job/hour.\n2) Combined rate = 1/4 + 1/6 = 5/12 job/hour.\n3) Time = 1 ÷ (combined rate) = 12/5 hours.\nFinal answer: $\\frac{12}{5}$", "final_answer": "\\frac{12}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "406fc25a8f5f7be4", "problem_id": "43495be8e9653dbf", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Worker A finishes a job in 4 hours and Worker B finishes it in 6 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/4 job/hour. B's rate = 1/6 job/hour.\nCombined rate = 1/4 + 1/6 = 5/12 job/hour.\nTime = 1 ÷ (combined rate) = 12/5 hours.\nFinal answer: $\\frac{12}{5}$", "final_answer": "\\frac{12}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ce4baf909a056040", "problem_id": "94e35b34faab03bf", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$- 3 x - y$ = $8$\n$- x + y$ = $4$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $3 x + y$ = $-8$.\n3) Multiply the second equation by $-3$: $3 x - 3 y$ = $-12$.\n4) Subtract to eliminate x: $4 y$ = $4$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, 1\\right)$", "final_answer": "(x,y)=\\left(-3, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6ccc43582ee62174", "problem_id": "94e35b34faab03bf", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the linear system:\n$- 3 x - y$ = $8$\n$- x + y$ = $4$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $3 x + y$ = $-8$.\nMultiply the second equation by $-3$: $3 x - 3 y$ = $-12$.\nFinal answer: $(x,y)=\\left(-3, 1\\right)$", "final_answer": "(x,y)=\\left(-3, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "155617e06777d376", "problem_id": "61597971a8e26afc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - x^{2} - 2 x - 3$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} - 3 x - 3.\n3) Then the limit is -21.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: -18.\nFinal answer: $-18$", "final_answer": "-18", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "41f1b08b1c712602", "problem_id": "c822293f20c06925", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Worker A finishes a job in 17 hours and Worker B finishes it in 20 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/17 job/hour. B's rate = 1/20 job/hour.\n2) Combined rate = 1/17 + 1/20 = 37/340 job/hour.\n3) Time = 1 ÷ (combined rate) = 340/37 hours.\nFinal answer: $\\frac{340}{37}$", "final_answer": "\\frac{340}{37}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6567a4282213c60d", "problem_id": "b2a46ce958ab7f40", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A store sells a backpack for $82. It is discounted by 20%. You buy 8 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 82 × (1 − 20/100) = 82 × 80/100 = 328/5.\n2) Subtotal = 8 × 328/5 = 2624/5.\n3) Total with tax = 2624/5 × 105/100 = 13776/25.\nFinal answer: $\\frac{13776}{25}$", "final_answer": "\\frac{13776}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72138a9dd05b6e1c", "problem_id": "399ece830e72a8be", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - 2 x^{3} - 5 x^{2} - 21 x - 83$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{3} - 5 x^{2} - 22 x - 83.\n3) Then the limit is -379.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: -375.\nFinal answer: $-375$", "final_answer": "-375", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5fb972d5fd5a8f8f", "problem_id": "ad4dfe43a139295a", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 13 red balls and 11 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(24,8).\nFavorable samples: choose k reds and n−k blues: C(13,1)·C(11,7).\nCompute totals: C(24,8) = 735471, and favorable = 4290.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{130}{22287}$.\nFinal answer: $\\frac{130}{22287}$", "final_answer": "\\frac{130}{22287}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5f0c88e29fe3b1c6", "problem_id": "ad4dfe43a139295a", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "An urn has 13 red balls and 11 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(24,8).\n3) Favorable samples: choose k reds and n−k blues: C(13,1)·C(11,7).\n4) Compute totals: C(24,8) = 735471, and favorable = 4290.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{130}{22287}$", "final_answer": "\\frac{130}{22287}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "13d56d91de510227", "problem_id": "34e5a758da7f92fc", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} - 2 x^{4} - 3 x^{3} - 2 x^{2} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{4} - 3 x^{3} - 2 x^{2} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{33 \\pi^{5}}{80} - \\frac{3 \\pi^{3}}{4} - \\frac{3 \\pi^{2}}{2} + 5 + 3 \\pi + \\frac{45 \\pi^{4}}{64}$.\nFinal answer: $- \\frac{33\\cdot \\pi^{5}}{80} - \\frac{3\\cdot \\pi^{3}}{4} - \\frac{3\\cdot \\pi^{2}}{2} + 5 + 3\\cdot \\pi + \\frac{45\\cdot \\pi^{4}}{64}$", "final_answer": "- \\frac{33\\cdot \\pi^{5}}{80} - \\frac{3\\cdot \\pi^{3}}{4} - \\frac{3\\cdot \\pi^{2}}{2} + 5 + 3\\cdot \\pi + \\frac{45\\cdot \\pi^{4}}{64}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ef713565fdc44fde", "problem_id": "34e5a758da7f92fc", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} - 2 x^{4} - 3 x^{3} - 2 x^{2} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{4} - 3 x^{3} - 2 x^{2} + 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{33 \\pi^{5}}{80} - \\frac{3 \\pi^{3}}{4} - \\frac{3 \\pi^{2}}{2} + 5 + 3 \\pi + \\frac{45 \\pi^{4}}{64}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{33\\cdot \\pi^{5}}{80} - \\frac{3\\cdot \\pi^{3}}{4} - \\frac{3\\cdot \\pi^{2}}{2} + 5 + 3\\cdot \\pi + \\frac{45\\cdot \\pi^{4}}{64}$", "final_answer": "- \\frac{33\\cdot \\pi^{5}}{80} - \\frac{3\\cdot \\pi^{3}}{4} - \\frac{3\\cdot \\pi^{2}}{2} + 5 + 3\\cdot \\pi + \\frac{45\\cdot \\pi^{4}}{64}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f8c58e7dd7f63800", "problem_id": "f429bbaf4fd2c334", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$- 4 x + 9 y$ = $69$\n$- x + 2 y$ = $15$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $4 x - 9 y$ = $-69$.\n3) Multiply the second equation by $-4$: $4 x - 8 y$ = $-60$.\n4) Subtract to eliminate x: $- y$ = $-9$.\n5) Solve for y: y = $9$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, 9\\right)$", "final_answer": "(x,y)=\\left(3, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "78336a977edb476a", "problem_id": "f429bbaf4fd2c334", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the linear system:\n$- 4 x + 9 y$ = $69$\n$- x + 2 y$ = $15$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $4 x - 9 y$ = $-69$.\nMultiply the second equation by $-4$: $4 x - 8 y$ = $-60$.\nFinal answer: $(x,y)=\\left(3, 9\\right)$", "final_answer": "(x,y)=\\left(3, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "556f6ec130b023ba", "problem_id": "5ee675c3b0f00cc7", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 3 x^{3} + 9 x^{2} - 24 x + 70$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{3} + 9 x^{2} - 25 x + 70.\n3) Then the limit is 307.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: 304.\nFinal answer: $304$", "final_answer": "304", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8e729901fd64a825", "problem_id": "4bba580fb5bc4b89", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the multiplicative inverse of 18 modulo 31.", "output": "1) We want x such that 18x ≡ 1 (mod 31).\n2) Use extended Euclid to find s,t with 18s + 31t = 1.\n3) One Bezout identity is: 18·(-12) + 31·(7) = 1.\n4) Reduce s modulo 31: x ≡ 19 (mod 31).\nFinal answer: $x\\equiv 19\\pmod{31}$", "final_answer": "x\\equiv 19\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0a100187b2bbcda9", "problem_id": "0d100a38039c2985", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Two friends split $1296 in the ratio 5:3. How much does each receive?", "output": "1) Total ratio parts = 5+3 = 8.\n2) Each part = 1296 ÷ 8 = 162.\n3) First share = 5 × 162 = 810.\n4) Second share = 3 × 162 = 486.\nFinal answer: $486$", "final_answer": "486", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bc6eefcb988fcee2", "problem_id": "6bbf3d3938c041e5", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} x^{3} + x^{2} + 7 x + 25$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to x^{3} + x^{2} + 6 x + 25.\n3) Then the limit is 129.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: 133.\nFinal answer: $133$", "final_answer": "133", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e0c356967084eb50", "problem_id": "ffb89c5ca57e0a18", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "You want 58 liters of a 30% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 58−x liters is 70% solution.\n2) Salt equation: 20x + 70(58−x) = 30·58.\n3) (20-70)x = (30-70)·58 ⇒ x = 232/5.\n4) Then 58−x = 58/5.\nFinal answer: $232/5 L of 20% and 58/5 L of 70%$", "final_answer": "232/5 L of 20% and 58/5 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "53987ac70810f420", "problem_id": "b708677091c5e5b4", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(2 x^{3} - x^{2} + x - 2\\right) \\cos{\\left(2 x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 2 \\cos{\\left(2 x + 3 \\right)}) = 2 \\sin{\\left(2 x + 3 \\right)}$.\n3) So $f'(x) = - 4 x^{3} \\sin{\\left(2 x + 3 \\right)} + 2 x^{2} \\sin{\\left(2 x + 3 \\right)} + 6 x^{2} \\cos{\\left(2 x + 3 \\right)} + 2 x e^{x} - 2 x \\sin{\\left(2 x + 3 \\right)} - 2 x \\cos{\\left(2 x + 3 \\right)} + 6 e^{x} + 2 \\sin{\\left(2 x + 3 \\right)} + \\cos{\\left(2 x + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 2 \\cos{\\left(2 x + 3 \\right)}) = 4 \\sin{\\left(2 x + 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x + 2\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) + 1\\right) - 2\\right) \\sin{\\left(2 x + 3 \\right)} + \\left(6 x^{2} - 2 x + 1\\right) \\cos{\\left(2 x + 3 \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(x + 2\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) + 1\\right) - 2\\right) \\sin{\\left(2 x + 3 \\right)} + \\left(6 x^{2} - 2 x + 1\\right) \\cos{\\left(2 x + 3 \\right)} + 2 e^{x}$", "final_answer": "2 \\left(x + 2\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) + 1\\right) - 2\\right) \\sin{\\left(2 x + 3 \\right)} + \\left(6 x^{2} - 2 x + 1\\right) \\cos{\\left(2 x + 3 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "34f21cef4bb57416", "problem_id": "a364f72434d525c8", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Two friends split $420 in the ratio 4:6. How much does each receive?", "output": "Total ratio parts = 4+6 = 10.\nEach part = 420 ÷ 10 = 42.\nFirst share = 4 × 42 = 168.\nFinal answer: $252$", "final_answer": "252", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "afec07867bce5725", "problem_id": "a364f72434d525c8", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Two friends split $420 in the ratio 4:6. How much does each receive?", "output": "1) Total ratio parts = 4+6 = 10.\n2) Each part = 420 ÷ 10 = 42.\n3) First share = 4 × 42 = 168.\n4) Second share = 6 × 42 = 252.\nFinal answer: $252$", "final_answer": "252", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f05d7a8d6b51bd7d", "problem_id": "fd37383f123c3281", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many binary strings of length 23 contain exactly 8 ones with no two ones adjacent?", "output": "Reserve 7 mandatory zeros between the 8 ones.\nThis leaves n−k+1 = 16 valid slots for the ones.\nChoose the slots: C(16,8) = 12870.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $12870$", "final_answer": "12870", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "56c2868b1ee196d3", "problem_id": "fd37383f123c3281", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "How many binary strings of length 23 contain exactly 8 ones with no two ones adjacent?", "output": "Reserve 7 mandatory zeros between the 8 ones.\nThis leaves n−k+1 = 16 valid slots for the ones.\nChoose the slots: C(16,8) = 12870.\nFinal answer: $12870$", "final_answer": "12870", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6816efb9aaaccf7a", "problem_id": "431439e6e8cb6cda", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - \\frac{11 x}{2} + 2$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$- \\frac{11}{2}$, c=$2$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{25}{4}$, so √Δ = $\\frac{5}{2}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{4}{3}$.\n7) x2 = $\\frac{1}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{4}{3}, \\frac{1}{2}\\}$", "final_answer": "x \\in \\{\\frac{4}{3}, \\frac{1}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b5a8c6ebceed7f07", "problem_id": "431439e6e8cb6cda", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - \\frac{11 x}{2} + 2$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$- \\frac{11}{2}$, c=$2$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{25}{4}$, so √Δ = $\\frac{5}{2}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{4}{3}$.\nx2 = $\\frac{1}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{4}{3}, \\frac{1}{2}\\}$", "final_answer": "x \\in \\{\\frac{4}{3}, \\frac{1}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d2b8fee933e4848c", "problem_id": "cbc3630026ba10d8", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$- 3 x + 2 y$ = $-4$\n$x$ = $-2$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $- 3 x + 2 y$ = $-4$.\n3) Multiply the second equation by $-3$: $- 3 x$ = $6$.\n4) Subtract to eliminate x: $2 y$ = $-10$.\n5) Solve for y: y = $-5$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, -5\\right)$", "final_answer": "(x,y)=\\left(-2, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "42cef6eafa4be954", "problem_id": "cbc3630026ba10d8", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$- 3 x + 2 y$ = $-4$\n$x$ = $-2$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $- 3 x + 2 y$ = $-4$.\nMultiply the second equation by $-3$: $- 3 x$ = $6$.\nSubtract to eliminate x: $2 y$ = $-10$.\nSolve for y: y = $-5$.\nSubstitute back to find x: x = $-2$.\nCheck: plug (x,y)=(-2,-5) into both equations.\nFinal answer: $(x,y)=\\left(-2, -5\\right)$", "final_answer": "(x,y)=\\left(-2, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dc25a6f998619c81", "problem_id": "49837cfc3bf25074", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - 2 x - 6$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to - 3 x - 6.\n3) Then the limit is -15.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: -12.\nFinal answer: $-12$", "final_answer": "-12", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "98f77ecfecd3fa5d", "problem_id": "a0c7f35c42eba568", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} 4 x^{4} + 3 x^{3} - 4 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $4 x^{4} + 3 x^{3} - 4 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1$.\n2) Evaluate at the bounds: F(\\pi) and F(0).\n3) Subtract: F(\\pi) − F(0) = $\\frac{\\pi \\left(- 80 \\pi^{2} - 90 \\pi - 60 + 45 \\pi^{3} + 48 \\pi^{4}\\right)}{60}$.\n4) Simplify the result.\nFinal answer: $\\frac{\\pi\\cdot \\left(- 80\\cdot \\pi^{2} - 90\\cdot \\pi - 60 + 45\\cdot \\pi^{3} + 48\\cdot \\pi^{4}\\right)}{60}$", "final_answer": "\\frac{\\pi\\cdot \\left(- 80\\cdot \\pi^{2} - 90\\cdot \\pi - 60 + 45\\cdot \\pi^{3} + 48\\cdot \\pi^{4}\\right)}{60}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0c865c2f8a59ed11", "problem_id": "d2c3f52a0804d2ac", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Two friends split $1608 in the ratio 5:7. How much does each receive?", "output": "1) Total ratio parts = 5+7 = 12.\n2) Each part = 1608 ÷ 12 = 134.\n3) First share = 5 × 134 = 670.\n4) Second share = 7 × 134 = 938.\nFinal answer: $938$", "final_answer": "938", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "711e064ec302b188", "problem_id": "d2c3f52a0804d2ac", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $1608 in the ratio 5:7. How much does each receive?", "output": "Total ratio parts = 5+7 = 12.\nEach part = 1608 ÷ 12 = 134.\nFirst share = 5 × 134 = 670.\nSecond share = 7 × 134 = 938.\nCheck: 670+938=1608.\nFinal answer: $938$", "final_answer": "938", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f1a90524f619d119", "problem_id": "57847887b88135a7", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "An urn has 23 red balls and 21 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(44,8).\n3) Favorable samples: choose k reds and n−k blues: C(23,2)·C(21,6).\n4) Compute totals: C(44,8) = 177232627, and favorable = 13728792.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{65688}{848003}$", "final_answer": "\\frac{65688}{848003}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bc3a0e03fde49672", "problem_id": "57847887b88135a7", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 23 red balls and 21 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(44,8).\nFavorable samples: choose k reds and n−k blues: C(23,2)·C(21,6).\nCompute totals: C(44,8) = 177232627, and favorable = 13728792.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{65688}{848003}$.\nFinal answer: $\\frac{65688}{848003}$", "final_answer": "\\frac{65688}{848003}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d6eab05db962fac2", "problem_id": "bae1c69538978b13", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Two friends split $648 in the ratio 1:3. How much does each receive?", "output": "1) Total ratio parts = 1+3 = 4.\n2) Each part = 648 ÷ 4 = 162.\n3) First share = 1 × 162 = 162.\n4) Second share = 3 × 162 = 486.\nFinal answer: $486$", "final_answer": "486", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bb9a5f869754aa0f", "problem_id": "bae1c69538978b13", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Two friends split $648 in the ratio 1:3. How much does each receive?", "output": "Total ratio parts = 1+3 = 4.\nEach part = 648 ÷ 4 = 162.\nFirst share = 1 × 162 = 162.\nFinal answer: $486$", "final_answer": "486", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ad992c73e7630e4e", "problem_id": "1d09f1980dd6bc70", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=31$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=31 and k=5.\n4) Compute C(35,4) = 52360.\nFinal answer: $52360$", "final_answer": "52360", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d5425c36f8e1effd", "problem_id": "4c2405f43e56810f", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 3 x - 4$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to 2 x - 4.\n3) Then the limit is -6.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: -7.\nFinal answer: $-7$", "final_answer": "-7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a4fc33a61fe23050", "problem_id": "dbce9d9acab91b35", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 13 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=13, k=3, p=$\\frac{2}{3}$ and simplify.\nFinal answer: $\\frac{2288}{1594323}$", "final_answer": "\\frac{2288}{1594323}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5e644c445fa1789e", "problem_id": "dbce9d9acab91b35", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 13 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=13, k=3, p=$\\frac{2}{3}$ and simplify.\n4) Here C(13,3) = 286.\nFinal answer: $\\frac{2288}{1594323}$", "final_answer": "\\frac{2288}{1594323}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "44cd7614cab4540c", "problem_id": "6938d8d343661ecb", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 15 red balls and 13 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(28,8).\n3) Favorable samples: choose k reds and n−k blues: C(15,1)·C(13,7).\n4) Compute totals: C(28,8) = 3108105, and favorable = 25740.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{4}{483}$", "final_answer": "\\frac{4}{483}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fe06e44fa4b30df3", "problem_id": "6938d8d343661ecb", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 15 red balls and 13 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(28,8).\nFavorable samples: choose k reds and n−k blues: C(15,1)·C(13,7).\nCompute totals: C(28,8) = 3108105, and favorable = 25740.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{4}{483}$.\nFinal answer: $\\frac{4}{483}$", "final_answer": "\\frac{4}{483}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c639ba5ecad9aa5b", "problem_id": "4d03021993636817", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{4} + 4 x^{2} - 2 x + 3\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 3 x^{4} + 4 x^{2} - 2 x + 3$.\nCompute an antiderivative: F(x) = $- \\frac{3 x^{5}}{5} + \\frac{4 x^{3}}{3} - x^{2} + 3 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - \\frac{3\\cdot x^{5}}{5} + \\frac{4\\cdot x^{3}}{3} - x^{2} + 3\\cdot x$", "final_answer": "C - \\frac{3\\cdot x^{5}}{5} + \\frac{4\\cdot x^{3}}{3} - x^{2} + 3\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e41d23b703486c01", "problem_id": "4d03021993636817", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{4} + 4 x^{2} - 2 x + 3\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 3 x^{4} + 4 x^{2} - 2 x + 3$.\n3) Compute an antiderivative: F(x) = $- \\frac{3 x^{5}}{5} + \\frac{4 x^{3}}{3} - x^{2} + 3 x$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{3\\cdot x^{5}}{5} + \\frac{4\\cdot x^{3}}{3} - x^{2} + 3\\cdot x$", "final_answer": "C - \\frac{3\\cdot x^{5}}{5} + \\frac{4\\cdot x^{3}}{3} - x^{2} + 3\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2296887075eae8a1", "problem_id": "c501f6c1f86284d1", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - x^{3} + 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- x^{3} + 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1$.\n3) Compute an antiderivative: F(x) = $- \\frac{x^{4}}{4} + \\frac{2 x^{3}}{3} + \\frac{3 x^{2}}{2} - x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{x^{4}}{4} + \\frac{2\\cdot x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{x^{4}}{4} + \\frac{2\\cdot x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "053495817596058b", "problem_id": "c501f6c1f86284d1", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - x^{3} + 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- x^{3} + 2 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1$.\nCompute an antiderivative: F(x) = $- \\frac{x^{4}}{4} + \\frac{2 x^{3}}{3} + \\frac{3 x^{2}}{2} - x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $C - \\frac{x^{4}}{4} + \\frac{2\\cdot x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{x^{4}}{4} + \\frac{2\\cdot x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "79ae182d170f80e7", "problem_id": "e5ed28071b92e7e3", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "An urn has 26 red balls and 23 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(49,11).\n3) Favorable samples: choose k reds and n−k blues: C(26,2)·C(23,9).\n4) Compute totals: C(49,11) = 29135916264, and favorable = 265586750.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{444125}{48722268}$", "final_answer": "\\frac{444125}{48722268}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "343c249f2bc0de8a", "problem_id": "e5ed28071b92e7e3", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 26 red balls and 23 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(49,11).\nFavorable samples: choose k reds and n−k blues: C(26,2)·C(23,9).\nCompute totals: C(49,11) = 29135916264, and favorable = 265586750.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{444125}{48722268}$.\nFinal answer: $\\frac{444125}{48722268}$", "final_answer": "\\frac{444125}{48722268}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8d665595a2260bcc", "problem_id": "b6f250b72967a5f9", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} + 3 x^{2} + x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)}$ at $x=-1$.", "output": "Differentiate: f'(x) = $9 x^{2} + 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 1$.\nSlope at x=-1: m = f'(-1) = $\\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)} + 4$.\nPoint on curve: (-1, f(-1)) = (-1, $-1 + \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -1 + \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} = \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)} + 4(x − -1).\nOptionally expand: y = $\\frac{x}{2 e^{\\frac{1}{2}}} + 2 x \\sin{\\left(2 \\right)} + 4 x + \\cos{\\left(2 \\right)} + \\frac{3}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)} + 3$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + 4\\cdot x + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)} + 3$", "final_answer": "y = \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + 4\\cdot x + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)} + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2fa44281ca4743ff", "problem_id": "da68ca0158e4bdcc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{4} - 2 x^{2} - 2 x + 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x) = - 2 x$.\n3) So $f'(x) = - 4 x^{3} - 6 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x) = -2$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- 2 x^{3} - 2 x - 1\\right)$.\nFinal answer: $2 \\left(- 2 x^{3} - 2 x - 1\\right)$", "final_answer": "2 \\left(- 2 x^{3} - 2 x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8e4c332daea765ad", "problem_id": "c9595881da58dcee", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the linear system:\n$- 3 x - 4 y$ = $-24$\n$- x + y$ = $-1$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $3 x + 4 y$ = $24$.\nMultiply the second equation by $-3$: $3 x - 3 y$ = $3$.\nFinal answer: $(x,y)=\\left(4, 3\\right)$", "final_answer": "(x,y)=\\left(4, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "118ae7b04e43255a", "problem_id": "c9595881da58dcee", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$- 3 x - 4 y$ = $-24$\n$- x + y$ = $-1$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $3 x + 4 y$ = $24$.\nMultiply the second equation by $-3$: $3 x - 3 y$ = $3$.\nSubtract to eliminate x: $7 y$ = $21$.\nSolve for y: y = $3$.\nSubstitute back to find x: x = $4$.\nCheck: plug (x,y)=(4,3) into both equations.\nFinal answer: $(x,y)=\\left(4, 3\\right)$", "final_answer": "(x,y)=\\left(4, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c5d4ae6965af2cb1", "problem_id": "8ace5bf22ab7eae3", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{3} - x^{2} - x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1}{x^{2} + 2}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 2}) = \\frac{2 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 2}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 8 x^{2} + 8} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 8 x^{2} + 8} - \\frac{6 x^{4}}{2 x^{4} + 8 x^{2} + 8} + \\frac{4 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 8 x^{2} + 8} + \\frac{8 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 8 x^{2} + 8} - \\frac{34 x^{2}}{2 x^{4} + 8 x^{2} + 8} - \\frac{4 x}{2 x^{4} + 8 x^{2} + 8} + \\frac{4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 4 x^{2} + 4} + \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 8 x^{2} + 8} + \\frac{8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 8 x^{2} + 8} - \\frac{4}{2 x^{4} + 8 x^{2} + 8} + \\frac{2 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 2} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 2} - \\frac{2 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 2}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 2}) = - \\frac{4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 2\\right)^{2}} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 2} + \\frac{2 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 2}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 4 x \\left(- 3 x^{3} - x^{2} - x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) + \\left(x^{2} + 2\\right) \\left(- 18 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4 x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x^{2} + 2\\right)^{2}}$.\nFinal answer: $\\frac{- 4 x \\left(- 3 x^{3} - x^{2} - x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) + \\left(x^{2} + 2\\right) \\left(- 18 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4 x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x^{2} + 2\\right)^{2}}$", "final_answer": "\\frac{- 4 x \\left(- 3 x^{3} - x^{2} - x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) + \\left(x^{2} + 2\\right) \\left(- 18 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4 x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x^{2} + 2\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3b9e75db42037cc2", "problem_id": "038f3965ccfca8ac", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡6 (mod 11), x≡0 (mod 7). Give the solution modulo 77.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 77.\n3) Compute the CRT solution to get x ≡ 28 (mod 77).\nFinal answer: $x\\equiv 28\\pmod{77}$", "final_answer": "x\\equiv 28\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a98cee44b89183f6", "problem_id": "331742ad7366406c", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 3 x^{2} - 5 x - 12$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 3 x^{3} + x^{2} - 2 x + 2$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-34$", "final_answer": "-34", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4359108a387f0960", "problem_id": "331742ad7366406c", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 3 x^{2} - 5 x - 12$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 3 x^{3} + x^{2} - 2 x + 2$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 9 x^{2} + 2 x - 2$.\nEvaluate at a: p'(2) = $-34$.\nCheck: the identity gives $-34$.\nFinal answer: $-34$", "final_answer": "-34", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2589a6230b046a3a", "problem_id": "16332d235e75d4ca", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A cyclist rides at 74 miles per hour for 1/2 hours, then at 40 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 74 × 1/2 = 37 miles.\nSecond leg distance = 40 × 5/2 = 100 miles.\nTotal distance = 37 + 100 = 137 miles.\nCheck: adding both legs gives 137.\nFinal answer: $137$", "final_answer": "137", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c41767f2955daf16", "problem_id": "16332d235e75d4ca", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A cyclist rides at 74 miles per hour for 1/2 hours, then at 40 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 74 × 1/2 = 37 miles.\n2) Second leg distance = 40 × 5/2 = 100 miles.\n3) Total distance = 37 + 100 = 137 miles.\nFinal answer: $137$", "final_answer": "137", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "726f59dbfb52b274", "problem_id": "a04dfc12c636f2ab", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Two friends split $1332 in the ratio 1:8. How much does each receive?", "output": "Total ratio parts = 1+8 = 9.\nEach part = 1332 ÷ 9 = 148.\nFirst share = 1 × 148 = 148.\nFinal answer: $1184$", "final_answer": "1184", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ce4f3315eea70c57", "problem_id": "a04dfc12c636f2ab", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Two friends split $1332 in the ratio 1:8. How much does each receive?", "output": "1) Total ratio parts = 1+8 = 9.\n2) Each part = 1332 ÷ 9 = 148.\n3) First share = 1 × 148 = 148.\n4) Second share = 8 × 148 = 1184.\nFinal answer: $1184$", "final_answer": "1184", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e49d25c1c9f5c1d5", "problem_id": "3c5d640b4e777cd7", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A store sells a backpack for $440. It is discounted by 10%. You buy 10 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 440 × (1 − 10/100) = 440 × 90/100 = 396.\n2) Subtotal = 10 × 396 = 3960.\n3) Total with tax = 3960 × 107/100 = 21186/5.\nFinal answer: $\\frac{21186}{5}$", "final_answer": "\\frac{21186}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dd43aa92507b81e8", "problem_id": "3c5d640b4e777cd7", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A store sells a backpack for $440. It is discounted by 10%. You buy 10 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 440 × (1 − 10/100) = 440 × 90/100 = 396.\nSubtotal = 10 × 396 = 3960.\nTotal with tax = 3960 × 107/100 = 21186/5.\nFinal answer: $\\frac{21186}{5}$", "final_answer": "\\frac{21186}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "769fe2d677d34629", "problem_id": "5dd51a4812113ce8", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Worker A finishes a job in 15 hours and Worker B finishes it in 17 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/15 job/hour. B's rate = 1/17 job/hour.\n2) Combined rate = 1/15 + 1/17 = 32/255 job/hour.\n3) Time = 1 ÷ (combined rate) = 255/32 hours.\nFinal answer: $\\frac{255}{32}$", "final_answer": "\\frac{255}{32}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a3d31396197da7ca", "problem_id": "0954e66808e05141", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(2 x^{3} - x^{2}\\right) \\cos{\\left(3 x^{2} + 2 x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x^{2} \\cos{\\left(\\left(3 x^{2} + 2 x\\right) + 3 \\right)}) = x^{2} \\sin{\\left(3 x^{2} + 2 x + 3 \\right)}$.\n3) So $f'(x) = - 12 x^{4} \\sin{\\left(\\left(3 x^{2} + 2 x\\right) + 3 \\right)} - 4 x^{3} \\sin{\\left(\\left(3 x^{2} + 2 x\\right) + 3 \\right)} + x^{2} \\sin{\\left(3 x^{2} + 2 x + 3 \\right)} + 6 x^{2} \\cos{\\left(\\left(3 x^{2} + 2 x\\right) + 3 \\right)} + 2 x e^{x} + 6 e^{x}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x^{2} \\cos{\\left(\\left(3 x^{2} + 2 x\\right) + 3 \\right)}) = x^{2} \\left(6 x + 2\\right) \\sin{\\left(\\left(3 x^{2} + 2 x\\right) + 3 \\right)} - 2 x \\cos{\\left(\\left(3 x^{2} + 2 x\\right) + 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- x^{2} \\left(2 x - 1\\right) \\left(3 x + 1\\right) \\sin{\\left(\\left(3 x^{2} + 2 x\\right) + 3 \\right)} + x \\left(3 x - 1\\right) \\cos{\\left(\\left(3 x^{2} + 2 x\\right) + 3 \\right)} + \\left(x + 2\\right) e^{x} + e^{x}\\right)$.\nFinal answer: $2 \\left(- x^{2} \\left(2 x - 1\\right) \\left(3 x + 1\\right) \\sin{\\left(\\left(3 x^{2} + 2 x\\right) + 3 \\right)} + x \\left(3 x - 1\\right) \\cos{\\left(\\left(3 x^{2} + 2 x\\right) + 3 \\right)} + \\left(x + 2\\right) e^{x} + e^{x}\\right)$", "final_answer": "2 \\left(- x^{2} \\left(2 x - 1\\right) \\left(3 x + 1\\right) \\sin{\\left(\\left(3 x^{2} + 2 x\\right) + 3 \\right)} + x \\left(3 x - 1\\right) \\cos{\\left(\\left(3 x^{2} + 2 x\\right) + 3 \\right)} + \\left(x + 2\\right) e^{x} + e^{x}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "713e06346bc5ccff", "problem_id": "5d1d9aee6cf3e8b1", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 2 x^{2} + 8 x - 21$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{2} + 7 x - 21.\n3) Then the limit is -60.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -63.\nFinal answer: $-63$", "final_answer": "-63", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "58157bbe1e27d332", "problem_id": "04ceb8b46ffa402d", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "An urn has 14 red balls and 14 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(28,7).\n3) Favorable samples: choose k reds and n−k blues: C(14,3)·C(14,4).\n4) Compute totals: C(28,7) = 1184040, and favorable = 364364.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{637}{2070}$", "final_answer": "\\frac{637}{2070}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f7aa1b71d925ba6b", "problem_id": "5275780ed3aed0b3", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int x^{2} + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $x^{2} + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1$.\n3) Compute an antiderivative: F(x) = $\\frac{x^{3}}{3} - x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{x^{3}}{3} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{3}}{3} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "908784b427c7d1b2", "problem_id": "5275780ed3aed0b3", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int x^{2} + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $x^{2} + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 1$.\nCompute an antiderivative: F(x) = $\\frac{x^{3}}{3} - x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{x^{3}}{3} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{3}}{3} - x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "63ca5d217750ce25", "problem_id": "988e81489daf95db", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 20 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=20, k=4, p=$\\frac{1}{2}$ and simplify.\n4) Here C(20,4) = 4845.\nFinal answer: $\\frac{4845}{1048576}$", "final_answer": "\\frac{4845}{1048576}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7d10fcca9a343dec", "problem_id": "a296c28adb8b96c6", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 4 - 3 x$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 4 - 4 x.\n3) Then the limit is 12.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 10.\nFinal answer: $10$", "final_answer": "10", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3a341a13274b6dc4", "problem_id": "809ddb5f36aa2186", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$- 3 x + 8 y$ = $-13$\n$11 x + 10 y$ = $-31$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $11$: $- 33 x + 88 y$ = $-143$.\n3) Multiply the second equation by $-3$: $- 33 x - 30 y$ = $93$.\n4) Subtract to eliminate x: $118 y$ = $-236$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, -2\\right)$", "final_answer": "(x,y)=\\left(-1, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ab50ae182d863c0", "problem_id": "809ddb5f36aa2186", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$- 3 x + 8 y$ = $-13$\n$11 x + 10 y$ = $-31$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $11$: $- 33 x + 88 y$ = $-143$.\nMultiply the second equation by $-3$: $- 33 x - 30 y$ = $93$.\nSubtract to eliminate x: $118 y$ = $-236$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $-1$.\nCheck: plug (x,y)=(-1,-2) into both equations.\nFinal answer: $(x,y)=\\left(-1, -2\\right)$", "final_answer": "(x,y)=\\left(-1, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e7469d025eb5268", "problem_id": "51cda4a125be539b", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Two friends split $203 in the ratio 1:6. How much does each receive?", "output": "1) Total ratio parts = 1+6 = 7.\n2) Each part = 203 ÷ 7 = 29.\n3) First share = 1 × 29 = 29.\n4) Second share = 6 × 29 = 174.\nFinal answer: $174$", "final_answer": "174", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a32b1ba024e530c8", "problem_id": "3e52b4d472b1c18d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 2 \\sin{\\left(x \\right)}) = - 2 \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 2 \\sin{\\left(x \\right)}) = - 2 \\cos{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $- x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$", "final_answer": "- x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "dd4410a2d243d84f", "problem_id": "8f63eda47bc3a605", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 6 x - 189$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} + 6 x - 189$ = $3 \\left(x - 7\\right) \\left(x + 9\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $7$.\nFrom the second factor: x = $-9$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{7, -9\\}$", "final_answer": "x \\in \\{7, -9\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4a133f845d98a20e", "problem_id": "8f63eda47bc3a605", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 6 x - 189$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} + 6 x - 189$ = $3 \\left(x - 7\\right) \\left(x + 9\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $7$.\n5) From the second factor: x = $-9$.\nFinal answer: $x \\in \\{7, -9\\}$", "final_answer": "x \\in \\{7, -9\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8a84d80694475dd1", "problem_id": "d4ef7de249030ef3", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 12 times, what is the probability of exactly 8 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=8, p=$\\frac{1}{8}$ and simplify.\n4) Here C(12,8) = 495.\nFinal answer: $\\frac{1188495}{68719476736}$", "final_answer": "\\frac{1188495}{68719476736}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f1a42236666e34a9", "problem_id": "d4ef7de249030ef3", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 12 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=12, k=8, p=$\\frac{1}{8}$ and simplify.\nFinal answer: $\\frac{1188495}{68719476736}$", "final_answer": "\\frac{1188495}{68719476736}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fd6d903ba8f81215", "problem_id": "0c67ac015774890e", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A store sells a backpack for $150. It is discounted by 35%. You buy 6 backpacks. What is the total cost?", "output": "1) Discounted price = 150 × (1 − 35/100) = 150 × 65/100 = 195/2.\n2) Subtotal = 6 × 195/2 = 585.\nFinal answer: $585$", "final_answer": "585", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e1d8267360542d6", "problem_id": "127845561a322a32", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{3} + 4 x^{2} + 7 x + 11$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} + 4 x^{2} + 6 x + 11.\n3) Then the limit is 55.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 57.\nFinal answer: $57$", "final_answer": "57", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b2fd9cdc3fab8941", "problem_id": "5a5d9c1cc5be799f", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Two friends split $234 in the ratio 1:5. How much does each receive?", "output": "1) Total ratio parts = 1+5 = 6.\n2) Each part = 234 ÷ 6 = 39.\n3) First share = 1 × 39 = 39.\n4) Second share = 5 × 39 = 195.\nFinal answer: $195$", "final_answer": "195", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "709dcaac64e323af", "problem_id": "5a5d9c1cc5be799f", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $234 in the ratio 1:5. How much does each receive?", "output": "Total ratio parts = 1+5 = 6.\nEach part = 234 ÷ 6 = 39.\nFirst share = 1 × 39 = 39.\nSecond share = 5 × 39 = 195.\nCheck: 39+195=234.\nFinal answer: $195$", "final_answer": "195", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f31a622ca6b162cb", "problem_id": "93e4523faf344684", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Two friends split $136 in the ratio 1:1. How much does each receive?", "output": "1) Total ratio parts = 1+1 = 2.\n2) Each part = 136 ÷ 2 = 68.\n3) First share = 1 × 68 = 68.\n4) Second share = 1 × 68 = 68.\nFinal answer: $68$", "final_answer": "68", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "87e84f50062d2006", "problem_id": "93e4523faf344684", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $136 in the ratio 1:1. How much does each receive?", "output": "Total ratio parts = 1+1 = 2.\nEach part = 136 ÷ 2 = 68.\nFirst share = 1 × 68 = 68.\nSecond share = 1 × 68 = 68.\nCheck: 68+68=136.\nFinal answer: $68$", "final_answer": "68", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "75617040d6349aae", "problem_id": "89d6f9983667a940", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 16 times, what is the probability of exactly 7 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=16, k=7, p=$\\frac{2}{3}$ and simplify.\n4) Here C(16,7) = 11440.\nFinal answer: $\\frac{1464320}{43046721}$", "final_answer": "\\frac{1464320}{43046721}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6bbc2b90f545ab57", "problem_id": "89d6f9983667a940", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 16 times, what is the probability of exactly 7 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=16, k=7, p=$\\frac{2}{3}$ and simplify.\nHere C(16,7) = 11440.\nCheck: expression simplifies to $\\frac{1464320}{43046721}$.\nFinal answer: $\\frac{1464320}{43046721}$", "final_answer": "\\frac{1464320}{43046721}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "35cd46372470f7e6", "problem_id": "95d1724145ae7f77", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the multiplicative inverse of 36 modulo 41.", "output": "We want x such that 36x ≡ 1 (mod 41).\nUse extended Euclid to find s,t with 36s + 41t = 1.\nOne Bezout identity is: 36·(8) + 41·(-7) = 1.\nReduce s modulo 41: x ≡ 8 (mod 41).\nCheck: 36·8 mod 41 = 1.\nFinal answer: $x\\equiv 8\\pmod{41}$", "final_answer": "x\\equiv 8\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d390d98abce14817", "problem_id": "95d1724145ae7f77", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Find the multiplicative inverse of 36 modulo 41.", "output": "We want x such that 36x ≡ 1 (mod 41).\nUse extended Euclid to find s,t with 36s + 41t = 1.\nOne Bezout identity is: 36·(8) + 41·(-7) = 1.\nFinal answer: $x\\equiv 8\\pmod{41}$", "final_answer": "x\\equiv 8\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b05cc80d38205b9f", "problem_id": "2b21156c7d504cac", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} \\sin{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $- 2 x^{3} \\sin{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "06246446af245a50", "problem_id": "2b21156c7d504cac", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} \\sin{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $- 2 x^{3} \\sin{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fb0084f9ab9e01ab", "problem_id": "c426713789e9661c", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int \\left(x + 2\\right) e^{x} + \\left(2 x + 1\\right) \\cos{\\left(x \\right)}\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $\\left(x + 2\\right) e^{x} + \\left(2 x + 1\\right) \\cos{\\left(x \\right)}$.\n3) Compute an antiderivative: F(x) = $2 x \\sin{\\left(x \\right)} + \\left(x + 1\\right) e^{x} + \\sin{\\left(x \\right)} + 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\left(x + 1\\right)\\cdot e^{x} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\left(x + 1\\right)\\cdot e^{x} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d60dfec2e2b12d4", "problem_id": "4e5e51f53bf24457", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} 3 x^{3} + 10 x^{2} + 39 x + 153$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} + 10 x^{2} + 38 x + 153.\n3) Then the limit is 657.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: 661.\nFinal answer: $661$", "final_answer": "661", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2f9a8c5a8b862c92", "problem_id": "4544c4ac0e9fee9a", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} + 3 x^{2} + 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 2$ at $x=0$.", "output": "1) Differentiate: f'(x) = $6 x^{2} + 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 3$.\n2) Slope at x=0: m = f'(0) = $\\frac{7}{2}$.\n3) Point on curve: (0, f(0)) = (0, $4$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 4 = \\frac{7}{2}(x − 0).\n6) Optionally expand: y = $\\frac{7 x}{2} + 4$.\nFinal answer: $y = \\frac{7\\cdot x}{2} + 4$", "final_answer": "y = \\frac{7\\cdot x}{2} + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b09b4f8e7de091af", "problem_id": "4544c4ac0e9fee9a", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} + 3 x^{2} + 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 2$ at $x=0$.", "output": "Differentiate: f'(x) = $6 x^{2} + 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 3$.\nSlope at x=0: m = f'(0) = $\\frac{7}{2}$.\nPoint on curve: (0, f(0)) = (0, $4$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 4 = \\frac{7}{2}(x − 0).\nOptionally expand: y = $\\frac{7 x}{2} + 4$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = \\frac{7\\cdot x}{2} + 4$", "final_answer": "y = \\frac{7\\cdot x}{2} + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d03ff6d5d7939b95", "problem_id": "2a557cb6324a4387", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡2 (mod 7), x≡3 (mod 9). Give the solution modulo 63.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 63.\n3) Compute the CRT solution to get x ≡ 30 (mod 63).\nFinal answer: $x\\equiv 30\\pmod{63}$", "final_answer": "x\\equiv 30\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6466fa3f1b699839", "problem_id": "2a557cb6324a4387", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the congruences: x≡2 (mod 7), x≡3 (mod 9). Give the solution modulo 63.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 63.\nCompute the CRT solution to get x ≡ 30 (mod 63).\nFinal answer: $x\\equiv 30\\pmod{63}$", "final_answer": "x\\equiv 30\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6697f17f9976c88a", "problem_id": "2d7e3350e7114741", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{2} + 3 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{x - 4}$.", "output": "1) Let f(x) = $\\frac{- 3 x^{2} + 3 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{x - 4}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{6 x^{2} - 6 x + \\left(x - 4\\right) \\left(- 12 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4}{2 \\left(x - 4\\right)^{2}}$.\nFinal answer: $\\frac{6\\cdot x^{2} - 6\\cdot x + \\left(x - 4\\right)\\cdot \\left(- 12\\cdot x + \\left(x - 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x - 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6\\right) - \\left(2\\cdot x - 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 4}{2\\cdot \\left(x - 4\\right)^{2}}$", "final_answer": "\\frac{6\\cdot x^{2} - 6\\cdot x + \\left(x - 4\\right)\\cdot \\left(- 12\\cdot x + \\left(x - 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x - 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6\\right) - \\left(2\\cdot x - 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 4}{2\\cdot \\left(x - 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "07330c8d18ec49ca", "problem_id": "2d7e3350e7114741", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{2} + 3 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{x - 4}$.", "output": "Let f(x) = $\\frac{- 3 x^{2} + 3 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{x - 4}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{6 x^{2} - 6 x + \\left(x - 4\\right) \\left(- 12 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4}{2 \\left(x - 4\\right)^{2}}$.\nFinal answer: $\\frac{6\\cdot x^{2} - 6\\cdot x + \\left(x - 4\\right)\\cdot \\left(- 12\\cdot x + \\left(x - 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x - 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6\\right) - \\left(2\\cdot x - 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 4}{2\\cdot \\left(x - 4\\right)^{2}}$", "final_answer": "\\frac{6\\cdot x^{2} - 6\\cdot x + \\left(x - 4\\right)\\cdot \\left(- 12\\cdot x + \\left(x - 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x - 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6\\right) - \\left(2\\cdot x - 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 4}{2\\cdot \\left(x - 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "48033c8c3b26b3c4", "problem_id": "7d685a0b089e8f17", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} + x^{2} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)}$ at $x=1$.", "output": "Differentiate: f'(x) = $- 6 x^{2} + 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\nSlope at x=1: m = f'(1) = $-6 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}$.\nPoint on curve: (1, f(1)) = (1, $-3 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -3 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} = -6 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}(x − 1).\nOptionally expand: y = $- 6 x - 2 x \\sin{\\left(2 \\right)} + \\frac{x e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2 \\sin{\\left(2 \\right)} + 3$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 6\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 3$", "final_answer": "y = - 6\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "11bbac0a5db9b24b", "problem_id": "7d685a0b089e8f17", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} + x^{2} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)}$ at $x=1$.", "output": "1) Differentiate: f'(x) = $- 6 x^{2} + 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\n2) Slope at x=1: m = f'(1) = $-6 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}$.\n3) Point on curve: (1, f(1)) = (1, $-3 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -3 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} = -6 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}(x − 1).\n6) Optionally expand: y = $- 6 x - 2 x \\sin{\\left(2 \\right)} + \\frac{x e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2 \\sin{\\left(2 \\right)} + 3$.\nFinal answer: $y = - 6\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 3$", "final_answer": "y = - 6\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c761dff6387af091", "problem_id": "a68d13c0f7ad52bf", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡1 (mod 11), x≡12 (mod 13). Give the solution modulo 143.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 143.\nCompute the CRT solution to get x ≡ 12 (mod 143).\nCheck: 12 mod 11=1 and mod 13=12.\nFinal answer: $x\\equiv 12\\pmod{143}$", "final_answer": "x\\equiv 12\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "267aee2376117a9b", "problem_id": "a68d13c0f7ad52bf", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡1 (mod 11), x≡12 (mod 13). Give the solution modulo 143.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 143.\n3) Compute the CRT solution to get x ≡ 12 (mod 143).\nFinal answer: $x\\equiv 12\\pmod{143}$", "final_answer": "x\\equiv 12\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c15c1cabe044f2eb", "problem_id": "0ce13ce9807cb363", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡3 (mod 9), x≡3 (mod 13). Give the solution modulo 117.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 117.\n3) Compute the CRT solution to get x ≡ 3 (mod 117).\nFinal answer: $x\\equiv 3\\pmod{117}$", "final_answer": "x\\equiv 3\\pmod{117}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "57f2f29c989c6702", "problem_id": "0ce13ce9807cb363", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡3 (mod 9), x≡3 (mod 13). Give the solution modulo 117.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 117.\nCompute the CRT solution to get x ≡ 3 (mod 117).\nCheck: 3 mod 9=3 and mod 13=3.\nFinal answer: $x\\equiv 3\\pmod{117}$", "final_answer": "x\\equiv 3\\pmod{117}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0a3fd7f29a381f8b", "problem_id": "efeea33b1dba98bf", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{2} - \\frac{2 x}{3} + \\left(x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - \\frac{1}{3}}{x + 1}$.", "output": "1) Let f(x) = $\\frac{x^{2} - \\frac{2 x}{3} + \\left(x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - \\frac{1}{3}}{x + 1}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{- 6 x^{2} + 4 x + \\left(x + 1\\right) \\left(12 x + 3 \\left(x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4\\right) - 6 \\left(x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{6 \\left(x + 1\\right)^{2}}$.\nFinal answer: $\\frac{- 6\\cdot x^{2} + 4\\cdot x + \\left(x + 1\\right)\\cdot \\left(12\\cdot x + \\left(3\\cdot x + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 4\\right) - \\left(6\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 2}{6\\cdot \\left(x + 1\\right)^{2}}$", "final_answer": "\\frac{- 6\\cdot x^{2} + 4\\cdot x + \\left(x + 1\\right)\\cdot \\left(12\\cdot x + \\left(3\\cdot x + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 4\\right) - \\left(6\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 2}{6\\cdot \\left(x + 1\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c2e38a6c7e7ac069", "problem_id": "efeea33b1dba98bf", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{2} - \\frac{2 x}{3} + \\left(x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - \\frac{1}{3}}{x + 1}$.", "output": "Let f(x) = $\\frac{x^{2} - \\frac{2 x}{3} + \\left(x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - \\frac{1}{3}}{x + 1}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{- 6 x^{2} + 4 x + \\left(x + 1\\right) \\left(12 x + 3 \\left(x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6 \\left(x + 1\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4\\right) - 6 \\left(x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{6 \\left(x + 1\\right)^{2}}$.\nFinal answer: $\\frac{- 6\\cdot x^{2} + 4\\cdot x + \\left(x + 1\\right)\\cdot \\left(12\\cdot x + \\left(3\\cdot x + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 4\\right) - \\left(6\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 2}{6\\cdot \\left(x + 1\\right)^{2}}$", "final_answer": "\\frac{- 6\\cdot x^{2} + 4\\cdot x + \\left(x + 1\\right)\\cdot \\left(12\\cdot x + \\left(3\\cdot x + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 4\\right) - \\left(6\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 2}{6\\cdot \\left(x + 1\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "62ebb88fdd8a266d", "problem_id": "b0e9997ef210f1c9", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the multiplicative inverse of 10 modulo 47.", "output": "1) We want x such that 10x ≡ 1 (mod 47).\n2) Use extended Euclid to find s,t with 10s + 47t = 1.\n3) One Bezout identity is: 10·(-14) + 47·(3) = 1.\n4) Reduce s modulo 47: x ≡ 33 (mod 47).\nFinal answer: $x\\equiv 33\\pmod{47}$", "final_answer": "x\\equiv 33\\pmod{47}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cb35ab7dbce2402f", "problem_id": "b0e9997ef210f1c9", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 10 modulo 47.", "output": "We want x such that 10x ≡ 1 (mod 47).\nUse extended Euclid to find s,t with 10s + 47t = 1.\nOne Bezout identity is: 10·(-14) + 47·(3) = 1.\nReduce s modulo 47: x ≡ 33 (mod 47).\nCheck: 10·33 mod 47 = 1.\nFinal answer: $x\\equiv 33\\pmod{47}$", "final_answer": "x\\equiv 33\\pmod{47}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "813837f727ad0a0c", "problem_id": "b6481bbfacab4bb6", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A cyclist rides at 46 miles per hour for 3/2 hours, then at 34 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 46 × 3/2 = 69 miles.\n2) Second leg distance = 34 × 2 = 68 miles.\n3) Total distance = 69 + 68 = 137 miles.\nFinal answer: $137$", "final_answer": "137", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f72090cf380d251d", "problem_id": "b6481bbfacab4bb6", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A cyclist rides at 46 miles per hour for 3/2 hours, then at 34 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 46 × 3/2 = 69 miles.\nSecond leg distance = 34 × 2 = 68 miles.\nTotal distance = 69 + 68 = 137 miles.\nFinal answer: $137$", "final_answer": "137", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4cc96e1aa4fb6429", "problem_id": "cc5bf0a30ffd57f9", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x^{3} - 2 x - 2$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to x^{3} - 3 x - 2.\n3) Then the limit is 0.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 2.\nFinal answer: $2$", "final_answer": "2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f8952ca7a1d34255", "problem_id": "d9f74c89c0839836", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} 3 x^{3} + 4 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{3} + 4 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- 5 \\pi - 1 + \\frac{45 \\pi^{2}}{8} + \\frac{65 \\pi^{3}}{6} + \\frac{765 \\pi^{4}}{64}$.\n4) Simplify the result.\nFinal answer: $- 5\\cdot \\pi - 1 + \\frac{45\\cdot \\pi^{2}}{8} + \\frac{65\\cdot \\pi^{3}}{6} + \\frac{765\\cdot \\pi^{4}}{64}$", "final_answer": "- 5\\cdot \\pi - 1 + \\frac{45\\cdot \\pi^{2}}{8} + \\frac{65\\cdot \\pi^{3}}{6} + \\frac{765\\cdot \\pi^{4}}{64}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5c8b308070946417", "problem_id": "c8349e2140794b52", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} - x^{2} + 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)}$ at $x=2$.", "output": "1) Differentiate: f'(x) = $6 x^{2} - 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 3$.\n2) Slope at x=2: m = f'(2) = $\\frac{e}{2} - 2 \\sin{\\left(4 \\right)} + 23$.\n3) Point on curve: (2, f(2)) = (2, $\\cos{\\left(4 \\right)} + e + 18$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(4 \\right)} + e + 18 = \\frac{e}{2} - 2 \\sin{\\left(4 \\right)} + 23(x − 2).\n6) Optionally expand: y = $\\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 23 x - 28 + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$.\nFinal answer: $y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 23\\cdot x - 28 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$", "final_answer": "y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 23\\cdot x - 28 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "897100e0eb85e988", "problem_id": "4166a418f2c3a4dd", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x^{2} \\sin{\\left(x \\right)}) = x^{2} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = x^{2} \\cos{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x^{2} \\sin{\\left(x \\right)}) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$", "final_answer": "x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "06350e0155276c6e", "problem_id": "d123c24e395ac785", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_9=30$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=30 and k=9.\n4) Compute C(38,8) = 48903492.\nFinal answer: $48903492$", "final_answer": "48903492", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b79b9ebdaeebf88", "problem_id": "d123c24e395ac785", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_9=30$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=30 and k=9.\nCompute C(38,8) = 48903492.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $48903492$", "final_answer": "48903492", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "68d91ffb431f8574", "problem_id": "8a8723c9471595fc", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many binary strings of length 22 contain exactly 9 ones with no two ones adjacent?", "output": "1) Reserve 8 mandatory zeros between the 9 ones.\n2) This leaves n−k+1 = 14 valid slots for the ones.\n3) Choose the slots: C(14,9) = 2002.\nFinal answer: $2002$", "final_answer": "2002", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9311f1057404d052", "problem_id": "8a8723c9471595fc", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many binary strings of length 22 contain exactly 9 ones with no two ones adjacent?", "output": "Reserve 8 mandatory zeros between the 9 ones.\nThis leaves n−k+1 = 14 valid slots for the ones.\nChoose the slots: C(14,9) = 2002.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $2002$", "final_answer": "2002", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3282fd1026206dfa", "problem_id": "68ee3569e1adc983", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} + x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 3$ at $x=2$.", "output": "Differentiate: f'(x) = $- 6 x^{2} + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 1$.\nSlope at x=2: m = f'(2) = $-23 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\nPoint on curve: (2, f(2)) = (2, $-11 + \\cos{\\left(4 \\right)} + e$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -11 + \\cos{\\left(4 \\right)} + e = -23 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\nOptionally expand: y = $- 23 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 35$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 23\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 35$", "final_answer": "y = - 23\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 35", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b8e1a996344e5bfe", "problem_id": "68ee3569e1adc983", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} + x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 3$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 6 x^{2} + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 1$.\n2) Slope at x=2: m = f'(2) = $-23 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\n3) Point on curve: (2, f(2)) = (2, $-11 + \\cos{\\left(4 \\right)} + e$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -11 + \\cos{\\left(4 \\right)} + e = -23 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\n6) Optionally expand: y = $- 23 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 35$.\nFinal answer: $y = - 23\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 35$", "final_answer": "y = - 23\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 35", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8863c4be9de78b9f", "problem_id": "dceaa21b048efbbf", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$- 2 x + y$ = $-5$\n$2 x + 3 y$ = $-15$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $- 4 x + 2 y$ = $-10$.\n3) Multiply the second equation by $-2$: $- 4 x - 6 y$ = $30$.\n4) Subtract to eliminate x: $8 y$ = $-40$.\n5) Solve for y: y = $-5$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, -5\\right)$", "final_answer": "(x,y)=\\left(0, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9c1e90ef8cdbc242", "problem_id": "9e84074b80b7df79", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(- x^{2} - 2 x - 1\\right) \\cos{\\left(x^{2} + x - 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\cos{\\left(\\left(x^{2} + x\\right) - 1 \\right)}) = \\sin{\\left(x^{2} + x - 1 \\right)}$.\n3) So $f'(x) = 2 x^{3} \\sin{\\left(x^{2} + x - 1 \\right)} + 5 x^{2} \\sin{\\left(x^{2} + x - 1 \\right)} + x e^{x} + 2 x \\sin{\\left(x^{2} + x - 1 \\right)} - 2 x \\cos{\\left(\\left(x^{2} + x\\right) - 1 \\right)} + 3 e^{x} + \\sin{\\left(x^{2} + x - 1 \\right)} - 2 \\cos{\\left(\\left(x^{2} + x\\right) - 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\cos{\\left(\\left(x^{2} + x\\right) - 1 \\right)}) = \\left(2 x + 1\\right) \\sin{\\left(x^{2} + x - 1 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- x - 1\\right) \\cos{\\left(\\left(x^{2} + x\\right) - 1 \\right)} + \\left(x + 2\\right) e^{x} - \\left(2 x + 1\\right) \\left(x \\left(- x - 2\\right) - 1\\right) \\sin{\\left(x^{2} + x - 1 \\right)} + e^{x}$.\nFinal answer: $2 \\left(- x - 1\\right) \\cos{\\left(\\left(x^{2} + x\\right) - 1 \\right)} + \\left(x + 2\\right) e^{x} - \\left(2 x + 1\\right) \\left(x \\left(- x - 2\\right) - 1\\right) \\sin{\\left(x^{2} + x - 1 \\right)} + e^{x}$", "final_answer": "2 \\left(- x - 1\\right) \\cos{\\left(\\left(x^{2} + x\\right) - 1 \\right)} + \\left(x + 2\\right) e^{x} - \\left(2 x + 1\\right) \\left(x \\left(- x - 2\\right) - 1\\right) \\sin{\\left(x^{2} + x - 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "24a0f0bd8cb1b207", "problem_id": "8a79e7f40d6cc220", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the multiplicative inverse of 10 modulo 31.", "output": "We want x such that 10x ≡ 1 (mod 31).\nUse extended Euclid to find s,t with 10s + 31t = 1.\nOne Bezout identity is: 10·(-3) + 31·(1) = 1.\nReduce s modulo 31: x ≡ 28 (mod 31).\nCheck: 10·28 mod 31 = 1.\nFinal answer: $x\\equiv 28\\pmod{31}$", "final_answer": "x\\equiv 28\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aefd1bd1d041e0f6", "problem_id": "8a79e7f40d6cc220", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Find the multiplicative inverse of 10 modulo 31.", "output": "We want x such that 10x ≡ 1 (mod 31).\nUse extended Euclid to find s,t with 10s + 31t = 1.\nOne Bezout identity is: 10·(-3) + 31·(1) = 1.\nFinal answer: $x\\equiv 28\\pmod{31}$", "final_answer": "x\\equiv 28\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4504c5e9966b65e6", "problem_id": "62ad3172a079c701", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} - x^{2} + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 3$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 6 x^{2} - 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)}$.\n2) Slope at x=2: m = f'(2) = $-28 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\n3) Point on curve: (2, f(2)) = (2, $-23 + \\cos{\\left(4 \\right)} + e$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -23 + \\cos{\\left(4 \\right)} + e = -28 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\n6) Optionally expand: y = $- 28 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 33$.\nFinal answer: $y = - 28\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 33$", "final_answer": "y = - 28\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 33", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bd6ad966794db41e", "problem_id": "62ad3172a079c701", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} - x^{2} + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 3$ at $x=2$.", "output": "Differentiate: f'(x) = $- 6 x^{2} - 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)}$.\nSlope at x=2: m = f'(2) = $-28 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\nPoint on curve: (2, f(2)) = (2, $-23 + \\cos{\\left(4 \\right)} + e$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -23 + \\cos{\\left(4 \\right)} + e = -28 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\nOptionally expand: y = $- 28 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 33$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 28\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 33$", "final_answer": "y = - 28\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 33", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a013387a6d2fddb", "problem_id": "d9f4b85add393e72", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 98$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} - 98$ = $2 \\left(x - 7\\right) \\left(x + 7\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $7$.\nFrom the second factor: x = $-7$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{7, -7\\}$", "final_answer": "x \\in \\{7, -7\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9edbc1c5d5cfb08d", "problem_id": "d9f4b85add393e72", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 98$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 98$ = $2 \\left(x - 7\\right) \\left(x + 7\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $7$.\n5) From the second factor: x = $-7$.\nFinal answer: $x \\in \\{7, -7\\}$", "final_answer": "x \\in \\{7, -7\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fe1cfa846ac7675e", "problem_id": "dfd2b6c2bd691370", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many binary strings of length 24 contain exactly 6 ones with no two ones adjacent?", "output": "Reserve 5 mandatory zeros between the 6 ones.\nThis leaves n−k+1 = 19 valid slots for the ones.\nChoose the slots: C(19,6) = 27132.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $27132$", "final_answer": "27132", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5808d1848625e685", "problem_id": "dfd2b6c2bd691370", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "How many binary strings of length 24 contain exactly 6 ones with no two ones adjacent?", "output": "Reserve 5 mandatory zeros between the 6 ones.\nThis leaves n−k+1 = 19 valid slots for the ones.\nChoose the slots: C(19,6) = 27132.\nFinal answer: $27132$", "final_answer": "27132", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f23324fedcd612c7", "problem_id": "92b29d3a51b979a9", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the congruences: x≡1 (mod 11), x≡2 (mod 5). Give the solution modulo 55.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 55.\n3) Compute the CRT solution to get x ≡ 12 (mod 55).\nFinal answer: $x\\equiv 12\\pmod{55}$", "final_answer": "x\\equiv 12\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a710d81d54e6bc5d", "problem_id": "92b29d3a51b979a9", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the congruences: x≡1 (mod 11), x≡2 (mod 5). Give the solution modulo 55.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 55.\nCompute the CRT solution to get x ≡ 12 (mod 55).\nFinal answer: $x\\equiv 12\\pmod{55}$", "final_answer": "x\\equiv 12\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bdc3af771346a0ec", "problem_id": "d98f6ac1a404f87b", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=35$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=35 and k=10.\n4) Compute C(44,9) = 708930508.\nFinal answer: $708930508$", "final_answer": "708930508", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1307a5f4ba8d4bb7", "problem_id": "d98f6ac1a404f87b", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=35$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=35 and k=10.\nCompute C(44,9) = 708930508.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $708930508$", "final_answer": "708930508", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fb691a57964853e9", "problem_id": "818f8b1509568b49", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "How many binary strings of length 24 contain exactly 3 ones with no two ones adjacent?", "output": "1) Reserve 2 mandatory zeros between the 3 ones.\n2) This leaves n−k+1 = 22 valid slots for the ones.\n3) Choose the slots: C(22,3) = 1540.\nFinal answer: $1540$", "final_answer": "1540", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "97c22a81a75200cf", "problem_id": "818f8b1509568b49", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many binary strings of length 24 contain exactly 3 ones with no two ones adjacent?", "output": "Reserve 2 mandatory zeros between the 3 ones.\nThis leaves n−k+1 = 22 valid slots for the ones.\nChoose the slots: C(22,3) = 1540.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $1540$", "final_answer": "1540", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "04399b5ee341dd03", "problem_id": "d15a8fdc53fd63b6", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A cyclist rides at 80 miles per hour for 1/2 hours, then at 56 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 80 × 1/2 = 40 miles.\n2) Second leg distance = 56 × 3 = 168 miles.\n3) Total distance = 40 + 168 = 208 miles.\nFinal answer: $208$", "final_answer": "208", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5277627073b0d3dc", "problem_id": "d15a8fdc53fd63b6", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A cyclist rides at 80 miles per hour for 1/2 hours, then at 56 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 80 × 1/2 = 40 miles.\nSecond leg distance = 56 × 3 = 168 miles.\nTotal distance = 40 + 168 = 208 miles.\nCheck: adding both legs gives 208.\nFinal answer: $208$", "final_answer": "208", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5f4d3f4429a8c3a1", "problem_id": "abf95d5fa5b004de", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$5 x - 7 y$ = $-42$\n$3 x + 2 y$ = $-19$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $15 x - 21 y$ = $-126$.\nMultiply the second equation by $5$: $15 x + 10 y$ = $-95$.\nSubtract to eliminate x: $- 31 y$ = $-31$.\nSolve for y: y = $1$.\nSubstitute back to find x: x = $-7$.\nCheck: plug (x,y)=(-7,1) into both equations.\nFinal answer: $(x,y)=\\left(-7, 1\\right)$", "final_answer": "(x,y)=\\left(-7, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e132d3c491d34bc0", "problem_id": "abf95d5fa5b004de", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$5 x - 7 y$ = $-42$\n$3 x + 2 y$ = $-19$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $3$: $15 x - 21 y$ = $-126$.\n3) Multiply the second equation by $5$: $15 x + 10 y$ = $-95$.\n4) Subtract to eliminate x: $- 31 y$ = $-31$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $-7$.\nFinal answer: $(x,y)=\\left(-7, 1\\right)$", "final_answer": "(x,y)=\\left(-7, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "47d465c6c9941344", "problem_id": "7cc6cbb0c7f66d66", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "An urn has 14 red balls and 11 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(25,4).\n3) Favorable samples: choose k reds and n−k blues: C(14,1)·C(11,3).\n4) Compute totals: C(25,4) = 12650, and favorable = 2310.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{21}{115}$", "final_answer": "\\frac{21}{115}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d32bc286e0269c60", "problem_id": "7cc6cbb0c7f66d66", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "An urn has 14 red balls and 11 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(25,4).\nFavorable samples: choose k reds and n−k blues: C(14,1)·C(11,3).\nFinal answer: $\\frac{21}{115}$", "final_answer": "\\frac{21}{115}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d456c9e7dbb347bd", "problem_id": "11bde156b03bac64", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{3} x + x - \\frac{15}{4} + \\frac{3 \\sqrt{3}}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$1 - \\sqrt{3}$, c=$- \\frac{15}{4} + \\frac{3 \\sqrt{3}}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $19 - 8 \\sqrt{3}$, so √Δ = $\\sqrt{19 - 8 \\sqrt{3}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{1}{2} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{19 - 8 \\sqrt{3}}}{2}$.\n7) x2 = $- \\frac{\\sqrt{19 - 8 \\sqrt{3}}}{2} - \\frac{1}{2} + \\frac{\\sqrt{3}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{1}{2} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{19 - 8\\cdot \\sqrt{3}}}{2}, - \\frac{\\sqrt{19 - 8\\cdot \\sqrt{3}}}{2} - \\frac{1}{2} + \\frac{\\sqrt{3}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{1}{2} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{19 - 8\\cdot \\sqrt{3}}}{2}, - \\frac{\\sqrt{19 - 8\\cdot \\sqrt{3}}}{2} - \\frac{1}{2} + \\frac{\\sqrt{3}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6efbd9b592bb5cc0", "problem_id": "11bde156b03bac64", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{3} x + x - \\frac{15}{4} + \\frac{3 \\sqrt{3}}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$1 - \\sqrt{3}$, c=$- \\frac{15}{4} + \\frac{3 \\sqrt{3}}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{- \\frac{1}{2} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{19 - 8\\cdot \\sqrt{3}}}{2}, - \\frac{\\sqrt{19 - 8\\cdot \\sqrt{3}}}{2} - \\frac{1}{2} + \\frac{\\sqrt{3}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{1}{2} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{19 - 8\\cdot \\sqrt{3}}}{2}, - \\frac{\\sqrt{19 - 8\\cdot \\sqrt{3}}}{2} - \\frac{1}{2} + \\frac{\\sqrt{3}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5471f4ed708a30d2", "problem_id": "1ea09fb271aa4bc0", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A store sells a backpack for $40. It is discounted by 35%. You buy 11 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 40 × (1 − 35/100) = 40 × 65/100 = 26.\n2) Subtotal = 11 × 26 = 286.\n3) Total with tax = 286 × 109/100 = 15587/50.\nFinal answer: $\\frac{15587}{50}$", "final_answer": "\\frac{15587}{50}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "791d2771c71d1421", "problem_id": "686bb2946a7217ea", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} + 2 x - 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(2 x^{2}) = 4 x^{2}$.\n3) So $f'(x) = 4 x^{2} + 2$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(2 x^{2}) = 4 x$.\nTherefore the correct derivative is $f'(x) = 2 \\left(2 x + 1\\right)$.\nFinal answer: $2 \\left(2 x + 1\\right)$", "final_answer": "2 \\left(2 x + 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "885831bfbe6d2298", "problem_id": "30393832575ec955", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the multiplicative inverse of 19 modulo 59.", "output": "1) We want x such that 19x ≡ 1 (mod 59).\n2) Use extended Euclid to find s,t with 19s + 59t = 1.\n3) One Bezout identity is: 19·(28) + 59·(-9) = 1.\n4) Reduce s modulo 59: x ≡ 28 (mod 59).\nFinal answer: $x\\equiv 28\\pmod{59}$", "final_answer": "x\\equiv 28\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "13bc469a6cb6f32a", "problem_id": "30393832575ec955", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Find the multiplicative inverse of 19 modulo 59.", "output": "We want x such that 19x ≡ 1 (mod 59).\nUse extended Euclid to find s,t with 19s + 59t = 1.\nOne Bezout identity is: 19·(28) + 59·(-9) = 1.\nFinal answer: $x\\equiv 28\\pmod{59}$", "final_answer": "x\\equiv 28\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "550805153b9176d9", "problem_id": "0dc2f8488645f051", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{2} + 3 x + 5$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $2 x^{3} - x^{2} - x + 2$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $6 x^{2} - 2 x - 1$.\n5) Evaluate at a: p'(2) = $19$.\nFinal answer: $19$", "final_answer": "19", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8fafaa9212fa240e", "problem_id": "0dc2f8488645f051", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{2} + 3 x + 5$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{3} - x^{2} - x + 2$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $19$", "final_answer": "19", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4e55a31fc2ceeef8", "problem_id": "d26f10f35f204cfa", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} + 3 x - 1$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-1) = 0.\n3) d/dx(x^{2}) = 2 x.\n4) d/dx(3 x) = 3.\n5) Combine: f'(x) = 2 x + 3.\nFinal answer: $2\\cdot x + 3$", "final_answer": "2\\cdot x + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "248d3ddd712f9231", "problem_id": "d26f10f35f204cfa", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} + 3 x - 1$.", "output": "Differentiate term-by-term:\nd/dx(-1) = 0.\nd/dx(x^{2}) = 2 x.\nd/dx(3 x) = 3.\nCombine: f'(x) = 2 x + 3.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $2\\cdot x + 3$", "final_answer": "2\\cdot x + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4c13a1ecccca199e", "problem_id": "dfe45db6d235951c", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $- 2 x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - x\\cdot \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - x\\cdot \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e924f52370eb68f5", "problem_id": "dfe45db6d235951c", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $- 2 x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - x\\cdot \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - x\\cdot \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6cbab2ed6e3803da", "problem_id": "1bb5aa2ca6fcdb89", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=27$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=27 and k=7.\n4) Compute C(33,6) = 1107568.\nFinal answer: $1107568$", "final_answer": "1107568", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b3ee81a8a459d393", "problem_id": "1bb5aa2ca6fcdb89", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=27$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=27 and k=7.\nCompute C(33,6) = 1107568.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $1107568$", "final_answer": "1107568", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a3347b5e2dcd9cb3", "problem_id": "26eafb4fa6424c10", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡5 (mod 11), x≡3 (mod 8). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 27 (mod 88).\nFinal answer: $x\\equiv 27\\pmod{88}$", "final_answer": "x\\equiv 27\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b572f6d57041b09a", "problem_id": "26eafb4fa6424c10", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡5 (mod 11), x≡3 (mod 8). Give the solution modulo 88.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 88.\nCompute the CRT solution to get x ≡ 27 (mod 88).\nCheck: 27 mod 11=5 and mod 8=3.\nFinal answer: $x\\equiv 27\\pmod{88}$", "final_answer": "x\\equiv 27\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f5fd53fba4a27e9c", "problem_id": "486ff695c648f711", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$3 x - 3 y$ = $-3$\n$- x - y$ = $1$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $- 3 x + 3 y$ = $3$.\n3) Multiply the second equation by $3$: $- 3 x - 3 y$ = $3$.\n4) Subtract to eliminate x: $6 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, 0\\right)$", "final_answer": "(x,y)=\\left(-1, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3049617ae97a9133", "problem_id": "486ff695c648f711", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the linear system:\n$3 x - 3 y$ = $-3$\n$- x - y$ = $1$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $- 3 x + 3 y$ = $3$.\nMultiply the second equation by $3$: $- 3 x - 3 y$ = $3$.\nFinal answer: $(x,y)=\\left(-1, 0\\right)$", "final_answer": "(x,y)=\\left(-1, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "145a7ae5e4ae27c6", "problem_id": "936e13033f401138", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 3 x^{2} + 9 x + 19$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $3 x^{3} + 3 x^{2} + x + 1$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $9 x^{2} + 6 x + 1$.\nEvaluate at a: p'(2) = $49$.\nCheck: the identity gives $49$.\nFinal answer: $49$", "final_answer": "49", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7f7ac2a6414d1fc6", "problem_id": "e87817263420430d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(x^{3} + 2 x^{2} + 2 x + 3\\right) \\cos{\\left(3 x - 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 \\cos{\\left(3 x - 2 \\right)}) = - 3 \\sin{\\left(3 x - 2 \\right)}$.\n3) So $f'(x) = - 3 x^{3} \\sin{\\left(3 x - 2 \\right)} - 6 x^{2} \\sin{\\left(3 x - 2 \\right)} + 3 x^{2} \\cos{\\left(3 x - 2 \\right)} + x e^{x} - 6 x \\sin{\\left(3 x - 2 \\right)} + 4 x \\cos{\\left(3 x - 2 \\right)} + 2 e^{x} - 3 \\sin{\\left(3 x - 2 \\right)} + 2 \\cos{\\left(3 x - 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 \\cos{\\left(3 x - 2 \\right)}) = - 9 \\sin{\\left(3 x - 2 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 1\\right) e^{x} - 3 \\left(x \\left(x \\left(x + 2\\right) + 2\\right) + 3\\right) \\sin{\\left(3 x - 2 \\right)} + \\left(3 x^{2} + 4 x + 2\\right) \\cos{\\left(3 x - 2 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 1\\right) e^{x} - 3 \\left(x \\left(x \\left(x + 2\\right) + 2\\right) + 3\\right) \\sin{\\left(3 x - 2 \\right)} + \\left(3 x^{2} + 4 x + 2\\right) \\cos{\\left(3 x - 2 \\right)} + e^{x}$", "final_answer": "\\left(x + 1\\right) e^{x} - 3 \\left(x \\left(x \\left(x + 2\\right) + 2\\right) + 3\\right) \\sin{\\left(3 x - 2 \\right)} + \\left(3 x^{2} + 4 x + 2\\right) \\cos{\\left(3 x - 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "229e3976b2aff281", "problem_id": "cebd3e276f26dbca", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A cyclist rides at 49 miles per hour for 3/2 hours, then at 72 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 49 × 3/2 = 147/2 miles.\n2) Second leg distance = 72 × 1 = 72 miles.\n3) Total distance = 147/2 + 72 = 291/2 miles.\nFinal answer: $\\frac{291}{2}$", "final_answer": "\\frac{291}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0aea25587079e3de", "problem_id": "cebd3e276f26dbca", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A cyclist rides at 49 miles per hour for 3/2 hours, then at 72 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 49 × 3/2 = 147/2 miles.\nSecond leg distance = 72 × 1 = 72 miles.\nTotal distance = 147/2 + 72 = 291/2 miles.\nFinal answer: $\\frac{291}{2}$", "final_answer": "\\frac{291}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "31fe2aec1dd55016", "problem_id": "681d12f6f2b3fcb4", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the congruences: x≡4 (mod 9), x≡1 (mod 5). Give the solution modulo 45.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 45.\n3) Compute the CRT solution to get x ≡ 31 (mod 45).\nFinal answer: $x\\equiv 31\\pmod{45}$", "final_answer": "x\\equiv 31\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5aa2c4ca2d6ece25", "problem_id": "681d12f6f2b3fcb4", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡4 (mod 9), x≡1 (mod 5). Give the solution modulo 45.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 45.\nCompute the CRT solution to get x ≡ 31 (mod 45).\nCheck: 31 mod 9=4 and mod 5=1.\nFinal answer: $x\\equiv 31\\pmod{45}$", "final_answer": "x\\equiv 31\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8972c0e4edeb8862", "problem_id": "827d129cfc0feeda", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "You want 25 liters of a 45% salt solution. You have a 10% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 25−x liters is 70% solution.\nSalt equation: 10x + 70(25−x) = 45·25.\n(10-70)x = (45-70)·25 ⇒ x = 125/12.\nThen 25−x = 175/12.\nCheck: 10·125/12 + 70·175/12 = 45·25.\nFinal answer: $125/12 L of 10% and 175/12 L of 70%$", "final_answer": "125/12 L of 10% and 175/12 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e03e569571cff6d1", "problem_id": "827d129cfc0feeda", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "You want 25 liters of a 45% salt solution. You have a 10% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 25−x liters is 70% solution.\n2) Salt equation: 10x + 70(25−x) = 45·25.\n3) (10-70)x = (45-70)·25 ⇒ x = 125/12.\n4) Then 25−x = 175/12.\nFinal answer: $125/12 L of 10% and 175/12 L of 70%$", "final_answer": "125/12 L of 10% and 175/12 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c6ef60911d82c956", "problem_id": "35231e72955d70b2", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 10 x - 48$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} + 10 x - 48$ = $2 \\left(x - 3\\right) \\left(x + 8\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $3$.\n5) From the second factor: x = $-8$.\nFinal answer: $x \\in \\{3, -8\\}$", "final_answer": "x \\in \\{3, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b91619d405ef5fb9", "problem_id": "35231e72955d70b2", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 10 x - 48$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} + 10 x - 48$ = $2 \\left(x - 3\\right) \\left(x + 8\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $3$.\nFrom the second factor: x = $-8$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{3, -8\\}$", "final_answer": "x \\in \\{3, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6bb3874ad641b511", "problem_id": "c9561810e59437b4", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "You want 58 liters of a 30% salt solution. You have a 20% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 58−x liters is 40% solution.\nSalt equation: 20x + 40(58−x) = 30·58.\n(20-40)x = (30-40)·58 ⇒ x = 29.\nFinal answer: $29 L of 20% and 29 L of 40%$", "final_answer": "29 L of 20% and 29 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c8a225943fa206f7", "problem_id": "c9561810e59437b4", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "You want 58 liters of a 30% salt solution. You have a 20% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 58−x liters is 40% solution.\n2) Salt equation: 20x + 40(58−x) = 30·58.\n3) (20-40)x = (30-40)·58 ⇒ x = 29.\n4) Then 58−x = 29.\nFinal answer: $29 L of 20% and 29 L of 40%$", "final_answer": "29 L of 20% and 29 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5ab714d4690b7f4f", "problem_id": "460283c9813b0a6c", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 4 x^{3} + 3 x^{2} + 3 x - 2$.", "output": "Differentiate term-by-term:\nd/dx(-2) = 0.\nd/dx(- 4 x^{3}) = - 12 x^{2}.\nd/dx(3 x) = 3.\nd/dx(3 x^{2}) = 6 x.\nCombine: f'(x) = 3 \\left(- 4 x^{2} + 2 x + 1\\right).\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 12\\cdot x^{2} + 6\\cdot x + 3$", "final_answer": "- 12\\cdot x^{2} + 6\\cdot x + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "da5e4a74397bdb0a", "problem_id": "460283c9813b0a6c", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 4 x^{3} + 3 x^{2} + 3 x - 2$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-2) = 0.\n3) d/dx(- 4 x^{3}) = - 12 x^{2}.\n4) d/dx(3 x) = 3.\n5) d/dx(3 x^{2}) = 6 x.\n6) Combine: f'(x) = 3 \\left(- 4 x^{2} + 2 x + 1\\right).\nFinal answer: $- 12\\cdot x^{2} + 6\\cdot x + 3$", "final_answer": "- 12\\cdot x^{2} + 6\\cdot x + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "83366caf10f4a6ad", "problem_id": "ede2db2adb30ce3b", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A store sells a backpack for $144. It is discounted by 35%. You buy 12 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 144 × (1 − 35/100) = 144 × 65/100 = 468/5.\n2) Subtotal = 12 × 468/5 = 5616/5.\n3) Total with tax = 5616/5 × 107/100 = 150228/125.\nFinal answer: $\\frac{150228}{125}$", "final_answer": "\\frac{150228}{125}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a0d8d96df51225ff", "problem_id": "ede2db2adb30ce3b", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $144. It is discounted by 35%. You buy 12 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 144 × (1 − 35/100) = 144 × 65/100 = 468/5.\nSubtotal = 12 × 468/5 = 5616/5.\nTotal with tax = 5616/5 × 107/100 = 150228/125.\nCheck: recomputing the formula yields 150228/125.\nFinal answer: $\\frac{150228}{125}$", "final_answer": "\\frac{150228}{125}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4d05677829416782", "problem_id": "325cfdb151709f01", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$9 x - 9 y$ = $-108$\n$- 5 x + 10 y$ = $95$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-5$: $- 45 x + 45 y$ = $540$.\n3) Multiply the second equation by $9$: $- 45 x + 90 y$ = $855$.\n4) Subtract to eliminate x: $- 45 y$ = $-315$.\n5) Solve for y: y = $7$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, 7\\right)$", "final_answer": "(x,y)=\\left(-5, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ca37a409d6cca870", "problem_id": "325cfdb151709f01", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the linear system:\n$9 x - 9 y$ = $-108$\n$- 5 x + 10 y$ = $95$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-5$: $- 45 x + 45 y$ = $540$.\nMultiply the second equation by $9$: $- 45 x + 90 y$ = $855$.\nFinal answer: $(x,y)=\\left(-5, 7\\right)$", "final_answer": "(x,y)=\\left(-5, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "35b5a567efbd9dc6", "problem_id": "ceecab0fe9f72f3a", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Two friends split $510 in the ratio 6:9. How much does each receive?", "output": "1) Total ratio parts = 6+9 = 15.\n2) Each part = 510 ÷ 15 = 34.\n3) First share = 6 × 34 = 204.\n4) Second share = 9 × 34 = 306.\nFinal answer: $306$", "final_answer": "306", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6e0bfd5db5ff7364", "problem_id": "ceecab0fe9f72f3a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $510 in the ratio 6:9. How much does each receive?", "output": "Total ratio parts = 6+9 = 15.\nEach part = 510 ÷ 15 = 34.\nFirst share = 6 × 34 = 204.\nSecond share = 9 × 34 = 306.\nCheck: 204+306=510.\nFinal answer: $306$", "final_answer": "306", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9d768fce7d26d482", "problem_id": "584e24b5c0e285ee", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A cyclist rides at 27 miles per hour for 1 hours, then at 71 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 27 × 1 = 27 miles.\n2) Second leg distance = 71 × 1 = 71 miles.\n3) Total distance = 27 + 71 = 98 miles.\nFinal answer: $98$", "final_answer": "98", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fde6ebd250017aa9", "problem_id": "584e24b5c0e285ee", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 27 miles per hour for 1 hours, then at 71 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 27 × 1 = 27 miles.\nSecond leg distance = 71 × 1 = 71 miles.\nTotal distance = 27 + 71 = 98 miles.\nCheck: adding both legs gives 98.\nFinal answer: $98$", "final_answer": "98", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "10e81c23cc320802", "problem_id": "5a3f507045cec5b3", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A cyclist rides at 32 miles per hour for 1 hours, then at 21 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 32 × 1 = 32 miles.\n2) Second leg distance = 21 × 1 = 21 miles.\n3) Total distance = 32 + 21 = 53 miles.\nFinal answer: $53$", "final_answer": "53", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "629ab7ffe1e1fc4e", "problem_id": "82f49cd34d98cb42", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x^{3} - x^{2} - 3 x - 3$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - x^{3} - x^{2} - 4 x - 3.\n3) Then the limit is -23.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -21.\nFinal answer: $-21$", "final_answer": "-21", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "78a050df74f9dac0", "problem_id": "dfe45db6d235951c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 \\sin{\\left(x \\right)}) = 2 \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 \\sin{\\left(x \\right)}) = 2 \\cos{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$", "final_answer": "- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "55e2030f550f4dae", "problem_id": "7770ea4d83c748b7", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$3 x + 3 y$ = $-24$\n$- 6 x - y$ = $43$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-6$: $- 18 x - 18 y$ = $144$.\n3) Multiply the second equation by $3$: $- 18 x - 3 y$ = $129$.\n4) Subtract to eliminate x: $- 15 y$ = $15$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $-7$.\nFinal answer: $(x,y)=\\left(-7, -1\\right)$", "final_answer": "(x,y)=\\left(-7, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "705df8dccb6d0a34", "problem_id": "94588ba7d62138fe", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$- 7 x + 4 y$ = $39$\n$- 8 x + 8 y$ = $24$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-8$: $56 x - 32 y$ = $-312$.\nMultiply the second equation by $-7$: $56 x - 56 y$ = $-168$.\nSubtract to eliminate x: $24 y$ = $-144$.\nSolve for y: y = $-6$.\nSubstitute back to find x: x = $-9$.\nCheck: plug (x,y)=(-9,-6) into both equations.\nFinal answer: $(x,y)=\\left(-9, -6\\right)$", "final_answer": "(x,y)=\\left(-9, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e8f2fb752897006d", "problem_id": "94588ba7d62138fe", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$- 7 x + 4 y$ = $39$\n$- 8 x + 8 y$ = $24$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-8$: $56 x - 32 y$ = $-312$.\n3) Multiply the second equation by $-7$: $56 x - 56 y$ = $-168$.\n4) Subtract to eliminate x: $24 y$ = $-144$.\n5) Solve for y: y = $-6$.\n6) Substitute back to find x: x = $-9$.\nFinal answer: $(x,y)=\\left(-9, -6\\right)$", "final_answer": "(x,y)=\\left(-9, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6aefa1e869a72a4b", "problem_id": "372b44139cb0bed3", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 5 x + 3$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$-5$, c=$3$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $1$, so √Δ = $1$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{3}{2}$.\n7) x2 = $1$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{3}{2}, 1\\}$", "final_answer": "x \\in \\{\\frac{3}{2}, 1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fc5a78ba7b9512d8", "problem_id": "372b44139cb0bed3", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 5 x + 3$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$-5$, c=$3$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{3}{2}, 1\\}$", "final_answer": "x \\in \\{\\frac{3}{2}, 1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "652d5f13818c4a7a", "problem_id": "d6c8be7de0bc7c8a", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - 3 x^{2} + x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 3$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $- 9 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 1$.\n2) Slope at x=-1: m = f'(-1) = $-2 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\n3) Point on curve: (-1, f(-1)) = (-1, $\\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 2$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 2 = -2 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}(x − -1).\n6) Optionally expand: y = $- 2 x + \\frac{x}{2 e^{\\frac{1}{2}}} + 2 x \\sin{\\left(2 \\right)} + \\cos{\\left(2 \\right)} + \\frac{3}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = - 2\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 2\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f22ae91f37c95f06", "problem_id": "d6c8be7de0bc7c8a", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - 3 x^{2} + x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 3$ at $x=-1$.", "output": "Differentiate: f'(x) = $- 9 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 1$.\nSlope at x=-1: m = f'(-1) = $-2 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nPoint on curve: (-1, f(-1)) = (-1, $\\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 2$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 2 = -2 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}(x − -1).\nOptionally expand: y = $- 2 x + \\frac{x}{2 e^{\\frac{1}{2}}} + 2 x \\sin{\\left(2 \\right)} + \\cos{\\left(2 \\right)} + \\frac{3}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 2\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 2\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f3be72b3ea13b9eb", "problem_id": "8bb9cf79dab694d9", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 4 x^{3} - 2 x^{2} + 3 x + 4\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $4 x^{3} - 2 x^{2} + 3 x + 4$.\n3) Compute an antiderivative: F(x) = $x^{4} - \\frac{2 x^{3}}{3} + \\frac{3 x^{2}}{2} + 4 x$.\n4) Include the constant of integration C.\nFinal answer: $C + x^{4} - \\frac{2\\cdot x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} + 4\\cdot x$", "final_answer": "C + x^{4} - \\frac{2\\cdot x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} + 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "901fa76ba3b15756", "problem_id": "8bb9cf79dab694d9", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 4 x^{3} - 2 x^{2} + 3 x + 4\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $4 x^{3} - 2 x^{2} + 3 x + 4$.\nCompute an antiderivative: F(x) = $x^{4} - \\frac{2 x^{3}}{3} + \\frac{3 x^{2}}{2} + 4 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + x^{4} - \\frac{2\\cdot x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} + 4\\cdot x$", "final_answer": "C + x^{4} - \\frac{2\\cdot x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} + 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "22e3ec47db1be747", "problem_id": "90b5c9913946f531", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- x^{2} - x - 2\\right) \\cos{\\left(2 x^{2} + 2 x \\right)}$.", "output": "1) Let f(x) = $2 \\left(x + 1\\right) e^{x} + \\left(- x^{2} - x - 2\\right) \\cos{\\left(2 x^{2} + 2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\left(- 2 x - 1\\right) \\cos{\\left(2 x^{2} + 2 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 2 \\left(2 x + 1\\right) \\left(- x^{2} - x - 2\\right) \\sin{\\left(2 x^{2} + 2 x \\right)} + 2 e^{x}$.\nFinal answer: $\\left(- 2\\cdot x - 1\\right)\\cdot \\cos{\\left(2\\cdot x^{2} + 2\\cdot x \\right)} + \\left(2\\cdot x + 2\\right)\\cdot e^{x} - \\left(4\\cdot x + 2\\right)\\cdot \\left(- x^{2} - x - 2\\right)\\cdot \\sin{\\left(2\\cdot x^{2} + 2\\cdot x \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(- 2\\cdot x - 1\\right)\\cdot \\cos{\\left(2\\cdot x^{2} + 2\\cdot x \\right)} + \\left(2\\cdot x + 2\\right)\\cdot e^{x} - \\left(4\\cdot x + 2\\right)\\cdot \\left(- x^{2} - x - 2\\right)\\cdot \\sin{\\left(2\\cdot x^{2} + 2\\cdot x \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e3bb97da770b8adb", "problem_id": "90b5c9913946f531", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- x^{2} - x - 2\\right) \\cos{\\left(2 x^{2} + 2 x \\right)}$.", "output": "Let f(x) = $2 \\left(x + 1\\right) e^{x} + \\left(- x^{2} - x - 2\\right) \\cos{\\left(2 x^{2} + 2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\left(- 2 x - 1\\right) \\cos{\\left(2 x^{2} + 2 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 2 \\left(2 x + 1\\right) \\left(- x^{2} - x - 2\\right) \\sin{\\left(2 x^{2} + 2 x \\right)} + 2 e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\left(- 2\\cdot x - 1\\right)\\cdot \\cos{\\left(2\\cdot x^{2} + 2\\cdot x \\right)} + \\left(2\\cdot x + 2\\right)\\cdot e^{x} - \\left(4\\cdot x + 2\\right)\\cdot \\left(- x^{2} - x - 2\\right)\\cdot \\sin{\\left(2\\cdot x^{2} + 2\\cdot x \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(- 2\\cdot x - 1\\right)\\cdot \\cos{\\left(2\\cdot x^{2} + 2\\cdot x \\right)} + \\left(2\\cdot x + 2\\right)\\cdot e^{x} - \\left(4\\cdot x + 2\\right)\\cdot \\left(- x^{2} - x - 2\\right)\\cdot \\sin{\\left(2\\cdot x^{2} + 2\\cdot x \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f16c8f58ddf9205a", "problem_id": "dbe34a8b224711d0", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 x^{3} - 4 x^{2} + 11 x - 19$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} - 4 x^{2} + 10 x - 19.\n3) Then the limit is -79.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -81.\nFinal answer: $-81$", "final_answer": "-81", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a63782976604fed2", "problem_id": "772eb7524a94615d", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{10} x - \\frac{x}{3} - \\frac{4}{3} + \\frac{4 \\sqrt{10}}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\sqrt{10} - \\frac{1}{3}$, c=$- \\frac{4}{3} + \\frac{4 \\sqrt{10}}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{139}{9} - \\frac{14 \\sqrt{10}}{3}$, so √Δ = $\\sqrt{\\frac{139}{9} - \\frac{14 \\sqrt{10}}{3}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{1}{6} + \\frac{\\sqrt{139 - 42 \\sqrt{10}}}{6} + \\frac{\\sqrt{10}}{2}$.\nx2 = $- \\frac{\\sqrt{139 - 42 \\sqrt{10}}}{6} + \\frac{1}{6} + \\frac{\\sqrt{10}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{1}{6} + \\frac{\\sqrt{139 - 42\\cdot \\sqrt{10}}}{6} + \\frac{\\sqrt{10}}{2}, - \\frac{\\sqrt{139 - 42\\cdot \\sqrt{10}}}{6} + \\frac{1}{6} + \\frac{\\sqrt{10}}{2}\\}$", "final_answer": "x \\in \\{\\frac{1}{6} + \\frac{\\sqrt{139 - 42\\cdot \\sqrt{10}}}{6} + \\frac{\\sqrt{10}}{2}, - \\frac{\\sqrt{139 - 42\\cdot \\sqrt{10}}}{6} + \\frac{1}{6} + \\frac{\\sqrt{10}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ceae5bcff4b3e3cc", "problem_id": "e1f15c32f670b3cc", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - \\sqrt{3} x - 12 - 2 \\sqrt{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$- \\sqrt{3}$, c=$-12 - 2 \\sqrt{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $24 \\sqrt{3} + 147$, so √Δ = $\\sqrt{24 \\sqrt{3} + 147}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{3} \\left(1 + \\sqrt{8 \\sqrt{3} + 49}\\right)}{6}$.\n7) x2 = $\\frac{\\sqrt{3} \\left(1 - \\sqrt{8 \\sqrt{3} + 49}\\right)}{6}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{3}\\cdot \\left(1 + \\sqrt{8\\cdot \\sqrt{3} + 49}\\right)}{6}, \\frac{\\sqrt{3}\\cdot \\left(1 - \\sqrt{8\\cdot \\sqrt{3} + 49}\\right)}{6}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{3}\\cdot \\left(1 + \\sqrt{8\\cdot \\sqrt{3} + 49}\\right)}{6}, \\frac{\\sqrt{3}\\cdot \\left(1 - \\sqrt{8\\cdot \\sqrt{3} + 49}\\right)}{6}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "21b66635c457e173", "problem_id": "e1f15c32f670b3cc", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - \\sqrt{3} x - 12 - 2 \\sqrt{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$- \\sqrt{3}$, c=$-12 - 2 \\sqrt{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $24 \\sqrt{3} + 147$, so √Δ = $\\sqrt{24 \\sqrt{3} + 147}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{3} \\left(1 + \\sqrt{8 \\sqrt{3} + 49}\\right)}{6}$.\nx2 = $\\frac{\\sqrt{3} \\left(1 - \\sqrt{8 \\sqrt{3} + 49}\\right)}{6}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{3}\\cdot \\left(1 + \\sqrt{8\\cdot \\sqrt{3} + 49}\\right)}{6}, \\frac{\\sqrt{3}\\cdot \\left(1 - \\sqrt{8\\cdot \\sqrt{3} + 49}\\right)}{6}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{3}\\cdot \\left(1 + \\sqrt{8\\cdot \\sqrt{3} + 49}\\right)}{6}, \\frac{\\sqrt{3}\\cdot \\left(1 - \\sqrt{8\\cdot \\sqrt{3} + 49}\\right)}{6}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8e77eb79b455cc09", "problem_id": "69ad04dd891ad81e", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the multiplicative inverse of 5 modulo 37.", "output": "1) We want x such that 5x ≡ 1 (mod 37).\n2) Use extended Euclid to find s,t with 5s + 37t = 1.\n3) One Bezout identity is: 5·(15) + 37·(-2) = 1.\n4) Reduce s modulo 37: x ≡ 15 (mod 37).\nFinal answer: $x\\equiv 15\\pmod{37}$", "final_answer": "x\\equiv 15\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "15982973983ce7c3", "problem_id": "69ad04dd891ad81e", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 5 modulo 37.", "output": "We want x such that 5x ≡ 1 (mod 37).\nUse extended Euclid to find s,t with 5s + 37t = 1.\nOne Bezout identity is: 5·(15) + 37·(-2) = 1.\nReduce s modulo 37: x ≡ 15 (mod 37).\nCheck: 5·15 mod 37 = 1.\nFinal answer: $x\\equiv 15\\pmod{37}$", "final_answer": "x\\equiv 15\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4b296e87bdd7a61a", "problem_id": "331a6d19e06cbeed", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_6=43$?", "output": "Let y_i=x_i-3, then y_i≥0 and y_1+…+y_6=25.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=25 and k=6.\nCompute C(30,5) = 142506.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $142506$", "final_answer": "142506", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ee6aa5e8c8b68bb8", "problem_id": "331a6d19e06cbeed", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_6=43$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_6=25.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=25 and k=6.\n4) Compute C(30,5) = 142506.\nFinal answer: $142506$", "final_answer": "142506", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3b9c96379f302cf8", "problem_id": "6f281415018dc9f9", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Worker A finishes a job in 21 hours and Worker B finishes it in 7 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/21 job/hour. B's rate = 1/7 job/hour.\n2) Combined rate = 1/21 + 1/7 = 4/21 job/hour.\n3) Time = 1 ÷ (combined rate) = 21/4 hours.\nFinal answer: $\\frac{21}{4}$", "final_answer": "\\frac{21}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "591df8ce9bedf776", "problem_id": "6f281415018dc9f9", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Worker A finishes a job in 21 hours and Worker B finishes it in 7 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/21 job/hour. B's rate = 1/7 job/hour.\nCombined rate = 1/21 + 1/7 = 4/21 job/hour.\nTime = 1 ÷ (combined rate) = 21/4 hours.\nCheck: (combined rate)×(time) = 4/21×21/4 = 1.\nFinal answer: $\\frac{21}{4}$", "final_answer": "\\frac{21}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e0f50f18fd17d224", "problem_id": "7383002178fca419", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A cyclist rides at 33 miles per hour for 1 hours, then at 79 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 33 × 1 = 33 miles.\n2) Second leg distance = 79 × 3/2 = 237/2 miles.\n3) Total distance = 33 + 237/2 = 303/2 miles.\nFinal answer: $\\frac{303}{2}$", "final_answer": "\\frac{303}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b0242d2bd6d231bf", "problem_id": "7383002178fca419", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A cyclist rides at 33 miles per hour for 1 hours, then at 79 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 33 × 1 = 33 miles.\nSecond leg distance = 79 × 3/2 = 237/2 miles.\nTotal distance = 33 + 237/2 = 303/2 miles.\nCheck: adding both legs gives 303/2.\nFinal answer: $\\frac{303}{2}$", "final_answer": "\\frac{303}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b0e091314c517348", "problem_id": "31444fb78dad34db", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Worker A finishes a job in 17 hours and Worker B finishes it in 22 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/17 job/hour. B's rate = 1/22 job/hour.\n2) Combined rate = 1/17 + 1/22 = 39/374 job/hour.\n3) Time = 1 ÷ (combined rate) = 374/39 hours.\nFinal answer: $\\frac{374}{39}$", "final_answer": "\\frac{374}{39}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d916e9e83a3fb10", "problem_id": "72a1763b38dfa521", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(- x^{3} - x^{2}\\right) \\cos{\\left(x^{2} + 2 x - 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x^{2} \\cos{\\left(\\left(x^{2} + 2 x\\right) - 2 \\right)}) = x^{2} \\sin{\\left(x^{2} + 2 x - 2 \\right)}$.\n3) So $f'(x) = 2 x^{4} \\sin{\\left(x^{2} + 2 x - 2 \\right)} + 2 x^{3} \\sin{\\left(x^{2} + 2 x - 2 \\right)} + x^{2} \\sin{\\left(x^{2} + 2 x - 2 \\right)} - 3 x^{2} \\cos{\\left(\\left(x^{2} + 2 x\\right) - 2 \\right)} + x e^{x} - 2 e^{x}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x^{2} \\cos{\\left(\\left(x^{2} + 2 x\\right) - 2 \\right)}) = x^{2} \\left(2 x + 2\\right) \\sin{\\left(x^{2} + 2 x - 2 \\right)} - 2 x \\cos{\\left(\\left(x^{2} + 2 x\\right) - 2 \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{2} \\left(- x - 1\\right) \\left(x + 1\\right) \\sin{\\left(x^{2} + 2 x - 2 \\right)} + x \\left(- 3 x - 2\\right) \\cos{\\left(\\left(x^{2} + 2 x\\right) - 2 \\right)} + \\left(x - 3\\right) e^{x} + e^{x}$.\nFinal answer: $- 2 x^{2} \\left(- x - 1\\right) \\left(x + 1\\right) \\sin{\\left(x^{2} + 2 x - 2 \\right)} + x \\left(- 3 x - 2\\right) \\cos{\\left(\\left(x^{2} + 2 x\\right) - 2 \\right)} + \\left(x - 3\\right) e^{x} + e^{x}$", "final_answer": "- 2 x^{2} \\left(- x - 1\\right) \\left(x + 1\\right) \\sin{\\left(x^{2} + 2 x - 2 \\right)} + x \\left(- 3 x - 2\\right) \\cos{\\left(\\left(x^{2} + 2 x\\right) - 2 \\right)} + \\left(x - 3\\right) e^{x} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "566a5cfd47ae7df7", "problem_id": "ea856f0536cbf9ab", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} - 4 x^{4} + 3 x^{3} - 4 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 4 x^{4} + 3 x^{3} - 4 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\n2) Evaluate at the bounds: F(\\pi) and F(0).\n3) Subtract: F(\\pi) − F(0) = $\\frac{\\pi^{2} \\left(- 48 \\pi^{3} - 80 \\pi - 90 + 45 \\pi^{2}\\right)}{60}$.\n4) Simplify the result.\nFinal answer: $\\frac{\\pi^{2}\\cdot \\left(- 48\\cdot \\pi^{3} - 80\\cdot \\pi - 90 + 45\\cdot \\pi^{2}\\right)}{60}$", "final_answer": "\\frac{\\pi^{2}\\cdot \\left(- 48\\cdot \\pi^{3} - 80\\cdot \\pi - 90 + 45\\cdot \\pi^{2}\\right)}{60}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3a5f97de8c28a267", "problem_id": "ea856f0536cbf9ab", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} - 4 x^{4} + 3 x^{3} - 4 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 4 x^{4} + 3 x^{3} - 4 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nEvaluate at the bounds: F(\\pi) and F(0).\nSubtract: F(\\pi) − F(0) = $\\frac{\\pi^{2} \\left(- 48 \\pi^{3} - 80 \\pi - 90 + 45 \\pi^{2}\\right)}{60}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{\\pi^{2}\\cdot \\left(- 48\\cdot \\pi^{3} - 80\\cdot \\pi - 90 + 45\\cdot \\pi^{2}\\right)}{60}$", "final_answer": "\\frac{\\pi^{2}\\cdot \\left(- 48\\cdot \\pi^{3} - 80\\cdot \\pi - 90 + 45\\cdot \\pi^{2}\\right)}{60}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8c367ca58edbac7f", "problem_id": "f8be37dd6d45f889", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$3 y$ = $24$\n$7 x + 2 y$ = $-12$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $7$: $21 y$ = $168$.\n3) Multiply the second equation by $0$: $0$ = $0$.\n4) Subtract to eliminate x: $21 y$ = $168$.\n5) Solve for y: y = $8$.\n6) Substitute back to find x: x = $-4$.\nFinal answer: $(x,y)=\\left(-4, 8\\right)$", "final_answer": "(x,y)=\\left(-4, 8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "948fc5d112b25f67", "problem_id": "f8be37dd6d45f889", "style": "answer_only", "instruction": "Final answer only.", "input": "Solve the linear system:\n$3 y$ = $24$\n$7 x + 2 y$ = $-12$", "output": "$(x,y)=\\left(-4, 8\\right)$", "final_answer": "(x,y)=\\left(-4, 8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "812dd3d93ed47ce9", "problem_id": "b3445b7beadfecda", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the congruences: x≡4 (mod 9), x≡5 (mod 7). Give the solution modulo 63.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 63.\nCompute the CRT solution to get x ≡ 40 (mod 63).\nFinal answer: $x\\equiv 40\\pmod{63}$", "final_answer": "x\\equiv 40\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7618d86f37fa46d7", "problem_id": "b3445b7beadfecda", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡4 (mod 9), x≡5 (mod 7). Give the solution modulo 63.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 63.\n3) Compute the CRT solution to get x ≡ 40 (mod 63).\nFinal answer: $x\\equiv 40\\pmod{63}$", "final_answer": "x\\equiv 40\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dddcd4ee07dd80d6", "problem_id": "f66393517368d768", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x^{2} - 2 x - 2$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 2 x^{3} + 2 x^{2} + 2 x + 2$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 6 x^{2} + 4 x + 2$.\n5) Evaluate at a: p'(2) = $-14$.\nFinal answer: $-14$", "final_answer": "-14", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "41e2e86cb96dcda0", "problem_id": "f66393517368d768", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x^{2} - 2 x - 2$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 2 x^{3} + 2 x^{2} + 2 x + 2$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 6 x^{2} + 4 x + 2$.\nEvaluate at a: p'(2) = $-14$.\nCheck: the identity gives $-14$.\nFinal answer: $-14$", "final_answer": "-14", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3a656885536cdfce", "problem_id": "6948eacda878a363", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Worker A finishes a job in 21 hours and Worker B finishes it in 14 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/21 job/hour. B's rate = 1/14 job/hour.\n2) Combined rate = 1/21 + 1/14 = 5/42 job/hour.\n3) Time = 1 ÷ (combined rate) = 42/5 hours.\nFinal answer: $\\frac{42}{5}$", "final_answer": "\\frac{42}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ffbc60e6d93e8a79", "problem_id": "6948eacda878a363", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Worker A finishes a job in 21 hours and Worker B finishes it in 14 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/21 job/hour. B's rate = 1/14 job/hour.\nCombined rate = 1/21 + 1/14 = 5/42 job/hour.\nTime = 1 ÷ (combined rate) = 42/5 hours.\nCheck: (combined rate)×(time) = 5/42×42/5 = 1.\nFinal answer: $\\frac{42}{5}$", "final_answer": "\\frac{42}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "068961edba61ebfb", "problem_id": "5b7d6296da3f9db2", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{10}$. If it is flipped 12 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=2, p=$\\frac{3}{10}$ and simplify.\n4) Here C(12,2) = 66.\nFinal answer: $\\frac{83895148953}{500000000000}$", "final_answer": "\\frac{83895148953}{500000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dbd63097f0d95583", "problem_id": "434fb9ff87e5e1ff", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{3} - 3 x^{2} + 2 x + \\left(x^{2} + 3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{x^{2} + 3 x + 7}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) + 7}) = \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3 x + 7}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} - \\frac{2 x^{4}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{6 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{12 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} - \\frac{12 x^{3}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 6 x^{3} + 23 x^{2} + 42 x + 49} + \\frac{23 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{46 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} - \\frac{64 x^{2}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{3 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 6 x^{3} + 23 x^{2} + 42 x + 49} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 6 x + 14} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3 x + 7} - \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) + 7} + \\frac{42 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{84 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} - \\frac{92 x}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} - \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) + 7} + \\frac{49 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{98 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{16}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) + 7}) = \\frac{x^{2} \\left(- 2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(x^{2} + 3 x\\right) + 7\\right)^{2}} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(x^{2} + 3 x\\right) + 7\\right)} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) + 7} + \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) + 7}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(- 2 x - 3\\right) \\left(- x^{3} - 3 x^{2} + 2 x + \\left(x \\left(x + 3\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2\\right) + \\left(x \\left(x + 3\\right) + 7\\right) \\left(- 6 x^{2} - 12 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x + 3\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x + 3\\right) + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x + 3\\right) + 7\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(- 2 x - 3\\right) \\left(- x^{3} - 3 x^{2} + 2 x + \\left(x \\left(x + 3\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2\\right) + \\left(x \\left(x + 3\\right) + 7\\right) \\left(- 6 x^{2} - 12 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x + 3\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x + 3\\right) + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x + 3\\right) + 7\\right)^{2}}$", "final_answer": "\\frac{2 \\left(- 2 x - 3\\right) \\left(- x^{3} - 3 x^{2} + 2 x + \\left(x \\left(x + 3\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2\\right) + \\left(x \\left(x + 3\\right) + 7\\right) \\left(- 6 x^{2} - 12 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x + 3\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x + 3\\right) + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x + 3\\right) + 7\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "9c1b5a6c4f5ddcc5", "problem_id": "25628b5d82b50023", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A store sells a backpack for $164. It is discounted by 25%. You buy 5 backpacks. What is the total cost?", "output": "1) Discounted price = 164 × (1 − 25/100) = 164 × 75/100 = 123.\n2) Subtotal = 5 × 123 = 615.\nFinal answer: $615$", "final_answer": "615", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "207c208f35c78e4b", "problem_id": "25628b5d82b50023", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $164. It is discounted by 25%. You buy 5 backpacks. What is the total cost?", "output": "Discounted price = 164 × (1 − 25/100) = 164 × 75/100 = 123.\nSubtotal = 5 × 123 = 615.\nCheck: recomputing the formula yields 615.\nFinal answer: $615$", "final_answer": "615", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d3ae7919a8fee17", "problem_id": "9e7625479df02b8f", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 9 times, what is the probability of exactly 7 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=9, k=7, p=$\\frac{1}{5}$ and simplify.\nHere C(9,7) = 36.\nCheck: expression simplifies to $\\frac{576}{1953125}$.\nFinal answer: $\\frac{576}{1953125}$", "final_answer": "\\frac{576}{1953125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "63580e092575e3b5", "problem_id": "9e7625479df02b8f", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 9 times, what is the probability of exactly 7 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=9, k=7, p=$\\frac{1}{5}$ and simplify.\n4) Here C(9,7) = 36.\nFinal answer: $\\frac{576}{1953125}$", "final_answer": "\\frac{576}{1953125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7dbef76bb37d0e62", "problem_id": "6e6e469d9e5d5bc2", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 13 times, what is the probability of exactly 9 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=13, k=9, p=$\\frac{3}{8}$ and simplify.\n4) Here C(13,9) = 715.\nFinal answer: $\\frac{8795840625}{549755813888}$", "final_answer": "\\frac{8795840625}{549755813888}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ca07e7e3e92b57fa", "problem_id": "6e6e469d9e5d5bc2", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 13 times, what is the probability of exactly 9 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=13, k=9, p=$\\frac{3}{8}$ and simplify.\nHere C(13,9) = 715.\nCheck: expression simplifies to $\\frac{8795840625}{549755813888}$.\nFinal answer: $\\frac{8795840625}{549755813888}$", "final_answer": "\\frac{8795840625}{549755813888}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dc656c1aeb43213a", "problem_id": "db84843392b0d6f8", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡4 (mod 9), x≡6 (mod 11). Give the solution modulo 99.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 99.\nCompute the CRT solution to get x ≡ 94 (mod 99).\nCheck: 94 mod 9=4 and mod 11=6.\nFinal answer: $x\\equiv 94\\pmod{99}$", "final_answer": "x\\equiv 94\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2c74e1ade12fface", "problem_id": "db84843392b0d6f8", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡4 (mod 9), x≡6 (mod 11). Give the solution modulo 99.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 99.\n3) Compute the CRT solution to get x ≡ 94 (mod 99).\nFinal answer: $x\\equiv 94\\pmod{99}$", "final_answer": "x\\equiv 94\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9e18fdae51f235ea", "problem_id": "bb468fac20432fbd", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡7 (mod 11), x≡5 (mod 9). Give the solution modulo 99.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 99.\nCompute the CRT solution to get x ≡ 95 (mod 99).\nCheck: 95 mod 11=7 and mod 9=5.\nFinal answer: $x\\equiv 95\\pmod{99}$", "final_answer": "x\\equiv 95\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0f66cbf6c8b43d2a", "problem_id": "bb468fac20432fbd", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the congruences: x≡7 (mod 11), x≡5 (mod 9). Give the solution modulo 99.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 99.\n3) Compute the CRT solution to get x ≡ 95 (mod 99).\nFinal answer: $x\\equiv 95\\pmod{99}$", "final_answer": "x\\equiv 95\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "222fdc4a761395ed", "problem_id": "0b02fa7f2099f57b", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - 2 x - 8$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 2 x^{2}$ and a = 4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 4 x$.\n5) Evaluate at a: p'(4) = $-16$.\nFinal answer: $-16$", "final_answer": "-16", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6c2893b37610c524", "problem_id": "0b02fa7f2099f57b", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - 2 x - 8$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 2 x^{2}$ and a = 4.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 4 x$.\nEvaluate at a: p'(4) = $-16$.\nCheck: the identity gives $-16$.\nFinal answer: $-16$", "final_answer": "-16", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9013c1b1e3a98607", "problem_id": "5500a0c96e7359c2", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A cyclist rides at 38 miles per hour for 2 hours, then at 79 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 38 × 2 = 76 miles.\nSecond leg distance = 79 × 2 = 158 miles.\nTotal distance = 76 + 158 = 234 miles.\nFinal answer: $234$", "final_answer": "234", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "20aa473a5b5d2d31", "problem_id": "5500a0c96e7359c2", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A cyclist rides at 38 miles per hour for 2 hours, then at 79 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 38 × 2 = 76 miles.\nSecond leg distance = 79 × 2 = 158 miles.\nTotal distance = 76 + 158 = 234 miles.\nCheck: adding both legs gives 234.\nFinal answer: $234$", "final_answer": "234", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "33222ffa8eade0a5", "problem_id": "31f51934106c2b48", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 18 times, what is the probability of exactly 11 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=18, k=11, p=$\\frac{1}{5}$ and simplify.\n4) Here C(18,11) = 31824.\nFinal answer: $\\frac{521404416}{3814697265625}$", "final_answer": "\\frac{521404416}{3814697265625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4984a4da67f5755f", "problem_id": "31f51934106c2b48", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 18 times, what is the probability of exactly 11 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=18, k=11, p=$\\frac{1}{5}$ and simplify.\nHere C(18,11) = 31824.\nCheck: expression simplifies to $\\frac{521404416}{3814697265625}$.\nFinal answer: $\\frac{521404416}{3814697265625}$", "final_answer": "\\frac{521404416}{3814697265625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c77d51dc9d461ee2", "problem_id": "5cccfcf9275976bb", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A store sells a backpack for $164. It is discounted by 35%. You buy 8 backpacks. What is the total cost?", "output": "1) Discounted price = 164 × (1 − 35/100) = 164 × 65/100 = 533/5.\n2) Subtotal = 8 × 533/5 = 4264/5.\nFinal answer: $\\frac{4264}{5}$", "final_answer": "\\frac{4264}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "70f438327a1386fc", "problem_id": "5cccfcf9275976bb", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $164. It is discounted by 35%. You buy 8 backpacks. What is the total cost?", "output": "Discounted price = 164 × (1 − 35/100) = 164 × 65/100 = 533/5.\nSubtotal = 8 × 533/5 = 4264/5.\nCheck: recomputing the formula yields 4264/5.\nFinal answer: $\\frac{4264}{5}$", "final_answer": "\\frac{4264}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "71f2735bbdc3f3fe", "problem_id": "205c58973853b7ec", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $3 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2$.\n3) Compute an antiderivative: F(x) = $x^{3} - x^{2} + 2 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + x^{3} - x^{2} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + x^{3} - x^{2} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "635fbed2426396d9", "problem_id": "205c58973853b7ec", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 3 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $3 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2$.\nCompute an antiderivative: F(x) = $x^{3} - x^{2} + 2 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + x^{3} - x^{2} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + x^{3} - x^{2} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e19b431cfbebcf39", "problem_id": "ca0bb1f0ded3436e", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $2249 in the ratio 9:4. How much does each receive?", "output": "1) Total ratio parts = 9+4 = 13.\n2) Each part = 2249 ÷ 13 = 173.\n3) First share = 9 × 173 = 1557.\n4) Second share = 4 × 173 = 692.\nFinal answer: $692$", "final_answer": "692", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dc411eae7bac14ec", "problem_id": "ca0bb1f0ded3436e", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $2249 in the ratio 9:4. How much does each receive?", "output": "Total ratio parts = 9+4 = 13.\nEach part = 2249 ÷ 13 = 173.\nFirst share = 9 × 173 = 1557.\nSecond share = 4 × 173 = 692.\nCheck: 1557+692=2249.\nFinal answer: $692$", "final_answer": "692", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f00163e4d736cbf8", "problem_id": "6ca4b9c0c325f74b", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} 15 - 3 x$.", "output": "$27$", "final_answer": "27", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "17cf17566bf9de2f", "problem_id": "6ca4b9c0c325f74b", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} 15 - 3 x$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 3 x^{2} + 3 x + 1$ and a = -4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $3 - 6 x$.\n5) Evaluate at a: p'(-4) = $27$.\nFinal answer: $27$", "final_answer": "27", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "180fc3be27e7e1d3", "problem_id": "3cdf6ccdf6e07da0", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $48. It is discounted by 20%. You buy 12 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 48 × (1 − 20/100) = 48 × 80/100 = 192/5.\nSubtotal = 12 × 192/5 = 2304/5.\nTotal with tax = 2304/5 × 107/100 = 61632/125.\nCheck: recomputing the formula yields 61632/125.\nFinal answer: $\\frac{61632}{125}$", "final_answer": "\\frac{61632}{125}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a42a6673f45b2639", "problem_id": "3cdf6ccdf6e07da0", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A store sells a backpack for $48. It is discounted by 20%. You buy 12 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 48 × (1 − 20/100) = 48 × 80/100 = 192/5.\n2) Subtotal = 12 × 192/5 = 2304/5.\n3) Total with tax = 2304/5 × 107/100 = 61632/125.\nFinal answer: $\\frac{61632}{125}$", "final_answer": "\\frac{61632}{125}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "19e9ce41f0ccb90f", "problem_id": "145328d3f21fe968", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(- 3 x^{2} - 2 x - 2\\right) \\cos{\\left(2 x^{2} - x + 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(e^{x}) = e^{x}$.\n3) So $f'(x) = 12 x^{3} \\sin{\\left(\\left(2 x^{2} - x\\right) + 1 \\right)} + 5 x^{2} \\sin{\\left(\\left(2 x^{2} - x\\right) + 1 \\right)} + x e^{x} + 6 x \\sin{\\left(\\left(2 x^{2} - x\\right) + 1 \\right)} - 6 x \\cos{\\left(\\left(2 x^{2} - x\\right) + 1 \\right)} + 2 e^{x} - 2 \\sin{\\left(\\left(2 x^{2} - x\\right) + 1 \\right)} - 2 \\cos{\\left(\\left(2 x^{2} - x\\right) + 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(e^{x}) = e^{x}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- 3 x - 1\\right) \\cos{\\left(\\left(2 x^{2} - x\\right) + 1 \\right)} + \\left(x + 1\\right) e^{x} - \\left(4 x - 1\\right) \\left(x \\left(- 3 x - 2\\right) - 2\\right) \\sin{\\left(\\left(2 x^{2} - x\\right) + 1 \\right)} + e^{x}$.\nFinal answer: $2 \\left(- 3 x - 1\\right) \\cos{\\left(\\left(2 x^{2} - x\\right) + 1 \\right)} + \\left(x + 1\\right) e^{x} - \\left(4 x - 1\\right) \\left(x \\left(- 3 x - 2\\right) - 2\\right) \\sin{\\left(\\left(2 x^{2} - x\\right) + 1 \\right)} + e^{x}$", "final_answer": "2 \\left(- 3 x - 1\\right) \\cos{\\left(\\left(2 x^{2} - x\\right) + 1 \\right)} + \\left(x + 1\\right) e^{x} - \\left(4 x - 1\\right) \\left(x \\left(- 3 x - 2\\right) - 2\\right) \\sin{\\left(\\left(2 x^{2} - x\\right) + 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "4c4e873ac539cf18", "problem_id": "d16433ba9cd1a265", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A cyclist rides at 22 miles per hour for 1 hours, then at 83 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 22 × 1 = 22 miles.\n2) Second leg distance = 83 × 2 = 166 miles.\n3) Total distance = 22 + 166 = 188 miles.\nFinal answer: $188$", "final_answer": "188", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0c8e07bb2092901b", "problem_id": "8f14797b9bef4663", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{2} - x - 4$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $- 4 x - 1$.\n2) Slope at x=-1: m = f'(-1) = $3$.\n3) Point on curve: (-1, f(-1)) = (-1, $-5$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -5 = 3(x − -1).\n6) Optionally expand: y = $3 x - 2$.\nFinal answer: $y = 3\\cdot x - 2$", "final_answer": "y = 3\\cdot x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "92f06f5e5abb663f", "problem_id": "8f14797b9bef4663", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{2} - x - 4$ at $x=-1$.", "output": "Differentiate: f'(x) = $- 4 x - 1$.\nSlope at x=-1: m = f'(-1) = $3$.\nPoint on curve: (-1, f(-1)) = (-1, $-5$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -5 = 3(x − -1).\nOptionally expand: y = $3 x - 2$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 3\\cdot x - 2$", "final_answer": "y = 3\\cdot x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9425b55b960ac927", "problem_id": "c6d02680d1ed8f9e", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_6=31$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_6=25.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=25 and k=6.\n4) Compute C(30,5) = 142506.\nFinal answer: $142506$", "final_answer": "142506", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1256f576ea15d31e", "problem_id": "a8b7f4ccc19b327f", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 2 x^{2} + 3 x - 12$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{2} + 2 x - 12.\n3) Then the limit is -36.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -39.\nFinal answer: $-39$", "final_answer": "-39", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "be2adcdf6f1d2786", "problem_id": "33f99155006ea1bf", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡0 (mod 3), x≡6 (mod 10). Give the solution modulo 30.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 30.\n3) Compute the CRT solution to get x ≡ 6 (mod 30).\nFinal answer: $x\\equiv 6\\pmod{30}$", "final_answer": "x\\equiv 6\\pmod{30}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a90284aa8e7ddcd", "problem_id": "33f99155006ea1bf", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡0 (mod 3), x≡6 (mod 10). Give the solution modulo 30.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 30.\nCompute the CRT solution to get x ≡ 6 (mod 30).\nCheck: 6 mod 3=0 and mod 10=6.\nFinal answer: $x\\equiv 6\\pmod{30}$", "final_answer": "x\\equiv 6\\pmod{30}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "849b01ba8d20d90e", "problem_id": "b25da1383a89c6c7", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{2} - 3 x + \\left(- x^{2} + 2 x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{- x^{2} + 2 x + 6}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- x^{2} + 2 x\\right) + 6}) = \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{- x^{2} + 2 x + 6}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{4 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{8 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{8 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{16 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{18 x^{2}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} + \\frac{24 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} + \\frac{48 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{68 x}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{12 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 4 x^{3} - 8 x^{2} + 24 x + 36} + \\frac{36 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} + \\frac{72 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{40}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} + \\frac{12 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 4 x^{3} - 8 x^{2} + 24 x + 36} + \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{- x^{2} + 2 x + 6} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- x^{2} + 2 x\\right) + 6} - \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(- x^{2} + 2 x\\right) + 6}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- x^{2} + 2 x\\right) + 6}) = \\frac{6 \\left(2 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(- x^{2} + 2 x\\right) + 6\\right)^{2}} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- x^{2} + 2 x\\right) + 6} + \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(- x^{2} + 2 x\\right) + 6}$.\nTherefore the correct derivative is $f'(x) = \\frac{4 \\left(x - 1\\right) \\left(- 3 x^{2} - 3 x + \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(x \\left(2 - x\\right) + 6\\right) \\left(- 12 x + 4 \\left(1 - x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x \\left(2 - x\\right) + 6\\right)^{2}}$.\nFinal answer: $\\frac{4 \\left(x - 1\\right) \\left(- 3 x^{2} - 3 x + \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(x \\left(2 - x\\right) + 6\\right) \\left(- 12 x + 4 \\left(1 - x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x \\left(2 - x\\right) + 6\\right)^{2}}$", "final_answer": "\\frac{4 \\left(x - 1\\right) \\left(- 3 x^{2} - 3 x + \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(x \\left(2 - x\\right) + 6\\right) \\left(- 12 x + 4 \\left(1 - x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x \\left(2 - x\\right) + 6\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "769a5ae07635720b", "problem_id": "b0be4f0589bbd9ac", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$3 x - 7 y$ = $-10$\n$- 8 x + y$ = $-44$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-8$: $- 24 x + 56 y$ = $80$.\n3) Multiply the second equation by $3$: $- 24 x + 3 y$ = $-132$.\n4) Subtract to eliminate x: $53 y$ = $212$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $6$.\nFinal answer: $(x,y)=\\left(6, 4\\right)$", "final_answer": "(x,y)=\\left(6, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6ce5afb7d1ab671e", "problem_id": "66e520fe24ec8147", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$- 3 x - 4 y$ = $-9$\n$- 4 x + y$ = $-12$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $12 x + 16 y$ = $36$.\nMultiply the second equation by $-3$: $12 x - 3 y$ = $36$.\nSubtract to eliminate x: $19 y$ = $0$.\nSolve for y: y = $0$.\nSubstitute back to find x: x = $3$.\nCheck: plug (x,y)=(3,0) into both equations.\nFinal answer: $(x,y)=\\left(3, 0\\right)$", "final_answer": "(x,y)=\\left(3, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "53b58d166f226175", "problem_id": "66e520fe24ec8147", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$- 3 x - 4 y$ = $-9$\n$- 4 x + y$ = $-12$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $12 x + 16 y$ = $36$.\n3) Multiply the second equation by $-3$: $12 x - 3 y$ = $36$.\n4) Subtract to eliminate x: $19 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, 0\\right)$", "final_answer": "(x,y)=\\left(3, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3fd766b53b511724", "problem_id": "76fff5015cbabd2c", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the multiplicative inverse of 16 modulo 19.", "output": "1) We want x such that 16x ≡ 1 (mod 19).\n2) Use extended Euclid to find s,t with 16s + 19t = 1.\n3) One Bezout identity is: 16·(6) + 19·(-5) = 1.\n4) Reduce s modulo 19: x ≡ 6 (mod 19).\nFinal answer: $x\\equiv 6\\pmod{19}$", "final_answer": "x\\equiv 6\\pmod{19}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "27698fe13e57b21c", "problem_id": "76fff5015cbabd2c", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the multiplicative inverse of 16 modulo 19.", "output": "We want x such that 16x ≡ 1 (mod 19).\nUse extended Euclid to find s,t with 16s + 19t = 1.\nOne Bezout identity is: 16·(6) + 19·(-5) = 1.\nReduce s modulo 19: x ≡ 6 (mod 19).\nCheck: 16·6 mod 19 = 1.\nFinal answer: $x\\equiv 6\\pmod{19}$", "final_answer": "x\\equiv 6\\pmod{19}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6e5261d57f4a9fe3", "problem_id": "c4b81e555e266eab", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 3 x - 168$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} + 3 x - 168$ = $3 \\left(x - 7\\right) \\left(x + 8\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $7$.\n5) From the second factor: x = $-8$.\nFinal answer: $x \\in \\{7, -8\\}$", "final_answer": "x \\in \\{7, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bc361f078fde0500", "problem_id": "c4b81e555e266eab", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 3 x - 168$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} + 3 x - 168$ = $3 \\left(x - 7\\right) \\left(x + 8\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $7$.\nFrom the second factor: x = $-8$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{7, -8\\}$", "final_answer": "x \\in \\{7, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "64bc2bb875e8c910", "problem_id": "8fadf2bd808928ac", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 8 red balls and 8 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(16,8).\nFavorable samples: choose k reds and n−k blues: C(8,1)·C(8,7).\nCompute totals: C(16,8) = 12870, and favorable = 64.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{32}{6435}$.\nFinal answer: $\\frac{32}{6435}$", "final_answer": "\\frac{32}{6435}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7692e3621ff94ad3", "problem_id": "8fadf2bd808928ac", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 8 red balls and 8 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(16,8).\n3) Favorable samples: choose k reds and n−k blues: C(8,1)·C(8,7).\n4) Compute totals: C(16,8) = 12870, and favorable = 64.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{32}{6435}$", "final_answer": "\\frac{32}{6435}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e34518b59c12379a", "problem_id": "8eec1675945c8d14", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡7 (mod 9), x≡5 (mod 10). Give the solution modulo 90.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 90.\nCompute the CRT solution to get x ≡ 25 (mod 90).\nCheck: 25 mod 9=7 and mod 10=5.\nFinal answer: $x\\equiv 25\\pmod{90}$", "final_answer": "x\\equiv 25\\pmod{90}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d16046395933e21e", "problem_id": "8eec1675945c8d14", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡7 (mod 9), x≡5 (mod 10). Give the solution modulo 90.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 90.\n3) Compute the CRT solution to get x ≡ 25 (mod 90).\nFinal answer: $x\\equiv 25\\pmod{90}$", "final_answer": "x\\equiv 25\\pmod{90}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "737c37efb0803af6", "problem_id": "f678d9ad30c788f7", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{2} - x + 2 \\sin{\\left(x \\right)} + 1$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 6 x + 2 \\cos{\\left(x \\right)} - 1$.\n2) Slope at x=2: m = f'(2) = $-13 + 2 \\cos{\\left(2 \\right)}$.\n3) Point on curve: (2, f(2)) = (2, $-13 + 2 \\sin{\\left(2 \\right)}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -13 + 2 \\sin{\\left(2 \\right)} = -13 + 2 \\cos{\\left(2 \\right)}(x − 2).\n6) Optionally expand: y = $- 13 x + 2 x \\cos{\\left(2 \\right)} - 4 \\cos{\\left(2 \\right)} + 2 \\sin{\\left(2 \\right)} + 13$.\nFinal answer: $y = - 13\\cdot x + 2\\cdot x\\cdot \\cos{\\left(2 \\right)} - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)} + 13$", "final_answer": "y = - 13\\cdot x + 2\\cdot x\\cdot \\cos{\\left(2 \\right)} - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)} + 13", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2dead2c4ea745201", "problem_id": "f678d9ad30c788f7", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{2} - x + 2 \\sin{\\left(x \\right)} + 1$ at $x=2$.", "output": "Differentiate: f'(x) = $- 6 x + 2 \\cos{\\left(x \\right)} - 1$.\nSlope at x=2: m = f'(2) = $-13 + 2 \\cos{\\left(2 \\right)}$.\nPoint on curve: (2, f(2)) = (2, $-13 + 2 \\sin{\\left(2 \\right)}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -13 + 2 \\sin{\\left(2 \\right)} = -13 + 2 \\cos{\\left(2 \\right)}(x − 2).\nOptionally expand: y = $- 13 x + 2 x \\cos{\\left(2 \\right)} - 4 \\cos{\\left(2 \\right)} + 2 \\sin{\\left(2 \\right)} + 13$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 13\\cdot x + 2\\cdot x\\cdot \\cos{\\left(2 \\right)} - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)} + 13$", "final_answer": "y = - 13\\cdot x + 2\\cdot x\\cdot \\cos{\\left(2 \\right)} - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)} + 13", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7d10fcca9a343dec", "problem_id": "a296c28adb8b96c6", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 4 - 3 x$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 4 - 4 x.\n3) Then the limit is 12.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 10.\nFinal answer: $10$", "final_answer": "10", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "001c41c11b3f5659", "problem_id": "24ae50cb1aba80cf", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Worker A finishes a job in 19 hours and Worker B finishes it in 15 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/19 job/hour. B's rate = 1/15 job/hour.\nCombined rate = 1/19 + 1/15 = 34/285 job/hour.\nTime = 1 ÷ (combined rate) = 285/34 hours.\nCheck: (combined rate)×(time) = 34/285×285/34 = 1.\nFinal answer: $\\frac{285}{34}$", "final_answer": "\\frac{285}{34}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b4822bbbfefa1637", "problem_id": "491af4d09608353f", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{13 x}{2} + 9$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{13}{2}$, c=$9$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{25}{4}$, so √Δ = $\\frac{5}{2}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{9}{2}$.\n7) x2 = $2$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{9}{2}, 2\\}$", "final_answer": "x \\in \\{\\frac{9}{2}, 2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ed4f78b576778feb", "problem_id": "491af4d09608353f", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{13 x}{2} + 9$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{13}{2}$, c=$9$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{25}{4}$, so √Δ = $\\frac{5}{2}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{9}{2}$.\nx2 = $2$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{9}{2}, 2\\}$", "final_answer": "x \\in \\{\\frac{9}{2}, 2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e41d9943001dbea0", "problem_id": "2a3abab3e32ff561", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 2 x^{3} - 3 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{3} - 3 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $- 9 \\pi^{3} - 3 \\pi + \\frac{15 \\pi^{4}}{2}$.\n4) Simplify the result.\nFinal answer: $- 9\\cdot \\pi^{3} - 3\\cdot \\pi + \\frac{15\\cdot \\pi^{4}}{2}$", "final_answer": "- 9\\cdot \\pi^{3} - 3\\cdot \\pi + \\frac{15\\cdot \\pi^{4}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d1c3e5a806fbd968", "problem_id": "90254d9381cdeeed", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 x^{3} - 3 x^{2} + 9 x - 21$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} - 3 x^{2} + 8 x - 21.\n3) Then the limit is -73.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -75.\nFinal answer: $-75$", "final_answer": "-75", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8511cd66bf6de06a", "problem_id": "675f34ff153eaffc", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{2} - x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 3 x^{2} - x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2$.\n3) Compute an antiderivative: F(x) = $- x^{3} - \\frac{x^{2}}{2} + 2 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C - x^{3} - \\frac{x^{2}}{2} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - x^{3} - \\frac{x^{2}}{2} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a6e389b42f14fe5c", "problem_id": "675f34ff153eaffc", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 3 x^{2} - x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 3 x^{2} - x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 2$.\nCompute an antiderivative: F(x) = $- x^{3} - \\frac{x^{2}}{2} + 2 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - x^{3} - \\frac{x^{2}}{2} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - x^{3} - \\frac{x^{2}}{2} + 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ea1a2e57b5775f96", "problem_id": "6862b95554fa0fba", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$x - 4 y$ = $25$\n$- 4 x - 2 y$ = $-10$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $- 4 x + 16 y$ = $-100$.\n3) Multiply the second equation by $1$: $- 4 x - 2 y$ = $-10$.\n4) Subtract to eliminate x: $18 y$ = $-90$.\n5) Solve for y: y = $-5$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, -5\\right)$", "final_answer": "(x,y)=\\left(5, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a0282f4f2bc00188", "problem_id": "6862b95554fa0fba", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Solve the linear system:\n$x - 4 y$ = $25$\n$- 4 x - 2 y$ = $-10$", "output": "$(x,y)=\\left(5, -5\\right)$", "final_answer": "(x,y)=\\left(5, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "712b0773a7f6edcd", "problem_id": "689ea89b5678b53a", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡2 (mod 3), x≡5 (mod 7). Give the solution modulo 21.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 21.\n3) Compute the CRT solution to get x ≡ 5 (mod 21).\nFinal answer: $x\\equiv 5\\pmod{21}$", "final_answer": "x\\equiv 5\\pmod{21}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7fc077ce187d2aec", "problem_id": "689ea89b5678b53a", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 3), x≡5 (mod 7). Give the solution modulo 21.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 21.\nCompute the CRT solution to get x ≡ 5 (mod 21).\nCheck: 5 mod 3=2 and mod 7=5.\nFinal answer: $x\\equiv 5\\pmod{21}$", "final_answer": "x\\equiv 5\\pmod{21}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d9adc41c9c49b011", "problem_id": "79dd17ae1d4c35c0", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 14 times, what is the probability of exactly 12 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=14, k=12, p=$\\frac{3}{7}$ and simplify.\nHere C(14,12) = 91.\nCheck: expression simplifies to $\\frac{110539728}{96889010407}$.\nFinal answer: $\\frac{110539728}{96889010407}$", "final_answer": "\\frac{110539728}{96889010407}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c0ef3bde54d0a4a2", "problem_id": "79dd17ae1d4c35c0", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 14 times, what is the probability of exactly 12 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=14, k=12, p=$\\frac{3}{7}$ and simplify.\n4) Here C(14,12) = 91.\nFinal answer: $\\frac{110539728}{96889010407}$", "final_answer": "\\frac{110539728}{96889010407}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c9e5fe22a23e61a1", "problem_id": "15433b7c4cc8822a", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 16 times, what is the probability of exactly 9 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=16, k=9, p=$\\frac{4}{5}$ and simplify.\n4) Here C(16,9) = 11440.\nFinal answer: $\\frac{599785472}{30517578125}$", "final_answer": "\\frac{599785472}{30517578125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f509130bfb2df652", "problem_id": "15433b7c4cc8822a", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 16 times, what is the probability of exactly 9 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=16, k=9, p=$\\frac{4}{5}$ and simplify.\nFinal answer: $\\frac{599785472}{30517578125}$", "final_answer": "\\frac{599785472}{30517578125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "79de8b28147d97d5", "problem_id": "e53584a00cbf3012", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{10}$. If it is flipped 11 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=6, p=$\\frac{3}{10}$ and simplify.\n4) Here C(11,6) = 462.\nFinal answer: $\\frac{2830281993}{50000000000}$", "final_answer": "\\frac{2830281993}{50000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fb219e18fccf7cb9", "problem_id": "e53584a00cbf3012", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{10}$. If it is flipped 11 times, what is the probability of exactly 6 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=11, k=6, p=$\\frac{3}{10}$ and simplify.\nHere C(11,6) = 462.\nCheck: expression simplifies to $\\frac{2830281993}{50000000000}$.\nFinal answer: $\\frac{2830281993}{50000000000}$", "final_answer": "\\frac{2830281993}{50000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9c8b5ff52578614d", "problem_id": "f655cce2ad1a09af", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $316. It is discounted by 25%. You buy 4 backpacks. What is the total cost?", "output": "Discounted price = 316 × (1 − 25/100) = 316 × 75/100 = 237.\nSubtotal = 4 × 237 = 948.\nCheck: recomputing the formula yields 948.\nFinal answer: $948$", "final_answer": "948", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "091beb0e41eaa737", "problem_id": "f655cce2ad1a09af", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A store sells a backpack for $316. It is discounted by 25%. You buy 4 backpacks. What is the total cost?", "output": "1) Discounted price = 316 × (1 − 25/100) = 316 × 75/100 = 237.\n2) Subtotal = 4 × 237 = 948.\nFinal answer: $948$", "final_answer": "948", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "53678ee27e7ff6d4", "problem_id": "f27f6efd2b0235cb", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{11 x}{5} - \\frac{12}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{11}{5}$, c=$- \\frac{12}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{361}{25}$, so √Δ = $\\frac{19}{5}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $3$.\n7) x2 = $- \\frac{4}{5}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{3, - \\frac{4}{5}\\}$", "final_answer": "x \\in \\{3, - \\frac{4}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "92d2fde11b8e1ecd", "problem_id": "f27f6efd2b0235cb", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{11 x}{5} - \\frac{12}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{11}{5}$, c=$- \\frac{12}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{361}{25}$, so √Δ = $\\frac{19}{5}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $3$.\nx2 = $- \\frac{4}{5}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{3, - \\frac{4}{5}\\}$", "final_answer": "x \\in \\{3, - \\frac{4}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dc32115a6662e88a", "problem_id": "fb958666915ac358", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} 3 x^{2} + 10 x + 27$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{3} + x^{2} - 3 x - 1$ and a = 3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $9 x^{2} + 2 x - 3$.\n5) Evaluate at a: p'(3) = $84$.\nFinal answer: $84$", "final_answer": "84", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1503715035823af1", "problem_id": "fb958666915ac358", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} 3 x^{2} + 10 x + 27$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $3 x^{3} + x^{2} - 3 x - 1$ and a = 3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $9 x^{2} + 2 x - 3$.\nEvaluate at a: p'(3) = $84$.\nCheck: the identity gives $84$.\nFinal answer: $84$", "final_answer": "84", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "39f1a38738702e69", "problem_id": "228edec1dd586676", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 20 times, what is the probability of exactly 6 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=20, k=6, p=$\\frac{2}{3}$ and simplify.\nHere C(20,6) = 38760.\nCheck: expression simplifies to $\\frac{826880}{1162261467}$.\nFinal answer: $\\frac{826880}{1162261467}$", "final_answer": "\\frac{826880}{1162261467}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0b4c0783ac2003c9", "problem_id": "228edec1dd586676", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 20 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=20, k=6, p=$\\frac{2}{3}$ and simplify.\n4) Here C(20,6) = 38760.\nFinal answer: $\\frac{826880}{1162261467}$", "final_answer": "\\frac{826880}{1162261467}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3e68d0d4012861a6", "problem_id": "e4a582f3f43ad001", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A cyclist rides at 48 miles per hour for 3/2 hours, then at 85 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 48 × 3/2 = 72 miles.\nSecond leg distance = 85 × 1 = 85 miles.\nTotal distance = 72 + 85 = 157 miles.\nCheck: adding both legs gives 157.\nFinal answer: $157$", "final_answer": "157", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ab86b0e511250c21", "problem_id": "e4a582f3f43ad001", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A cyclist rides at 48 miles per hour for 3/2 hours, then at 85 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 48 × 3/2 = 72 miles.\n2) Second leg distance = 85 × 1 = 85 miles.\n3) Total distance = 72 + 85 = 157 miles.\nFinal answer: $157$", "final_answer": "157", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f098d1dd22545223", "problem_id": "6fa8d838cfb2e227", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡9 (mod 11), x≡4 (mod 13). Give the solution modulo 143.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 143.\n3) Compute the CRT solution to get x ≡ 108 (mod 143).\nFinal answer: $x\\equiv 108\\pmod{143}$", "final_answer": "x\\equiv 108\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0b71e1383e539983", "problem_id": "6fa8d838cfb2e227", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡9 (mod 11), x≡4 (mod 13). Give the solution modulo 143.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 143.\nCompute the CRT solution to get x ≡ 108 (mod 143).\nCheck: 108 mod 11=9 and mod 13=4.\nFinal answer: $x\\equiv 108\\pmod{143}$", "final_answer": "x\\equiv 108\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "03d1fe8718c30c43", "problem_id": "26b86aee00441451", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡0 (mod 8), x≡6 (mod 13). Give the solution modulo 104.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 104.\n3) Compute the CRT solution to get x ≡ 32 (mod 104).\nFinal answer: $x\\equiv 32\\pmod{104}$", "final_answer": "x\\equiv 32\\pmod{104}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "26ea89d9335e42fa", "problem_id": "26b86aee00441451", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the congruences: x≡0 (mod 8), x≡6 (mod 13). Give the solution modulo 104.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 104.\nCompute the CRT solution to get x ≡ 32 (mod 104).\nFinal answer: $x\\equiv 32\\pmod{104}$", "final_answer": "x\\equiv 32\\pmod{104}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d8e700a55990a1e", "problem_id": "e4bdbe5d6712457b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{3 x^{2} - x + \\left(- 3 x^{2} + 2 x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{- 3 x^{2} + 2 x + 5}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- 3 x^{2} + 2 x\\right) + 5}) = \\frac{2 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{- 3 x^{2} + 2 x + 5}$.\n3) So $f'(x) = \\frac{9 x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} + \\frac{18 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} - \\frac{12 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} - \\frac{24 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} - \\frac{26 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} - \\frac{52 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} + \\frac{6 x^{2}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} - \\frac{12 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{4} - 12 x^{3} - 26 x^{2} + 20 x + 25} + \\frac{20 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} + \\frac{40 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} + \\frac{96 x}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} + \\frac{4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{4} - 12 x^{3} - 26 x^{2} + 20 x + 25} + \\frac{2 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{- 3 x^{2} + 2 x + 5} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- 3 x^{2} + 2 x\\right) + 5} - \\frac{2 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(- 3 x^{2} + 2 x\\right) + 5} + \\frac{25 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} + \\frac{50 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} - \\frac{22}{18 x^{4} - 24 x^{3} - 52 x^{2} + 40 x + 50} - \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- 3 x^{2} + 2 x\\right) + 5}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- 3 x^{2} + 2 x\\right) + 5}) = \\frac{2 x \\left(6 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(- 3 x^{2} + 2 x\\right) + 5\\right)^{2}} + \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- 3 x^{2} + 2 x\\right) + 5} + \\frac{2 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(- 3 x^{2} + 2 x\\right) + 5} + \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- 3 x^{2} + 2 x\\right) + 5}$.\nTherefore the correct derivative is $f'(x) = \\frac{4 \\left(3 x - 1\\right) \\left(3 x^{2} - x + \\left(x \\left(2 - 3 x\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(2 - 3 x\\right) + 5\\right) \\left(12 x + 4 \\left(1 - 3 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 - 3 x\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 - 3 x\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x \\left(2 - 3 x\\right) + 5\\right)^{2}}$.\nFinal answer: $\\frac{4 \\left(3 x - 1\\right) \\left(3 x^{2} - x + \\left(x \\left(2 - 3 x\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(2 - 3 x\\right) + 5\\right) \\left(12 x + 4 \\left(1 - 3 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 - 3 x\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 - 3 x\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x \\left(2 - 3 x\\right) + 5\\right)^{2}}$", "final_answer": "\\frac{4 \\left(3 x - 1\\right) \\left(3 x^{2} - x + \\left(x \\left(2 - 3 x\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(2 - 3 x\\right) + 5\\right) \\left(12 x + 4 \\left(1 - 3 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 - 3 x\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 - 3 x\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x \\left(2 - 3 x\\right) + 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2feecef1e0e918d0", "problem_id": "069bb540bda82f1f", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡3 (mod 8), x≡4 (mod 7). Give the solution modulo 56.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 56.\n3) Compute the CRT solution to get x ≡ 11 (mod 56).\nFinal answer: $x\\equiv 11\\pmod{56}$", "final_answer": "x\\equiv 11\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d61ed0eea52b831", "problem_id": "879a44a9a7b4d8ba", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A cyclist rides at 48 miles per hour for 1 hours, then at 55 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 48 × 1 = 48 miles.\n2) Second leg distance = 55 × 3 = 165 miles.\n3) Total distance = 48 + 165 = 213 miles.\nFinal answer: $213$", "final_answer": "213", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "401abd3756925d55", "problem_id": "879a44a9a7b4d8ba", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "A cyclist rides at 48 miles per hour for 1 hours, then at 55 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 48 × 1 = 48 miles.\nSecond leg distance = 55 × 3 = 165 miles.\nTotal distance = 48 + 165 = 213 miles.\nFinal answer: $213$", "final_answer": "213", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fc011966b5a68021", "problem_id": "4dc60f4aaf2b57ec", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "You want 25 liters of a 37% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 25−x liters is 50% solution.\n2) Salt equation: 25x + 50(25−x) = 37·25.\n3) (25-50)x = (37-50)·25 ⇒ x = 13.\n4) Then 25−x = 12.\nFinal answer: $13 L of 25% and 12 L of 50%$", "final_answer": "13 L of 25% and 12 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72fd7fd4512c4ea5", "problem_id": "4dc60f4aaf2b57ec", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "You want 25 liters of a 37% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 25−x liters is 50% solution.\nSalt equation: 25x + 50(25−x) = 37·25.\n(25-50)x = (37-50)·25 ⇒ x = 13.\nFinal answer: $13 L of 25% and 12 L of 50%$", "final_answer": "13 L of 25% and 12 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ada45d2e2f67ffa5", "problem_id": "8a89a12b0af480fa", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 3), x≡1 (mod 5). Give the solution modulo 15.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 15.\nCompute the CRT solution to get x ≡ 11 (mod 15).\nCheck: 11 mod 3=2 and mod 5=1.\nFinal answer: $x\\equiv 11\\pmod{15}$", "final_answer": "x\\equiv 11\\pmod{15}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bc705d7e9f0cde2e", "problem_id": "8a89a12b0af480fa", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡2 (mod 3), x≡1 (mod 5). Give the solution modulo 15.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 15.\n3) Compute the CRT solution to get x ≡ 11 (mod 15).\nFinal answer: $x\\equiv 11\\pmod{15}$", "final_answer": "x\\equiv 11\\pmod{15}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "22d7e9f6d139c15b", "problem_id": "83f0808dd896b436", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Two friends split $1888 in the ratio 8:8. How much does each receive?", "output": "1) Total ratio parts = 8+8 = 16.\n2) Each part = 1888 ÷ 16 = 118.\n3) First share = 8 × 118 = 944.\n4) Second share = 8 × 118 = 944.\nFinal answer: $944$", "final_answer": "944", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a1523de4b72674af", "problem_id": "9226cf6badd482d3", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(- x^{2} - x\\right) \\cos{\\left(3 x^{2} - x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x \\cos{\\left(\\left(3 x^{2} - x\\right) + 3 \\right)}) = x \\sin{\\left(3 x^{2} - x + 3 \\right)}$.\n3) So $f'(x) = 6 x^{3} \\sin{\\left(\\left(3 x^{2} - x\\right) + 3 \\right)} - x^{2} \\sin{\\left(\\left(3 x^{2} - x\\right) + 3 \\right)} + x e^{x} + x \\sin{\\left(3 x^{2} - x + 3 \\right)} - 2 x \\cos{\\left(\\left(3 x^{2} - x\\right) + 3 \\right)} + 2 e^{x}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x \\cos{\\left(\\left(3 x^{2} - x\\right) + 3 \\right)}) = x \\left(6 x - 1\\right) \\sin{\\left(\\left(3 x^{2} - x\\right) + 3 \\right)} - \\cos{\\left(\\left(3 x^{2} - x\\right) + 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = - x \\left(- x - 1\\right) \\left(6 x - 1\\right) \\sin{\\left(\\left(3 x^{2} - x\\right) + 3 \\right)} + \\left(- 2 x - 1\\right) \\cos{\\left(\\left(3 x^{2} - x\\right) + 3 \\right)} + \\left(x + 1\\right) e^{x} + e^{x}$.\nFinal answer: $- x \\left(- x - 1\\right) \\left(6 x - 1\\right) \\sin{\\left(\\left(3 x^{2} - x\\right) + 3 \\right)} + \\left(- 2 x - 1\\right) \\cos{\\left(\\left(3 x^{2} - x\\right) + 3 \\right)} + \\left(x + 1\\right) e^{x} + e^{x}$", "final_answer": "- x \\left(- x - 1\\right) \\left(6 x - 1\\right) \\sin{\\left(\\left(3 x^{2} - x\\right) + 3 \\right)} + \\left(- 2 x - 1\\right) \\cos{\\left(\\left(3 x^{2} - x\\right) + 3 \\right)} + \\left(x + 1\\right) e^{x} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5f8684e551dacb55", "problem_id": "5ddc1fb587c43c85", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the multiplicative inverse of 26 modulo 37.", "output": "We want x such that 26x ≡ 1 (mod 37).\nUse extended Euclid to find s,t with 26s + 37t = 1.\nOne Bezout identity is: 26·(10) + 37·(-7) = 1.\nReduce s modulo 37: x ≡ 10 (mod 37).\nCheck: 26·10 mod 37 = 1.\nFinal answer: $x\\equiv 10\\pmod{37}$", "final_answer": "x\\equiv 10\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6d72d171550d8dd0", "problem_id": "5ddc1fb587c43c85", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Find the multiplicative inverse of 26 modulo 37.", "output": "$x\\equiv 10\\pmod{37}$", "final_answer": "x\\equiv 10\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "85ff62734dc01a9a", "problem_id": "30b64681f6e02e0e", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_10=44$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_10=14.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=14 and k=10.\n4) Compute C(23,9) = 817190.\nFinal answer: $817190$", "final_answer": "817190", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "34593ceea0df44a1", "problem_id": "782934a1915a228b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} + 4 x - 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x^{2}) = - 4 x^{2}$.\n3) So $f'(x) = 4 - 4 x^{2}$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x^{2}) = - 4 x$.\nTherefore the correct derivative is $f'(x) = 4 \\left(1 - x\\right)$.\nFinal answer: $4 \\left(1 - x\\right)$", "final_answer": "4 \\left(1 - x\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "34685076287a1734", "problem_id": "a0247240555962dc", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$- 3 x + 3 y$ = $-18$\n$4 x + 2 y$ = $18$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $- 12 x + 12 y$ = $-72$.\n3) Multiply the second equation by $-3$: $- 12 x - 6 y$ = $-54$.\n4) Subtract to eliminate x: $18 y$ = $-18$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, -1\\right)$", "final_answer": "(x,y)=\\left(5, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "634313fb6d655872", "problem_id": "a0247240555962dc", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the linear system:\n$- 3 x + 3 y$ = $-18$\n$4 x + 2 y$ = $18$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $- 12 x + 12 y$ = $-72$.\nMultiply the second equation by $-3$: $- 12 x - 6 y$ = $-54$.\nFinal answer: $(x,y)=\\left(5, -1\\right)$", "final_answer": "(x,y)=\\left(5, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a5bbf89d0ab60fda", "problem_id": "8348edd404a24880", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $- 3 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $- 3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9d5d8c9963d4b604", "problem_id": "8348edd404a24880", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $- 3 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "892c2007d7d7e09d", "problem_id": "61220bdb0e4785c7", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡2 (mod 8), x≡8 (mod 11). Give the solution modulo 88.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 88.\nCompute the CRT solution to get x ≡ 74 (mod 88).\nCheck: 74 mod 8=2 and mod 11=8.\nFinal answer: $x\\equiv 74\\pmod{88}$", "final_answer": "x\\equiv 74\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a42cf7bec44cad50", "problem_id": "61220bdb0e4785c7", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡2 (mod 8), x≡8 (mod 11). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 74 (mod 88).\nFinal answer: $x\\equiv 74\\pmod{88}$", "final_answer": "x\\equiv 74\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f5bb2f1f6d4d6d69", "problem_id": "f591dfb6d9b583f5", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - \\frac{12 x}{5} - \\frac{36}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$- \\frac{12}{5}$, c=$- \\frac{36}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{2304}{25}$, so √Δ = $\\frac{48}{5}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $2$.\n7) x2 = $- \\frac{6}{5}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{2, - \\frac{6}{5}\\}$", "final_answer": "x \\in \\{2, - \\frac{6}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "138d58e7661d85ce", "problem_id": "f591dfb6d9b583f5", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - \\frac{12 x}{5} - \\frac{36}{5}$ = 0.", "output": "$x \\in \\{2, - \\frac{6}{5}\\}$", "final_answer": "x \\in \\{2, - \\frac{6}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "11522d6ae3017461", "problem_id": "a76d79b2451827fb", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - x - 6$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$-1$, c=$-6$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $49$, so √Δ = $7$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $2$.\n7) x2 = $- \\frac{3}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{2, - \\frac{3}{2}\\}$", "final_answer": "x \\in \\{2, - \\frac{3}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e31ecc2a948443da", "problem_id": "a76d79b2451827fb", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - x - 6$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$-1$, c=$-6$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{2, - \\frac{3}{2}\\}$", "final_answer": "x \\in \\{2, - \\frac{3}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1b3146431255e98c", "problem_id": "8c24045b36ec41c9", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} x^{3} + 4 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $x^{3} + 4 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $3 \\pi + \\frac{15 \\pi^{4}}{4} + 12 \\pi^{3}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $3\\cdot \\pi + \\frac{15\\cdot \\pi^{4}}{4} + 12\\cdot \\pi^{3}$", "final_answer": "3\\cdot \\pi + \\frac{15\\cdot \\pi^{4}}{4} + 12\\cdot \\pi^{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "86113690283a1d39", "problem_id": "8c24045b36ec41c9", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} x^{3} + 4 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $x^{3} + 4 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $3 \\pi + \\frac{15 \\pi^{4}}{4} + 12 \\pi^{3}$.\n4) Simplify the result.\nFinal answer: $3\\cdot \\pi + \\frac{15\\cdot \\pi^{4}}{4} + 12\\cdot \\pi^{3}$", "final_answer": "3\\cdot \\pi + \\frac{15\\cdot \\pi^{4}}{4} + 12\\cdot \\pi^{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8811e50c80eeea28", "problem_id": "106b17991f2bb3f3", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Two friends split $1650 in the ratio 6:5. How much does each receive?", "output": "1) Total ratio parts = 6+5 = 11.\n2) Each part = 1650 ÷ 11 = 150.\n3) First share = 6 × 150 = 900.\n4) Second share = 5 × 150 = 750.\nFinal answer: $750$", "final_answer": "750", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db087dd2094c625f", "problem_id": "106b17991f2bb3f3", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Two friends split $1650 in the ratio 6:5. How much does each receive?", "output": "Total ratio parts = 6+5 = 11.\nEach part = 1650 ÷ 11 = 150.\nFirst share = 6 × 150 = 900.\nSecond share = 5 × 150 = 750.\nCheck: 900+750=1650.\nFinal answer: $750$", "final_answer": "750", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0019079c96748971", "problem_id": "426dbb38f6605fdd", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + \\frac{7 x}{6} + \\frac{1}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{7}{6}$, c=$\\frac{1}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{- \\frac{1}{2}, - \\frac{2}{3}\\}$", "final_answer": "x \\in \\{- \\frac{1}{2}, - \\frac{2}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b07e6f3e98131880", "problem_id": "426dbb38f6605fdd", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + \\frac{7 x}{6} + \\frac{1}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{7}{6}$, c=$\\frac{1}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{1}{36}$, so √Δ = $\\frac{1}{6}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{1}{2}$.\nx2 = $- \\frac{2}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{1}{2}, - \\frac{2}{3}\\}$", "final_answer": "x \\in \\{- \\frac{1}{2}, - \\frac{2}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5bc3ac1a4d088218", "problem_id": "4467cfc70d837b1e", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A cyclist rides at 83 miles per hour for 2 hours, then at 62 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 83 × 2 = 166 miles.\n2) Second leg distance = 62 × 3 = 186 miles.\n3) Total distance = 166 + 186 = 352 miles.\nFinal answer: $352$", "final_answer": "352", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ab82aea7482a59f6", "problem_id": "4467cfc70d837b1e", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A cyclist rides at 83 miles per hour for 2 hours, then at 62 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 83 × 2 = 166 miles.\nSecond leg distance = 62 × 3 = 186 miles.\nTotal distance = 166 + 186 = 352 miles.\nCheck: adding both legs gives 352.\nFinal answer: $352$", "final_answer": "352", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a681d786f1afcce0", "problem_id": "c15b2ca94672c568", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Two friends split $1140 in the ratio 7:8. How much does each receive?", "output": "1) Total ratio parts = 7+8 = 15.\n2) Each part = 1140 ÷ 15 = 76.\n3) First share = 7 × 76 = 532.\n4) Second share = 8 × 76 = 608.\nFinal answer: $608$", "final_answer": "608", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "962d589fd6c2a4cc", "problem_id": "c15b2ca94672c568", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $1140 in the ratio 7:8. How much does each receive?", "output": "Total ratio parts = 7+8 = 15.\nEach part = 1140 ÷ 15 = 76.\nFirst share = 7 × 76 = 532.\nSecond share = 8 × 76 = 608.\nCheck: 532+608=1140.\nFinal answer: $608$", "final_answer": "608", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "07bc040a233e8fc4", "problem_id": "fcd7b72ba29fbe59", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "You want 33 liters of a 35% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 33−x liters is 70% solution.\n2) Salt equation: 25x + 70(33−x) = 35·33.\n3) (25-70)x = (35-70)·33 ⇒ x = 77/3.\n4) Then 33−x = 22/3.\nFinal answer: $77/3 L of 25% and 22/3 L of 70%$", "final_answer": "77/3 L of 25% and 22/3 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "41ddf302674de09d", "problem_id": "fcd7b72ba29fbe59", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 33 liters of a 35% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 33−x liters is 70% solution.\nSalt equation: 25x + 70(33−x) = 35·33.\n(25-70)x = (35-70)·33 ⇒ x = 77/3.\nThen 33−x = 22/3.\nCheck: 25·77/3 + 70·22/3 = 35·33.\nFinal answer: $77/3 L of 25% and 22/3 L of 70%$", "final_answer": "77/3 L of 25% and 22/3 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f5f2470b6e538fa0", "problem_id": "8d88fb3d54dbc50b", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "An urn has 23 red balls and 20 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(43,8).\n3) Favorable samples: choose k reds and n−k blues: C(23,2)·C(20,6).\n4) Compute totals: C(43,8) = 145008513, and favorable = 9806280.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{172040}{2544009}$", "final_answer": "\\frac{172040}{2544009}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "469f1cfdf5ab39bb", "problem_id": "8d88fb3d54dbc50b", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 23 red balls and 20 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(43,8).\nFavorable samples: choose k reds and n−k blues: C(23,2)·C(20,6).\nCompute totals: C(43,8) = 145008513, and favorable = 9806280.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{172040}{2544009}$.\nFinal answer: $\\frac{172040}{2544009}$", "final_answer": "\\frac{172040}{2544009}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a33c2926ecc730dd", "problem_id": "afa5952512988534", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 62 miles per hour for 3/2 hours, then at 72 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 62 × 3/2 = 93 miles.\nSecond leg distance = 72 × 2 = 144 miles.\nTotal distance = 93 + 144 = 237 miles.\nCheck: adding both legs gives 237.\nFinal answer: $237$", "final_answer": "237", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a912c249ded67b22", "problem_id": "afa5952512988534", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A cyclist rides at 62 miles per hour for 3/2 hours, then at 72 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 62 × 3/2 = 93 miles.\n2) Second leg distance = 72 × 2 = 144 miles.\n3) Total distance = 93 + 144 = 237 miles.\nFinal answer: $237$", "final_answer": "237", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "704f07e506f01870", "problem_id": "dbfdbf9a0ef83d9a", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "An urn has 19 red balls and 20 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(39,9).\n3) Favorable samples: choose k reds and n−k blues: C(19,1)·C(20,8).\n4) Compute totals: C(39,9) = 211915132, and favorable = 2393430.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{285}{25234}$", "final_answer": "\\frac{285}{25234}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "05e873d0e4ec2f51", "problem_id": "dbfdbf9a0ef83d9a", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "An urn has 19 red balls and 20 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(39,9).\nFavorable samples: choose k reds and n−k blues: C(19,1)·C(20,8).\nFinal answer: $\\frac{285}{25234}$", "final_answer": "\\frac{285}{25234}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3c42ce1a98e45672", "problem_id": "d26f10f35f204cfa", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} + 3 x - 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x^{2}) = 2 x^{2}$.\n3) So $f'(x) = 2 x^{2} + 3$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x^{2}) = 2 x$.\nTherefore the correct derivative is $f'(x) = 2 x + 3$.\nFinal answer: $2 x + 3$", "final_answer": "2 x + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "99a891dd11ac528c", "problem_id": "21db5a7ea5973fb2", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{10 x}{3} - \\frac{2 \\sqrt{6} x}{3} - \\frac{8}{3} + \\frac{8 \\sqrt{6}}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{10}{3} - \\frac{2 \\sqrt{6}}{3}$, c=$- \\frac{8}{3} + \\frac{8 \\sqrt{6}}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{220}{9} - \\frac{56 \\sqrt{6}}{9}$, so √Δ = $\\sqrt{\\frac{220}{9} - \\frac{56 \\sqrt{6}}{9}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{6}}{3} + \\frac{\\sqrt{55 - 14 \\sqrt{6}}}{3} + \\frac{5}{3}$.\n7) x2 = $- \\frac{\\sqrt{55 - 14 \\sqrt{6}}}{3} + \\frac{\\sqrt{6}}{3} + \\frac{5}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{6}}{3} + \\frac{\\sqrt{55 - 14\\cdot \\sqrt{6}}}{3} + \\frac{5}{3}, - \\frac{\\sqrt{55 - 14\\cdot \\sqrt{6}}}{3} + \\frac{\\sqrt{6}}{3} + \\frac{5}{3}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{6}}{3} + \\frac{\\sqrt{55 - 14\\cdot \\sqrt{6}}}{3} + \\frac{5}{3}, - \\frac{\\sqrt{55 - 14\\cdot \\sqrt{6}}}{3} + \\frac{\\sqrt{6}}{3} + \\frac{5}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a0a594f4b3665e8e", "problem_id": "97ebe4ff8be36362", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 88 miles per hour for 1/2 hours, then at 33 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 88 × 1/2 = 44 miles.\nSecond leg distance = 33 × 2 = 66 miles.\nTotal distance = 44 + 66 = 110 miles.\nCheck: adding both legs gives 110.\nFinal answer: $110$", "final_answer": "110", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "39290de486b95872", "problem_id": "97ebe4ff8be36362", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A cyclist rides at 88 miles per hour for 1/2 hours, then at 33 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 88 × 1/2 = 44 miles.\n2) Second leg distance = 33 × 2 = 66 miles.\n3) Total distance = 44 + 66 = 110 miles.\nFinal answer: $110$", "final_answer": "110", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c42852619ae19bc5", "problem_id": "18b487ccd0e5202e", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$12 x + 11 y$ = $163$\n$- 12 x + 8 y$ = $-68$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-12$: $- 144 x - 132 y$ = $-1956$.\n3) Multiply the second equation by $12$: $- 144 x + 96 y$ = $-816$.\n4) Subtract to eliminate x: $- 228 y$ = $-1140$.\n5) Solve for y: y = $5$.\n6) Substitute back to find x: x = $9$.\nFinal answer: $(x,y)=\\left(9, 5\\right)$", "final_answer": "(x,y)=\\left(9, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f2d4c38a08659074", "problem_id": "18b487ccd0e5202e", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$12 x + 11 y$ = $163$\n$- 12 x + 8 y$ = $-68$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-12$: $- 144 x - 132 y$ = $-1956$.\nMultiply the second equation by $12$: $- 144 x + 96 y$ = $-816$.\nSubtract to eliminate x: $- 228 y$ = $-1140$.\nSolve for y: y = $5$.\nSubstitute back to find x: x = $9$.\nCheck: plug (x,y)=(9,5) into both equations.\nFinal answer: $(x,y)=\\left(9, 5\\right)$", "final_answer": "(x,y)=\\left(9, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "185c9dcbd0d93987", "problem_id": "55e160bbb2b6badb", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡2 (mod 7), x≡1 (mod 13). Give the solution modulo 91.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 91.\n3) Compute the CRT solution to get x ≡ 79 (mod 91).\nFinal answer: $x\\equiv 79\\pmod{91}$", "final_answer": "x\\equiv 79\\pmod{91}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "18a4f8c9694b4325", "problem_id": "55e160bbb2b6badb", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡2 (mod 7), x≡1 (mod 13). Give the solution modulo 91.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 91.\nCompute the CRT solution to get x ≡ 79 (mod 91).\nCheck: 79 mod 7=2 and mod 13=1.\nFinal answer: $x\\equiv 79\\pmod{91}$", "final_answer": "x\\equiv 79\\pmod{91}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d9caf2512acddef3", "problem_id": "853d20611c2214e1", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$7 x - 6 y$ = $-5$\n$- 2 x - 4 y$ = $-10$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $- 14 x + 12 y$ = $10$.\nMultiply the second equation by $7$: $- 14 x - 28 y$ = $-70$.\nSubtract to eliminate x: $40 y$ = $80$.\nSolve for y: y = $2$.\nSubstitute back to find x: x = $1$.\nCheck: plug (x,y)=(1,2) into both equations.\nFinal answer: $(x,y)=\\left(1, 2\\right)$", "final_answer": "(x,y)=\\left(1, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "03f230bc22cc484b", "problem_id": "853d20611c2214e1", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$7 x - 6 y$ = $-5$\n$- 2 x - 4 y$ = $-10$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $- 14 x + 12 y$ = $10$.\n3) Multiply the second equation by $7$: $- 14 x - 28 y$ = $-70$.\n4) Subtract to eliminate x: $40 y$ = $80$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $1$.\nFinal answer: $(x,y)=\\left(1, 2\\right)$", "final_answer": "(x,y)=\\left(1, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b839e72d05bf4175", "problem_id": "492a1f5aa09259a7", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$- 5 x - 7 y$ = $-108$\n$x + 7 y$ = $72$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $- 5 x - 7 y$ = $-108$.\n3) Multiply the second equation by $-5$: $- 5 x - 35 y$ = $-360$.\n4) Subtract to eliminate x: $28 y$ = $252$.\n5) Solve for y: y = $9$.\n6) Substitute back to find x: x = $9$.\nFinal answer: $(x,y)=\\left(9, 9\\right)$", "final_answer": "(x,y)=\\left(9, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "830bc23be099e691", "problem_id": "492a1f5aa09259a7", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Solve the linear system:\n$- 5 x - 7 y$ = $-108$\n$x + 7 y$ = $72$", "output": "$(x,y)=\\left(9, 9\\right)$", "final_answer": "(x,y)=\\left(9, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ea56adcb90bb86e8", "problem_id": "a3d224eb06bd2a0d", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡7 (mod 13). Give the solution modulo 39.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 39.\n3) Compute the CRT solution to get x ≡ 7 (mod 39).\nFinal answer: $x\\equiv 7\\pmod{39}$", "final_answer": "x\\equiv 7\\pmod{39}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0185c470bf3abe02", "problem_id": "a3d224eb06bd2a0d", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡7 (mod 13). Give the solution modulo 39.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 39.\nCompute the CRT solution to get x ≡ 7 (mod 39).\nCheck: 7 mod 3=1 and mod 13=7.\nFinal answer: $x\\equiv 7\\pmod{39}$", "final_answer": "x\\equiv 7\\pmod{39}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "252cda2cdd22471d", "problem_id": "68107fe9a30d3efe", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$2 x - y$ = $-9$\n$6 x + 3 y$ = $3$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $6$: $12 x - 6 y$ = $-54$.\n3) Multiply the second equation by $2$: $12 x + 6 y$ = $6$.\n4) Subtract to eliminate x: $- 12 y$ = $-60$.\n5) Solve for y: y = $5$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, 5\\right)$", "final_answer": "(x,y)=\\left(-2, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a3d498dcaf0242b", "problem_id": "68107fe9a30d3efe", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$2 x - y$ = $-9$\n$6 x + 3 y$ = $3$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $6$: $12 x - 6 y$ = $-54$.\nMultiply the second equation by $2$: $12 x + 6 y$ = $6$.\nSubtract to eliminate x: $- 12 y$ = $-60$.\nSolve for y: y = $5$.\nSubstitute back to find x: x = $-2$.\nCheck: plug (x,y)=(-2,5) into both equations.\nFinal answer: $(x,y)=\\left(-2, 5\\right)$", "final_answer": "(x,y)=\\left(-2, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "92ec283c1957debb", "problem_id": "d036af4452493e7d", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "An urn has 27 red balls and 17 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(44,7).\n3) Favorable samples: choose k reds and n−k blues: C(27,3)·C(17,4).\n4) Compute totals: C(44,7) = 38320568, and favorable = 6961500.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{133875}{736934}$", "final_answer": "\\frac{133875}{736934}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bad5a66fd5a11f94", "problem_id": "d036af4452493e7d", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 27 red balls and 17 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(44,7).\nFavorable samples: choose k reds and n−k blues: C(27,3)·C(17,4).\nCompute totals: C(44,7) = 38320568, and favorable = 6961500.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{133875}{736934}$.\nFinal answer: $\\frac{133875}{736934}$", "final_answer": "\\frac{133875}{736934}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "22663b272270b63f", "problem_id": "bda48a2c50db7cae", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} \\sin{\\left(x \\right)} - 3 x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $x^{2} \\sin{\\left(x \\right)} - 3 x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.\nFinal answer: $x^{2}\\cdot \\cos{\\left(x \\right)} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} - 3\\cdot x\\cdot \\cos{\\left(x \\right)} - 3\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - \\cos{\\left(x \\right)}$", "final_answer": "x^{2}\\cdot \\cos{\\left(x \\right)} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} - 3\\cdot x\\cdot \\cos{\\left(x \\right)} - 3\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9947a91312260147", "problem_id": "bda48a2c50db7cae", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} \\sin{\\left(x \\right)} - 3 x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $x^{2} \\sin{\\left(x \\right)} - 3 x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $x^{2}\\cdot \\cos{\\left(x \\right)} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} - 3\\cdot x\\cdot \\cos{\\left(x \\right)} - 3\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - \\cos{\\left(x \\right)}$", "final_answer": "x^{2}\\cdot \\cos{\\left(x \\right)} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} - 3\\cdot x\\cdot \\cos{\\left(x \\right)} - 3\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b7046de7677cf40f", "problem_id": "414fde83bfa8e3ed", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 2 x^{2} + x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}$.", "output": "1) Let f(x) = $\\frac{- 2 x^{2} + x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{- 4 x \\left(- 2 x^{2} + x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right) \\left(4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8 x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{2 \\left(x^{2} - 3\\right)^{2}}$.\nFinal answer: $\\frac{- 4\\cdot x\\cdot \\left(- 2\\cdot x^{2} + x + \\left(x^{2} - 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right)\\cdot \\left(4\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 8\\cdot x + \\left(x^{2} - 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\right)}{2\\cdot \\left(x^{2} - 3\\right)^{2}}$", "final_answer": "\\frac{- 4\\cdot x\\cdot \\left(- 2\\cdot x^{2} + x + \\left(x^{2} - 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right)\\cdot \\left(4\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 8\\cdot x + \\left(x^{2} - 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\right)}{2\\cdot \\left(x^{2} - 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e03a2edb65eb4426", "problem_id": "414fde83bfa8e3ed", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 2 x^{2} + x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}$.", "output": "Let f(x) = $\\frac{- 2 x^{2} + x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{- 4 x \\left(- 2 x^{2} + x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right) \\left(4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8 x + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2\\right)}{2 \\left(x^{2} - 3\\right)^{2}}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\frac{- 4\\cdot x\\cdot \\left(- 2\\cdot x^{2} + x + \\left(x^{2} - 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right)\\cdot \\left(4\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 8\\cdot x + \\left(x^{2} - 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\right)}{2\\cdot \\left(x^{2} - 3\\right)^{2}}$", "final_answer": "\\frac{- 4\\cdot x\\cdot \\left(- 2\\cdot x^{2} + x + \\left(x^{2} - 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right)\\cdot \\left(4\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 8\\cdot x + \\left(x^{2} - 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\right)}{2\\cdot \\left(x^{2} - 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1c9627caf80bbc7a", "problem_id": "cd2fc4dee65651d5", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the linear system:\n$5 x$ = $-40$\n$8 x - y$ = $-73$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $8$: $40 x$ = $-320$.\n3) Multiply the second equation by $5$: $40 x - 5 y$ = $-365$.\n4) Subtract to eliminate x: $5 y$ = $45$.\n5) Solve for y: y = $9$.\n6) Substitute back to find x: x = $-8$.\nFinal answer: $(x,y)=\\left(-8, 9\\right)$", "final_answer": "(x,y)=\\left(-8, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bcf2eedf7c00206d", "problem_id": "a6d73985fdff9722", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(2 x^{3} - x^{2} - 3 x - 1\\right) \\cos{\\left(2 x - 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(e^{x}) = e^{x}$.\n3) So $f'(x) = - 4 x^{3} \\sin{\\left(2 x - 2 \\right)} + 2 x^{2} \\sin{\\left(2 x - 2 \\right)} + 6 x^{2} \\cos{\\left(2 x - 2 \\right)} + x e^{x} + 6 x \\sin{\\left(2 x - 2 \\right)} - 2 x \\cos{\\left(2 x - 2 \\right)} + 2 e^{x} + 2 \\sin{\\left(2 x - 2 \\right)} - 3 \\cos{\\left(2 x - 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(e^{x}) = e^{x}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 1\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) - 3\\right) - 1\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(6 x^{2} - 2 x - 3\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 1\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) - 3\\right) - 1\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(6 x^{2} - 2 x - 3\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}$", "final_answer": "\\left(x + 1\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) - 3\\right) - 1\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(6 x^{2} - 2 x - 3\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "167c2d7aa435e59f", "problem_id": "703027c8992664ca", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A store sells a backpack for $370. It is discounted by 10%. You buy 5 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 370 × (1 − 10/100) = 370 × 90/100 = 333.\n2) Subtotal = 5 × 333 = 1665.\n3) Total with tax = 1665 × 107/100 = 35631/20.\nFinal answer: $\\frac{35631}{20}$", "final_answer": "\\frac{35631}{20}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "635c80c5584807a8", "problem_id": "d903d488291fd67c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 2 x^{2} + 2 x + \\left(x^{2} + 3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{x^{2} + 3 x - 5}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 5}) = - \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3 x - 5}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{6 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{12 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{16 x^{2}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{30 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{60 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{48 x}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{10 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 6 x^{3} - x^{2} - 30 x + 25} + \\frac{25 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} + \\frac{50 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{8}{2 x^{4} + 12 x^{3} - 2 x^{2} - 60 x + 50} - \\frac{15 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 6 x^{3} - x^{2} - 30 x + 25} + \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 6 x - 10} - \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3 x - 5} + \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 5}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 5}) = - \\frac{5 \\left(- 2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(x^{2} + 3 x\\right) - 5\\right)^{2}} - \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(x^{2} + 3 x\\right) - 5\\right)} - \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) - 5}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(- 2 x - 3\\right) \\left(- 2 x^{2} + 2 x + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(x \\left(x + 3\\right) - 5\\right) \\left(- 8 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x + 3\\right) - 5\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(- 2 x - 3\\right) \\left(- 2 x^{2} + 2 x + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(x \\left(x + 3\\right) - 5\\right) \\left(- 8 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x + 3\\right) - 5\\right)^{2}}$", "final_answer": "\\frac{2 \\left(- 2 x - 3\\right) \\left(- 2 x^{2} + 2 x + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(x \\left(x + 3\\right) - 5\\right) \\left(- 8 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x + 3\\right) - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x + 3\\right) - 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "dc085253ace1ee4b", "problem_id": "c94653b55cacc193", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} \\sin{\\left(x \\right)} - x^{2} \\sin{\\left(x \\right)} - 3 x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $- 2 x^{3} \\sin{\\left(x \\right)} - x^{2} \\sin{\\left(x \\right)} - 3 x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - x^{2}\\cdot \\cos{\\left(x \\right)} - 2\\cdot x\\cdot \\sin{\\left(x \\right)} - 3\\cdot x\\cdot \\cos{\\left(x \\right)} - 3\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - x^{2}\\cdot \\cos{\\left(x \\right)} - 2\\cdot x\\cdot \\sin{\\left(x \\right)} - 3\\cdot x\\cdot \\cos{\\left(x \\right)} - 3\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ae94f401b54aee22", "problem_id": "c94653b55cacc193", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} \\sin{\\left(x \\right)} - x^{2} \\sin{\\left(x \\right)} - 3 x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $- 2 x^{3} \\sin{\\left(x \\right)} - x^{2} \\sin{\\left(x \\right)} - 3 x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - x^{2}\\cdot \\cos{\\left(x \\right)} - 2\\cdot x\\cdot \\sin{\\left(x \\right)} - 3\\cdot x\\cdot \\cos{\\left(x \\right)} - 3\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - x^{2}\\cdot \\cos{\\left(x \\right)} - 2\\cdot x\\cdot \\sin{\\left(x \\right)} - 3\\cdot x\\cdot \\cos{\\left(x \\right)} - 3\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "95b716c1492c8ded", "problem_id": "cd5f4efe7053e7ec", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the multiplicative inverse of 12 modulo 37.", "output": "1) We want x such that 12x ≡ 1 (mod 37).\n2) Use extended Euclid to find s,t with 12s + 37t = 1.\n3) One Bezout identity is: 12·(-3) + 37·(1) = 1.\n4) Reduce s modulo 37: x ≡ 34 (mod 37).\nFinal answer: $x\\equiv 34\\pmod{37}$", "final_answer": "x\\equiv 34\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "36386268cce31d5f", "problem_id": "cd5f4efe7053e7ec", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the multiplicative inverse of 12 modulo 37.", "output": "We want x such that 12x ≡ 1 (mod 37).\nUse extended Euclid to find s,t with 12s + 37t = 1.\nOne Bezout identity is: 12·(-3) + 37·(1) = 1.\nReduce s modulo 37: x ≡ 34 (mod 37).\nCheck: 12·34 mod 37 = 1.\nFinal answer: $x\\equiv 34\\pmod{37}$", "final_answer": "x\\equiv 34\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "82646e8c86b4c816", "problem_id": "4b3e3836d918e339", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the linear system:\n$- 4 x + 3 y$ = $-8$\n$- 6 x + y$ = $-26$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-6$: $24 x - 18 y$ = $48$.\nMultiply the second equation by $-4$: $24 x - 4 y$ = $104$.\nFinal answer: $(x,y)=\\left(5, 4\\right)$", "final_answer": "(x,y)=\\left(5, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f85ad3508f4cf25f", "problem_id": "4b3e3836d918e339", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$- 4 x + 3 y$ = $-8$\n$- 6 x + y$ = $-26$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-6$: $24 x - 18 y$ = $48$.\n3) Multiply the second equation by $-4$: $24 x - 4 y$ = $104$.\n4) Subtract to eliminate x: $- 14 y$ = $-56$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, 4\\right)$", "final_answer": "(x,y)=\\left(5, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ded8e5ed06b73952", "problem_id": "ab479f5ed737974e", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 20 times, what is the probability of exactly 18 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=20, k=18, p=$\\frac{3}{7}$ and simplify.\n4) Here C(20,18) = 190.\nFinal answer: $\\frac{1177758286560}{79792266297612001}$", "final_answer": "\\frac{1177758286560}{79792266297612001}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "28c6a9b4865075f3", "problem_id": "ab479f5ed737974e", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 20 times, what is the probability of exactly 18 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=20, k=18, p=$\\frac{3}{7}$ and simplify.\nHere C(20,18) = 190.\nCheck: expression simplifies to $\\frac{1177758286560}{79792266297612001}$.\nFinal answer: $\\frac{1177758286560}{79792266297612001}$", "final_answer": "\\frac{1177758286560}{79792266297612001}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9a2dcac6b4eef16e", "problem_id": "87b507a28cf97cf2", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_10=36$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_10=6.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=6 and k=10.\n4) Compute C(15,9) = 5005.\nFinal answer: $5005$", "final_answer": "5005", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "83bdad1837b1d2bc", "problem_id": "5f137f0736c7c320", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 3 x^{2} + 8 x - 24$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} + 7 x - 24.\n3) Then the limit is -72.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -75.\nFinal answer: $-75$", "final_answer": "-75", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6cfba3d5f7e7342f", "problem_id": "100920e1d6e35af4", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the multiplicative inverse of 6 modulo 59.", "output": "1) We want x such that 6x ≡ 1 (mod 59).\n2) Use extended Euclid to find s,t with 6s + 59t = 1.\n3) One Bezout identity is: 6·(10) + 59·(-1) = 1.\n4) Reduce s modulo 59: x ≡ 10 (mod 59).\nFinal answer: $x\\equiv 10\\pmod{59}$", "final_answer": "x\\equiv 10\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "155617e06777d376", "problem_id": "61597971a8e26afc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - x^{2} - 2 x - 3$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} - 3 x - 3.\n3) Then the limit is -21.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: -18.\nFinal answer: $-18$", "final_answer": "-18", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "9e45495dd2190c52", "problem_id": "526246a4f7ea153f", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡0 (mod 3), x≡3 (mod 11). Give the solution modulo 33.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 33.\n3) Compute the CRT solution to get x ≡ 3 (mod 33).\nFinal answer: $x\\equiv 3\\pmod{33}$", "final_answer": "x\\equiv 3\\pmod{33}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a5103ce12bd9031", "problem_id": "526246a4f7ea153f", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡0 (mod 3), x≡3 (mod 11). Give the solution modulo 33.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 33.\nCompute the CRT solution to get x ≡ 3 (mod 33).\nCheck: 3 mod 3=0 and mod 11=3.\nFinal answer: $x\\equiv 3\\pmod{33}$", "final_answer": "x\\equiv 3\\pmod{33}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a1e6f497e843a4c", "problem_id": "a6298ff0ccc04ae0", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 27 liters of a 20% salt solution. You have a 15% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 27−x liters is 50% solution.\nSalt equation: 15x + 50(27−x) = 20·27.\n(15-50)x = (20-50)·27 ⇒ x = 162/7.\nThen 27−x = 27/7.\nCheck: 15·162/7 + 50·27/7 = 20·27.\nFinal answer: $162/7 L of 15% and 27/7 L of 50%$", "final_answer": "162/7 L of 15% and 27/7 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4f96db1042448435", "problem_id": "a6298ff0ccc04ae0", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "You want 27 liters of a 20% salt solution. You have a 15% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 27−x liters is 50% solution.\n2) Salt equation: 15x + 50(27−x) = 20·27.\n3) (15-50)x = (20-50)·27 ⇒ x = 162/7.\n4) Then 27−x = 27/7.\nFinal answer: $162/7 L of 15% and 27/7 L of 50%$", "final_answer": "162/7 L of 15% and 27/7 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "11a5a42f8a1b9c81", "problem_id": "a11da1c72c31cfdd", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡4 (mod 8), x≡2 (mod 5). Give the solution modulo 40.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 40.\nCompute the CRT solution to get x ≡ 12 (mod 40).\nCheck: 12 mod 8=4 and mod 5=2.\nFinal answer: $x\\equiv 12\\pmod{40}$", "final_answer": "x\\equiv 12\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1f8d1a70f1783ec7", "problem_id": "a11da1c72c31cfdd", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡4 (mod 8), x≡2 (mod 5). Give the solution modulo 40.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 40.\n3) Compute the CRT solution to get x ≡ 12 (mod 40).\nFinal answer: $x\\equiv 12\\pmod{40}$", "final_answer": "x\\equiv 12\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "af37187768a89888", "problem_id": "d154b19c7ce23c0b", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A store sells a backpack for $192. It is discounted by 25%. You buy 10 backpacks. What is the total cost?", "output": "1) Discounted price = 192 × (1 − 25/100) = 192 × 75/100 = 144.\n2) Subtotal = 10 × 144 = 1440.\nFinal answer: $1440$", "final_answer": "1440", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e190b50eb7bead45", "problem_id": "d154b19c7ce23c0b", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "A store sells a backpack for $192. It is discounted by 25%. You buy 10 backpacks. What is the total cost?", "output": "Discounted price = 192 × (1 − 25/100) = 192 × 75/100 = 144.\nSubtotal = 10 × 144 = 1440.\nFinal answer: $1440$", "final_answer": "1440", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c5b9f7256a50efad", "problem_id": "69e797abd465fec0", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$x$ = $-5$\n$- 4 x - 4 y$ = $40$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $- 4 x$ = $20$.\n3) Multiply the second equation by $1$: $- 4 x - 4 y$ = $40$.\n4) Subtract to eliminate x: $4 y$ = $-20$.\n5) Solve for y: y = $-5$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, -5\\right)$", "final_answer": "(x,y)=\\left(-5, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "25e5f42a0dbf967a", "problem_id": "34496964d84262b4", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(3 x^{3} - 3 x^{2} + x + 1\\right) \\cos{\\left(x^{2} - x + 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 3 x^{2} \\cos{\\left(\\left(x^{2} - x\\right) + 1 \\right)}) = 3 x^{2} \\sin{\\left(x^{2} - x + 1 \\right)}$.\n3) So $f'(x) = - 6 x^{4} \\sin{\\left(\\left(x^{2} - x\\right) + 1 \\right)} + 3 x^{3} \\sin{\\left(\\left(x^{2} - x\\right) + 1 \\right)} - 2 x^{2} \\sin{\\left(\\left(x^{2} - x\\right) + 1 \\right)} + 3 x^{2} \\sin{\\left(x^{2} - x + 1 \\right)} + 9 x^{2} \\cos{\\left(\\left(x^{2} - x\\right) + 1 \\right)} + x e^{x} - x \\sin{\\left(\\left(x^{2} - x\\right) + 1 \\right)} + 2 e^{x} + \\sin{\\left(\\left(x^{2} - x\\right) + 1 \\right)} + \\cos{\\left(\\left(x^{2} - x\\right) + 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 3 x^{2} \\cos{\\left(\\left(x^{2} - x\\right) + 1 \\right)}) = 3 x^{2} \\left(2 x - 1\\right) \\sin{\\left(\\left(x^{2} - x\\right) + 1 \\right)} - 6 x \\cos{\\left(\\left(x^{2} - x\\right) + 1 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 1\\right) e^{x} - \\left(2 x - 1\\right) \\left(x \\left(3 x \\left(x - 1\\right) + 1\\right) + 1\\right) \\sin{\\left(\\left(x^{2} - x\\right) + 1 \\right)} + \\left(9 x^{2} - 6 x + 1\\right) \\cos{\\left(\\left(x^{2} - x\\right) + 1 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 1\\right) e^{x} - \\left(2 x - 1\\right) \\left(x \\left(3 x \\left(x - 1\\right) + 1\\right) + 1\\right) \\sin{\\left(\\left(x^{2} - x\\right) + 1 \\right)} + \\left(9 x^{2} - 6 x + 1\\right) \\cos{\\left(\\left(x^{2} - x\\right) + 1 \\right)} + e^{x}$", "final_answer": "\\left(x + 1\\right) e^{x} - \\left(2 x - 1\\right) \\left(x \\left(3 x \\left(x - 1\\right) + 1\\right) + 1\\right) \\sin{\\left(\\left(x^{2} - x\\right) + 1 \\right)} + \\left(9 x^{2} - 6 x + 1\\right) \\cos{\\left(\\left(x^{2} - x\\right) + 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b8945b9be5108dfd", "problem_id": "727bec7ffd87c9ba", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} + 2 x + \\left(2 x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{2 x^{2} - x + 5}$.", "output": "Let f(x) = $\\frac{- x^{2} + 2 x + \\left(2 x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{2 x^{2} - x + 5}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{2 \\left(1 - 4 x\\right) \\left(- x^{2} + 2 x + \\left(2 x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(2 x^{2} - x + 5\\right) \\left(- 4 x + 2 \\left(4 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(2 x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(2 x^{2} - x + 5\\right)^{2}}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\frac{\\left(2 - 8\\cdot x\\right)\\cdot \\left(- x^{2} + 2\\cdot x + \\left(2\\cdot x^{2} - x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right) + \\left(2\\cdot x^{2} - x + 5\\right)\\cdot \\left(- 4\\cdot x + \\left(8\\cdot x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} - x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x^{2} - 2\\cdot x + 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\right)}{2\\cdot \\left(2\\cdot x^{2} - x + 5\\right)^{2}}$", "final_answer": "\\frac{\\left(2 - 8\\cdot x\\right)\\cdot \\left(- x^{2} + 2\\cdot x + \\left(2\\cdot x^{2} - x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right) + \\left(2\\cdot x^{2} - x + 5\\right)\\cdot \\left(- 4\\cdot x + \\left(8\\cdot x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} - x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x^{2} - 2\\cdot x + 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\right)}{2\\cdot \\left(2\\cdot x^{2} - x + 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fc8d91a040dc021c", "problem_id": "727bec7ffd87c9ba", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} + 2 x + \\left(2 x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{2 x^{2} - x + 5}$.", "output": "1) Let f(x) = $\\frac{- x^{2} + 2 x + \\left(2 x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{2 x^{2} - x + 5}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{2 \\left(1 - 4 x\\right) \\left(- x^{2} + 2 x + \\left(2 x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(2 x^{2} - x + 5\\right) \\left(- 4 x + 2 \\left(4 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(2 x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(2 x^{2} - x + 5\\right)^{2}}$.\nFinal answer: $\\frac{\\left(2 - 8\\cdot x\\right)\\cdot \\left(- x^{2} + 2\\cdot x + \\left(2\\cdot x^{2} - x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right) + \\left(2\\cdot x^{2} - x + 5\\right)\\cdot \\left(- 4\\cdot x + \\left(8\\cdot x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} - x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x^{2} - 2\\cdot x + 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\right)}{2\\cdot \\left(2\\cdot x^{2} - x + 5\\right)^{2}}$", "final_answer": "\\frac{\\left(2 - 8\\cdot x\\right)\\cdot \\left(- x^{2} + 2\\cdot x + \\left(2\\cdot x^{2} - x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right) + \\left(2\\cdot x^{2} - x + 5\\right)\\cdot \\left(- 4\\cdot x + \\left(8\\cdot x - 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} - x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x^{2} - 2\\cdot x + 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\right)}{2\\cdot \\left(2\\cdot x^{2} - x + 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9e1f556f92bba771", "problem_id": "b84cd07eba6a01ac", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$x - 12 y$ = $111$\n$2 x - 3 y$ = $33$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $2 x - 24 y$ = $222$.\n3) Multiply the second equation by $1$: $2 x - 3 y$ = $33$.\n4) Subtract to eliminate x: $- 21 y$ = $189$.\n5) Solve for y: y = $-9$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, -9\\right)$", "final_answer": "(x,y)=\\left(3, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "47590308fc36f5ab", "problem_id": "b84cd07eba6a01ac", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$x - 12 y$ = $111$\n$2 x - 3 y$ = $33$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $2$: $2 x - 24 y$ = $222$.\nMultiply the second equation by $1$: $2 x - 3 y$ = $33$.\nSubtract to eliminate x: $- 21 y$ = $189$.\nSolve for y: y = $-9$.\nSubstitute back to find x: x = $3$.\nCheck: plug (x,y)=(3,-9) into both equations.\nFinal answer: $(x,y)=\\left(3, -9\\right)$", "final_answer": "(x,y)=\\left(3, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "46d87610d78ef819", "problem_id": "62230e3f308bfa9d", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡1 (mod 8), x≡2 (mod 11). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 57 (mod 88).\nFinal answer: $x\\equiv 57\\pmod{88}$", "final_answer": "x\\equiv 57\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "57b6153cf15fca67", "problem_id": "62230e3f308bfa9d", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡1 (mod 8), x≡2 (mod 11). Give the solution modulo 88.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 88.\nCompute the CRT solution to get x ≡ 57 (mod 88).\nCheck: 57 mod 8=1 and mod 11=2.\nFinal answer: $x\\equiv 57\\pmod{88}$", "final_answer": "x\\equiv 57\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f9e876146cea0dbb", "problem_id": "b5e62c50c193f72f", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} - 3 x^{3} - x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{3} - x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\nEvaluate at the bounds: F(\\pi) and F(- \\pi).\nSubtract: F(\\pi) − F(- \\pi) = $\\frac{2 \\pi \\left(- \\pi^{2} - 9\\right)}{3}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(- \\pi^{2} - 9\\right)}{3}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(- \\pi^{2} - 9\\right)}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "41f2381dcff7b946", "problem_id": "b5e62c50c193f72f", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} - 3 x^{3} - x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{3} - x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\nEvaluate at the bounds: F(\\pi) and F(- \\pi).\nSubtract: F(\\pi) − F(- \\pi) = $\\frac{2 \\pi \\left(- \\pi^{2} - 9\\right)}{3}$.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(- \\pi^{2} - 9\\right)}{3}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(- \\pi^{2} - 9\\right)}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "508f719e935bddf9", "problem_id": "1d6c50be8178d326", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A cyclist rides at 90 miles per hour for 1 hours, then at 88 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 90 × 1 = 90 miles.\n2) Second leg distance = 88 × 1 = 88 miles.\n3) Total distance = 90 + 88 = 178 miles.\nFinal answer: $178$", "final_answer": "178", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "18e76dd34a010562", "problem_id": "1d6c50be8178d326", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A cyclist rides at 90 miles per hour for 1 hours, then at 88 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 90 × 1 = 90 miles.\nSecond leg distance = 88 × 1 = 88 miles.\nTotal distance = 90 + 88 = 178 miles.\nCheck: adding both legs gives 178.\nFinal answer: $178$", "final_answer": "178", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "63ce5f645cf6469a", "problem_id": "f15e8b1a177ea248", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 13 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=13, k=6, p=$\\frac{1}{8}$ and simplify.\n4) Here C(13,6) = 1716.\nFinal answer: $\\frac{353299947}{137438953472}$", "final_answer": "\\frac{353299947}{137438953472}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0a4ab527139f077b", "problem_id": "f15e8b1a177ea248", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 13 times, what is the probability of exactly 6 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=13, k=6, p=$\\frac{1}{8}$ and simplify.\nHere C(13,6) = 1716.\nCheck: expression simplifies to $\\frac{353299947}{137438953472}$.\nFinal answer: $\\frac{353299947}{137438953472}$", "final_answer": "\\frac{353299947}{137438953472}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "76161927510f51b2", "problem_id": "4c89293736fdcd1e", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡3 (mod 11), x≡0 (mod 8). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 80 (mod 88).\nFinal answer: $x\\equiv 80\\pmod{88}$", "final_answer": "x\\equiv 80\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8d920af934892550", "problem_id": "4c89293736fdcd1e", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡3 (mod 11), x≡0 (mod 8). Give the solution modulo 88.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 88.\nCompute the CRT solution to get x ≡ 80 (mod 88).\nCheck: 80 mod 11=3 and mod 8=0.\nFinal answer: $x\\equiv 80\\pmod{88}$", "final_answer": "x\\equiv 80\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "91d840025fb94ba5", "problem_id": "1bea1c540a0aa966", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(- 2 x^{3} + x^{2} - x - 3\\right) \\cos{\\left(x + 2 \\right)}$.", "output": "Let f(x) = $\\left(x + 1\\right) e^{x} + \\left(- 2 x^{3} + x^{2} - x - 3\\right) \\cos{\\left(x + 2 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\left(x + 1\\right) e^{x} + \\left(- 6 x^{2} + 2 x - 1\\right) \\cos{\\left(x + 2 \\right)} - \\left(- 2 x^{3} + x^{2} - x - 3\\right) \\sin{\\left(x + 2 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 1\\right)\\cdot e^{x} + \\left(- 6\\cdot x^{2} + 2\\cdot x - 1\\right)\\cdot \\cos{\\left(x + 2 \\right)} - \\left(- 2\\cdot x^{3} + x^{2} - x - 3\\right)\\cdot \\sin{\\left(x + 2 \\right)} + e^{x}$", "final_answer": "\\left(x + 1\\right)\\cdot e^{x} + \\left(- 6\\cdot x^{2} + 2\\cdot x - 1\\right)\\cdot \\cos{\\left(x + 2 \\right)} - \\left(- 2\\cdot x^{3} + x^{2} - x - 3\\right)\\cdot \\sin{\\left(x + 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "95953eeb2f9bc854", "problem_id": "1bea1c540a0aa966", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(- 2 x^{3} + x^{2} - x - 3\\right) \\cos{\\left(x + 2 \\right)}$.", "output": "1) Let f(x) = $\\left(x + 1\\right) e^{x} + \\left(- 2 x^{3} + x^{2} - x - 3\\right) \\cos{\\left(x + 2 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\left(x + 1\\right) e^{x} + \\left(- 6 x^{2} + 2 x - 1\\right) \\cos{\\left(x + 2 \\right)} - \\left(- 2 x^{3} + x^{2} - x - 3\\right) \\sin{\\left(x + 2 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 1\\right)\\cdot e^{x} + \\left(- 6\\cdot x^{2} + 2\\cdot x - 1\\right)\\cdot \\cos{\\left(x + 2 \\right)} - \\left(- 2\\cdot x^{3} + x^{2} - x - 3\\right)\\cdot \\sin{\\left(x + 2 \\right)} + e^{x}$", "final_answer": "\\left(x + 1\\right)\\cdot e^{x} + \\left(- 6\\cdot x^{2} + 2\\cdot x - 1\\right)\\cdot \\cos{\\left(x + 2 \\right)} - \\left(- 2\\cdot x^{3} + x^{2} - x - 3\\right)\\cdot \\sin{\\left(x + 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cebc4b1bcff8ead0", "problem_id": "0b53d65a0d61112c", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} - 2 x^{2} - 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)}$ at $x=0$.", "output": "1) Differentiate: f'(x) = $- 3 x^{2} - 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 3$.\n2) Slope at x=0: m = f'(0) = $- \\frac{5}{2}$.\n3) Point on curve: (0, f(0)) = (0, $2$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 2 = - \\frac{5}{2}(x − 0).\n6) Optionally expand: y = $2 - \\frac{5 x}{2}$.\nFinal answer: $y = 2 - \\frac{5\\cdot x}{2}$", "final_answer": "y = 2 - \\frac{5\\cdot x}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "396c32175572afd3", "problem_id": "0b53d65a0d61112c", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} - 2 x^{2} - 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)}$ at $x=0$.", "output": "Differentiate: f'(x) = $- 3 x^{2} - 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 3$.\nSlope at x=0: m = f'(0) = $- \\frac{5}{2}$.\nPoint on curve: (0, f(0)) = (0, $2$).\nFinal answer: $y = 2 - \\frac{5\\cdot x}{2}$", "final_answer": "y = 2 - \\frac{5\\cdot x}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "24f4e6940b47e79b", "problem_id": "dde2d4953b80c452", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$8 x + 2 y$ = $26$\n$- x - 7 y$ = $-37$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $- 8 x - 2 y$ = $-26$.\n3) Multiply the second equation by $8$: $- 8 x - 56 y$ = $-296$.\n4) Subtract to eliminate x: $54 y$ = $270$.\n5) Solve for y: y = $5$.\n6) Substitute back to find x: x = $2$.\nFinal answer: $(x,y)=\\left(2, 5\\right)$", "final_answer": "(x,y)=\\left(2, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8febe588170e187a", "problem_id": "d2266df0b4a0f3d1", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(2 x^{3} - 3 x^{2} - 3 x\\right) \\cos{\\left(3 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x^{3} \\cos{\\left(3 x \\right)}) = - 2 x^{3} \\sin{\\left(3 x \\right)}$.\n3) So $f'(x) = - 2 x^{3} \\sin{\\left(3 x \\right)} + 9 x^{2} \\sin{\\left(3 x \\right)} + 2 x e^{x} + 9 x \\sin{\\left(3 x \\right)} - 6 x \\cos{\\left(3 x \\right)} - 4 e^{x} - 3 \\cos{\\left(3 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x^{3} \\cos{\\left(3 x \\right)}) = - 6 x^{3} \\sin{\\left(3 x \\right)} + 6 x^{2} \\cos{\\left(3 x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 3 x \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x \\right)} + 2 \\left(x - 3\\right) e^{x} + 3 \\left(2 x^{2} - 2 x - 1\\right) \\cos{\\left(3 x \\right)} + 2 e^{x}$.\nFinal answer: $- 3 x \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x \\right)} + 2 \\left(x - 3\\right) e^{x} + 3 \\left(2 x^{2} - 2 x - 1\\right) \\cos{\\left(3 x \\right)} + 2 e^{x}$", "final_answer": "- 3 x \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x \\right)} + 2 \\left(x - 3\\right) e^{x} + 3 \\left(2 x^{2} - 2 x - 1\\right) \\cos{\\left(3 x \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e9d313cd6746f456", "problem_id": "a7ce9b486339d81f", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{8 x}{3} - \\frac{\\sqrt{10} x}{2} + \\frac{\\sqrt{10}}{3} + \\frac{4}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{8}{3} - \\frac{\\sqrt{10}}{2}$, c=$\\frac{\\sqrt{10}}{3} + \\frac{4}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{4 \\sqrt{10}}{3} + \\frac{77}{18}$, so √Δ = $\\sqrt{\\frac{4 \\sqrt{10}}{3} + \\frac{77}{18}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{10}}{4} + \\frac{4}{3} + \\frac{\\sqrt{48 \\sqrt{10} + 154}}{12}$.\n7) x2 = $- \\frac{\\sqrt{48 \\sqrt{10} + 154}}{12} + \\frac{\\sqrt{10}}{4} + \\frac{4}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{10}}{4} + \\frac{4}{3} + \\frac{\\sqrt{48\\cdot \\sqrt{10} + 154}}{12}, - \\frac{\\sqrt{48\\cdot \\sqrt{10} + 154}}{12} + \\frac{\\sqrt{10}}{4} + \\frac{4}{3}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{10}}{4} + \\frac{4}{3} + \\frac{\\sqrt{48\\cdot \\sqrt{10} + 154}}{12}, - \\frac{\\sqrt{48\\cdot \\sqrt{10} + 154}}{12} + \\frac{\\sqrt{10}}{4} + \\frac{4}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "01efb479a43bf114", "problem_id": "a7ce9b486339d81f", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{8 x}{3} - \\frac{\\sqrt{10} x}{2} + \\frac{\\sqrt{10}}{3} + \\frac{4}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{8}{3} - \\frac{\\sqrt{10}}{2}$, c=$\\frac{\\sqrt{10}}{3} + \\frac{4}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{4 \\sqrt{10}}{3} + \\frac{77}{18}$, so √Δ = $\\sqrt{\\frac{4 \\sqrt{10}}{3} + \\frac{77}{18}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{10}}{4} + \\frac{4}{3} + \\frac{\\sqrt{48 \\sqrt{10} + 154}}{12}$.\nx2 = $- \\frac{\\sqrt{48 \\sqrt{10} + 154}}{12} + \\frac{\\sqrt{10}}{4} + \\frac{4}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{10}}{4} + \\frac{4}{3} + \\frac{\\sqrt{48\\cdot \\sqrt{10} + 154}}{12}, - \\frac{\\sqrt{48\\cdot \\sqrt{10} + 154}}{12} + \\frac{\\sqrt{10}}{4} + \\frac{4}{3}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{10}}{4} + \\frac{4}{3} + \\frac{\\sqrt{48\\cdot \\sqrt{10} + 154}}{12}, - \\frac{\\sqrt{48\\cdot \\sqrt{10} + 154}}{12} + \\frac{\\sqrt{10}}{4} + \\frac{4}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e144dbfb8cbab621", "problem_id": "d5692682e8ca7e5a", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - x^{3} + x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- x^{3} + x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}$.\n3) Compute an antiderivative: F(x) = $- \\frac{x^{4}}{4} + \\frac{x^{3}}{3} + x^{2} - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{x^{4}}{4} + \\frac{x^{3}}{3} + x^{2} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{x^{4}}{4} + \\frac{x^{3}}{3} + x^{2} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "57346efe783dce2a", "problem_id": "d5692682e8ca7e5a", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - x^{3} + x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- x^{3} + x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}$.\nCompute an antiderivative: F(x) = $- \\frac{x^{4}}{4} + \\frac{x^{3}}{3} + x^{2} - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $C - \\frac{x^{4}}{4} + \\frac{x^{3}}{3} + x^{2} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C - \\frac{x^{4}}{4} + \\frac{x^{3}}{3} + x^{2} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a5c0f74d2a2d260d", "problem_id": "0170a83251b70270", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{10}$. If it is flipped 13 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=13, k=8, p=$\\frac{3}{10}$ and simplify.\nFinal answer: $\\frac{141918425649}{10000000000000}$", "final_answer": "\\frac{141918425649}{10000000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e6c1af0ca96c0201", "problem_id": "0170a83251b70270", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{10}$. If it is flipped 13 times, what is the probability of exactly 8 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=13, k=8, p=$\\frac{3}{10}$ and simplify.\n4) Here C(13,8) = 1287.\nFinal answer: $\\frac{141918425649}{10000000000000}$", "final_answer": "\\frac{141918425649}{10000000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2ba773a064464e3e", "problem_id": "01c836bc7356e73c", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A cyclist rides at 57 miles per hour for 1/2 hours, then at 55 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 57 × 1/2 = 57/2 miles.\nSecond leg distance = 55 × 2 = 110 miles.\nTotal distance = 57/2 + 110 = 277/2 miles.\nCheck: adding both legs gives 277/2.\nFinal answer: $\\frac{277}{2}$", "final_answer": "\\frac{277}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "078d9de184c625ab", "problem_id": "01c836bc7356e73c", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A cyclist rides at 57 miles per hour for 1/2 hours, then at 55 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 57 × 1/2 = 57/2 miles.\n2) Second leg distance = 55 × 2 = 110 miles.\n3) Total distance = 57/2 + 110 = 277/2 miles.\nFinal answer: $\\frac{277}{2}$", "final_answer": "\\frac{277}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fbeef4d8c55e2557", "problem_id": "96d70d275380604f", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "An urn has 13 red balls and 10 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(23,7).\n3) Favorable samples: choose k reds and n−k blues: C(13,3)·C(10,4).\n4) Compute totals: C(23,7) = 245157, and favorable = 60060.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1820}{7429}$", "final_answer": "\\frac{1820}{7429}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e8ee03cdd2eaa3dc", "problem_id": "96d70d275380604f", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 13 red balls and 10 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(23,7).\nFavorable samples: choose k reds and n−k blues: C(13,3)·C(10,4).\nCompute totals: C(23,7) = 245157, and favorable = 60060.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1820}{7429}$.\nFinal answer: $\\frac{1820}{7429}$", "final_answer": "\\frac{1820}{7429}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "408fbb76175d80f4", "problem_id": "dbc3fa2efaaeeaba", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{19 x}{2} + \\frac{35}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$\\frac{19}{2}$, c=$\\frac{35}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{81}{4}$, so √Δ = $\\frac{9}{2}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{5}{4}$.\n7) x2 = $- \\frac{7}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{5}{4}, - \\frac{7}{2}\\}$", "final_answer": "x \\in \\{- \\frac{5}{4}, - \\frac{7}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3722578a0a823079", "problem_id": "dbc3fa2efaaeeaba", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{19 x}{2} + \\frac{35}{4}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$\\frac{19}{2}$, c=$\\frac{35}{4}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{81}{4}$, so √Δ = $\\frac{9}{2}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{5}{4}$.\nx2 = $- \\frac{7}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{5}{4}, - \\frac{7}{2}\\}$", "final_answer": "x \\in \\{- \\frac{5}{4}, - \\frac{7}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f3f3c154f7428fc9", "problem_id": "37ecf5f605acfe9d", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{3 x^{3} + 2 x^{2} - x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{3 x + 4}$.", "output": "1) Let f(x) = $\\frac{3 x^{3} + 2 x^{2} - x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{3 x + 4}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{- 18 x^{3} - 12 x^{2} + 6 x + \\left(3 x + 4\\right) \\left(18 x^{2} + 8 x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 6 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 12}{2 \\left(3 x + 4\\right)^{2}}$.\nFinal answer: $\\frac{- 18\\cdot x^{3} - 12\\cdot x^{2} + 6\\cdot x + \\left(3\\cdot x + 4\\right)\\cdot \\left(18\\cdot x^{2} + 8\\cdot x + \\left(3\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x + 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right) - \\left(18\\cdot x + 24\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 12}{2\\cdot \\left(3\\cdot x + 4\\right)^{2}}$", "final_answer": "\\frac{- 18\\cdot x^{3} - 12\\cdot x^{2} + 6\\cdot x + \\left(3\\cdot x + 4\\right)\\cdot \\left(18\\cdot x^{2} + 8\\cdot x + \\left(3\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x + 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right) - \\left(18\\cdot x + 24\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 12}{2\\cdot \\left(3\\cdot x + 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cb67e70ec4394202", "problem_id": "37ecf5f605acfe9d", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{3 x^{3} + 2 x^{2} - x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{3 x + 4}$.", "output": "Let f(x) = $\\frac{3 x^{3} + 2 x^{2} - x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{3 x + 4}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{- 18 x^{3} - 12 x^{2} + 6 x + \\left(3 x + 4\\right) \\left(18 x^{2} + 8 x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 6 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 12}{2 \\left(3 x + 4\\right)^{2}}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\frac{- 18\\cdot x^{3} - 12\\cdot x^{2} + 6\\cdot x + \\left(3\\cdot x + 4\\right)\\cdot \\left(18\\cdot x^{2} + 8\\cdot x + \\left(3\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x + 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right) - \\left(18\\cdot x + 24\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 12}{2\\cdot \\left(3\\cdot x + 4\\right)^{2}}$", "final_answer": "\\frac{- 18\\cdot x^{3} - 12\\cdot x^{2} + 6\\cdot x + \\left(3\\cdot x + 4\\right)\\cdot \\left(18\\cdot x^{2} + 8\\cdot x + \\left(3\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x + 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right) - \\left(18\\cdot x + 24\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 12}{2\\cdot \\left(3\\cdot x + 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a40275c0ea8e4af5", "problem_id": "61455693a2990bb1", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} - 2 x + 3$ at $x=1$.", "output": "Differentiate: f'(x) = $- 6 x^{2} - 2$.\nSlope at x=1: m = f'(1) = $-8$.\nPoint on curve: (1, f(1)) = (1, $-1$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -1 = -8(x − 1).\nOptionally expand: y = $7 - 8 x$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 7 - 8\\cdot x$", "final_answer": "y = 7 - 8\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5da7d44d1e6f0895", "problem_id": "61455693a2990bb1", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} - 2 x + 3$ at $x=1$.", "output": "1) Differentiate: f'(x) = $- 6 x^{2} - 2$.\n2) Slope at x=1: m = f'(1) = $-8$.\n3) Point on curve: (1, f(1)) = (1, $-1$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -1 = -8(x − 1).\n6) Optionally expand: y = $7 - 8 x$.\nFinal answer: $y = 7 - 8\\cdot x$", "final_answer": "y = 7 - 8\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1a264fd259b8cf9b", "problem_id": "8bcff8dace367d5c", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=14$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=14 and k=5.\nFinal answer: $3060$", "final_answer": "3060", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6ca4e7ba94738463", "problem_id": "8bcff8dace367d5c", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=14$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=14 and k=5.\n4) Compute C(18,4) = 3060.\nFinal answer: $3060$", "final_answer": "3060", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3a26c3a153927846", "problem_id": "82acac71f27cfb09", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 23 red balls and 11 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(34,8).\nFavorable samples: choose k reds and n−k blues: C(23,1)·C(11,7).\nCompute totals: C(34,8) = 18156204, and favorable = 7590.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{115}{275094}$.\nFinal answer: $\\frac{115}{275094}$", "final_answer": "\\frac{115}{275094}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "daf9f83254ad2250", "problem_id": "82acac71f27cfb09", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "An urn has 23 red balls and 11 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(34,8).\n3) Favorable samples: choose k reds and n−k blues: C(23,1)·C(11,7).\n4) Compute totals: C(34,8) = 18156204, and favorable = 7590.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{115}{275094}$", "final_answer": "\\frac{115}{275094}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a7e9c2c5b5059a24", "problem_id": "aaa146a38fe8ea5e", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{2} + 2 x - 3$.", "output": "Differentiate term-by-term:\nd/dx(-3) = 0.\nd/dx(2 x) = 2.\nd/dx(4 x^{2}) = 8 x.\nCombine: f'(x) = 2 \\left(4 x + 1\\right).\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $8\\cdot x + 2$", "final_answer": "8\\cdot x + 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f6d6687276383993", "problem_id": "aaa146a38fe8ea5e", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{2} + 2 x - 3$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-3) = 0.\n3) d/dx(2 x) = 2.\n4) d/dx(4 x^{2}) = 8 x.\n5) Combine: f'(x) = 2 \\left(4 x + 1\\right).\nFinal answer: $8\\cdot x + 2$", "final_answer": "8\\cdot x + 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cb049033c73dfb75", "problem_id": "684cd892106e074c", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_8=28$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=28 and k=8.\n4) Compute C(35,7) = 6724520.\nFinal answer: $6724520$", "final_answer": "6724520", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1520b1a722428c29", "problem_id": "684cd892106e074c", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_8=28$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=28 and k=8.\nCompute C(35,7) = 6724520.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $6724520$", "final_answer": "6724520", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "421b66f3c5d5c40b", "problem_id": "ef5023bf6c9ca0e9", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the congruences: x≡4 (mod 7), x≡3 (mod 8). Give the solution modulo 56.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 56.\nCompute the CRT solution to get x ≡ 11 (mod 56).\nFinal answer: $x\\equiv 11\\pmod{56}$", "final_answer": "x\\equiv 11\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dda0de753f061d19", "problem_id": "ef5023bf6c9ca0e9", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡4 (mod 7), x≡3 (mod 8). Give the solution modulo 56.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 56.\n3) Compute the CRT solution to get x ≡ 11 (mod 56).\nFinal answer: $x\\equiv 11\\pmod{56}$", "final_answer": "x\\equiv 11\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "55354b3fed45b275", "problem_id": "71b2a8a8aeb57740", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 52 liters of a 30% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 52−x liters is 70% solution.\nSalt equation: 25x + 70(52−x) = 30·52.\n(25-70)x = (30-70)·52 ⇒ x = 416/9.\nThen 52−x = 52/9.\nCheck: 25·416/9 + 70·52/9 = 30·52.\nFinal answer: $416/9 L of 25% and 52/9 L of 70%$", "final_answer": "416/9 L of 25% and 52/9 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9f3c812d64b3e18e", "problem_id": "71b2a8a8aeb57740", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "You want 52 liters of a 30% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 52−x liters is 70% solution.\n2) Salt equation: 25x + 70(52−x) = 30·52.\n3) (25-70)x = (30-70)·52 ⇒ x = 416/9.\n4) Then 52−x = 52/9.\nFinal answer: $416/9 L of 25% and 52/9 L of 70%$", "final_answer": "416/9 L of 25% and 52/9 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c82e31e185616077", "problem_id": "c67c57cfd55f30cf", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 11 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=5, p=$\\frac{1}{2}$ and simplify.\n4) Here C(11,5) = 462.\nFinal answer: $\\frac{231}{1024}$", "final_answer": "\\frac{231}{1024}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "29ca646d50a56284", "problem_id": "1c45b1a20f6a39de", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 \\cos{\\left(2 x \\right)}) = - 3 \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 3 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 \\cos{\\left(2 x \\right)}) = - 6 \\sin{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$", "final_answer": "3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "9e9386b9bcc2fe98", "problem_id": "97ea977def1433f7", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{13} x + 4 x - 12 + 2 \\sqrt{13}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$4 - \\sqrt{13}$, c=$-12 + 2 \\sqrt{13}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $77 - 16 \\sqrt{13}$, so √Δ = $\\sqrt{77 - 16 \\sqrt{13}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $-2 + \\frac{\\sqrt{13}}{2} + \\frac{\\sqrt{77 - 16 \\sqrt{13}}}{2}$.\n7) x2 = $- \\frac{\\sqrt{77 - 16 \\sqrt{13}}}{2} - 2 + \\frac{\\sqrt{13}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{-2 + \\frac{\\sqrt{13}}{2} + \\frac{\\sqrt{77 - 16\\cdot \\sqrt{13}}}{2}, - \\frac{\\sqrt{77 - 16\\cdot \\sqrt{13}}}{2} - 2 + \\frac{\\sqrt{13}}{2}\\}$", "final_answer": "x \\in \\{-2 + \\frac{\\sqrt{13}}{2} + \\frac{\\sqrt{77 - 16\\cdot \\sqrt{13}}}{2}, - \\frac{\\sqrt{77 - 16\\cdot \\sqrt{13}}}{2} - 2 + \\frac{\\sqrt{13}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4930fbce3d280489", "problem_id": "97ea977def1433f7", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{13} x + 4 x - 12 + 2 \\sqrt{13}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$4 - \\sqrt{13}$, c=$-12 + 2 \\sqrt{13}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{-2 + \\frac{\\sqrt{13}}{2} + \\frac{\\sqrt{77 - 16\\cdot \\sqrt{13}}}{2}, - \\frac{\\sqrt{77 - 16\\cdot \\sqrt{13}}}{2} - 2 + \\frac{\\sqrt{13}}{2}\\}$", "final_answer": "x \\in \\{-2 + \\frac{\\sqrt{13}}{2} + \\frac{\\sqrt{77 - 16\\cdot \\sqrt{13}}}{2}, - \\frac{\\sqrt{77 - 16\\cdot \\sqrt{13}}}{2} - 2 + \\frac{\\sqrt{13}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "440975eea7e00d81", "problem_id": "8e0ddc9f3bc1cce5", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 2 x^{3} - 2 x^{2} + 5 x - 12$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $2 x^{4} + 2 x^{3} + x^{2} - 2 x + 1$ and a = -2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $8 x^{3} + 6 x^{2} + 2 x - 2$.\n5) Evaluate at a: p'(-2) = $-46$.\nFinal answer: $-46$", "final_answer": "-46", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "04295d915d6b030e", "problem_id": "8e0ddc9f3bc1cce5", "style": "answer_only", "instruction": "Final answer only.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 2 x^{3} - 2 x^{2} + 5 x - 12$.", "output": "$-46$", "final_answer": "-46", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a9c373e0d623a63f", "problem_id": "486d83b780a98bc1", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Two friends split $1440 in the ratio 6:2. How much does each receive?", "output": "Total ratio parts = 6+2 = 8.\nEach part = 1440 ÷ 8 = 180.\nFirst share = 6 × 180 = 1080.\nSecond share = 2 × 180 = 360.\nCheck: 1080+360=1440.\nFinal answer: $360$", "final_answer": "360", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "090c172706cec7c8", "problem_id": "486d83b780a98bc1", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Two friends split $1440 in the ratio 6:2. How much does each receive?", "output": "Total ratio parts = 6+2 = 8.\nEach part = 1440 ÷ 8 = 180.\nFirst share = 6 × 180 = 1080.\nFinal answer: $360$", "final_answer": "360", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b4166e66de9b1662", "problem_id": "7a3d5ad19521d839", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the congruences: x≡7 (mod 8), x≡1 (mod 7). Give the solution modulo 56.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 56.\n3) Compute the CRT solution to get x ≡ 15 (mod 56).\nFinal answer: $x\\equiv 15\\pmod{56}$", "final_answer": "x\\equiv 15\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3809c82c54fadc85", "problem_id": "7a3d5ad19521d839", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the congruences: x≡7 (mod 8), x≡1 (mod 7). Give the solution modulo 56.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 56.\nCompute the CRT solution to get x ≡ 15 (mod 56).\nFinal answer: $x\\equiv 15\\pmod{56}$", "final_answer": "x\\equiv 15\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "afb4ebb06c2ed8e8", "problem_id": "e712e5bea65f0839", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{3}$. If it is flipped 10 times, what is the probability of exactly 7 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=7, p=$\\frac{1}{3}$ and simplify.\nHere C(10,7) = 120.\nCheck: expression simplifies to $\\frac{320}{19683}$.\nFinal answer: $\\frac{320}{19683}$", "final_answer": "\\frac{320}{19683}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d53b150d019ac0a6", "problem_id": "e712e5bea65f0839", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{3}$. If it is flipped 10 times, what is the probability of exactly 7 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=7, p=$\\frac{1}{3}$ and simplify.\n4) Here C(10,7) = 120.\nFinal answer: $\\frac{320}{19683}$", "final_answer": "\\frac{320}{19683}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a0fcb8221c5571d", "problem_id": "d3547b73874d4919", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_7=27$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_7=13.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=13 and k=7.\n4) Compute C(19,6) = 27132.\nFinal answer: $27132$", "final_answer": "27132", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d6214b78f35ea81f", "problem_id": "d3547b73874d4919", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_7=27$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_7=13.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=13 and k=7.\nCompute C(19,6) = 27132.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $27132$", "final_answer": "27132", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "808b9c14bcb2924f", "problem_id": "8aab6c22ef72e1e3", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 23 red balls and 16 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(39,10).\n3) Favorable samples: choose k reds and n−k blues: C(23,4)·C(16,6).\n4) Compute totals: C(39,10) = 635745396, and favorable = 70910840.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{123970}{1111443}$", "final_answer": "\\frac{123970}{1111443}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "54c75b329a6bdb8f", "problem_id": "8aab6c22ef72e1e3", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 23 red balls and 16 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(39,10).\nFavorable samples: choose k reds and n−k blues: C(23,4)·C(16,6).\nCompute totals: C(39,10) = 635745396, and favorable = 70910840.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{123970}{1111443}$.\nFinal answer: $\\frac{123970}{1111443}$", "final_answer": "\\frac{123970}{1111443}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "56dc9d5829d4d3d4", "problem_id": "450765c6cdbf096c", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡4 (mod 10). Give the solution modulo 30.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 30.\n3) Compute the CRT solution to get x ≡ 4 (mod 30).\nFinal answer: $x\\equiv 4\\pmod{30}$", "final_answer": "x\\equiv 4\\pmod{30}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "644e78fa5a14133e", "problem_id": "450765c6cdbf096c", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡4 (mod 10). Give the solution modulo 30.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 30.\nCompute the CRT solution to get x ≡ 4 (mod 30).\nCheck: 4 mod 3=1 and mod 10=4.\nFinal answer: $x\\equiv 4\\pmod{30}$", "final_answer": "x\\equiv 4\\pmod{30}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "80fa4b151726f15a", "problem_id": "2b540d56d3f629eb", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "An urn has 20 red balls and 26 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(46,7).\nFavorable samples: choose k reds and n−k blues: C(20,1)·C(26,6).\nFinal answer: $\\frac{455}{5289}$", "final_answer": "\\frac{455}{5289}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "07ad45def03f964f", "problem_id": "2b540d56d3f629eb", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 20 red balls and 26 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(46,7).\n3) Favorable samples: choose k reds and n−k blues: C(20,1)·C(26,6).\n4) Compute totals: C(46,7) = 53524680, and favorable = 4604600.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{455}{5289}$", "final_answer": "\\frac{455}{5289}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "630d2d83e36cf946", "problem_id": "0a6d7a9859ba8c40", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{11 x}{4} - \\frac{\\sqrt{11} x}{2} + \\frac{7}{4} + \\frac{7 \\sqrt{11}}{8}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{11}{4} - \\frac{\\sqrt{11}}{2}$, c=$\\frac{7}{4} + \\frac{7 \\sqrt{11}}{8}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{\\sqrt{53 - 12\\cdot \\sqrt{11}}}{8} + \\frac{\\sqrt{11}}{4} + \\frac{11}{8}, - \\frac{\\sqrt{53 - 12\\cdot \\sqrt{11}}}{8} + \\frac{\\sqrt{11}}{4} + \\frac{11}{8}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{53 - 12\\cdot \\sqrt{11}}}{8} + \\frac{\\sqrt{11}}{4} + \\frac{11}{8}, - \\frac{\\sqrt{53 - 12\\cdot \\sqrt{11}}}{8} + \\frac{\\sqrt{11}}{4} + \\frac{11}{8}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b4fde773ebff7059", "problem_id": "0a6d7a9859ba8c40", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{11 x}{4} - \\frac{\\sqrt{11} x}{2} + \\frac{7}{4} + \\frac{7 \\sqrt{11}}{8}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{11}{4} - \\frac{\\sqrt{11}}{2}$, c=$\\frac{7}{4} + \\frac{7 \\sqrt{11}}{8}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{53}{16} - \\frac{3 \\sqrt{11}}{4}$, so √Δ = $\\sqrt{\\frac{53}{16} - \\frac{3 \\sqrt{11}}{4}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{53 - 12 \\sqrt{11}}}{8} + \\frac{\\sqrt{11}}{4} + \\frac{11}{8}$.\n7) x2 = $- \\frac{\\sqrt{53 - 12 \\sqrt{11}}}{8} + \\frac{\\sqrt{11}}{4} + \\frac{11}{8}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{53 - 12\\cdot \\sqrt{11}}}{8} + \\frac{\\sqrt{11}}{4} + \\frac{11}{8}, - \\frac{\\sqrt{53 - 12\\cdot \\sqrt{11}}}{8} + \\frac{\\sqrt{11}}{4} + \\frac{11}{8}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{53 - 12\\cdot \\sqrt{11}}}{8} + \\frac{\\sqrt{11}}{4} + \\frac{11}{8}, - \\frac{\\sqrt{53 - 12\\cdot \\sqrt{11}}}{8} + \\frac{\\sqrt{11}}{4} + \\frac{11}{8}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f49f0f3ca0fd9215", "problem_id": "546ae3fa5da626e6", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{9 x}{5} - \\sqrt{3} x - \\frac{18}{5} - \\frac{6 \\sqrt{3}}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{9}{5} - \\sqrt{3}$, c=$- \\frac{18}{5} - \\frac{6 \\sqrt{3}}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{42 \\sqrt{3}}{5} + \\frac{516}{25}$, so √Δ = $\\sqrt{\\frac{42 \\sqrt{3}}{5} + \\frac{516}{25}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{3}}{2} + \\frac{9}{10} + \\frac{\\sqrt{210 \\sqrt{3} + 516}}{10}$.\nx2 = $- \\frac{\\sqrt{210 \\sqrt{3} + 516}}{10} + \\frac{\\sqrt{3}}{2} + \\frac{9}{10}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{3}}{2} + \\frac{9}{10} + \\frac{\\sqrt{210\\cdot \\sqrt{3} + 516}}{10}, - \\frac{\\sqrt{210\\cdot \\sqrt{3} + 516}}{10} + \\frac{\\sqrt{3}}{2} + \\frac{9}{10}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{3}}{2} + \\frac{9}{10} + \\frac{\\sqrt{210\\cdot \\sqrt{3} + 516}}{10}, - \\frac{\\sqrt{210\\cdot \\sqrt{3} + 516}}{10} + \\frac{\\sqrt{3}}{2} + \\frac{9}{10}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9e82ef1d6f970018", "problem_id": "546ae3fa5da626e6", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{9 x}{5} - \\sqrt{3} x - \\frac{18}{5} - \\frac{6 \\sqrt{3}}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{9}{5} - \\sqrt{3}$, c=$- \\frac{18}{5} - \\frac{6 \\sqrt{3}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{42 \\sqrt{3}}{5} + \\frac{516}{25}$, so √Δ = $\\sqrt{\\frac{42 \\sqrt{3}}{5} + \\frac{516}{25}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{3}}{2} + \\frac{9}{10} + \\frac{\\sqrt{210 \\sqrt{3} + 516}}{10}$.\n7) x2 = $- \\frac{\\sqrt{210 \\sqrt{3} + 516}}{10} + \\frac{\\sqrt{3}}{2} + \\frac{9}{10}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{3}}{2} + \\frac{9}{10} + \\frac{\\sqrt{210\\cdot \\sqrt{3} + 516}}{10}, - \\frac{\\sqrt{210\\cdot \\sqrt{3} + 516}}{10} + \\frac{\\sqrt{3}}{2} + \\frac{9}{10}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{3}}{2} + \\frac{9}{10} + \\frac{\\sqrt{210\\cdot \\sqrt{3} + 516}}{10}, - \\frac{\\sqrt{210\\cdot \\sqrt{3} + 516}}{10} + \\frac{\\sqrt{3}}{2} + \\frac{9}{10}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b2ce3c6de85c6573", "problem_id": "91c99363e54a688d", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{7}$. If it is flipped 16 times, what is the probability of exactly 10 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=16, k=10, p=$\\frac{4}{7}$ and simplify.\nHere C(16,10) = 8008.\nCheck: expression simplifies to $\\frac{874487218176}{4747561509943}$.\nFinal answer: $\\frac{874487218176}{4747561509943}$", "final_answer": "\\frac{874487218176}{4747561509943}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dbb3f36d85c13b23", "problem_id": "91c99363e54a688d", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{7}$. If it is flipped 16 times, what is the probability of exactly 10 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=16, k=10, p=$\\frac{4}{7}$ and simplify.\n4) Here C(16,10) = 8008.\nFinal answer: $\\frac{874487218176}{4747561509943}$", "final_answer": "\\frac{874487218176}{4747561509943}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c143cd61a4b67c46", "problem_id": "fec462ec36edfd45", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the multiplicative inverse of 9 modulo 59.", "output": "1) We want x such that 9x ≡ 1 (mod 59).\n2) Use extended Euclid to find s,t with 9s + 59t = 1.\n3) One Bezout identity is: 9·(-13) + 59·(2) = 1.\n4) Reduce s modulo 59: x ≡ 46 (mod 59).\nFinal answer: $x\\equiv 46\\pmod{59}$", "final_answer": "x\\equiv 46\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7a44ba56ea81695a", "problem_id": "fec462ec36edfd45", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the multiplicative inverse of 9 modulo 59.", "output": "We want x such that 9x ≡ 1 (mod 59).\nUse extended Euclid to find s,t with 9s + 59t = 1.\nOne Bezout identity is: 9·(-13) + 59·(2) = 1.\nReduce s modulo 59: x ≡ 46 (mod 59).\nCheck: 9·46 mod 59 = 1.\nFinal answer: $x\\equiv 46\\pmod{59}$", "final_answer": "x\\equiv 46\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1df2ce2470e83303", "problem_id": "a158a911006cbd54", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "You want 55 liters of a 40% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 55−x liters is 50% solution.\n2) Salt equation: 25x + 50(55−x) = 40·55.\n3) (25-50)x = (40-50)·55 ⇒ x = 22.\n4) Then 55−x = 33.\nFinal answer: $22 L of 25% and 33 L of 50%$", "final_answer": "22 L of 25% and 33 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eb0f6839d809354d", "problem_id": "a158a911006cbd54", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "You want 55 liters of a 40% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "22 L of 25% and 33 L of 50%", "final_answer": "22 L of 25% and 33 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2fa11b386e2b4e7a", "problem_id": "a3e8b2673ada02be", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A disease has prevalence $\\frac{1}{200}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{83}{800}$.\n3) Compute P(D|+) = $\\frac{17}{415}$.\nFinal answer: $\\frac{17}{415}$", "final_answer": "\\frac{17}{415}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e7ed02fceb07a767", "problem_id": "a3e8b2673ada02be", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "A disease has prevalence $\\frac{1}{200}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{83}{800}$.\nCompute P(D|+) = $\\frac{17}{415}$.\nFinal answer: $\\frac{17}{415}$", "final_answer": "\\frac{17}{415}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a3a20aa8f740ae51", "problem_id": "d6bb2ee279f828cb", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} + x + 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(3 x^{2}) = 6 x^{2}$.\n3) So $f'(x) = 6 x^{2} + 1$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(3 x^{2}) = 6 x$.\nTherefore the correct derivative is $f'(x) = 6 x + 1$.\nFinal answer: $6 x + 1$", "final_answer": "6 x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2cd4a2ee151850ca", "problem_id": "bdd2170f323d33b2", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$5 x - 4 y$ = $12$\n$- 4 x - y$ = $24$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $- 20 x + 16 y$ = $-48$.\nMultiply the second equation by $5$: $- 20 x - 5 y$ = $120$.\nSubtract to eliminate x: $21 y$ = $-168$.\nSolve for y: y = $-8$.\nSubstitute back to find x: x = $-4$.\nCheck: plug (x,y)=(-4,-8) into both equations.\nFinal answer: $(x,y)=\\left(-4, -8\\right)$", "final_answer": "(x,y)=\\left(-4, -8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "00f6562c4bc9f969", "problem_id": "72a1763b38dfa521", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(- x^{3} - x^{2}\\right) \\cos{\\left(x^{2} + 2 x - 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x^{3} \\cos{\\left(\\left(x^{2} + 2 x\\right) - 2 \\right)}) = x^{3} \\sin{\\left(x^{2} + 2 x - 2 \\right)}$.\n3) So $f'(x) = 3 x^{3} \\sin{\\left(x^{2} + 2 x - 2 \\right)} + 2 x^{2} \\sin{\\left(x^{2} + 2 x - 2 \\right)} + x e^{x} - 2 x \\cos{\\left(\\left(x^{2} + 2 x\\right) - 2 \\right)} - 2 e^{x}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x^{3} \\cos{\\left(\\left(x^{2} + 2 x\\right) - 2 \\right)}) = x^{3} \\left(2 x + 2\\right) \\sin{\\left(x^{2} + 2 x - 2 \\right)} - 3 x^{2} \\cos{\\left(\\left(x^{2} + 2 x\\right) - 2 \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{2} \\left(- x - 1\\right) \\left(x + 1\\right) \\sin{\\left(x^{2} + 2 x - 2 \\right)} + x \\left(- 3 x - 2\\right) \\cos{\\left(\\left(x^{2} + 2 x\\right) - 2 \\right)} + \\left(x - 3\\right) e^{x} + e^{x}$.\nFinal answer: $- 2 x^{2} \\left(- x - 1\\right) \\left(x + 1\\right) \\sin{\\left(x^{2} + 2 x - 2 \\right)} + x \\left(- 3 x - 2\\right) \\cos{\\left(\\left(x^{2} + 2 x\\right) - 2 \\right)} + \\left(x - 3\\right) e^{x} + e^{x}$", "final_answer": "- 2 x^{2} \\left(- x - 1\\right) \\left(x + 1\\right) \\sin{\\left(x^{2} + 2 x - 2 \\right)} + x \\left(- 3 x - 2\\right) \\cos{\\left(\\left(x^{2} + 2 x\\right) - 2 \\right)} + \\left(x - 3\\right) e^{x} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f72058dd51c2581f", "problem_id": "15cf95f0b79f95a1", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 18 times, what is the probability of exactly 8 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=18, k=8, p=$\\frac{2}{7}$ and simplify.\n4) Here C(18,8) = 43758.\nFinal answer: $\\frac{109395000000000}{1628413597910449}$", "final_answer": "\\frac{109395000000000}{1628413597910449}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "efc7d074af2e31ce", "problem_id": "d91a625ed0de584d", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 2 \\sqrt{10} x - \\frac{5 x}{2} - \\sqrt{10} - \\frac{3}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- 2 \\sqrt{10} - \\frac{5}{2}$, c=$- \\sqrt{10} - \\frac{3}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $14 \\sqrt{10} + \\frac{209}{4}$, so √Δ = $\\sqrt{14 \\sqrt{10} + \\frac{209}{4}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{5}{4} + \\sqrt{10} + \\frac{\\sqrt{56 \\sqrt{10} + 209}}{4}$.\n7) x2 = $- \\frac{\\sqrt{56 \\sqrt{10} + 209}}{4} + \\frac{5}{4} + \\sqrt{10}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{5}{4} + \\sqrt{10} + \\frac{\\sqrt{56\\cdot \\sqrt{10} + 209}}{4}, - \\frac{\\sqrt{56\\cdot \\sqrt{10} + 209}}{4} + \\frac{5}{4} + \\sqrt{10}\\}$", "final_answer": "x \\in \\{\\frac{5}{4} + \\sqrt{10} + \\frac{\\sqrt{56\\cdot \\sqrt{10} + 209}}{4}, - \\frac{\\sqrt{56\\cdot \\sqrt{10} + 209}}{4} + \\frac{5}{4} + \\sqrt{10}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "96fb4f131ff38e76", "problem_id": "56037c522cbd4301", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 2 x^{2} - 4 x + 15$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{2} - 5 x + 15.\n3) Then the limit is 48.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: 45.\nFinal answer: $45$", "final_answer": "45", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b4e6c79cab91d152", "problem_id": "abf1a8a1940879ef", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 19 times, what is the probability of exactly 6 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=19, k=6, p=$\\frac{2}{5}$ and simplify.\nHere C(19,6) = 27132.\nCheck: expression simplifies to $\\frac{2768458984704}{19073486328125}$.\nFinal answer: $\\frac{2768458984704}{19073486328125}$", "final_answer": "\\frac{2768458984704}{19073486328125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "59581347b64c6b18", "problem_id": "abf1a8a1940879ef", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 19 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=19, k=6, p=$\\frac{2}{5}$ and simplify.\n4) Here C(19,6) = 27132.\nFinal answer: $\\frac{2768458984704}{19073486328125}$", "final_answer": "\\frac{2768458984704}{19073486328125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e2369ea60e908776", "problem_id": "d2e589a6d6e8b92a", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A store sells a backpack for $102. It is discounted by 20%. You buy 5 backpacks. What is the total cost?", "output": "1) Discounted price = 102 × (1 − 20/100) = 102 × 80/100 = 408/5.\n2) Subtotal = 5 × 408/5 = 408.\nFinal answer: $408$", "final_answer": "408", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d92a65e657d73fbe", "problem_id": "0d1ed7df721c833d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(3 x^{2} + x + 1\\right) \\cos{\\left(x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 x^{2} \\cos{\\left(x + 3 \\right)}) = - 3 x^{2} \\sin{\\left(x + 3 \\right)}$.\n3) So $f'(x) = - 3 x^{2} \\sin{\\left(x + 3 \\right)} + 2 x e^{x} - x \\sin{\\left(x + 3 \\right)} - 4 e^{x} - \\sin{\\left(x + 3 \\right)} + \\cos{\\left(x + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 x^{2} \\cos{\\left(x + 3 \\right)}) = - 3 x^{2} \\sin{\\left(x + 3 \\right)} + 6 x \\cos{\\left(x + 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x - 3\\right) e^{x} + \\left(6 x + 1\\right) \\cos{\\left(x + 3 \\right)} - \\left(x \\left(3 x + 1\\right) + 1\\right) \\sin{\\left(x + 3 \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(x - 3\\right) e^{x} + \\left(6 x + 1\\right) \\cos{\\left(x + 3 \\right)} - \\left(x \\left(3 x + 1\\right) + 1\\right) \\sin{\\left(x + 3 \\right)} + 2 e^{x}$", "final_answer": "2 \\left(x - 3\\right) e^{x} + \\left(6 x + 1\\right) \\cos{\\left(x + 3 \\right)} - \\left(x \\left(3 x + 1\\right) + 1\\right) \\sin{\\left(x + 3 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "1200cad0e2e1a99b", "problem_id": "084f0e42069068fa", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "An urn has 12 red balls and 10 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(22,11).\nFavorable samples: choose k reds and n−k blues: C(12,1)·C(10,10).\nFinal answer: $\\frac{1}{58786}$", "final_answer": "\\frac{1}{58786}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1f58b04a7aa9c631", "problem_id": "084f0e42069068fa", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 12 red balls and 10 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(22,11).\n3) Favorable samples: choose k reds and n−k blues: C(12,1)·C(10,10).\n4) Compute totals: C(22,11) = 705432, and favorable = 12.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1}{58786}$", "final_answer": "\\frac{1}{58786}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "60ce995701f594f9", "problem_id": "efb81a8fb5c189f9", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "An urn has 13 red balls and 27 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(40,8).\n3) Favorable samples: choose k reds and n−k blues: C(13,4)·C(27,4).\n4) Compute totals: C(40,8) = 76904685, and favorable = 12548250.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1950}{11951}$", "final_answer": "\\frac{1950}{11951}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0a47f6e32291858a", "problem_id": "77bf007d2ad0bb7e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} x - 3$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to -3.\n3) Then the limit is -3.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -6.\nFinal answer: $-6$", "final_answer": "-6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "eb4117d7560124d0", "problem_id": "f1893a4d4b51e08a", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 11 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=2, p=$\\frac{2}{3}$ and simplify.\n4) Here C(11,2) = 55.\nFinal answer: $\\frac{220}{177147}$", "final_answer": "\\frac{220}{177147}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "260ccd2a612b2ed0", "problem_id": "f1893a4d4b51e08a", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 11 times, what is the probability of exactly 2 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=11, k=2, p=$\\frac{2}{3}$ and simplify.\nFinal answer: $\\frac{220}{177147}$", "final_answer": "\\frac{220}{177147}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "575f0cea446819b5", "problem_id": "f78a30833131ecba", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 36 x + 96$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} + 36 x + 96$ = $3 \\left(x + 4\\right) \\left(x + 8\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-4$.\n5) From the second factor: x = $-8$.\nFinal answer: $x \\in \\{-4, -8\\}$", "final_answer": "x \\in \\{-4, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "770a6e2246032d09", "problem_id": "f78a30833131ecba", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 36 x + 96$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} + 36 x + 96$ = $3 \\left(x + 4\\right) \\left(x + 8\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $-4$.\nFrom the second factor: x = $-8$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-4, -8\\}$", "final_answer": "x \\in \\{-4, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2cc6343aea542257", "problem_id": "0c0b9250a2690b91", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} - x^{2} + 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 3 x^{2} - 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 3$.\n2) Slope at x=2: m = f'(2) = $-13 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\n3) Point on curve: (2, f(2)) = (2, $-5 + \\cos{\\left(4 \\right)} + e$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -5 + \\cos{\\left(4 \\right)} + e = -13 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\n6) Optionally expand: y = $- 13 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 21$.\nFinal answer: $y = - 13\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 21$", "final_answer": "y = - 13\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 21", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bfb01d6add0dd09f", "problem_id": "0c0b9250a2690b91", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} - x^{2} + 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=2$.", "output": "Differentiate: f'(x) = $- 3 x^{2} - 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 3$.\nSlope at x=2: m = f'(2) = $-13 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\nPoint on curve: (2, f(2)) = (2, $-5 + \\cos{\\left(4 \\right)} + e$).\nFinal answer: $y = - 13\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 21$", "final_answer": "y = - 13\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 21", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e34e0bf0d7305d10", "problem_id": "49f21da7ba95c98b", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(x^{2} - 2 x + 3\\right) \\cos{\\left(x^{2} + x - 3 \\right)}$.", "output": "Let f(x) = $\\left(x + 2\\right) e^{x} + \\left(x^{2} - 2 x + 3\\right) \\cos{\\left(x^{2} + x - 3 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $2 \\left(x - 1\\right) \\cos{\\left(x^{2} + x - 3 \\right)} + \\left(x + 2\\right) e^{x} - \\left(2 x + 1\\right) \\left(x^{2} - 2 x + 3\\right) \\sin{\\left(x^{2} + x - 3 \\right)} + e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\left(x + 2\\right)\\cdot e^{x} + \\left(2\\cdot x - 2\\right)\\cdot \\cos{\\left(x^{2} + x - 3 \\right)} - \\left(2\\cdot x + 1\\right)\\cdot \\left(x^{2} - 2\\cdot x + 3\\right)\\cdot \\sin{\\left(x^{2} + x - 3 \\right)} + e^{x}$", "final_answer": "\\left(x + 2\\right)\\cdot e^{x} + \\left(2\\cdot x - 2\\right)\\cdot \\cos{\\left(x^{2} + x - 3 \\right)} - \\left(2\\cdot x + 1\\right)\\cdot \\left(x^{2} - 2\\cdot x + 3\\right)\\cdot \\sin{\\left(x^{2} + x - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "406a40bcbd9b46bc", "problem_id": "49f21da7ba95c98b", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(x^{2} - 2 x + 3\\right) \\cos{\\left(x^{2} + x - 3 \\right)}$.", "output": "1) Let f(x) = $\\left(x + 2\\right) e^{x} + \\left(x^{2} - 2 x + 3\\right) \\cos{\\left(x^{2} + x - 3 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $2 \\left(x - 1\\right) \\cos{\\left(x^{2} + x - 3 \\right)} + \\left(x + 2\\right) e^{x} - \\left(2 x + 1\\right) \\left(x^{2} - 2 x + 3\\right) \\sin{\\left(x^{2} + x - 3 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 2\\right)\\cdot e^{x} + \\left(2\\cdot x - 2\\right)\\cdot \\cos{\\left(x^{2} + x - 3 \\right)} - \\left(2\\cdot x + 1\\right)\\cdot \\left(x^{2} - 2\\cdot x + 3\\right)\\cdot \\sin{\\left(x^{2} + x - 3 \\right)} + e^{x}$", "final_answer": "\\left(x + 2\\right)\\cdot e^{x} + \\left(2\\cdot x - 2\\right)\\cdot \\cos{\\left(x^{2} + x - 3 \\right)} - \\left(2\\cdot x + 1\\right)\\cdot \\left(x^{2} - 2\\cdot x + 3\\right)\\cdot \\sin{\\left(x^{2} + x - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "12c2c53cd012c58b", "problem_id": "7925ecfb2378e3a3", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int \\left(2 x + 1\\right) \\cos{\\left(x \\right)} + \\left(3 x + 4\\right) e^{3 x}\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $\\left(2 x + 1\\right) \\cos{\\left(x \\right)} + \\left(3 x + 4\\right) e^{3 x}$.\n3) Compute an antiderivative: F(x) = $2 x \\sin{\\left(x \\right)} + \\left(x + 1\\right) e^{3 x} + \\sin{\\left(x \\right)} + 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\left(x + 1\\right)\\cdot e^{3\\cdot x} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\left(x + 1\\right)\\cdot e^{3\\cdot x} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1dbc11c5599ee97f", "problem_id": "7925ecfb2378e3a3", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int \\left(2 x + 1\\right) \\cos{\\left(x \\right)} + \\left(3 x + 4\\right) e^{3 x}\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $\\left(2 x + 1\\right) \\cos{\\left(x \\right)} + \\left(3 x + 4\\right) e^{3 x}$.\nCompute an antiderivative: F(x) = $2 x \\sin{\\left(x \\right)} + \\left(x + 1\\right) e^{3 x} + \\sin{\\left(x \\right)} + 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\left(x + 1\\right)\\cdot e^{3\\cdot x} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + 2\\cdot x\\cdot \\sin{\\left(x \\right)} + \\left(x + 1\\right)\\cdot e^{3\\cdot x} + \\sin{\\left(x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "38a16bdd3eaa0298", "problem_id": "236cae29f9281ed6", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 5 x + 4$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$5$, c=$4$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $9$, so √Δ = $3$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $-1$.\n7) x2 = $-4$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{-1, -4\\}$", "final_answer": "x \\in \\{-1, -4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e4323513d6e40f60", "problem_id": "236cae29f9281ed6", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 5 x + 4$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$5$, c=$4$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{-1, -4\\}$", "final_answer": "x \\in \\{-1, -4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e37cfad0ae22f964", "problem_id": "0badaf53d1be6996", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} - x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{9 \\pi}{2} - \\frac{3 \\pi^{3}}{8} + 2 + \\frac{3 \\pi^{2}}{8}$.\n4) Simplify the result.\nFinal answer: $- \\frac{9\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{3}}{8} + 2 + \\frac{3\\cdot \\pi^{2}}{8}$", "final_answer": "- \\frac{9\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{3}}{8} + 2 + \\frac{3\\cdot \\pi^{2}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f86810d3be7d6815", "problem_id": "d8cc166d19e77d96", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x^{3} - 5 x^{2} - 8 x - 14$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{3} - 5 x^{2} - 9 x - 14.\n3) Then the limit is -68.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -66.\nFinal answer: $-66$", "final_answer": "-66", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "008c780dba9d61c2", "problem_id": "ae5956a2cdc2ae36", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡1 (mod 4), x≡10 (mod 11). Give the solution modulo 44.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 44.\n3) Compute the CRT solution to get x ≡ 21 (mod 44).\nFinal answer: $x\\equiv 21\\pmod{44}$", "final_answer": "x\\equiv 21\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "983512b3e8d3f144", "problem_id": "ae5956a2cdc2ae36", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the congruences: x≡1 (mod 4), x≡10 (mod 11). Give the solution modulo 44.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 44.\nCompute the CRT solution to get x ≡ 21 (mod 44).\nFinal answer: $x\\equiv 21\\pmod{44}$", "final_answer": "x\\equiv 21\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b7576250ac51ff17", "problem_id": "5ccc0b57681e96e0", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 4 x - 4$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$-4$, c=$-4$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $64$, so √Δ = $8$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $2$.\n7) x2 = $- \\frac{2}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{2, - \\frac{2}{3}\\}$", "final_answer": "x \\in \\{2, - \\frac{2}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "80976827afea9dfc", "problem_id": "5ccc0b57681e96e0", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 4 x - 4$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$-4$, c=$-4$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $64$, so √Δ = $8$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $2$.\nx2 = $- \\frac{2}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{2, - \\frac{2}{3}\\}$", "final_answer": "x \\in \\{2, - \\frac{2}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "59b12518787cb7c4", "problem_id": "343e406ca438fb43", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_5=24$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_5=14.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=14 and k=5.\n4) Compute C(18,4) = 3060.\nFinal answer: $3060$", "final_answer": "3060", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "11f409f351d995a6", "problem_id": "343e406ca438fb43", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_5=24$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_5=14.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=14 and k=5.\nCompute C(18,4) = 3060.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $3060$", "final_answer": "3060", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ca443d3b6680d6b", "problem_id": "bbb84c5a3c718191", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "You want 62 liters of a 35% salt solution. You have a 25% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 62−x liters is 40% solution.\n2) Salt equation: 25x + 40(62−x) = 35·62.\n3) (25-40)x = (35-40)·62 ⇒ x = 62/3.\n4) Then 62−x = 124/3.\nFinal answer: $62/3 L of 25% and 124/3 L of 40%$", "final_answer": "62/3 L of 25% and 124/3 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9832df4eb14cb792", "problem_id": "bbb84c5a3c718191", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "You want 62 liters of a 35% salt solution. You have a 25% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 62−x liters is 40% solution.\nSalt equation: 25x + 40(62−x) = 35·62.\n(25-40)x = (35-40)·62 ⇒ x = 62/3.\nThen 62−x = 124/3.\nCheck: 25·62/3 + 40·124/3 = 35·62.\nFinal answer: $62/3 L of 25% and 124/3 L of 40%$", "final_answer": "62/3 L of 25% and 124/3 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6971c1cec0bea349", "problem_id": "8c1c512b808052e0", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - 3 x^{2} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=-1$.", "output": "Differentiate: f'(x) = $- 9 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\nSlope at x=-1: m = f'(-1) = $-5 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nPoint on curve: (-1, f(-1)) = (-1, $\\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 3$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 3 = -5 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}(x − -1).\nOptionally expand: y = $- 5 x + \\frac{x}{2 e^{\\frac{1}{2}}} + 2 x \\sin{\\left(2 \\right)} - 2 + \\cos{\\left(2 \\right)} + \\frac{3}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 5\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 2 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 5\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 2 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "acac1e17acaebb61", "problem_id": "8c1c512b808052e0", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - 3 x^{2} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $- 9 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\n2) Slope at x=-1: m = f'(-1) = $-5 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\n3) Point on curve: (-1, f(-1)) = (-1, $\\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 3$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 3 = -5 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}(x − -1).\n6) Optionally expand: y = $- 5 x + \\frac{x}{2 e^{\\frac{1}{2}}} + 2 x \\sin{\\left(2 \\right)} - 2 + \\cos{\\left(2 \\right)} + \\frac{3}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = - 5\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 2 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 5\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 2 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "71a519913111944f", "problem_id": "7133e55317a90e54", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} - 2 x^{2} - x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $- 6 x^{2} - 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 1$.\n2) Slope at x=-2: m = f'(-2) = $-17 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}$.\n3) Point on curve: (-2, f(-2)) = (-2, $\\cos{\\left(4 \\right)} + e^{-1} + 8$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(4 \\right)} + e^{-1} + 8 = -17 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}(x − -2).\n6) Optionally expand: y = $- 17 x + 2 x \\sin{\\left(4 \\right)} + \\frac{x}{2 e} - 26 + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$.\nFinal answer: $y = - 17\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 26 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$", "final_answer": "y = - 17\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 26 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f46fa535f4db79e3", "problem_id": "7133e55317a90e54", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} - 2 x^{2} - x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=-2$.", "output": "Differentiate: f'(x) = $- 6 x^{2} - 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 1$.\nSlope at x=-2: m = f'(-2) = $-17 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}$.\nPoint on curve: (-2, f(-2)) = (-2, $\\cos{\\left(4 \\right)} + e^{-1} + 8$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − \\cos{\\left(4 \\right)} + e^{-1} + 8 = -17 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}(x − -2).\nOptionally expand: y = $- 17 x + 2 x \\sin{\\left(4 \\right)} + \\frac{x}{2 e} - 26 + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 17\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 26 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$", "final_answer": "y = - 17\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 26 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f5ddbc093b741197", "problem_id": "b2fb1eabe5862b60", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} - 3 x^{2} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 3$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 3 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\n2) Slope at x=2: m = f'(2) = $-26 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\n3) Point on curve: (2, f(2)) = (2, $-27 + \\cos{\\left(4 \\right)} + e$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -27 + \\cos{\\left(4 \\right)} + e = -26 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\n6) Optionally expand: y = $- 26 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 25$.\nFinal answer: $y = - 26\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 25$", "final_answer": "y = - 26\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 25", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c4faf3727bae1dde", "problem_id": "b2fb1eabe5862b60", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} - 3 x^{2} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 3$ at $x=2$.", "output": "Differentiate: f'(x) = $- 3 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\nSlope at x=2: m = f'(2) = $-26 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\nPoint on curve: (2, f(2)) = (2, $-27 + \\cos{\\left(4 \\right)} + e$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -27 + \\cos{\\left(4 \\right)} + e = -26 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\nOptionally expand: y = $- 26 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 25$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 26\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 25$", "final_answer": "y = - 26\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 25", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "64ccd0f84342b544", "problem_id": "d072e87522171c6f", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$3 x$ = $0$\n$5 x - 4 y$ = $4$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $5$: $15 x$ = $0$.\nMultiply the second equation by $3$: $15 x - 12 y$ = $12$.\nSubtract to eliminate x: $12 y$ = $-12$.\nSolve for y: y = $-1$.\nSubstitute back to find x: x = $0$.\nCheck: plug (x,y)=(0,-1) into both equations.\nFinal answer: $(x,y)=\\left(0, -1\\right)$", "final_answer": "(x,y)=\\left(0, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8aa6a922b24664b0", "problem_id": "d072e87522171c6f", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$3 x$ = $0$\n$5 x - 4 y$ = $4$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $5$: $15 x$ = $0$.\n3) Multiply the second equation by $3$: $15 x - 12 y$ = $12$.\n4) Subtract to eliminate x: $12 y$ = $-12$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, -1\\right)$", "final_answer": "(x,y)=\\left(0, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0baebe4e52fc11fa", "problem_id": "53af66c1ef4959ab", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Two friends split $1111 in the ratio 9:2. How much does each receive?", "output": "Total ratio parts = 9+2 = 11.\nEach part = 1111 ÷ 11 = 101.\nFirst share = 9 × 101 = 909.\nFinal answer: $202$", "final_answer": "202", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e9c96d533d831f4c", "problem_id": "53af66c1ef4959ab", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Two friends split $1111 in the ratio 9:2. How much does each receive?", "output": "1) Total ratio parts = 9+2 = 11.\n2) Each part = 1111 ÷ 11 = 101.\n3) First share = 9 × 101 = 909.\n4) Second share = 2 × 101 = 202.\nFinal answer: $202$", "final_answer": "202", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b435d5a3b56ebcaa", "problem_id": "39a036e8e4b3525b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 3 x - 12$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to 2 x - 12.\n3) Then the limit is -18.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -21.\nFinal answer: $-21$", "final_answer": "-21", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e14eb3a7cee0a7dc", "problem_id": "2dc0ce46ed817f70", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡1 (mod 4), x≡0 (mod 9). Give the solution modulo 36.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 36.\n3) Compute the CRT solution to get x ≡ 9 (mod 36).\nFinal answer: $x\\equiv 9\\pmod{36}$", "final_answer": "x\\equiv 9\\pmod{36}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "80734088e872356a", "problem_id": "b9548f4c18494ff3", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} - 3 x^{2} - 1$ at $x=1$.", "output": "1) Differentiate: f'(x) = $9 x^{2} - 6 x$.\n2) Slope at x=1: m = f'(1) = $3$.\n3) Point on curve: (1, f(1)) = (1, $-1$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -1 = 3(x − 1).\n6) Optionally expand: y = $3 x - 4$.\nFinal answer: $y = 3\\cdot x - 4$", "final_answer": "y = 3\\cdot x - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "75ba7803391fb432", "problem_id": "b9548f4c18494ff3", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} - 3 x^{2} - 1$ at $x=1$.", "output": "Differentiate: f'(x) = $9 x^{2} - 6 x$.\nSlope at x=1: m = f'(1) = $3$.\nPoint on curve: (1, f(1)) = (1, $-1$).\nFinal answer: $y = 3\\cdot x - 4$", "final_answer": "y = 3\\cdot x - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b2f03299b1b0ddee", "problem_id": "055b1c94456ca4c7", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{2} x}{2} + \\frac{11 x}{2} - \\frac{27 \\sqrt{2}}{4} + \\frac{9}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{11}{2} - \\frac{3 \\sqrt{2}}{2}$, c=$\\frac{9}{2} - \\frac{27 \\sqrt{2}}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{21 \\sqrt{2}}{2} + \\frac{67}{4}$, so √Δ = $\\sqrt{\\frac{21 \\sqrt{2}}{2} + \\frac{67}{4}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{11}{4} + \\frac{3 \\sqrt{2}}{4} + \\frac{\\sqrt{42 \\sqrt{2} + 67}}{4}$.\n7) x2 = $- \\frac{\\sqrt{42 \\sqrt{2} + 67}}{4} - \\frac{11}{4} + \\frac{3 \\sqrt{2}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{11}{4} + \\frac{3\\cdot \\sqrt{2}}{4} + \\frac{\\sqrt{42\\cdot \\sqrt{2} + 67}}{4}, - \\frac{\\sqrt{42\\cdot \\sqrt{2} + 67}}{4} - \\frac{11}{4} + \\frac{3\\cdot \\sqrt{2}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{11}{4} + \\frac{3\\cdot \\sqrt{2}}{4} + \\frac{\\sqrt{42\\cdot \\sqrt{2} + 67}}{4}, - \\frac{\\sqrt{42\\cdot \\sqrt{2} + 67}}{4} - \\frac{11}{4} + \\frac{3\\cdot \\sqrt{2}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6bbd56de6f6bd7c2", "problem_id": "055b1c94456ca4c7", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{2} x}{2} + \\frac{11 x}{2} - \\frac{27 \\sqrt{2}}{4} + \\frac{9}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{11}{2} - \\frac{3 \\sqrt{2}}{2}$, c=$\\frac{9}{2} - \\frac{27 \\sqrt{2}}{4}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{21 \\sqrt{2}}{2} + \\frac{67}{4}$, so √Δ = $\\sqrt{\\frac{21 \\sqrt{2}}{2} + \\frac{67}{4}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{11}{4} + \\frac{3 \\sqrt{2}}{4} + \\frac{\\sqrt{42 \\sqrt{2} + 67}}{4}$.\nx2 = $- \\frac{\\sqrt{42 \\sqrt{2} + 67}}{4} - \\frac{11}{4} + \\frac{3 \\sqrt{2}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{11}{4} + \\frac{3\\cdot \\sqrt{2}}{4} + \\frac{\\sqrt{42\\cdot \\sqrt{2} + 67}}{4}, - \\frac{\\sqrt{42\\cdot \\sqrt{2} + 67}}{4} - \\frac{11}{4} + \\frac{3\\cdot \\sqrt{2}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{11}{4} + \\frac{3\\cdot \\sqrt{2}}{4} + \\frac{\\sqrt{42\\cdot \\sqrt{2} + 67}}{4}, - \\frac{\\sqrt{42\\cdot \\sqrt{2} + 67}}{4} - \\frac{11}{4} + \\frac{3\\cdot \\sqrt{2}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "901cde0854921daf", "problem_id": "5d873daace097042", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many binary strings of length 15 contain exactly 3 ones with no two ones adjacent?", "output": "Reserve 2 mandatory zeros between the 3 ones.\nThis leaves n−k+1 = 13 valid slots for the ones.\nChoose the slots: C(13,3) = 286.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $286$", "final_answer": "286", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e5ecc2db64b01b77", "problem_id": "5d873daace097042", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "How many binary strings of length 15 contain exactly 3 ones with no two ones adjacent?", "output": "$286$", "final_answer": "286", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "246781e311a0fe60", "problem_id": "74424d12706a4ae8", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Two friends split $216 in the ratio 5:1. How much does each receive?", "output": "1) Total ratio parts = 5+1 = 6.\n2) Each part = 216 ÷ 6 = 36.\n3) First share = 5 × 36 = 180.\n4) Second share = 1 × 36 = 36.\nFinal answer: $36$", "final_answer": "36", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bbc7cc75ccba1282", "problem_id": "74424d12706a4ae8", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $216 in the ratio 5:1. How much does each receive?", "output": "Total ratio parts = 5+1 = 6.\nEach part = 216 ÷ 6 = 36.\nFirst share = 5 × 36 = 180.\nSecond share = 1 × 36 = 36.\nCheck: 180+36=216.\nFinal answer: $36$", "final_answer": "36", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1fe91d848ed2b8f7", "problem_id": "fcc51ab40431bdc9", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 3 x - 270$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} + 3 x - 270$ = $3 \\left(x - 9\\right) \\left(x + 10\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $9$.\nFrom the second factor: x = $-10$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{9, -10\\}$", "final_answer": "x \\in \\{9, -10\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "86d256c7a4cf0870", "problem_id": "fcc51ab40431bdc9", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 3 x - 270$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} + 3 x - 270$ = $3 \\left(x - 9\\right) \\left(x + 10\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $9$.\n5) From the second factor: x = $-10$.\nFinal answer: $x \\in \\{9, -10\\}$", "final_answer": "x \\in \\{9, -10\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "78917b7f7919f3d6", "problem_id": "0da0b6bd4cc0cf93", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the multiplicative inverse of 10 modulo 29.", "output": "We want x such that 10x ≡ 1 (mod 29).\nUse extended Euclid to find s,t with 10s + 29t = 1.\nOne Bezout identity is: 10·(3) + 29·(-1) = 1.\nReduce s modulo 29: x ≡ 3 (mod 29).\nCheck: 10·3 mod 29 = 1.\nFinal answer: $x\\equiv 3\\pmod{29}$", "final_answer": "x\\equiv 3\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "57419cff0ea19ba7", "problem_id": "0da0b6bd4cc0cf93", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the multiplicative inverse of 10 modulo 29.", "output": "1) We want x such that 10x ≡ 1 (mod 29).\n2) Use extended Euclid to find s,t with 10s + 29t = 1.\n3) One Bezout identity is: 10·(3) + 29·(-1) = 1.\n4) Reduce s modulo 29: x ≡ 3 (mod 29).\nFinal answer: $x\\equiv 3\\pmod{29}$", "final_answer": "x\\equiv 3\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8246c3e067e9bb00", "problem_id": "86df158ebe362c45", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "An urn has 30 red balls and 11 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(41,11).\n3) Favorable samples: choose k reds and n−k blues: C(30,1)·C(11,10).\n4) Compute totals: C(41,11) = 3159461968, and favorable = 330.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{165}{1579730984}$", "final_answer": "\\frac{165}{1579730984}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "30407e57c485ab42", "problem_id": "86df158ebe362c45", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "An urn has 30 red balls and 11 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(41,11).\nFavorable samples: choose k reds and n−k blues: C(30,1)·C(11,10).\nFinal answer: $\\frac{165}{1579730984}$", "final_answer": "\\frac{165}{1579730984}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "88dbcc266cd07ee0", "problem_id": "b4805ca98668463a", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 4 x^{3} + 4 x^{2} - 4 x\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $4 x^{3} + 4 x^{2} - 4 x$.\n3) Compute an antiderivative: F(x) = $x^{4} + \\frac{4 x^{3}}{3} - 2 x^{2}$.\n4) Include the constant of integration C.\nFinal answer: $C + x^{4} + \\frac{4\\cdot x^{3}}{3} - 2\\cdot x^{2}$", "final_answer": "C + x^{4} + \\frac{4\\cdot x^{3}}{3} - 2\\cdot x^{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5c69821a788c2d0f", "problem_id": "888ac461b1afacb4", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} 2 x^{2} + 4 x + 1$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{2} + 3 x + 1.\n3) Then the limit is 6.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: 7.\nFinal answer: $7$", "final_answer": "7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7e66c25b233ab1c6", "problem_id": "d2266df0b4a0f3d1", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(2 x^{3} - 3 x^{2} - 3 x\\right) \\cos{\\left(3 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 3 x \\cos{\\left(3 x \\right)}) = 3 x \\sin{\\left(3 x \\right)}$.\n3) So $f'(x) = - 6 x^{3} \\sin{\\left(3 x \\right)} + 9 x^{2} \\sin{\\left(3 x \\right)} + 6 x^{2} \\cos{\\left(3 x \\right)} + 2 x e^{x} + 3 x \\sin{\\left(3 x \\right)} - 6 x \\cos{\\left(3 x \\right)} - 4 e^{x}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 3 x \\cos{\\left(3 x \\right)}) = 9 x \\sin{\\left(3 x \\right)} - 3 \\cos{\\left(3 x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 3 x \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x \\right)} + 2 \\left(x - 3\\right) e^{x} + 3 \\left(2 x^{2} - 2 x - 1\\right) \\cos{\\left(3 x \\right)} + 2 e^{x}$.\nFinal answer: $- 3 x \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x \\right)} + 2 \\left(x - 3\\right) e^{x} + 3 \\left(2 x^{2} - 2 x - 1\\right) \\cos{\\left(3 x \\right)} + 2 e^{x}$", "final_answer": "- 3 x \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x \\right)} + 2 \\left(x - 3\\right) e^{x} + 3 \\left(2 x^{2} - 2 x - 1\\right) \\cos{\\left(3 x \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c7ff0de11a67f9f0", "problem_id": "dd39dc3ef5d18978", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + \\frac{x}{2} - 1$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$\\frac{1}{2}$, c=$-1$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{49}{4}$, so √Δ = $\\frac{7}{2}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{1}{2}$.\n7) x2 = $- \\frac{2}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{1}{2}, - \\frac{2}{3}\\}$", "final_answer": "x \\in \\{\\frac{1}{2}, - \\frac{2}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b291cf0b97456601", "problem_id": "c853195e56904c17", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡1 (mod 8), x≡5 (mod 9). Give the solution modulo 72.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 72.\n3) Compute the CRT solution to get x ≡ 41 (mod 72).\nFinal answer: $x\\equiv 41\\pmod{72}$", "final_answer": "x\\equiv 41\\pmod{72}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8b015231f245a24a", "problem_id": "c853195e56904c17", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡1 (mod 8), x≡5 (mod 9). Give the solution modulo 72.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 72.\nCompute the CRT solution to get x ≡ 41 (mod 72).\nCheck: 41 mod 8=1 and mod 9=5.\nFinal answer: $x\\equiv 41\\pmod{72}$", "final_answer": "x\\equiv 41\\pmod{72}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aac2e78645f8c3c3", "problem_id": "737bdfe6c3054e15", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} x^{3} - 2 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $x^{3} - 2 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $- 6 \\pi^{3} - 4 + \\frac{3 \\pi^{2}}{2} + 9 \\pi + \\frac{15 \\pi^{4}}{4}$.\n4) Simplify the result.\nFinal answer: $- 6\\cdot \\pi^{3} - 4 + \\frac{3\\cdot \\pi^{2}}{2} + 9\\cdot \\pi + \\frac{15\\cdot \\pi^{4}}{4}$", "final_answer": "- 6\\cdot \\pi^{3} - 4 + \\frac{3\\cdot \\pi^{2}}{2} + 9\\cdot \\pi + \\frac{15\\cdot \\pi^{4}}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c560460f381a708a", "problem_id": "737bdfe6c3054e15", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} x^{3} - 2 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $x^{3} - 2 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $- 6 \\pi^{3} - 4 + \\frac{3 \\pi^{2}}{2} + 9 \\pi + \\frac{15 \\pi^{4}}{4}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- 6\\cdot \\pi^{3} - 4 + \\frac{3\\cdot \\pi^{2}}{2} + 9\\cdot \\pi + \\frac{15\\cdot \\pi^{4}}{4}$", "final_answer": "- 6\\cdot \\pi^{3} - 4 + \\frac{3\\cdot \\pi^{2}}{2} + 9\\cdot \\pi + \\frac{15\\cdot \\pi^{4}}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "82990b40f8c814f1", "problem_id": "0a32c90434213268", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 18 red balls and 19 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(37,11).\n3) Favorable samples: choose k reds and n−k blues: C(18,5)·C(19,6).\n4) Compute totals: C(37,11) = 854992152, and favorable = 232466976.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{9044}{33263}$", "final_answer": "\\frac{9044}{33263}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "007d067f6b137b89", "problem_id": "0a32c90434213268", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 18 red balls and 19 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(37,11).\nFavorable samples: choose k reds and n−k blues: C(18,5)·C(19,6).\nCompute totals: C(37,11) = 854992152, and favorable = 232466976.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{9044}{33263}$.\nFinal answer: $\\frac{9044}{33263}$", "final_answer": "\\frac{9044}{33263}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2968e9708d977443", "problem_id": "fc9175b190aa08b6", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$- 3 x$ = $-3$\n$6 x + 2 y$ = $2$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $6$: $- 18 x$ = $-18$.\n3) Multiply the second equation by $-3$: $- 18 x - 6 y$ = $-6$.\n4) Subtract to eliminate x: $6 y$ = $-12$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $1$.\nFinal answer: $(x,y)=\\left(1, -2\\right)$", "final_answer": "(x,y)=\\left(1, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1f49ecadb033936e", "problem_id": "fc9175b190aa08b6", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$- 3 x$ = $-3$\n$6 x + 2 y$ = $2$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $6$: $- 18 x$ = $-18$.\nMultiply the second equation by $-3$: $- 18 x - 6 y$ = $-6$.\nSubtract to eliminate x: $6 y$ = $-12$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $1$.\nCheck: plug (x,y)=(1,-2) into both equations.\nFinal answer: $(x,y)=\\left(1, -2\\right)$", "final_answer": "(x,y)=\\left(1, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e7a6135960852e1c", "problem_id": "adfe202ac964bfdc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{4} + 3 x^{3} + 3 x^{2} - 3 x + 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(3 x^{3}) = 9 x^{3}$.\n3) So $f'(x) = 25 x^{3} + 6 x - 3$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(3 x^{3}) = 9 x^{2}$.\nTherefore the correct derivative is $f'(x) = 16 x^{3} + 9 x^{2} + 6 x - 3$.\nFinal answer: $16 x^{3} + 9 x^{2} + 6 x - 3$", "final_answer": "16 x^{3} + 9 x^{2} + 6 x - 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7d4eadd3ae593a5e", "problem_id": "9140892983ef172d", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 7 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=7, k=3, p=$\\frac{1}{6}$ and simplify.\n4) Here C(7,3) = 35.\nFinal answer: $\\frac{21875}{279936}$", "final_answer": "\\frac{21875}{279936}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8f6af936ec79fb05", "problem_id": "faa9d72d2cb45121", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} x^{2} - 2$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{3} + 2 x^{2} - 2 x$ and a = -2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $3 x^{2} + 4 x - 2$.\n5) Evaluate at a: p'(-2) = $2$.\nFinal answer: $2$", "final_answer": "2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b8a6b421a91fb19f", "problem_id": "faa9d72d2cb45121", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} x^{2} - 2$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $x^{3} + 2 x^{2} - 2 x$ and a = -2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $2$", "final_answer": "2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dacbefbc42fa0cd0", "problem_id": "8bf568c4e6eaadff", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$- x - 3 y$ = $16$\n$- x + y$ = $-4$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $x + 3 y$ = $-16$.\n3) Multiply the second equation by $-1$: $x - y$ = $4$.\n4) Subtract to eliminate x: $4 y$ = $-20$.\n5) Solve for y: y = $-5$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, -5\\right)$", "final_answer": "(x,y)=\\left(-1, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "86234e9042be6602", "problem_id": "8bf568c4e6eaadff", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$- x - 3 y$ = $16$\n$- x + y$ = $-4$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $x + 3 y$ = $-16$.\nMultiply the second equation by $-1$: $x - y$ = $4$.\nSubtract to eliminate x: $4 y$ = $-20$.\nSolve for y: y = $-5$.\nSubstitute back to find x: x = $-1$.\nCheck: plug (x,y)=(-1,-5) into both equations.\nFinal answer: $(x,y)=\\left(-1, -5\\right)$", "final_answer": "(x,y)=\\left(-1, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c7f56323a4edf263", "problem_id": "0dce5ef0979aeddc", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{5} x - 2 x - \\frac{7}{9} + \\frac{7 \\sqrt{5}}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\sqrt{5} - 2$, c=$- \\frac{7}{9} + \\frac{7 \\sqrt{5}}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{109}{9} - \\frac{16 \\sqrt{5}}{3}$, so √Δ = $\\sqrt{\\frac{109}{9} - \\frac{16 \\sqrt{5}}{3}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{109 - 48 \\sqrt{5}}}{6} + 1 + \\frac{\\sqrt{5}}{2}$.\n7) x2 = $- \\frac{\\sqrt{109 - 48 \\sqrt{5}}}{6} + 1 + \\frac{\\sqrt{5}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{109 - 48\\cdot \\sqrt{5}}}{6} + 1 + \\frac{\\sqrt{5}}{2}, - \\frac{\\sqrt{109 - 48\\cdot \\sqrt{5}}}{6} + 1 + \\frac{\\sqrt{5}}{2}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{109 - 48\\cdot \\sqrt{5}}}{6} + 1 + \\frac{\\sqrt{5}}{2}, - \\frac{\\sqrt{109 - 48\\cdot \\sqrt{5}}}{6} + 1 + \\frac{\\sqrt{5}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ab46ce18fc1b20c", "problem_id": "0dce5ef0979aeddc", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{5} x - 2 x - \\frac{7}{9} + \\frac{7 \\sqrt{5}}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\sqrt{5} - 2$, c=$- \\frac{7}{9} + \\frac{7 \\sqrt{5}}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{109}{9} - \\frac{16 \\sqrt{5}}{3}$, so √Δ = $\\sqrt{\\frac{109}{9} - \\frac{16 \\sqrt{5}}{3}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{109 - 48 \\sqrt{5}}}{6} + 1 + \\frac{\\sqrt{5}}{2}$.\nx2 = $- \\frac{\\sqrt{109 - 48 \\sqrt{5}}}{6} + 1 + \\frac{\\sqrt{5}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{109 - 48\\cdot \\sqrt{5}}}{6} + 1 + \\frac{\\sqrt{5}}{2}, - \\frac{\\sqrt{109 - 48\\cdot \\sqrt{5}}}{6} + 1 + \\frac{\\sqrt{5}}{2}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{109 - 48\\cdot \\sqrt{5}}}{6} + 1 + \\frac{\\sqrt{5}}{2}, - \\frac{\\sqrt{109 - 48\\cdot \\sqrt{5}}}{6} + 1 + \\frac{\\sqrt{5}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7b60f88bdb6d5276", "problem_id": "7c048c3ccb4265e4", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$4 x + 2 y$ = $16$\n$- 4 x$ = $-8$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $- 16 x - 8 y$ = $-64$.\n3) Multiply the second equation by $4$: $- 16 x$ = $-32$.\n4) Subtract to eliminate x: $- 8 y$ = $-32$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $2$.\nFinal answer: $(x,y)=\\left(2, 4\\right)$", "final_answer": "(x,y)=\\left(2, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d525582d6b8a7587", "problem_id": "2024877873f6879b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - 3 x^{3} + 8 x^{2} - 19 x + 35$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{3} + 8 x^{2} - 20 x + 35.\n3) Then the limit is 131.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 129.\nFinal answer: $129$", "final_answer": "129", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "cf20d40cd0eb9f5f", "problem_id": "92418557c132824c", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 4 x - 160$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} + 4 x - 160$ = $2 \\left(x - 8\\right) \\left(x + 10\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $8$.\n5) From the second factor: x = $-10$.\nFinal answer: $x \\in \\{8, -10\\}$", "final_answer": "x \\in \\{8, -10\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8559ba6e485631b3", "problem_id": "fb66368f2415621d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(- 3 x^{3} - x + 1\\right) \\cos{\\left(- x^{2} + x + 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\cos{\\left(\\left(- x^{2} + x\\right) + 2 \\right)}) = - \\sin{\\left(- x^{2} + x + 2 \\right)}$.\n3) So $f'(x) = - 6 x^{4} \\sin{\\left(- x^{2} + x + 2 \\right)} + 3 x^{3} \\sin{\\left(- x^{2} + x + 2 \\right)} - 2 x^{2} \\sin{\\left(- x^{2} + x + 2 \\right)} - 9 x^{2} \\cos{\\left(\\left(- x^{2} + x\\right) + 2 \\right)} + 2 x e^{x} + x \\sin{\\left(- x^{2} + x + 2 \\right)} + 6 e^{x} - \\sin{\\left(- x^{2} + x + 2 \\right)} - \\cos{\\left(\\left(- x^{2} + x\\right) + 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\cos{\\left(\\left(- x^{2} + x\\right) + 2 \\right)}) = - \\left(1 - 2 x\\right) \\sin{\\left(- x^{2} + x + 2 \\right)}$.\nTherefore the correct derivative is $f'(x) = - \\left(1 - 2 x\\right) \\left(x \\left(- 3 x^{2} - 1\\right) + 1\\right) \\sin{\\left(- x^{2} + x + 2 \\right)} + 2 \\left(x + 2\\right) e^{x} + \\left(- 9 x^{2} - 1\\right) \\cos{\\left(\\left(- x^{2} + x\\right) + 2 \\right)} + 2 e^{x}$.\nFinal answer: $- \\left(1 - 2 x\\right) \\left(x \\left(- 3 x^{2} - 1\\right) + 1\\right) \\sin{\\left(- x^{2} + x + 2 \\right)} + 2 \\left(x + 2\\right) e^{x} + \\left(- 9 x^{2} - 1\\right) \\cos{\\left(\\left(- x^{2} + x\\right) + 2 \\right)} + 2 e^{x}$", "final_answer": "- \\left(1 - 2 x\\right) \\left(x \\left(- 3 x^{2} - 1\\right) + 1\\right) \\sin{\\left(- x^{2} + x + 2 \\right)} + 2 \\left(x + 2\\right) e^{x} + \\left(- 9 x^{2} - 1\\right) \\cos{\\left(\\left(- x^{2} + x\\right) + 2 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3e831a40c36cb73d", "problem_id": "bbc14bb32256766d", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{5} x - 2 x - 3 - \\sqrt{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\sqrt{5} - 2$, c=$-3 - \\sqrt{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $8 \\sqrt{5} + 21$, so √Δ = $\\sqrt{8 \\sqrt{5} + 21}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $1 + \\frac{\\sqrt{5}}{2} + \\frac{\\sqrt{8 \\sqrt{5} + 21}}{2}$.\n7) x2 = $- \\frac{\\sqrt{8 \\sqrt{5} + 21}}{2} + 1 + \\frac{\\sqrt{5}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{1 + \\frac{\\sqrt{5}}{2} + \\frac{\\sqrt{8\\cdot \\sqrt{5} + 21}}{2}, - \\frac{\\sqrt{8\\cdot \\sqrt{5} + 21}}{2} + 1 + \\frac{\\sqrt{5}}{2}\\}$", "final_answer": "x \\in \\{1 + \\frac{\\sqrt{5}}{2} + \\frac{\\sqrt{8\\cdot \\sqrt{5} + 21}}{2}, - \\frac{\\sqrt{8\\cdot \\sqrt{5} + 21}}{2} + 1 + \\frac{\\sqrt{5}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c20f7091ab09f1e5", "problem_id": "bbc14bb32256766d", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{5} x - 2 x - 3 - \\sqrt{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\sqrt{5} - 2$, c=$-3 - \\sqrt{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $8 \\sqrt{5} + 21$, so √Δ = $\\sqrt{8 \\sqrt{5} + 21}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $1 + \\frac{\\sqrt{5}}{2} + \\frac{\\sqrt{8 \\sqrt{5} + 21}}{2}$.\nx2 = $- \\frac{\\sqrt{8 \\sqrt{5} + 21}}{2} + 1 + \\frac{\\sqrt{5}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{1 + \\frac{\\sqrt{5}}{2} + \\frac{\\sqrt{8\\cdot \\sqrt{5} + 21}}{2}, - \\frac{\\sqrt{8\\cdot \\sqrt{5} + 21}}{2} + 1 + \\frac{\\sqrt{5}}{2}\\}$", "final_answer": "x \\in \\{1 + \\frac{\\sqrt{5}}{2} + \\frac{\\sqrt{8\\cdot \\sqrt{5} + 21}}{2}, - \\frac{\\sqrt{8\\cdot \\sqrt{5} + 21}}{2} + 1 + \\frac{\\sqrt{5}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "24da2d387bae03c5", "problem_id": "0e0f8151ec74f163", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{10} x + \\frac{4 x}{3} - \\frac{7 \\sqrt{10}}{3} - \\frac{7}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{4}{3} - \\sqrt{10}$, c=$- \\frac{7 \\sqrt{10}}{3} - \\frac{7}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{20 \\sqrt{10}}{3} + \\frac{190}{9}$, so √Δ = $\\sqrt{\\frac{20 \\sqrt{10}}{3} + \\frac{190}{9}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{2}{3} + \\frac{\\sqrt{10}}{2} + \\frac{\\sqrt{60 \\sqrt{10} + 190}}{6}$.\n7) x2 = $- \\frac{\\sqrt{60 \\sqrt{10} + 190}}{6} - \\frac{2}{3} + \\frac{\\sqrt{10}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{2}{3} + \\frac{\\sqrt{10}}{2} + \\frac{\\sqrt{60\\cdot \\sqrt{10} + 190}}{6}, - \\frac{\\sqrt{60\\cdot \\sqrt{10} + 190}}{6} - \\frac{2}{3} + \\frac{\\sqrt{10}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{2}{3} + \\frac{\\sqrt{10}}{2} + \\frac{\\sqrt{60\\cdot \\sqrt{10} + 190}}{6}, - \\frac{\\sqrt{60\\cdot \\sqrt{10} + 190}}{6} - \\frac{2}{3} + \\frac{\\sqrt{10}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cd0a5ca258543e85", "problem_id": "0e0f8151ec74f163", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{10} x + \\frac{4 x}{3} - \\frac{7 \\sqrt{10}}{3} - \\frac{7}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{4}{3} - \\sqrt{10}$, c=$- \\frac{7 \\sqrt{10}}{3} - \\frac{7}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{20 \\sqrt{10}}{3} + \\frac{190}{9}$, so √Δ = $\\sqrt{\\frac{20 \\sqrt{10}}{3} + \\frac{190}{9}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{2}{3} + \\frac{\\sqrt{10}}{2} + \\frac{\\sqrt{60 \\sqrt{10} + 190}}{6}$.\nx2 = $- \\frac{\\sqrt{60 \\sqrt{10} + 190}}{6} - \\frac{2}{3} + \\frac{\\sqrt{10}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{2}{3} + \\frac{\\sqrt{10}}{2} + \\frac{\\sqrt{60\\cdot \\sqrt{10} + 190}}{6}, - \\frac{\\sqrt{60\\cdot \\sqrt{10} + 190}}{6} - \\frac{2}{3} + \\frac{\\sqrt{10}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{2}{3} + \\frac{\\sqrt{10}}{2} + \\frac{\\sqrt{60\\cdot \\sqrt{10} + 190}}{6}, - \\frac{\\sqrt{60\\cdot \\sqrt{10} + 190}}{6} - \\frac{2}{3} + \\frac{\\sqrt{10}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "82ada980654bd63c", "problem_id": "293e42be639621f1", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Worker A finishes a job in 6 hours and Worker B finishes it in 10 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/6 job/hour. B's rate = 1/10 job/hour.\nCombined rate = 1/6 + 1/10 = 4/15 job/hour.\nTime = 1 ÷ (combined rate) = 15/4 hours.\nCheck: (combined rate)×(time) = 4/15×15/4 = 1.\nFinal answer: $\\frac{15}{4}$", "final_answer": "\\frac{15}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "55de2ab967e54e56", "problem_id": "293e42be639621f1", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Worker A finishes a job in 6 hours and Worker B finishes it in 10 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/6 job/hour. B's rate = 1/10 job/hour.\n2) Combined rate = 1/6 + 1/10 = 4/15 job/hour.\n3) Time = 1 ÷ (combined rate) = 15/4 hours.\nFinal answer: $\\frac{15}{4}$", "final_answer": "\\frac{15}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6fe0cd7e1d50ed54", "problem_id": "b762b9881ef35937", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_9=30$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_9=12.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=12 and k=9.\n4) Compute C(20,8) = 125970.\nFinal answer: $125970$", "final_answer": "125970", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d6a51e7ee7ce127", "problem_id": "b762b9881ef35937", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_9=30$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_9=12.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=12 and k=9.\nCompute C(20,8) = 125970.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $125970$", "final_answer": "125970", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0f7c73bebc57bfca", "problem_id": "5ac586fa41c5f207", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $808 in the ratio 6:2. How much does each receive?", "output": "1) Total ratio parts = 6+2 = 8.\n2) Each part = 808 ÷ 8 = 101.\n3) First share = 6 × 101 = 606.\n4) Second share = 2 × 101 = 202.\nFinal answer: $202$", "final_answer": "202", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7c9466c17944f3b8", "problem_id": "5ac586fa41c5f207", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Two friends split $808 in the ratio 6:2. How much does each receive?", "output": "Total ratio parts = 6+2 = 8.\nEach part = 808 ÷ 8 = 101.\nFirst share = 6 × 101 = 606.\nSecond share = 2 × 101 = 202.\nCheck: 606+202=808.\nFinal answer: $202$", "final_answer": "202", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "50a70adbb10932bc", "problem_id": "046a9d74b334f76e", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 11 times, what is the probability of exactly 2 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=11, k=2, p=$\\frac{1}{4}$ and simplify.\nHere C(11,2) = 55.\nCheck: expression simplifies to $\\frac{1082565}{4194304}$.\nFinal answer: $\\frac{1082565}{4194304}$", "final_answer": "\\frac{1082565}{4194304}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3fe2d2f8de535110", "problem_id": "046a9d74b334f76e", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 11 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=2, p=$\\frac{1}{4}$ and simplify.\n4) Here C(11,2) = 55.\nFinal answer: $\\frac{1082565}{4194304}$", "final_answer": "\\frac{1082565}{4194304}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "64dbd14dbe3cd369", "problem_id": "70ffec10349a23f3", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡1 (mod 5), x≡0 (mod 13). Give the solution modulo 65.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 65.\nCompute the CRT solution to get x ≡ 26 (mod 65).\nCheck: 26 mod 5=1 and mod 13=0.\nFinal answer: $x\\equiv 26\\pmod{65}$", "final_answer": "x\\equiv 26\\pmod{65}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "587a79b069ea8b8a", "problem_id": "70ffec10349a23f3", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡1 (mod 5), x≡0 (mod 13). Give the solution modulo 65.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 65.\n3) Compute the CRT solution to get x ≡ 26 (mod 65).\nFinal answer: $x\\equiv 26\\pmod{65}$", "final_answer": "x\\equiv 26\\pmod{65}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2e77287d953c35ee", "problem_id": "7b3dabea4cafeb75", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} x^{4} + 4 x^{3} + x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $x^{4} + 4 x^{3} + x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\n3) Subtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{\\pi^{2}}{2} + \\frac{\\pi^{3}}{24} + \\frac{\\pi^{5}}{160} + \\frac{3 \\pi}{2} + 5 + \\frac{\\pi^{4}}{16}$.\n4) Simplify the result.\nFinal answer: $- \\frac{\\pi^{2}}{2} + \\frac{\\pi^{3}}{24} + \\frac{\\pi^{5}}{160} + \\frac{3\\cdot \\pi}{2} + 5 + \\frac{\\pi^{4}}{16}$", "final_answer": "- \\frac{\\pi^{2}}{2} + \\frac{\\pi^{3}}{24} + \\frac{\\pi^{5}}{160} + \\frac{3\\cdot \\pi}{2} + 5 + \\frac{\\pi^{4}}{16}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9a3501f8290df3a2", "problem_id": "7b3dabea4cafeb75", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} x^{4} + 4 x^{3} + x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $x^{4} + 4 x^{3} + x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\nSubtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{\\pi^{2}}{2} + \\frac{\\pi^{3}}{24} + \\frac{\\pi^{5}}{160} + \\frac{3 \\pi}{2} + 5 + \\frac{\\pi^{4}}{16}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{\\pi^{2}}{2} + \\frac{\\pi^{3}}{24} + \\frac{\\pi^{5}}{160} + \\frac{3\\cdot \\pi}{2} + 5 + \\frac{\\pi^{4}}{16}$", "final_answer": "- \\frac{\\pi^{2}}{2} + \\frac{\\pi^{3}}{24} + \\frac{\\pi^{5}}{160} + \\frac{3\\cdot \\pi}{2} + 5 + \\frac{\\pi^{4}}{16}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d5ce4275f9d4e6b2", "problem_id": "89fd47801e4cadbf", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A store sells a backpack for $21. It is discounted by 5%. You buy 7 backpacks. What is the total cost?", "output": "1) Discounted price = 21 × (1 − 5/100) = 21 × 95/100 = 399/20.\n2) Subtotal = 7 × 399/20 = 2793/20.\nFinal answer: $\\frac{2793}{20}$", "final_answer": "\\frac{2793}{20}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "738328a4a2269b84", "problem_id": "89fd47801e4cadbf", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $21. It is discounted by 5%. You buy 7 backpacks. What is the total cost?", "output": "Discounted price = 21 × (1 − 5/100) = 21 × 95/100 = 399/20.\nSubtotal = 7 × 399/20 = 2793/20.\nCheck: recomputing the formula yields 2793/20.\nFinal answer: $\\frac{2793}{20}$", "final_answer": "\\frac{2793}{20}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "62d288a859544484", "problem_id": "7172cada9d361ab2", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the congruences: x≡3 (mod 4), x≡5 (mod 11). Give the solution modulo 44.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 44.\n3) Compute the CRT solution to get x ≡ 27 (mod 44).\nFinal answer: $x\\equiv 27\\pmod{44}$", "final_answer": "x\\equiv 27\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "15ffec19188b9b21", "problem_id": "7172cada9d361ab2", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡3 (mod 4), x≡5 (mod 11). Give the solution modulo 44.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 44.\nCompute the CRT solution to get x ≡ 27 (mod 44).\nCheck: 27 mod 4=3 and mod 11=5.\nFinal answer: $x\\equiv 27\\pmod{44}$", "final_answer": "x\\equiv 27\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f9cd222ecdc33aad", "problem_id": "147c7235f1be50db", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "An urn has 21 red balls and 29 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(50,7).\nFavorable samples: choose k reds and n−k blues: C(21,1)·C(29,6).\nFinal answer: $\\frac{23751}{237820}$", "final_answer": "\\frac{23751}{237820}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f8bcb24838b04252", "problem_id": "147c7235f1be50db", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 21 red balls and 29 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(50,7).\nFavorable samples: choose k reds and n−k blues: C(21,1)·C(29,6).\nCompute totals: C(50,7) = 99884400, and favorable = 9975420.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{23751}{237820}$.\nFinal answer: $\\frac{23751}{237820}$", "final_answer": "\\frac{23751}{237820}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e2a24cdee9562d4d", "problem_id": "ea915c96a1b41b2d", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡3 (mod 8), x≡4 (mod 5). Give the solution modulo 40.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 40.\n3) Compute the CRT solution to get x ≡ 19 (mod 40).\nFinal answer: $x\\equiv 19\\pmod{40}$", "final_answer": "x\\equiv 19\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "504e23a982f82266", "problem_id": "ea915c96a1b41b2d", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡3 (mod 8), x≡4 (mod 5). Give the solution modulo 40.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 40.\nCompute the CRT solution to get x ≡ 19 (mod 40).\nCheck: 19 mod 8=3 and mod 5=4.\nFinal answer: $x\\equiv 19\\pmod{40}$", "final_answer": "x\\equiv 19\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d1bb4062e55a1390", "problem_id": "03058dee2662ab8a", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{7} x + \\frac{5 x}{3} - \\frac{2 \\sqrt{7}}{3} + \\frac{2}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{5}{3} - \\sqrt{7}$, c=$\\frac{2}{3} - \\frac{2 \\sqrt{7}}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{64}{9} - \\frac{2 \\sqrt{7}}{3}$, so √Δ = $\\sqrt{\\frac{64}{9} - \\frac{2 \\sqrt{7}}{3}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{5}{6} + \\frac{\\sqrt{64 - 6 \\sqrt{7}}}{6} + \\frac{\\sqrt{7}}{2}$.\n7) x2 = $- \\frac{\\sqrt{64 - 6 \\sqrt{7}}}{6} - \\frac{5}{6} + \\frac{\\sqrt{7}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{5}{6} + \\frac{\\sqrt{64 - 6\\cdot \\sqrt{7}}}{6} + \\frac{\\sqrt{7}}{2}, - \\frac{\\sqrt{64 - 6\\cdot \\sqrt{7}}}{6} - \\frac{5}{6} + \\frac{\\sqrt{7}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{5}{6} + \\frac{\\sqrt{64 - 6\\cdot \\sqrt{7}}}{6} + \\frac{\\sqrt{7}}{2}, - \\frac{\\sqrt{64 - 6\\cdot \\sqrt{7}}}{6} - \\frac{5}{6} + \\frac{\\sqrt{7}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6e06bfec61df9920", "problem_id": "03058dee2662ab8a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{7} x + \\frac{5 x}{3} - \\frac{2 \\sqrt{7}}{3} + \\frac{2}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{5}{3} - \\sqrt{7}$, c=$\\frac{2}{3} - \\frac{2 \\sqrt{7}}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{64}{9} - \\frac{2 \\sqrt{7}}{3}$, so √Δ = $\\sqrt{\\frac{64}{9} - \\frac{2 \\sqrt{7}}{3}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{5}{6} + \\frac{\\sqrt{64 - 6 \\sqrt{7}}}{6} + \\frac{\\sqrt{7}}{2}$.\nx2 = $- \\frac{\\sqrt{64 - 6 \\sqrt{7}}}{6} - \\frac{5}{6} + \\frac{\\sqrt{7}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{5}{6} + \\frac{\\sqrt{64 - 6\\cdot \\sqrt{7}}}{6} + \\frac{\\sqrt{7}}{2}, - \\frac{\\sqrt{64 - 6\\cdot \\sqrt{7}}}{6} - \\frac{5}{6} + \\frac{\\sqrt{7}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{5}{6} + \\frac{\\sqrt{64 - 6\\cdot \\sqrt{7}}}{6} + \\frac{\\sqrt{7}}{2}, - \\frac{\\sqrt{64 - 6\\cdot \\sqrt{7}}}{6} - \\frac{5}{6} + \\frac{\\sqrt{7}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "339fc8d4eb4ad7b4", "problem_id": "ce763514af7477af", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 63 miles per hour for 1 hours, then at 68 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 63 × 1 = 63 miles.\nSecond leg distance = 68 × 3 = 204 miles.\nTotal distance = 63 + 204 = 267 miles.\nCheck: adding both legs gives 267.\nFinal answer: $267$", "final_answer": "267", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "501e914c2b77eb9a", "problem_id": "ce763514af7477af", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A cyclist rides at 63 miles per hour for 1 hours, then at 68 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 63 × 1 = 63 miles.\n2) Second leg distance = 68 × 3 = 204 miles.\n3) Total distance = 63 + 204 = 267 miles.\nFinal answer: $267$", "final_answer": "267", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ab502d76fcaa71c8", "problem_id": "25cd154ed2101e25", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the equation of the tangent line to $y=x^{3} + 3 x^{2} + 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=2$.", "output": "1) Differentiate: f'(x) = $3 x^{2} + 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 3$.\n2) Slope at x=2: m = f'(2) = $\\frac{e}{2} - 2 \\sin{\\left(4 \\right)} + 27$.\n3) Point on curve: (2, f(2)) = (2, $\\cos{\\left(4 \\right)} + e + 24$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(4 \\right)} + e + 24 = \\frac{e}{2} - 2 \\sin{\\left(4 \\right)} + 27(x − 2).\n6) Optionally expand: y = $\\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 27 x - 30 + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$.\nFinal answer: $y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 27\\cdot x - 30 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$", "final_answer": "y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 27\\cdot x - 30 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "45f0989e2ca83d82", "problem_id": "af72c159de6e5f1b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x + 2$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to x + 2.\n3) Then the limit is 4.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 6.\nFinal answer: $6$", "final_answer": "6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ec49e07d9e831cde", "problem_id": "1af1254f1dbc7854", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 12 x - 135$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} - 12 x - 135$ = $3 \\left(x - 9\\right) \\left(x + 5\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $9$.\n5) From the second factor: x = $-5$.\nFinal answer: $x \\in \\{9, -5\\}$", "final_answer": "x \\in \\{9, -5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a804a7b92078d67d", "problem_id": "1af1254f1dbc7854", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 12 x - 135$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} - 12 x - 135$ = $3 \\left(x - 9\\right) \\left(x + 5\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $9$.\nFrom the second factor: x = $-5$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{9, -5\\}$", "final_answer": "x \\in \\{9, -5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "66854fd2714aa7ab", "problem_id": "a104d810167684ee", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 22 times, what is the probability of exactly 14 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=22, k=14, p=$\\frac{4}{5}$ and simplify.\n4) Here C(22,14) = 319770.\nFinal answer: $\\frac{17167521153024}{476837158203125}$", "final_answer": "\\frac{17167521153024}{476837158203125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "467166fd3ac0c36a", "problem_id": "a104d810167684ee", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 22 times, what is the probability of exactly 14 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=22, k=14, p=$\\frac{4}{5}$ and simplify.\nHere C(22,14) = 319770.\nCheck: expression simplifies to $\\frac{17167521153024}{476837158203125}$.\nFinal answer: $\\frac{17167521153024}{476837158203125}$", "final_answer": "\\frac{17167521153024}{476837158203125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "88fa848b660480e1", "problem_id": "afbdcc8bfafc1955", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$- x - y$ = $2$\n$4 x - 2 y$ = $-8$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $- 4 x - 4 y$ = $8$.\nMultiply the second equation by $-1$: $- 4 x + 2 y$ = $8$.\nSubtract to eliminate x: $- 6 y$ = $0$.\nSolve for y: y = $0$.\nSubstitute back to find x: x = $-2$.\nCheck: plug (x,y)=(-2,0) into both equations.\nFinal answer: $(x,y)=\\left(-2, 0\\right)$", "final_answer": "(x,y)=\\left(-2, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e2e8a05ed64d3617", "problem_id": "afbdcc8bfafc1955", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$- x - y$ = $2$\n$4 x - 2 y$ = $-8$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $- 4 x - 4 y$ = $8$.\n3) Multiply the second equation by $-1$: $- 4 x + 2 y$ = $8$.\n4) Subtract to eliminate x: $- 6 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, 0\\right)$", "final_answer": "(x,y)=\\left(-2, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c23095f890720aaa", "problem_id": "281e87c996e31af6", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 61 miles per hour for 1 hours, then at 50 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 61 × 1 = 61 miles.\nSecond leg distance = 50 × 2 = 100 miles.\nTotal distance = 61 + 100 = 161 miles.\nCheck: adding both legs gives 161.\nFinal answer: $161$", "final_answer": "161", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b7a9b1801ead20cd", "problem_id": "281e87c996e31af6", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A cyclist rides at 61 miles per hour for 1 hours, then at 50 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 61 × 1 = 61 miles.\n2) Second leg distance = 50 × 2 = 100 miles.\n3) Total distance = 61 + 100 = 161 miles.\nFinal answer: $161$", "final_answer": "161", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d46990c695e90a52", "problem_id": "c6eb19d5c291a7ec", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 14 times, what is the probability of exactly 7 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=14, k=7, p=$\\frac{2}{5}$ and simplify.\n4) Here C(14,7) = 3432.\nFinal answer: $\\frac{960740352}{6103515625}$", "final_answer": "\\frac{960740352}{6103515625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e5a35b882e5e8187", "problem_id": "c6eb19d5c291a7ec", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 14 times, what is the probability of exactly 7 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=14, k=7, p=$\\frac{2}{5}$ and simplify.\nHere C(14,7) = 3432.\nCheck: expression simplifies to $\\frac{960740352}{6103515625}$.\nFinal answer: $\\frac{960740352}{6103515625}$", "final_answer": "\\frac{960740352}{6103515625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0a8d3851cd1a5bb9", "problem_id": "8d07be5f23752ba5", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 10 times, what is the probability of exactly 8 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=8, p=$\\frac{1}{2}$ and simplify.\n4) Here C(10,8) = 45.\nFinal answer: $\\frac{45}{1024}$", "final_answer": "\\frac{45}{1024}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c5e403db140dbd90", "problem_id": "8d07be5f23752ba5", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 10 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=8, p=$\\frac{1}{2}$ and simplify.\nHere C(10,8) = 45.\nCheck: expression simplifies to $\\frac{45}{1024}$.\nFinal answer: $\\frac{45}{1024}$", "final_answer": "\\frac{45}{1024}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "039ca6fdeba6ca1f", "problem_id": "658f0e8283559358", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{3}$. If it is flipped 18 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=18, k=5, p=$\\frac{1}{3}$ and simplify.\nHere C(18,5) = 8568.\nCheck: expression simplifies to $\\frac{7798784}{43046721}$.\nFinal answer: $\\frac{7798784}{43046721}$", "final_answer": "\\frac{7798784}{43046721}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "55e754658869a603", "problem_id": "658f0e8283559358", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{3}$. If it is flipped 18 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=18, k=5, p=$\\frac{1}{3}$ and simplify.\n4) Here C(18,5) = 8568.\nFinal answer: $\\frac{7798784}{43046721}$", "final_answer": "\\frac{7798784}{43046721}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8e141155428b964b", "problem_id": "d3faf64d698f0068", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} x + 5$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{2} + 2 x - 3$ and a = 3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $2 x + 2$.\n5) Evaluate at a: p'(3) = $8$.\nFinal answer: $8$", "final_answer": "8", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2891603dc26163e0", "problem_id": "d3faf64d698f0068", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} x + 5$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $x^{2} + 2 x - 3$ and a = 3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $2 x + 2$.\nEvaluate at a: p'(3) = $8$.\nCheck: the identity gives $8$.\nFinal answer: $8$", "final_answer": "8", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9673ceece9e10e58", "problem_id": "6b4a1eaabb3e71c0", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_6=29$?", "output": "Let y_i=x_i-3, then y_i≥0 and y_1+…+y_6=11.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=11 and k=6.\nFinal answer: $4368$", "final_answer": "4368", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0fdc6a8225343e4d", "problem_id": "6b4a1eaabb3e71c0", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_6=29$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_6=11.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=11 and k=6.\n4) Compute C(16,5) = 4368.\nFinal answer: $4368$", "final_answer": "4368", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "02d1e71f5ac0ab70", "problem_id": "a33e509c8260599e", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "An urn has 29 red balls and 12 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(41,10).\nFavorable samples: choose k reds and n−k blues: C(29,1)·C(12,9).\nFinal answer: $\\frac{145}{25479532}$", "final_answer": "\\frac{145}{25479532}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "34c6b01c22a340e7", "problem_id": "a33e509c8260599e", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 29 red balls and 12 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(41,10).\nFavorable samples: choose k reds and n−k blues: C(29,1)·C(12,9).\nCompute totals: C(41,10) = 1121099408, and favorable = 6380.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{145}{25479532}$.\nFinal answer: $\\frac{145}{25479532}$", "final_answer": "\\frac{145}{25479532}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f748579f27c6f6b9", "problem_id": "0d8d4ad00cd877fe", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{9 x}{2} - \\frac{9}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$- \\frac{9}{2}$, c=$- \\frac{9}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{225}{4}$, so √Δ = $\\frac{15}{2}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $3$.\n7) x2 = $- \\frac{3}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{3, - \\frac{3}{4}\\}$", "final_answer": "x \\in \\{3, - \\frac{3}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ccfd9af2094840b0", "problem_id": "0d8d4ad00cd877fe", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{9 x}{2} - \\frac{9}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$- \\frac{9}{2}$, c=$- \\frac{9}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{225}{4}$, so √Δ = $\\frac{15}{2}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $3$.\nx2 = $- \\frac{3}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{3, - \\frac{3}{4}\\}$", "final_answer": "x \\in \\{3, - \\frac{3}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8d2760e43e3fbda4", "problem_id": "96251958a899093c", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the congruences: x≡8 (mod 11), x≡1 (mod 10). Give the solution modulo 110.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 110.\n3) Compute the CRT solution to get x ≡ 41 (mod 110).\nFinal answer: $x\\equiv 41\\pmod{110}$", "final_answer": "x\\equiv 41\\pmod{110}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d5aaeb8fa9631fbd", "problem_id": "96251958a899093c", "style": "answer_only", "instruction": "Final answer only.", "input": "Solve the congruences: x≡8 (mod 11), x≡1 (mod 10). Give the solution modulo 110.", "output": "$x\\equiv 41\\pmod{110}$", "final_answer": "x\\equiv 41\\pmod{110}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3297a83a4d5067d2", "problem_id": "05c608361cb06927", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 8 x - 90$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} + 8 x - 90$ = $2 \\left(x - 5\\right) \\left(x + 9\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $5$.\n5) From the second factor: x = $-9$.\nFinal answer: $x \\in \\{5, -9\\}$", "final_answer": "x \\in \\{5, -9\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f24c1dd970db0040", "problem_id": "05c608361cb06927", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 8 x - 90$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} + 8 x - 90$ = $2 \\left(x - 5\\right) \\left(x + 9\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $5$.\nFrom the second factor: x = $-9$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{5, -9\\}$", "final_answer": "x \\in \\{5, -9\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "64710692a09a7d60", "problem_id": "6ba40e2dd8aeb918", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 8 red balls and 10 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(18,6).\n3) Favorable samples: choose k reds and n−k blues: C(8,2)·C(10,4).\n4) Compute totals: C(18,6) = 18564, and favorable = 5880.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{70}{221}$", "final_answer": "\\frac{70}{221}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4e5bcd491cf2587a", "problem_id": "1973eb501f26a2f2", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} + 2 x + \\left(x^{2} - x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{x^{2} - x + 7}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7}) = - \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 7}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{4 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{15 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{30 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{2 x^{2}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{2 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 15 x^{2} - 14 x + 49} - \\frac{14 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{28 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} - \\frac{40 x}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 15 x^{2} - 14 x + 49} + \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 2 x + 14} - \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 7} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7} + \\frac{49 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{98 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{34}{2 x^{4} - 4 x^{3} + 30 x^{2} - 28 x + 98} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7}) = - \\frac{x \\left(1 - 2 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(x^{2} - x\\right) + 7\\right)^{2}} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(x^{2} - x\\right) + 7\\right)} - \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 7}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(x - 1\\right) + 7\\right) \\left(- 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x - 1\\right) + 7\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(x - 1\\right) + 7\\right) \\left(- 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x - 1\\right) + 7\\right)^{2}}$", "final_answer": "\\frac{2 \\left(1 - 2 x\\right) \\left(- x^{2} + 2 x + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(x - 1\\right) + 7\\right) \\left(- 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x - 1\\right) + 7\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "1d97f419efafcc47", "problem_id": "a07697b46b7d9fa4", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$- 2 x + 4 y$ = $-24$\n$3 y$ = $-12$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $-2$: $- 6 y$ = $24$.\n4) Subtract to eliminate x: $6 y$ = $-24$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, -4\\right)$", "final_answer": "(x,y)=\\left(4, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "040c5e91b5d5be2c", "problem_id": "a07697b46b7d9fa4", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$- 2 x + 4 y$ = $-24$\n$3 y$ = $-12$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $-2$: $- 6 y$ = $24$.\nSubtract to eliminate x: $6 y$ = $-24$.\nSolve for y: y = $-4$.\nSubstitute back to find x: x = $4$.\nCheck: plug (x,y)=(4,-4) into both equations.\nFinal answer: $(x,y)=\\left(4, -4\\right)$", "final_answer": "(x,y)=\\left(4, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f19a33bb9c5acac7", "problem_id": "c04be937acd768a7", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_7=44$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_7=37.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=37 and k=7.\n4) Compute C(43,6) = 6096454.\nFinal answer: $6096454$", "final_answer": "6096454", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1a3167ef06f72d2b", "problem_id": "c04be937acd768a7", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_7=44$?", "output": "$6096454$", "final_answer": "6096454", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2695f76c8b78a862", "problem_id": "1e933fcf4883d391", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$- 4 x - 3 y$ = $-8$\n$x + 2 y$ = $-3$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $- 4 x - 3 y$ = $-8$.\n3) Multiply the second equation by $-4$: $- 4 x - 8 y$ = $12$.\n4) Subtract to eliminate x: $5 y$ = $-20$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, -4\\right)$", "final_answer": "(x,y)=\\left(5, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "acce6f36a100560b", "problem_id": "1e933fcf4883d391", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$- 4 x - 3 y$ = $-8$\n$x + 2 y$ = $-3$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $- 4 x - 3 y$ = $-8$.\nMultiply the second equation by $-4$: $- 4 x - 8 y$ = $12$.\nSubtract to eliminate x: $5 y$ = $-20$.\nSolve for y: y = $-4$.\nSubstitute back to find x: x = $5$.\nCheck: plug (x,y)=(5,-4) into both equations.\nFinal answer: $(x,y)=\\left(5, -4\\right)$", "final_answer": "(x,y)=\\left(5, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3adcd4a8bd59fc02", "problem_id": "bcffff676bac85c1", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{8 x}{3} + \\frac{16}{9}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{8}{3}$, c=$\\frac{16}{9}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $0$, so √Δ = $0$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{4}{3}$.\n7) x2 = $\\frac{4}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{4}{3}, \\frac{4}{3}\\}$", "final_answer": "x \\in \\{\\frac{4}{3}, \\frac{4}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "660cfc267f6fa874", "problem_id": "bcffff676bac85c1", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{8 x}{3} + \\frac{16}{9}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{8}{3}$, c=$\\frac{16}{9}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $0$, so √Δ = $0$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{4}{3}$.\nx2 = $\\frac{4}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{4}{3}, \\frac{4}{3}\\}$", "final_answer": "x \\in \\{\\frac{4}{3}, \\frac{4}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9e0bd0d7d4767f00", "problem_id": "ae1e25eaf7c0a431", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 20 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=20, k=3, p=$\\frac{1}{6}$ and simplify.\n4) Here C(20,3) = 1140.\nFinal answer: $\\frac{72479248046875}{304679870005248}$", "final_answer": "\\frac{72479248046875}{304679870005248}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a31c1c807cd40faa", "problem_id": "4130f56657689103", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(2 x^{2} - 3 x - 3\\right) \\cos{\\left(3 x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 3 \\cos{\\left(3 x + 3 \\right)}) = 3 \\sin{\\left(3 x + 3 \\right)}$.\n3) So $f'(x) = - 6 x^{2} \\sin{\\left(3 x + 3 \\right)} + x e^{x} + 9 x \\sin{\\left(3 x + 3 \\right)} + 4 x \\cos{\\left(3 x + 3 \\right)} - 2 e^{x} + 3 \\sin{\\left(3 x + 3 \\right)} - 3 \\cos{\\left(3 x + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 3 \\cos{\\left(3 x + 3 \\right)}) = 9 \\sin{\\left(3 x + 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x - 3\\right) e^{x} + \\left(4 x - 3\\right) \\cos{\\left(3 x + 3 \\right)} - 3 \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x + 3 \\right)} + e^{x}$.\nFinal answer: $\\left(x - 3\\right) e^{x} + \\left(4 x - 3\\right) \\cos{\\left(3 x + 3 \\right)} - 3 \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x + 3 \\right)} + e^{x}$", "final_answer": "\\left(x - 3\\right) e^{x} + \\left(4 x - 3\\right) \\cos{\\left(3 x + 3 \\right)} - 3 \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x + 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b84f3c714c622886", "problem_id": "1b094b9bfe77c602", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the linear system:\n$3 x + 2 y$ = $16$\n$2 x - 2 y$ = $-6$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $6 x + 4 y$ = $32$.\n3) Multiply the second equation by $3$: $6 x - 6 y$ = $-18$.\n4) Subtract to eliminate x: $10 y$ = $50$.\n5) Solve for y: y = $5$.\n6) Substitute back to find x: x = $2$.\nFinal answer: $(x,y)=\\left(2, 5\\right)$", "final_answer": "(x,y)=\\left(2, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7f48ff60ff02b1e3", "problem_id": "1b094b9bfe77c602", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the linear system:\n$3 x + 2 y$ = $16$\n$2 x - 2 y$ = $-6$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $2$: $6 x + 4 y$ = $32$.\nMultiply the second equation by $3$: $6 x - 6 y$ = $-18$.\nFinal answer: $(x,y)=\\left(2, 5\\right)$", "final_answer": "(x,y)=\\left(2, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b52cb0b4b87b5699", "problem_id": "01ae1cbcfa0f6cc0", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 21 times, what is the probability of exactly 11 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=21, k=11, p=$\\frac{1}{5}$ and simplify.\n4) Here C(21,11) = 352716.\nFinal answer: $\\frac{369849532416}{476837158203125}$", "final_answer": "\\frac{369849532416}{476837158203125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fe129df77b394d61", "problem_id": "01ae1cbcfa0f6cc0", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 21 times, what is the probability of exactly 11 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=21, k=11, p=$\\frac{1}{5}$ and simplify.\nHere C(21,11) = 352716.\nCheck: expression simplifies to $\\frac{369849532416}{476837158203125}$.\nFinal answer: $\\frac{369849532416}{476837158203125}$", "final_answer": "\\frac{369849532416}{476837158203125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "481ef709145e7a19", "problem_id": "ae5ebb9642461a0e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} 3 x^{3} - 2 x - 2$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} - 3 x - 2.\n3) Then the limit is -2.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: -1.\nFinal answer: $-1$", "final_answer": "-1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "48e91705e7b06a0e", "problem_id": "ea9e0bb1033befb8", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the multiplicative inverse of 28 modulo 43.", "output": "We want x such that 28x ≡ 1 (mod 43).\nUse extended Euclid to find s,t with 28s + 43t = 1.\nOne Bezout identity is: 28·(20) + 43·(-13) = 1.\nReduce s modulo 43: x ≡ 20 (mod 43).\nCheck: 28·20 mod 43 = 1.\nFinal answer: $x\\equiv 20\\pmod{43}$", "final_answer": "x\\equiv 20\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c9fc72513159a9cc", "problem_id": "42d39f5e704c25a4", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_6=22$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_6=10.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=10 and k=6.\n4) Compute C(15,5) = 3003.\nFinal answer: $3003$", "final_answer": "3003", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "49ad9a02cef46446", "problem_id": "42d39f5e704c25a4", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_6=22$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_6=10.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=10 and k=6.\nCompute C(15,5) = 3003.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $3003$", "final_answer": "3003", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ee7fcaac25357b8a", "problem_id": "7d5ddf75b128f6db", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Two friends split $2736 in the ratio 8:8. How much does each receive?", "output": "1) Total ratio parts = 8+8 = 16.\n2) Each part = 2736 ÷ 16 = 171.\n3) First share = 8 × 171 = 1368.\n4) Second share = 8 × 171 = 1368.\nFinal answer: $1368$", "final_answer": "1368", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "24508e7eb433475f", "problem_id": "7d5ddf75b128f6db", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Two friends split $2736 in the ratio 8:8. How much does each receive?", "output": "Total ratio parts = 8+8 = 16.\nEach part = 2736 ÷ 16 = 171.\nFirst share = 8 × 171 = 1368.\nSecond share = 8 × 171 = 1368.\nCheck: 1368+1368=2736.\nFinal answer: $1368$", "final_answer": "1368", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d247691323a23bb2", "problem_id": "dc41ff964f029ad6", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 23 red balls and 16 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(39,10).\n3) Favorable samples: choose k reds and n−k blues: C(23,2)·C(16,8).\n4) Compute totals: C(39,10) = 635745396, and favorable = 3256110.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{3795}{740962}$", "final_answer": "\\frac{3795}{740962}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "77664cbc7342bb49", "problem_id": "dc41ff964f029ad6", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "An urn has 23 red balls and 16 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(39,10).\nFavorable samples: choose k reds and n−k blues: C(23,2)·C(16,8).\nFinal answer: $\\frac{3795}{740962}$", "final_answer": "\\frac{3795}{740962}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ed00e03b94d05a68", "problem_id": "1b76e472115ebd3c", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Two friends split $612 in the ratio 9:8. How much does each receive?", "output": "1) Total ratio parts = 9+8 = 17.\n2) Each part = 612 ÷ 17 = 36.\n3) First share = 9 × 36 = 324.\n4) Second share = 8 × 36 = 288.\nFinal answer: $288$", "final_answer": "288", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f18e2cabd2619f1e", "problem_id": "1b76e472115ebd3c", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Two friends split $612 in the ratio 9:8. How much does each receive?", "output": "Total ratio parts = 9+8 = 17.\nEach part = 612 ÷ 17 = 36.\nFirst share = 9 × 36 = 324.\nFinal answer: $288$", "final_answer": "288", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a34b4438f23053d", "problem_id": "ac0202228275ae36", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the linear system:\n$x + 3 y$ = $-21$\n$- 5 x + 4 y$ = $10$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-5$: $- 5 x - 15 y$ = $105$.\n3) Multiply the second equation by $1$: $- 5 x + 4 y$ = $10$.\n4) Subtract to eliminate x: $- 19 y$ = $95$.\n5) Solve for y: y = $-5$.\n6) Substitute back to find x: x = $-6$.\nFinal answer: $(x,y)=\\left(-6, -5\\right)$", "final_answer": "(x,y)=\\left(-6, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "14f81627606d7912", "problem_id": "ac0202228275ae36", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$x + 3 y$ = $-21$\n$- 5 x + 4 y$ = $10$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-5$: $- 5 x - 15 y$ = $105$.\nMultiply the second equation by $1$: $- 5 x + 4 y$ = $10$.\nSubtract to eliminate x: $- 19 y$ = $95$.\nSolve for y: y = $-5$.\nSubstitute back to find x: x = $-6$.\nCheck: plug (x,y)=(-6,-5) into both equations.\nFinal answer: $(x,y)=\\left(-6, -5\\right)$", "final_answer": "(x,y)=\\left(-6, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2876b227eb32fd32", "problem_id": "67dbaf4ae146797a", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 29 red balls and 11 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(40,9).\n3) Favorable samples: choose k reds and n−k blues: C(29,4)·C(11,5).\n4) Compute totals: C(40,9) = 273438880, and favorable = 10972962.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{38367}{956080}$", "final_answer": "\\frac{38367}{956080}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "08b134cc9fc66fb6", "problem_id": "67dbaf4ae146797a", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "An urn has 29 red balls and 11 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(40,9).\nFavorable samples: choose k reds and n−k blues: C(29,4)·C(11,5).\nFinal answer: $\\frac{38367}{956080}$", "final_answer": "\\frac{38367}{956080}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0cbb8fdb48d4e27d", "problem_id": "e48cd0b84ad26284", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the multiplicative inverse of 4 modulo 17.", "output": "1) We want x such that 4x ≡ 1 (mod 17).\n2) Use extended Euclid to find s,t with 4s + 17t = 1.\n3) One Bezout identity is: 4·(-4) + 17·(1) = 1.\n4) Reduce s modulo 17: x ≡ 13 (mod 17).\nFinal answer: $x\\equiv 13\\pmod{17}$", "final_answer": "x\\equiv 13\\pmod{17}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "461f016bd61560ce", "problem_id": "e48cd0b84ad26284", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the multiplicative inverse of 4 modulo 17.", "output": "We want x such that 4x ≡ 1 (mod 17).\nUse extended Euclid to find s,t with 4s + 17t = 1.\nOne Bezout identity is: 4·(-4) + 17·(1) = 1.\nReduce s modulo 17: x ≡ 13 (mod 17).\nCheck: 4·13 mod 17 = 1.\nFinal answer: $x\\equiv 13\\pmod{17}$", "final_answer": "x\\equiv 13\\pmod{17}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cfe185c0769c83df", "problem_id": "ef85b0f08cd5bf68", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A cyclist rides at 33 miles per hour for 1/2 hours, then at 75 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 33 × 1/2 = 33/2 miles.\n2) Second leg distance = 75 × 5/2 = 375/2 miles.\n3) Total distance = 33/2 + 375/2 = 204 miles.\nFinal answer: $204$", "final_answer": "204", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b2e5d7e429c20b62", "problem_id": "ef85b0f08cd5bf68", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A cyclist rides at 33 miles per hour for 1/2 hours, then at 75 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 33 × 1/2 = 33/2 miles.\nSecond leg distance = 75 × 5/2 = 375/2 miles.\nTotal distance = 33/2 + 375/2 = 204 miles.\nCheck: adding both legs gives 204.\nFinal answer: $204$", "final_answer": "204", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ba3e843d7b47cbee", "problem_id": "bf19073bd3e12d32", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{11} x}{2} + \\frac{3 x}{2} - \\frac{3 \\sqrt{11}}{2} - \\frac{9}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{3}{2} - \\frac{\\sqrt{11}}{2}$, c=$- \\frac{3 \\sqrt{11}}{2} - \\frac{9}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{9 \\sqrt{11}}{2} + 23$, so √Δ = $\\sqrt{\\frac{9 \\sqrt{11}}{2} + 23}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{3}{4} + \\frac{\\sqrt{11}}{4} + \\frac{\\sqrt{18 \\sqrt{11} + 92}}{4}$.\n7) x2 = $- \\frac{\\sqrt{18 \\sqrt{11} + 92}}{4} - \\frac{3}{4} + \\frac{\\sqrt{11}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{3}{4} + \\frac{\\sqrt{11}}{4} + \\frac{\\sqrt{18\\cdot \\sqrt{11} + 92}}{4}, - \\frac{\\sqrt{18\\cdot \\sqrt{11} + 92}}{4} - \\frac{3}{4} + \\frac{\\sqrt{11}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{3}{4} + \\frac{\\sqrt{11}}{4} + \\frac{\\sqrt{18\\cdot \\sqrt{11} + 92}}{4}, - \\frac{\\sqrt{18\\cdot \\sqrt{11} + 92}}{4} - \\frac{3}{4} + \\frac{\\sqrt{11}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8bc076c30711ec4c", "problem_id": "bf19073bd3e12d32", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{11} x}{2} + \\frac{3 x}{2} - \\frac{3 \\sqrt{11}}{2} - \\frac{9}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{3}{2} - \\frac{\\sqrt{11}}{2}$, c=$- \\frac{3 \\sqrt{11}}{2} - \\frac{9}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{9 \\sqrt{11}}{2} + 23$, so √Δ = $\\sqrt{\\frac{9 \\sqrt{11}}{2} + 23}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{3}{4} + \\frac{\\sqrt{11}}{4} + \\frac{\\sqrt{18 \\sqrt{11} + 92}}{4}$.\nx2 = $- \\frac{\\sqrt{18 \\sqrt{11} + 92}}{4} - \\frac{3}{4} + \\frac{\\sqrt{11}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{3}{4} + \\frac{\\sqrt{11}}{4} + \\frac{\\sqrt{18\\cdot \\sqrt{11} + 92}}{4}, - \\frac{\\sqrt{18\\cdot \\sqrt{11} + 92}}{4} - \\frac{3}{4} + \\frac{\\sqrt{11}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{3}{4} + \\frac{\\sqrt{11}}{4} + \\frac{\\sqrt{18\\cdot \\sqrt{11} + 92}}{4}, - \\frac{\\sqrt{18\\cdot \\sqrt{11} + 92}}{4} - \\frac{3}{4} + \\frac{\\sqrt{11}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a95c809b5fede8f7", "problem_id": "dd502d34f0411a62", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{4} + 2 x^{3} + 3 x^{2} + 3 x - 2$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-2) = 0.\n3) d/dx(2 x^{3}) = 6 x^{2}.\n4) d/dx(2 x^{4}) = 8 x^{3}.\n5) d/dx(3 x) = 3.\n6) d/dx(3 x^{2}) = 6 x.\n7) Combine: f'(x) = 8 x^{3} + 6 x^{2} + 6 x + 3.\nFinal answer: $8\\cdot x^{3} + 6\\cdot x^{2} + 6\\cdot x + 3$", "final_answer": "8\\cdot x^{3} + 6\\cdot x^{2} + 6\\cdot x + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "04a02387f840b1ff", "problem_id": "dd502d34f0411a62", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Differentiate with respect to x:\n$f(x)=2 x^{4} + 2 x^{3} + 3 x^{2} + 3 x - 2$.", "output": "$8\\cdot x^{3} + 6\\cdot x^{2} + 6\\cdot x + 3$", "final_answer": "8\\cdot x^{3} + 6\\cdot x^{2} + 6\\cdot x + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cb79747a45d22008", "problem_id": "b8111cf0a33b330a", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the linear system:\n$6 x - 6 y$ = $72$\n$- 6 x - 5 y$ = $5$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-6$: $- 36 x + 36 y$ = $-432$.\nMultiply the second equation by $6$: $- 36 x - 30 y$ = $30$.\nFinal answer: $(x,y)=\\left(5, -7\\right)$", "final_answer": "(x,y)=\\left(5, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cedd2fac5c6bfea6", "problem_id": "b8111cf0a33b330a", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$6 x - 6 y$ = $72$\n$- 6 x - 5 y$ = $5$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-6$: $- 36 x + 36 y$ = $-432$.\nMultiply the second equation by $6$: $- 36 x - 30 y$ = $30$.\nSubtract to eliminate x: $66 y$ = $-462$.\nSolve for y: y = $-7$.\nSubstitute back to find x: x = $5$.\nCheck: plug (x,y)=(5,-7) into both equations.\nFinal answer: $(x,y)=\\left(5, -7\\right)$", "final_answer": "(x,y)=\\left(5, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "03a4b77f30759569", "problem_id": "153fb0e8a82bfbaa", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- 4 x - 3 y$ = $-10$\n$2 x + 3 y$ = $8$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $2$: $- 8 x - 6 y$ = $-20$.\nMultiply the second equation by $-4$: $- 8 x - 12 y$ = $-32$.\nSubtract to eliminate x: $6 y$ = $12$.\nSolve for y: y = $2$.\nSubstitute back to find x: x = $1$.\nCheck: plug (x,y)=(1,2) into both equations.\nFinal answer: $(x,y)=\\left(1, 2\\right)$", "final_answer": "(x,y)=\\left(1, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b9ede60f42321ee0", "problem_id": "153fb0e8a82bfbaa", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$- 4 x - 3 y$ = $-10$\n$2 x + 3 y$ = $8$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $- 8 x - 6 y$ = $-20$.\n3) Multiply the second equation by $-4$: $- 8 x - 12 y$ = $-32$.\n4) Subtract to eliminate x: $6 y$ = $12$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $1$.\nFinal answer: $(x,y)=\\left(1, 2\\right)$", "final_answer": "(x,y)=\\left(1, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "708ef20abbd84dd1", "problem_id": "21da3c4a2ba0d161", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Two friends split $600 in the ratio 3:9. How much does each receive?", "output": "1) Total ratio parts = 3+9 = 12.\n2) Each part = 600 ÷ 12 = 50.\n3) First share = 3 × 50 = 150.\n4) Second share = 9 × 50 = 450.\nFinal answer: $450$", "final_answer": "450", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "61c48500d55518d8", "problem_id": "21da3c4a2ba0d161", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Two friends split $600 in the ratio 3:9. How much does each receive?", "output": "Total ratio parts = 3+9 = 12.\nEach part = 600 ÷ 12 = 50.\nFirst share = 3 × 50 = 150.\nSecond share = 9 × 50 = 450.\nCheck: 150+450=600.\nFinal answer: $450$", "final_answer": "450", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9aa4e420451338e8", "problem_id": "62a97546af346132", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A cyclist rides at 52 miles per hour for 3/2 hours, then at 68 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 52 × 3/2 = 78 miles.\n2) Second leg distance = 68 × 1 = 68 miles.\n3) Total distance = 78 + 68 = 146 miles.\nFinal answer: $146$", "final_answer": "146", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "571eea9b0d151131", "problem_id": "62a97546af346132", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A cyclist rides at 52 miles per hour for 3/2 hours, then at 68 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 52 × 3/2 = 78 miles.\nSecond leg distance = 68 × 1 = 68 miles.\nTotal distance = 78 + 68 = 146 miles.\nFinal answer: $146$", "final_answer": "146", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7b6f97a2da684363", "problem_id": "db568e49e9c97c15", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$- 4 x - y$ = $-21$\n$4 x + 3 y$ = $31$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $- 16 x - 4 y$ = $-84$.\nMultiply the second equation by $-4$: $- 16 x - 12 y$ = $-124$.\nSubtract to eliminate x: $8 y$ = $40$.\nSolve for y: y = $5$.\nSubstitute back to find x: x = $4$.\nCheck: plug (x,y)=(4,5) into both equations.\nFinal answer: $(x,y)=\\left(4, 5\\right)$", "final_answer": "(x,y)=\\left(4, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e28bbbcca1086a0f", "problem_id": "db568e49e9c97c15", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$- 4 x - y$ = $-21$\n$4 x + 3 y$ = $31$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $- 16 x - 4 y$ = $-84$.\n3) Multiply the second equation by $-4$: $- 16 x - 12 y$ = $-124$.\n4) Subtract to eliminate x: $8 y$ = $40$.\n5) Solve for y: y = $5$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, 5\\right)$", "final_answer": "(x,y)=\\left(4, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c556cea01a7fa904", "problem_id": "15c257176f13b63c", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 4 \\sqrt{3} x - \\frac{3 x}{2} + 3 \\sqrt{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$- 4 \\sqrt{3} - \\frac{3}{2}$, c=$3 \\sqrt{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{201}{4} - 12 \\sqrt{3}$, so √Δ = $\\sqrt{\\frac{201}{4} - 12 \\sqrt{3}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{3}{8} + \\frac{\\sqrt{201 - 48 \\sqrt{3}}}{8} + \\sqrt{3}$.\n7) x2 = $- \\frac{\\sqrt{201 - 48 \\sqrt{3}}}{8} + \\frac{3}{8} + \\sqrt{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{3}{8} + \\frac{\\sqrt{201 - 48\\cdot \\sqrt{3}}}{8} + \\sqrt{3}, - \\frac{\\sqrt{201 - 48\\cdot \\sqrt{3}}}{8} + \\frac{3}{8} + \\sqrt{3}\\}$", "final_answer": "x \\in \\{\\frac{3}{8} + \\frac{\\sqrt{201 - 48\\cdot \\sqrt{3}}}{8} + \\sqrt{3}, - \\frac{\\sqrt{201 - 48\\cdot \\sqrt{3}}}{8} + \\frac{3}{8} + \\sqrt{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5de0ab88288d2df3", "problem_id": "15c257176f13b63c", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 4 \\sqrt{3} x - \\frac{3 x}{2} + 3 \\sqrt{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$- 4 \\sqrt{3} - \\frac{3}{2}$, c=$3 \\sqrt{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{201}{4} - 12 \\sqrt{3}$, so √Δ = $\\sqrt{\\frac{201}{4} - 12 \\sqrt{3}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{3}{8} + \\frac{\\sqrt{201 - 48 \\sqrt{3}}}{8} + \\sqrt{3}$.\nx2 = $- \\frac{\\sqrt{201 - 48 \\sqrt{3}}}{8} + \\frac{3}{8} + \\sqrt{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{3}{8} + \\frac{\\sqrt{201 - 48\\cdot \\sqrt{3}}}{8} + \\sqrt{3}, - \\frac{\\sqrt{201 - 48\\cdot \\sqrt{3}}}{8} + \\frac{3}{8} + \\sqrt{3}\\}$", "final_answer": "x \\in \\{\\frac{3}{8} + \\frac{\\sqrt{201 - 48\\cdot \\sqrt{3}}}{8} + \\sqrt{3}, - \\frac{\\sqrt{201 - 48\\cdot \\sqrt{3}}}{8} + \\frac{3}{8} + \\sqrt{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4a0b3e2e60b84159", "problem_id": "830e4a5d5a4bcc67", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Find the multiplicative inverse of 38 modulo 59.", "output": "We want x such that 38x ≡ 1 (mod 59).\nUse extended Euclid to find s,t with 38s + 59t = 1.\nOne Bezout identity is: 38·(14) + 59·(-9) = 1.\nFinal answer: $x\\equiv 14\\pmod{59}$", "final_answer": "x\\equiv 14\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e5cbecb7b0526da1", "problem_id": "830e4a5d5a4bcc67", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the multiplicative inverse of 38 modulo 59.", "output": "1) We want x such that 38x ≡ 1 (mod 59).\n2) Use extended Euclid to find s,t with 38s + 59t = 1.\n3) One Bezout identity is: 38·(14) + 59·(-9) = 1.\n4) Reduce s modulo 59: x ≡ 14 (mod 59).\nFinal answer: $x\\equiv 14\\pmod{59}$", "final_answer": "x\\equiv 14\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "221534b305a07e3c", "problem_id": "d5161946143d352a", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{2 x^{2} + 3 x + \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{x + 2}$.", "output": "1) Let f(x) = $\\frac{2 x^{2} + 3 x + \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{x + 2}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{- 4 x^{2} - 6 x + \\left(x + 2\\right) \\left(8 x + \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4}{2 \\left(x + 2\\right)^{2}}$.\nFinal answer: $\\frac{- 4\\cdot x^{2} - 6\\cdot x + \\left(x + 2\\right)\\cdot \\left(8\\cdot x + \\left(x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6\\right) - \\left(2\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 4}{2\\cdot \\left(x + 2\\right)^{2}}$", "final_answer": "\\frac{- 4\\cdot x^{2} - 6\\cdot x + \\left(x + 2\\right)\\cdot \\left(8\\cdot x + \\left(x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6\\right) - \\left(2\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 4}{2\\cdot \\left(x + 2\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c88cc6d8f8d37382", "problem_id": "d5161946143d352a", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{2 x^{2} + 3 x + \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{x + 2}$.", "output": "Let f(x) = $\\frac{2 x^{2} + 3 x + \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{x + 2}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{- 4 x^{2} - 6 x + \\left(x + 2\\right) \\left(8 x + \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4}{2 \\left(x + 2\\right)^{2}}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\frac{- 4\\cdot x^{2} - 6\\cdot x + \\left(x + 2\\right)\\cdot \\left(8\\cdot x + \\left(x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6\\right) - \\left(2\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 4}{2\\cdot \\left(x + 2\\right)^{2}}$", "final_answer": "\\frac{- 4\\cdot x^{2} - 6\\cdot x + \\left(x + 2\\right)\\cdot \\left(8\\cdot x + \\left(x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6\\right) - \\left(2\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 4}{2\\cdot \\left(x + 2\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "936e4bef535d683a", "problem_id": "7a20ea16e007c089", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡4 (mod 5), x≡3 (mod 11). Give the solution modulo 55.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 55.\n3) Compute the CRT solution to get x ≡ 14 (mod 55).\nFinal answer: $x\\equiv 14\\pmod{55}$", "final_answer": "x\\equiv 14\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b2aa899afe5f085b", "problem_id": "e8ea7e17163d28ac", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(3 x^{3} + 2 x^{2} + 2 x - 3\\right) \\cos{\\left(3 x^{2} + x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x^{2} \\cos{\\left(3 x^{2} + x \\right)}) = - 2 x^{2} \\sin{\\left(x \\left(3 x + 1\\right) \\right)}$.\n3) So $f'(x) = - 18 x^{4} \\sin{\\left(3 x^{2} + x \\right)} - 3 x^{3} \\sin{\\left(3 x^{2} + x \\right)} - 12 x^{2} \\sin{\\left(3 x^{2} + x \\right)} - 2 x^{2} \\sin{\\left(3 x^{2} + x \\right)} + 9 x^{2} \\cos{\\left(3 x^{2} + x \\right)} + 2 x e^{x} + 16 x \\sin{\\left(3 x^{2} + x \\right)} + 4 e^{x} + 3 \\sin{\\left(3 x^{2} + x \\right)} + 2 \\cos{\\left(3 x^{2} + x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x^{2} \\cos{\\left(3 x^{2} + x \\right)}) = - 2 x^{2} \\left(6 x + 1\\right) \\sin{\\left(3 x^{2} + x \\right)} + 4 x \\cos{\\left(3 x^{2} + x \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x + 1\\right) e^{x} - \\left(6 x + 1\\right) \\left(x \\left(x \\left(3 x + 2\\right) + 2\\right) - 3\\right) \\sin{\\left(3 x^{2} + x \\right)} + \\left(9 x^{2} + 4 x + 2\\right) \\cos{\\left(3 x^{2} + x \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(x + 1\\right) e^{x} - \\left(6 x + 1\\right) \\left(x \\left(x \\left(3 x + 2\\right) + 2\\right) - 3\\right) \\sin{\\left(3 x^{2} + x \\right)} + \\left(9 x^{2} + 4 x + 2\\right) \\cos{\\left(3 x^{2} + x \\right)} + 2 e^{x}$", "final_answer": "2 \\left(x + 1\\right) e^{x} - \\left(6 x + 1\\right) \\left(x \\left(x \\left(3 x + 2\\right) + 2\\right) - 3\\right) \\sin{\\left(3 x^{2} + x \\right)} + \\left(9 x^{2} + 4 x + 2\\right) \\cos{\\left(3 x^{2} + x \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e2940a602397668c", "problem_id": "c4fbd4d698f04647", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "You want 61 liters of a 35% salt solution. You have a 20% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 61−x liters is 40% solution.\n2) Salt equation: 20x + 40(61−x) = 35·61.\n3) (20-40)x = (35-40)·61 ⇒ x = 61/4.\n4) Then 61−x = 183/4.\nFinal answer: $61/4 L of 20% and 183/4 L of 40%$", "final_answer": "61/4 L of 20% and 183/4 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "83ca147ed4813bbf", "problem_id": "c4fbd4d698f04647", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "You want 61 liters of a 35% salt solution. You have a 20% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 61−x liters is 40% solution.\nSalt equation: 20x + 40(61−x) = 35·61.\n(20-40)x = (35-40)·61 ⇒ x = 61/4.\nFinal answer: $61/4 L of 20% and 183/4 L of 40%$", "final_answer": "61/4 L of 20% and 183/4 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4d7677d41f5d18e2", "problem_id": "84ef03e2676efdbb", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - x$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{2} + 3 x + 3$ and a = 3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $3 - 2 x$.\n5) Evaluate at a: p'(3) = $-3$.\nFinal answer: $-3$", "final_answer": "-3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "15c8212d7fd36eec", "problem_id": "84ef03e2676efdbb", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - x$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{2} + 3 x + 3$ and a = 3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $3 - 2 x$.\nEvaluate at a: p'(3) = $-3$.\nCheck: the identity gives $-3$.\nFinal answer: $-3$", "final_answer": "-3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d679e37843af6a06", "problem_id": "24b585b0476322c7", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the multiplicative inverse of 10 modulo 17.", "output": "1) We want x such that 10x ≡ 1 (mod 17).\n2) Use extended Euclid to find s,t with 10s + 17t = 1.\n3) One Bezout identity is: 10·(-5) + 17·(3) = 1.\n4) Reduce s modulo 17: x ≡ 12 (mod 17).\nFinal answer: $x\\equiv 12\\pmod{17}$", "final_answer": "x\\equiv 12\\pmod{17}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "60dce057f341d52a", "problem_id": "24b585b0476322c7", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the multiplicative inverse of 10 modulo 17.", "output": "We want x such that 10x ≡ 1 (mod 17).\nUse extended Euclid to find s,t with 10s + 17t = 1.\nOne Bezout identity is: 10·(-5) + 17·(3) = 1.\nReduce s modulo 17: x ≡ 12 (mod 17).\nCheck: 10·12 mod 17 = 1.\nFinal answer: $x\\equiv 12\\pmod{17}$", "final_answer": "x\\equiv 12\\pmod{17}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b4d55305da785cf3", "problem_id": "c0b63b96421cffde", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} - 2 x^{2} - x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=2$.", "output": "1) Differentiate: f'(x) = $9 x^{2} - 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 1$.\n2) Slope at x=2: m = f'(2) = $\\frac{e}{2} - 2 \\sin{\\left(4 \\right)} + 27$.\n3) Point on curve: (2, f(2)) = (2, $\\cos{\\left(4 \\right)} + e + 13$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(4 \\right)} + e + 13 = \\frac{e}{2} - 2 \\sin{\\left(4 \\right)} + 27(x − 2).\n6) Optionally expand: y = $\\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 27 x - 41 + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$.\nFinal answer: $y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 27\\cdot x - 41 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$", "final_answer": "y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 27\\cdot x - 41 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "061d614119931c21", "problem_id": "c0b63b96421cffde", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} - 2 x^{2} - x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=2$.", "output": "Differentiate: f'(x) = $9 x^{2} - 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 1$.\nSlope at x=2: m = f'(2) = $\\frac{e}{2} - 2 \\sin{\\left(4 \\right)} + 27$.\nPoint on curve: (2, f(2)) = (2, $\\cos{\\left(4 \\right)} + e + 13$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − \\cos{\\left(4 \\right)} + e + 13 = \\frac{e}{2} - 2 \\sin{\\left(4 \\right)} + 27(x − 2).\nOptionally expand: y = $\\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 27 x - 41 + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 27\\cdot x - 41 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$", "final_answer": "y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 27\\cdot x - 41 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0aa24faee67078a3", "problem_id": "861fdc22f6dc5902", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many binary strings of length 15 contain exactly 4 ones with no two ones adjacent?", "output": "1) Reserve 3 mandatory zeros between the 4 ones.\n2) This leaves n−k+1 = 12 valid slots for the ones.\n3) Choose the slots: C(12,4) = 495.\nFinal answer: $495$", "final_answer": "495", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "23f0e0b761a9143a", "problem_id": "861fdc22f6dc5902", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many binary strings of length 15 contain exactly 4 ones with no two ones adjacent?", "output": "Reserve 3 mandatory zeros between the 4 ones.\nThis leaves n−k+1 = 12 valid slots for the ones.\nChoose the slots: C(12,4) = 495.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $495$", "final_answer": "495", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0c113d79f8868160", "problem_id": "b06e0df40b96e16d", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A store sells a backpack for $68. It is discounted by 10%. You buy 6 backpacks. What is the total cost?", "output": "Discounted price = 68 × (1 − 10/100) = 68 × 90/100 = 306/5.\nSubtotal = 6 × 306/5 = 1836/5.\nFinal answer: $\\frac{1836}{5}$", "final_answer": "\\frac{1836}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "48abe5c36cc03e6f", "problem_id": "b06e0df40b96e16d", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A store sells a backpack for $68. It is discounted by 10%. You buy 6 backpacks. What is the total cost?", "output": "1) Discounted price = 68 × (1 − 10/100) = 68 × 90/100 = 306/5.\n2) Subtotal = 6 × 306/5 = 1836/5.\nFinal answer: $\\frac{1836}{5}$", "final_answer": "\\frac{1836}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b7d7656d0a92188", "problem_id": "90c8e5af9b67d0a8", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the multiplicative inverse of 24 modulo 43.", "output": "1) We want x such that 24x ≡ 1 (mod 43).\n2) Use extended Euclid to find s,t with 24s + 43t = 1.\n3) One Bezout identity is: 24·(9) + 43·(-5) = 1.\n4) Reduce s modulo 43: x ≡ 9 (mod 43).\nFinal answer: $x\\equiv 9\\pmod{43}$", "final_answer": "x\\equiv 9\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2c131bc36fdaf23d", "problem_id": "90c8e5af9b67d0a8", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the multiplicative inverse of 24 modulo 43.", "output": "We want x such that 24x ≡ 1 (mod 43).\nUse extended Euclid to find s,t with 24s + 43t = 1.\nOne Bezout identity is: 24·(9) + 43·(-5) = 1.\nReduce s modulo 43: x ≡ 9 (mod 43).\nCheck: 24·9 mod 43 = 1.\nFinal answer: $x\\equiv 9\\pmod{43}$", "final_answer": "x\\equiv 9\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "466a67157c624cf4", "problem_id": "9cbb4232bb6ce75c", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the congruences: x≡1 (mod 9), x≡3 (mod 7). Give the solution modulo 63.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 63.\nCompute the CRT solution to get x ≡ 10 (mod 63).\nFinal answer: $x\\equiv 10\\pmod{63}$", "final_answer": "x\\equiv 10\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0dd36e25cdc1c5fb", "problem_id": "9cbb4232bb6ce75c", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡1 (mod 9), x≡3 (mod 7). Give the solution modulo 63.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 63.\n3) Compute the CRT solution to get x ≡ 10 (mod 63).\nFinal answer: $x\\equiv 10\\pmod{63}$", "final_answer": "x\\equiv 10\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8b358fe94abfce59", "problem_id": "d56e76944314c694", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$- x + 8 y$ = $12$\n$7 x - y$ = $-29$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $7$: $- 7 x + 56 y$ = $84$.\n3) Multiply the second equation by $-1$: $- 7 x + y$ = $29$.\n4) Subtract to eliminate x: $55 y$ = $55$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $-4$.\nFinal answer: $(x,y)=\\left(-4, 1\\right)$", "final_answer": "(x,y)=\\left(-4, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a015bb0dfa86c9ff", "problem_id": "731291ed7828f8a8", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x + 4$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to x + 4.\n3) Then the limit is 6.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 8.\nFinal answer: $8$", "final_answer": "8", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7aa75077e9e9338d", "problem_id": "1c4e6fdf09820c45", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A store sells a backpack for $610. It is discounted by 30%. You buy 14 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 610 × (1 − 30/100) = 610 × 70/100 = 427.\n2) Subtotal = 14 × 427 = 5978.\n3) Total with tax = 5978 × 108/100 = 161406/25.\nFinal answer: $\\frac{161406}{25}$", "final_answer": "\\frac{161406}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5dc4b3c0285e4edb", "problem_id": "1c4e6fdf09820c45", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $610. It is discounted by 30%. You buy 14 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 610 × (1 − 30/100) = 610 × 70/100 = 427.\nSubtotal = 14 × 427 = 5978.\nTotal with tax = 5978 × 108/100 = 161406/25.\nCheck: recomputing the formula yields 161406/25.\nFinal answer: $\\frac{161406}{25}$", "final_answer": "\\frac{161406}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ad66f2bfea0afa13", "problem_id": "46e69b436e1869c4", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$- 2 x + y$ = $-7$\n$3 y$ = $-9$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $-2$: $- 6 y$ = $18$.\n4) Subtract to eliminate x: $6 y$ = $-18$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $2$.\nFinal answer: $(x,y)=\\left(2, -3\\right)$", "final_answer": "(x,y)=\\left(2, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7dbb316d2c27f254", "problem_id": "46e69b436e1869c4", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the linear system:\n$- 2 x + y$ = $-7$\n$3 y$ = $-9$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $-2$: $- 6 y$ = $18$.\nFinal answer: $(x,y)=\\left(2, -3\\right)$", "final_answer": "(x,y)=\\left(2, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d0fc1542a7d7fbfb", "problem_id": "30d6cf9a97495d00", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{x^{2} + 3}$.", "output": "1) Let f(x) = $\\frac{- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{x^{2} + 3}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{- 4 x \\left(- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} + 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x^{2} + 3\\right)^{2}}$.\nFinal answer: $\\frac{- 4\\cdot x\\cdot \\left(- x^{3} + 2\\cdot x^{2} - 3\\cdot x + \\left(x^{2} + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} + 3\\right)\\cdot \\left(- 6\\cdot x^{2} + 4\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 8\\cdot x + \\left(x^{2} + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 6\\right)}{2\\cdot \\left(x^{2} + 3\\right)^{2}}$", "final_answer": "\\frac{- 4\\cdot x\\cdot \\left(- x^{3} + 2\\cdot x^{2} - 3\\cdot x + \\left(x^{2} + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} + 3\\right)\\cdot \\left(- 6\\cdot x^{2} + 4\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 8\\cdot x + \\left(x^{2} + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 6\\right)}{2\\cdot \\left(x^{2} + 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dcafe53aa936aa79", "problem_id": "30d6cf9a97495d00", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{x^{2} + 3}$.", "output": "Let f(x) = $\\frac{- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{x^{2} + 3}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{- 4 x \\left(- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} + 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x^{2} + 3\\right)^{2}}$.\nFinal answer: $\\frac{- 4\\cdot x\\cdot \\left(- x^{3} + 2\\cdot x^{2} - 3\\cdot x + \\left(x^{2} + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} + 3\\right)\\cdot \\left(- 6\\cdot x^{2} + 4\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 8\\cdot x + \\left(x^{2} + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 6\\right)}{2\\cdot \\left(x^{2} + 3\\right)^{2}}$", "final_answer": "\\frac{- 4\\cdot x\\cdot \\left(- x^{3} + 2\\cdot x^{2} - 3\\cdot x + \\left(x^{2} + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} + 3\\right)\\cdot \\left(- 6\\cdot x^{2} + 4\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 8\\cdot x + \\left(x^{2} + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 6\\right)}{2\\cdot \\left(x^{2} + 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c39367aaca16cfb4", "problem_id": "98e765fee2d42e08", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} - 4 x + 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 4 x) = - 4 x$.\n3) So $f'(x) = - 8 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 4 x) = -4$.\nTherefore the correct derivative is $f'(x) = 4 \\left(- x - 1\\right)$.\nFinal answer: $4 \\left(- x - 1\\right)$", "final_answer": "4 \\left(- x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b8f5cf9d3f8449f4", "problem_id": "4dbb24bffeea42c3", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$- 3 x + 2 y$ = $-21$\n$- 4 x - 4 y$ = $12$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $12 x - 8 y$ = $84$.\n3) Multiply the second equation by $-3$: $12 x + 12 y$ = $-36$.\n4) Subtract to eliminate x: $- 20 y$ = $120$.\n5) Solve for y: y = $-6$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, -6\\right)$", "final_answer": "(x,y)=\\left(3, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a6af29ab430591c", "problem_id": "11e41a90b5af3da6", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_8=24$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=24 and k=8.\n4) Compute C(31,7) = 2629575.\nFinal answer: $2629575$", "final_answer": "2629575", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9abbda168b0d359b", "problem_id": "936e13033f401138", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 3 x^{2} + 9 x + 19$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{2} + 8 x + 19.\n3) Then the limit is 47.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 49.\nFinal answer: $49$", "final_answer": "49", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2a63835b173f3bc3", "problem_id": "989ebc491823d977", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + \\frac{5 x}{2} - 2$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$\\frac{5}{2}$, c=$-2$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{121}{4}$, so √Δ = $\\frac{11}{2}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{1}{2}$.\nx2 = $- \\frac{4}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{1}{2}, - \\frac{4}{3}\\}$", "final_answer": "x \\in \\{\\frac{1}{2}, - \\frac{4}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b263ef73bb8a70d8", "problem_id": "989ebc491823d977", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + \\frac{5 x}{2} - 2$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$\\frac{5}{2}$, c=$-2$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{121}{4}$, so √Δ = $\\frac{11}{2}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{1}{2}$.\n7) x2 = $- \\frac{4}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{1}{2}, - \\frac{4}{3}\\}$", "final_answer": "x \\in \\{\\frac{1}{2}, - \\frac{4}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2b794c320048ddab", "problem_id": "4093ce8183cef159", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=21$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=21 and k=5.\n4) Compute C(25,4) = 12650.\nFinal answer: $12650$", "final_answer": "12650", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "80c8fdc8c64a7270", "problem_id": "4093ce8183cef159", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=21$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=21 and k=5.\nCompute C(25,4) = 12650.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $12650$", "final_answer": "12650", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "379a969f90f67308", "problem_id": "2b2a2fd170e499c0", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many binary strings of length 14 contain exactly 2 ones with no two ones adjacent?", "output": "1) Reserve 1 mandatory zeros between the 2 ones.\n2) This leaves n−k+1 = 13 valid slots for the ones.\n3) Choose the slots: C(13,2) = 78.\nFinal answer: $78$", "final_answer": "78", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b7092d922a8f711a", "problem_id": "2b2a2fd170e499c0", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many binary strings of length 14 contain exactly 2 ones with no two ones adjacent?", "output": "Reserve 1 mandatory zeros between the 2 ones.\nThis leaves n−k+1 = 13 valid slots for the ones.\nChoose the slots: C(13,2) = 78.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $78$", "final_answer": "78", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4134c2587496f472", "problem_id": "3c8b282fb4643cec", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 30 x + 100$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 30 x + 100$ = $2 \\left(x - 10\\right) \\left(x - 5\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $10$.\n5) From the second factor: x = $5$.\nFinal answer: $x \\in \\{10, 5\\}$", "final_answer": "x \\in \\{10, 5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b3f131d18a3b650d", "problem_id": "2458bd55b3c9d678", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 2 x^{2} - 5 x + 16$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{2} - 6 x + 16.\n3) Then the limit is 52.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: 49.\nFinal answer: $49$", "final_answer": "49", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5028d97451084ec8", "problem_id": "4d08ee5193dc26cd", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 2 x - 4$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} + 2 x - 4$ = $2 \\left(x - 1\\right) \\left(x + 2\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $1$.\n5) From the second factor: x = $-2$.\nFinal answer: $x \\in \\{1, -2\\}$", "final_answer": "x \\in \\{1, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d31d88b892cb99c4", "problem_id": "4d08ee5193dc26cd", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 2 x - 4$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} + 2 x - 4$ = $2 \\left(x - 1\\right) \\left(x + 2\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $1$.\nFrom the second factor: x = $-2$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{1, -2\\}$", "final_answer": "x \\in \\{1, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "deb87fcd7ed7cdbb", "problem_id": "54b7a07ccaa8d8cf", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡3 (mod 11), x≡11 (mod 13). Give the solution modulo 143.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 143.\n3) Compute the CRT solution to get x ≡ 102 (mod 143).\nFinal answer: $x\\equiv 102\\pmod{143}$", "final_answer": "x\\equiv 102\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a2690ab3766bfe44", "problem_id": "54b7a07ccaa8d8cf", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡3 (mod 11), x≡11 (mod 13). Give the solution modulo 143.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 143.\nCompute the CRT solution to get x ≡ 102 (mod 143).\nCheck: 102 mod 11=3 and mod 13=11.\nFinal answer: $x\\equiv 102\\pmod{143}$", "final_answer": "x\\equiv 102\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a8fe9bda78b54635", "problem_id": "6edad1f8546f11c1", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Worker A finishes a job in 17 hours and Worker B finishes it in 21 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/17 job/hour. B's rate = 1/21 job/hour.\nCombined rate = 1/17 + 1/21 = 38/357 job/hour.\nTime = 1 ÷ (combined rate) = 357/38 hours.\nFinal answer: $\\frac{357}{38}$", "final_answer": "\\frac{357}{38}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0d0bba6bb9963e55", "problem_id": "6edad1f8546f11c1", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Worker A finishes a job in 17 hours and Worker B finishes it in 21 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/17 job/hour. B's rate = 1/21 job/hour.\nCombined rate = 1/17 + 1/21 = 38/357 job/hour.\nTime = 1 ÷ (combined rate) = 357/38 hours.\nCheck: (combined rate)×(time) = 38/357×357/38 = 1.\nFinal answer: $\\frac{357}{38}$", "final_answer": "\\frac{357}{38}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "991765e7e0d3c19b", "problem_id": "7b1bda6ede3f3fcc", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $1680 in the ratio 8:4. How much does each receive?", "output": "1) Total ratio parts = 8+4 = 12.\n2) Each part = 1680 ÷ 12 = 140.\n3) First share = 8 × 140 = 1120.\n4) Second share = 4 × 140 = 560.\nFinal answer: $560$", "final_answer": "560", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "492648852aae4524", "problem_id": "ca37e305e6636670", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{3 x^{2} - 3 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{2 x - 3}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 3}) = \\frac{2 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 3}$.\n3) So $f'(x) = \\frac{4 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{2} - 24 x + 18} + \\frac{8 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{2} - 24 x + 18} + \\frac{12 x^{2}}{8 x^{2} - 24 x + 18} - \\frac{12 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{2} - 24 x + 18} - \\frac{24 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{2} - 24 x + 18} - \\frac{36 x}{8 x^{2} - 24 x + 18} + \\frac{4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{4 x^{2} - 12 x + 9} + \\frac{2 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 3} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 3} - \\frac{2 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 3} + \\frac{9 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{2} - 24 x + 18} + \\frac{18 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{2} - 24 x + 18} + \\frac{26}{8 x^{2} - 24 x + 18} - \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 3}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 3}) = \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 3} + \\frac{2 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 3} - \\frac{4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x - 3\\right)^{2}} + \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 3}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 12 x^{2} + 12 x + \\left(2 x - 3\\right) \\left(12 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6\\right) - 4 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8}{2 \\left(2 x - 3\\right)^{2}}$.\nFinal answer: $\\frac{- 12 x^{2} + 12 x + \\left(2 x - 3\\right) \\left(12 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6\\right) - 4 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8}{2 \\left(2 x - 3\\right)^{2}}$", "final_answer": "\\frac{- 12 x^{2} + 12 x + \\left(2 x - 3\\right) \\left(12 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6\\right) - 4 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8}{2 \\left(2 x - 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "fac44a40d10b3d6e", "problem_id": "8aba13d005cbadf5", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Find the equation of the tangent line to $y=x^{3} + x^{2} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=1$.", "output": "1) Differentiate: f'(x) = $3 x^{2} + 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\n2) Slope at x=1: m = f'(1) = $- 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 3$.\n3) Point on curve: (1, f(1)) = (1, $\\cos{\\left(2 \\right)} + 1 + e^{\\frac{1}{2}}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(2 \\right)} + 1 + e^{\\frac{1}{2}} = - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 3(x − 1).\n6) Optionally expand: y = $- 2 x \\sin{\\left(2 \\right)} + \\frac{x e^{\\frac{1}{2}}}{2} + 3 x - 2 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 3\\cdot x - 2 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 3\\cdot x - 2 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "809d53cd8c1bee26", "problem_id": "8aba13d005cbadf5", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the equation of the tangent line to $y=x^{3} + x^{2} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=1$.", "output": "Differentiate: f'(x) = $3 x^{2} + 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\nSlope at x=1: m = f'(1) = $- 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 3$.\nPoint on curve: (1, f(1)) = (1, $\\cos{\\left(2 \\right)} + 1 + e^{\\frac{1}{2}}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − \\cos{\\left(2 \\right)} + 1 + e^{\\frac{1}{2}} = - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 3(x − 1).\nOptionally expand: y = $- 2 x \\sin{\\left(2 \\right)} + \\frac{x e^{\\frac{1}{2}}}{2} + 3 x - 2 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2 \\sin{\\left(2 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 3\\cdot x - 2 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 3\\cdot x - 2 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "26a27194dfa4ad12", "problem_id": "7fe60aba19f7900c", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_10=38$?", "output": "Let y_i=x_i-3, then y_i≥0 and y_1+…+y_10=8.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=8 and k=10.\nFinal answer: $24310$", "final_answer": "24310", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "30d4bf21a57a9324", "problem_id": "7fe60aba19f7900c", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_10=38$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_10=8.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=8 and k=10.\n4) Compute C(17,9) = 24310.\nFinal answer: $24310$", "final_answer": "24310", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "629ab7ffe1e1fc4e", "problem_id": "82f49cd34d98cb42", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x^{3} - x^{2} - 3 x - 3$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - x^{3} - x^{2} - 4 x - 3.\n3) Then the limit is -23.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -21.\nFinal answer: $-21$", "final_answer": "-21", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "849d1211957f0c37", "problem_id": "ff3c36a07abb0c27", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=43$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=43 and k=5.\n4) Compute C(47,4) = 178365.\nFinal answer: $178365$", "final_answer": "178365", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "831152b9669a8b83", "problem_id": "6854d03417f22b81", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} + 4 x + 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x^{2}) = 2 x^{2}$.\n3) So $f'(x) = 2 x^{2} + 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x^{2}) = 2 x$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x + 2\\right)$.\nFinal answer: $2 \\left(x + 2\\right)$", "final_answer": "2 \\left(x + 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "101138b318aa2c04", "problem_id": "1b896352067cd7d2", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - x^{4} - 4 x^{2} + 2 x - 4\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- x^{4} - 4 x^{2} + 2 x - 4$.\n3) Compute an antiderivative: F(x) = $- \\frac{x^{5}}{5} - \\frac{4 x^{3}}{3} + x^{2} - 4 x$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{x^{5}}{5} - \\frac{4\\cdot x^{3}}{3} + x^{2} - 4\\cdot x$", "final_answer": "C - \\frac{x^{5}}{5} - \\frac{4\\cdot x^{3}}{3} + x^{2} - 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "555fa19c2c19b990", "problem_id": "c2def7dcf4eecfe1", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{7}$. If it is flipped 13 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=13, k=3, p=$\\frac{4}{7}$ and simplify.\n4) Here C(13,3) = 286.\nFinal answer: $\\frac{1080832896}{96889010407}$", "final_answer": "\\frac{1080832896}{96889010407}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "966da050937bf9bd", "problem_id": "c2def7dcf4eecfe1", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{7}$. If it is flipped 13 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=13, k=3, p=$\\frac{4}{7}$ and simplify.\nHere C(13,3) = 286.\nCheck: expression simplifies to $\\frac{1080832896}{96889010407}$.\nFinal answer: $\\frac{1080832896}{96889010407}$", "final_answer": "\\frac{1080832896}{96889010407}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cd6e2ec08b4f358f", "problem_id": "2a3d0211988db459", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} - 3 x^{3} + 3 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{3} + 3 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $5 + \\frac{9 \\pi}{2} + \\frac{9 \\pi^{3}}{8} + \\frac{45 \\pi^{4}}{64}$.\n4) Simplify the result.\nFinal answer: $5 + \\frac{9\\cdot \\pi}{2} + \\frac{9\\cdot \\pi^{3}}{8} + \\frac{45\\cdot \\pi^{4}}{64}$", "final_answer": "5 + \\frac{9\\cdot \\pi}{2} + \\frac{9\\cdot \\pi^{3}}{8} + \\frac{45\\cdot \\pi^{4}}{64}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f577821a73f17f9", "problem_id": "2a3d0211988db459", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} - 3 x^{3} + 3 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{3} + 3 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $5 + \\frac{9 \\pi}{2} + \\frac{9 \\pi^{3}}{8} + \\frac{45 \\pi^{4}}{64}$.\nFinal answer: $5 + \\frac{9\\cdot \\pi}{2} + \\frac{9\\cdot \\pi^{3}}{8} + \\frac{45\\cdot \\pi^{4}}{64}$", "final_answer": "5 + \\frac{9\\cdot \\pi}{2} + \\frac{9\\cdot \\pi^{3}}{8} + \\frac{45\\cdot \\pi^{4}}{64}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c39a0b4aa1b2b5fa", "problem_id": "5ed61ac813819f28", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the multiplicative inverse of 5 modulo 19.", "output": "1) We want x such that 5x ≡ 1 (mod 19).\n2) Use extended Euclid to find s,t with 5s + 19t = 1.\n3) One Bezout identity is: 5·(4) + 19·(-1) = 1.\n4) Reduce s modulo 19: x ≡ 4 (mod 19).\nFinal answer: $x\\equiv 4\\pmod{19}$", "final_answer": "x\\equiv 4\\pmod{19}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3d4b0d9ec9db264d", "problem_id": "cb09f9741b7199b8", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(2 x^{2} - 3 x + 2\\right) \\cos{\\left(2 x + 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 \\cos{\\left(2 x + 1 \\right)}) = - 2 \\sin{\\left(2 x + 1 \\right)}$.\n3) So $f'(x) = - 4 x^{2} \\sin{\\left(2 x + 1 \\right)} + x e^{x} + 6 x \\sin{\\left(2 x + 1 \\right)} + 4 x \\cos{\\left(2 x + 1 \\right)} + 3 e^{x} - 2 \\sin{\\left(2 x + 1 \\right)} - 3 \\cos{\\left(2 x + 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 \\cos{\\left(2 x + 1 \\right)}) = - 4 \\sin{\\left(2 x + 1 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 2\\right) e^{x} + \\left(4 x - 3\\right) \\cos{\\left(2 x + 1 \\right)} - 2 \\left(x \\left(2 x - 3\\right) + 2\\right) \\sin{\\left(2 x + 1 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 2\\right) e^{x} + \\left(4 x - 3\\right) \\cos{\\left(2 x + 1 \\right)} - 2 \\left(x \\left(2 x - 3\\right) + 2\\right) \\sin{\\left(2 x + 1 \\right)} + e^{x}$", "final_answer": "\\left(x + 2\\right) e^{x} + \\left(4 x - 3\\right) \\cos{\\left(2 x + 1 \\right)} - 2 \\left(x \\left(2 x - 3\\right) + 2\\right) \\sin{\\left(2 x + 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "86c15f321679540b", "problem_id": "22390e44d8f6f830", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$x + y$ = $2$\n$- 4 x + 3 y$ = $6$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $- 4 x - 4 y$ = $-8$.\n3) Multiply the second equation by $1$: $- 4 x + 3 y$ = $6$.\n4) Subtract to eliminate x: $- 7 y$ = $-14$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, 2\\right)$", "final_answer": "(x,y)=\\left(0, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "68793a0d88ca01e3", "problem_id": "22390e44d8f6f830", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$x + y$ = $2$\n$- 4 x + 3 y$ = $6$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $- 4 x - 4 y$ = $-8$.\nMultiply the second equation by $1$: $- 4 x + 3 y$ = $6$.\nSubtract to eliminate x: $- 7 y$ = $-14$.\nSolve for y: y = $2$.\nSubstitute back to find x: x = $0$.\nCheck: plug (x,y)=(0,2) into both equations.\nFinal answer: $(x,y)=\\left(0, 2\\right)$", "final_answer": "(x,y)=\\left(0, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5af2a4911ccd16aa", "problem_id": "a79248f3f6e862f3", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 14 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=14, k=2, p=$\\frac{2}{3}$ and simplify.\n4) Here C(14,2) = 91.\nFinal answer: $\\frac{364}{4782969}$", "final_answer": "\\frac{364}{4782969}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "de3809ab03d11936", "problem_id": "4130f56657689103", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(2 x^{2} - 3 x - 3\\right) \\cos{\\left(3 x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 3 x \\cos{\\left(3 x + 3 \\right)}) = 3 x \\sin{\\left(3 x + 3 \\right)}$.\n3) So $f'(x) = - 6 x^{2} \\sin{\\left(3 x + 3 \\right)} + x e^{x} + 3 x \\sin{\\left(3 x + 3 \\right)} + 4 x \\cos{\\left(3 x + 3 \\right)} - 2 e^{x} + 9 \\sin{\\left(3 x + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 3 x \\cos{\\left(3 x + 3 \\right)}) = 9 x \\sin{\\left(3 x + 3 \\right)} - 3 \\cos{\\left(3 x + 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x - 3\\right) e^{x} + \\left(4 x - 3\\right) \\cos{\\left(3 x + 3 \\right)} - 3 \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x + 3 \\right)} + e^{x}$.\nFinal answer: $\\left(x - 3\\right) e^{x} + \\left(4 x - 3\\right) \\cos{\\left(3 x + 3 \\right)} - 3 \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x + 3 \\right)} + e^{x}$", "final_answer": "\\left(x - 3\\right) e^{x} + \\left(4 x - 3\\right) \\cos{\\left(3 x + 3 \\right)} - 3 \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x + 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0476f2bc86bcad57", "problem_id": "a6896293c0000514", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A store sells a backpack for $615. It is discounted by 40%. You buy 4 backpacks. What is the total cost?", "output": "1) Discounted price = 615 × (1 − 40/100) = 615 × 60/100 = 369.\n2) Subtotal = 4 × 369 = 1476.\nFinal answer: $1476$", "final_answer": "1476", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f28700267e608893", "problem_id": "a6896293c0000514", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $615. It is discounted by 40%. You buy 4 backpacks. What is the total cost?", "output": "Discounted price = 615 × (1 − 40/100) = 615 × 60/100 = 369.\nSubtotal = 4 × 369 = 1476.\nCheck: recomputing the formula yields 1476.\nFinal answer: $1476$", "final_answer": "1476", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "87a39db4bccac74b", "problem_id": "b7292bd35c92aa84", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - x^{4} - x^{3} - 4 x^{2} + 2 x - 3\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- x^{4} - x^{3} - 4 x^{2} + 2 x - 3$.\nCompute an antiderivative: F(x) = $- \\frac{x^{5}}{5} - \\frac{x^{4}}{4} - \\frac{4 x^{3}}{3} + x^{2} - 3 x$.\nFinal answer: $C - \\frac{x^{5}}{5} - \\frac{x^{4}}{4} - \\frac{4\\cdot x^{3}}{3} + x^{2} - 3\\cdot x$", "final_answer": "C - \\frac{x^{5}}{5} - \\frac{x^{4}}{4} - \\frac{4\\cdot x^{3}}{3} + x^{2} - 3\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9f3604b024c389a5", "problem_id": "b7292bd35c92aa84", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - x^{4} - x^{3} - 4 x^{2} + 2 x - 3\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- x^{4} - x^{3} - 4 x^{2} + 2 x - 3$.\nCompute an antiderivative: F(x) = $- \\frac{x^{5}}{5} - \\frac{x^{4}}{4} - \\frac{4 x^{3}}{3} + x^{2} - 3 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - \\frac{x^{5}}{5} - \\frac{x^{4}}{4} - \\frac{4\\cdot x^{3}}{3} + x^{2} - 3\\cdot x$", "final_answer": "C - \\frac{x^{5}}{5} - \\frac{x^{4}}{4} - \\frac{4\\cdot x^{3}}{3} + x^{2} - 3\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1086c10789f15d4f", "problem_id": "c0eac820ed13437c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} - 3 x^{2} + 10 x - 40$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} + 9 x - 40.\n3) Then the limit is -124.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: -128.\nFinal answer: $-128$", "final_answer": "-128", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b8c3ddae916cd9b6", "problem_id": "9e3c96fd2e398e5f", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{4 x}{3} - 14$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$\\frac{4}{3}$, c=$-14$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{1024}{9}$, so √Δ = $\\frac{32}{3}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{7}{3}$.\nx2 = $-3$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{7}{3}, -3\\}$", "final_answer": "x \\in \\{\\frac{7}{3}, -3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a0e4afdb2f66d934", "problem_id": "6e5f26e647e9189b", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 10 times, what is the probability of exactly 2 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=2, p=$\\frac{1}{4}$ and simplify.\nFinal answer: $\\frac{295245}{1048576}$", "final_answer": "\\frac{295245}{1048576}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3c82d5816b04ed81", "problem_id": "6e5f26e647e9189b", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 10 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=2, p=$\\frac{1}{4}$ and simplify.\n4) Here C(10,2) = 45.\nFinal answer: $\\frac{295245}{1048576}$", "final_answer": "\\frac{295245}{1048576}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3436428288c4df64", "problem_id": "a3f6c126117d348b", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡1 (mod 4), x≡1 (mod 11). Give the solution modulo 44.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 44.\n3) Compute the CRT solution to get x ≡ 1 (mod 44).\nFinal answer: $x\\equiv 1\\pmod{44}$", "final_answer": "x\\equiv 1\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8364c75638a7ad24", "problem_id": "a3f6c126117d348b", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡1 (mod 4), x≡1 (mod 11). Give the solution modulo 44.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 44.\nCompute the CRT solution to get x ≡ 1 (mod 44).\nCheck: 1 mod 4=1 and mod 11=1.\nFinal answer: $x\\equiv 1\\pmod{44}$", "final_answer": "x\\equiv 1\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4848eca663385dca", "problem_id": "eb8cba1f44cc36a0", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} - 2 x + \\left(3 x^{2} + x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{3 x^{2} + x - 4}$.", "output": "Let f(x) = $\\frac{- x^{2} - 2 x + \\left(3 x^{2} + x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{3 x^{2} + x - 4}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{2 \\left(- 6 x - 1\\right) \\left(- x^{2} - 2 x + \\left(3 x^{2} + x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(3 x^{2} + x - 4\\right) \\left(- 4 x + 2 \\left(6 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(3 x^{2} + x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x^{2} + x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 4\\right)}{2 \\left(3 x^{2} + x - 4\\right)^{2}}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\frac{\\left(- 12\\cdot x - 2\\right)\\cdot \\left(- x^{2} - 2\\cdot x + \\left(3\\cdot x^{2} + x - 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right) + \\left(3\\cdot x^{2} + x - 4\\right)\\cdot \\left(- 4\\cdot x + \\left(12\\cdot x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(3\\cdot x^{2} + x - 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x^{2} + 2\\cdot x - 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 4\\right)}{2\\cdot \\left(3\\cdot x^{2} + x - 4\\right)^{2}}$", "final_answer": "\\frac{\\left(- 12\\cdot x - 2\\right)\\cdot \\left(- x^{2} - 2\\cdot x + \\left(3\\cdot x^{2} + x - 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right) + \\left(3\\cdot x^{2} + x - 4\\right)\\cdot \\left(- 4\\cdot x + \\left(12\\cdot x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(3\\cdot x^{2} + x - 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x^{2} + 2\\cdot x - 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 4\\right)}{2\\cdot \\left(3\\cdot x^{2} + x - 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b536a622d38096b2", "problem_id": "6f74cc5326d881b1", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} 2 x^{3} - 11 x^{2} + 47 x - 191$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} - 11 x^{2} + 46 x - 191.\n3) Then the limit is -679.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: -683.\nFinal answer: $-683$", "final_answer": "-683", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ed1ebdeab2447252", "problem_id": "dc48dee3ed61e7cc", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{6} x + \\frac{13 x}{4} - 3 + \\frac{9 \\sqrt{6}}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{13}{4} - 3 \\sqrt{6}$, c=$-3 + \\frac{9 \\sqrt{6}}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{1225}{16} - \\frac{57 \\sqrt{6}}{2}$, so √Δ = $\\sqrt{\\frac{1225}{16} - \\frac{57 \\sqrt{6}}{2}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{13}{8} + \\frac{\\sqrt{1225 - 456 \\sqrt{6}}}{8} + \\frac{3 \\sqrt{6}}{2}$.\n7) x2 = $- \\frac{13}{8} - \\frac{\\sqrt{1225 - 456 \\sqrt{6}}}{8} + \\frac{3 \\sqrt{6}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{13}{8} + \\frac{\\sqrt{1225 - 456\\cdot \\sqrt{6}}}{8} + \\frac{3\\cdot \\sqrt{6}}{2}, - \\frac{13}{8} - \\frac{\\sqrt{1225 - 456\\cdot \\sqrt{6}}}{8} + \\frac{3\\cdot \\sqrt{6}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{13}{8} + \\frac{\\sqrt{1225 - 456\\cdot \\sqrt{6}}}{8} + \\frac{3\\cdot \\sqrt{6}}{2}, - \\frac{13}{8} - \\frac{\\sqrt{1225 - 456\\cdot \\sqrt{6}}}{8} + \\frac{3\\cdot \\sqrt{6}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a583cbfc13e4e9a7", "problem_id": "ba579723ea38e816", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} x^{2} + 4 x + 14$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to x^{2} + 3 x + 14.\n3) Then the limit is 32.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: 35.\nFinal answer: $35$", "final_answer": "35", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a335fbf38694d459", "problem_id": "8dafeffed0e67c36", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} 2 x - 9$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to x - 9.\n3) Then the limit is -13.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: -17.\nFinal answer: $-17$", "final_answer": "-17", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "208550242e26b1eb", "problem_id": "4c35a5b196a0cb85", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡0 (mod 3), x≡2 (mod 7). Give the solution modulo 21.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 21.\n3) Compute the CRT solution to get x ≡ 9 (mod 21).\nFinal answer: $x\\equiv 9\\pmod{21}$", "final_answer": "x\\equiv 9\\pmod{21}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7b27f2ea9c81754f", "problem_id": "133b8471053c54fd", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 33 x + 90$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} - 33 x + 90$ = $3 \\left(x - 6\\right) \\left(x - 5\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $6$.\n5) From the second factor: x = $5$.\nFinal answer: $x \\in \\{6, 5\\}$", "final_answer": "x \\in \\{6, 5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "272507d4da2caf5e", "problem_id": "133b8471053c54fd", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 33 x + 90$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} - 33 x + 90$ = $3 \\left(x - 6\\right) \\left(x - 5\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFinal answer: $x \\in \\{6, 5\\}$", "final_answer": "x \\in \\{6, 5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b48834bef77ef109", "problem_id": "90d29afc848f3c0e", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡1 (mod 9), x≡1 (mod 13). Give the solution modulo 117.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 117.\nCompute the CRT solution to get x ≡ 1 (mod 117).\nCheck: 1 mod 9=1 and mod 13=1.\nFinal answer: $x\\equiv 1\\pmod{117}$", "final_answer": "x\\equiv 1\\pmod{117}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "53695fa1e8b10903", "problem_id": "90d29afc848f3c0e", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡1 (mod 9), x≡1 (mod 13). Give the solution modulo 117.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 117.\n3) Compute the CRT solution to get x ≡ 1 (mod 117).\nFinal answer: $x\\equiv 1\\pmod{117}$", "final_answer": "x\\equiv 1\\pmod{117}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "17a9b07e2bea8604", "problem_id": "9bc1af7f441b1196", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} x^{4} - x^{3} - 3 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $x^{4} - x^{3} - 3 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $\\frac{3 \\pi \\left(- 25 \\pi^{3} - 60 \\pi^{2} - 20 \\pi + 40 + 44 \\pi^{4}\\right)}{20}$.\nFinal answer: $\\frac{3\\cdot \\pi\\cdot \\left(- 25\\cdot \\pi^{3} - 60\\cdot \\pi^{2} - 20\\cdot \\pi + 40 + 44\\cdot \\pi^{4}\\right)}{20}$", "final_answer": "\\frac{3\\cdot \\pi\\cdot \\left(- 25\\cdot \\pi^{3} - 60\\cdot \\pi^{2} - 20\\cdot \\pi + 40 + 44\\cdot \\pi^{4}\\right)}{20}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9b99fb3f17535b96", "problem_id": "9bc1af7f441b1196", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} x^{4} - x^{3} - 3 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $x^{4} - x^{3} - 3 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $\\frac{3 \\pi \\left(- 25 \\pi^{3} - 60 \\pi^{2} - 20 \\pi + 40 + 44 \\pi^{4}\\right)}{20}$.\n4) Simplify the result.\nFinal answer: $\\frac{3\\cdot \\pi\\cdot \\left(- 25\\cdot \\pi^{3} - 60\\cdot \\pi^{2} - 20\\cdot \\pi + 40 + 44\\cdot \\pi^{4}\\right)}{20}$", "final_answer": "\\frac{3\\cdot \\pi\\cdot \\left(- 25\\cdot \\pi^{3} - 60\\cdot \\pi^{2} - 20\\cdot \\pi + 40 + 44\\cdot \\pi^{4}\\right)}{20}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "82111b6313ebe814", "problem_id": "d05e6e09caef88ab", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=1$.", "output": "Differentiate: f'(x) = $9 x^{2} + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\nSlope at x=1: m = f'(1) = $- 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 7$.\nPoint on curve: (1, f(1)) = (1, $-1 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -1 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} = - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 7(x − 1).\nOptionally expand: y = $- 2 x \\sin{\\left(2 \\right)} + \\frac{x e^{\\frac{1}{2}}}{2} + 7 x - 8 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2 \\sin{\\left(2 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 7\\cdot x - 8 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 7\\cdot x - 8 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "adbaf6974a2bbf79", "problem_id": "d05e6e09caef88ab", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=1$.", "output": "Differentiate: f'(x) = $9 x^{2} + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\nSlope at x=1: m = f'(1) = $- 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 7$.\nPoint on curve: (1, f(1)) = (1, $-1 + \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}}$).\nFinal answer: $y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 7\\cdot x - 8 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + 7\\cdot x - 8 + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "71dbd9a88b09a810", "problem_id": "c3ac06b1f5cfea50", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$11 x + 3 y$ = $-64$\n$11 x - 8 y$ = $-31$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $11$: $121 x + 33 y$ = $-704$.\nMultiply the second equation by $11$: $121 x - 88 y$ = $-341$.\nSubtract to eliminate x: $121 y$ = $-363$.\nSolve for y: y = $-3$.\nSubstitute back to find x: x = $-5$.\nCheck: plug (x,y)=(-5,-3) into both equations.\nFinal answer: $(x,y)=\\left(-5, -3\\right)$", "final_answer": "(x,y)=\\left(-5, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6d0f75a2721021d0", "problem_id": "c3ac06b1f5cfea50", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$11 x + 3 y$ = $-64$\n$11 x - 8 y$ = $-31$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $11$: $121 x + 33 y$ = $-704$.\n3) Multiply the second equation by $11$: $121 x - 88 y$ = $-341$.\n4) Subtract to eliminate x: $121 y$ = $-363$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, -3\\right)$", "final_answer": "(x,y)=\\left(-5, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ce35aebd2ecd6716", "problem_id": "cf71f1bbdc595100", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A cyclist rides at 76 miles per hour for 1 hours, then at 84 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 76 × 1 = 76 miles.\nSecond leg distance = 84 × 1 = 84 miles.\nTotal distance = 76 + 84 = 160 miles.\nFinal answer: $160$", "final_answer": "160", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "024f5d8f4fc43cb6", "problem_id": "cf71f1bbdc595100", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A cyclist rides at 76 miles per hour for 1 hours, then at 84 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 76 × 1 = 76 miles.\n2) Second leg distance = 84 × 1 = 84 miles.\n3) Total distance = 76 + 84 = 160 miles.\nFinal answer: $160$", "final_answer": "160", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "971361f44a0f9367", "problem_id": "ac7aa19d28ca3e0c", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 2 x^{3} + 8 x^{2} - 22 x + 68$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 2 x^{4} + 2 x^{3} + 2 x^{2} + 2 x - 1$ and a = -3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 8 x^{3} + 6 x^{2} + 4 x + 2$.\nEvaluate at a: p'(-3) = $260$.\nCheck: the identity gives $260$.\nFinal answer: $260$", "final_answer": "260", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a6dc968ae8e51b1f", "problem_id": "e3905de450773855", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} - x - 4$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to - 2 x - 4.\n3) Then the limit is -6.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: -5.\nFinal answer: $-5$", "final_answer": "-5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ffd63f0e9899ec76", "problem_id": "bc75e80f20e6e3de", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} - x^{3} - 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{3} - 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\pi).\n3) Subtract: F(\\pi) − F(- \\pi) = $2 \\pi \\left(2 - \\pi^{2}\\right)$.\n4) Simplify the result.\nFinal answer: $2\\cdot \\pi\\cdot \\left(2 - \\pi^{2}\\right)$", "final_answer": "2\\cdot \\pi\\cdot \\left(2 - \\pi^{2}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "630e7cde582a51be", "problem_id": "bc75e80f20e6e3de", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} - x^{3} - 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{3} - 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\nEvaluate at the bounds: F(\\pi) and F(- \\pi).\nSubtract: F(\\pi) − F(- \\pi) = $2 \\pi \\left(2 - \\pi^{2}\\right)$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $2\\cdot \\pi\\cdot \\left(2 - \\pi^{2}\\right)$", "final_answer": "2\\cdot \\pi\\cdot \\left(2 - \\pi^{2}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f9e3622c42c68f5d", "problem_id": "ae74c66d8f3df446", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡7 (mod 11), x≡6 (mod 13). Give the solution modulo 143.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 143.\n3) Compute the CRT solution to get x ≡ 84 (mod 143).\nFinal answer: $x\\equiv 84\\pmod{143}$", "final_answer": "x\\equiv 84\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "acc18d8209764610", "problem_id": "208ef6ba63abd874", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{3} - 2 x^{2} + 3 x - 4$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x^{2}) = - 4 x^{2}$.\n3) So $f'(x) = 2 x^{2} + 3$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x^{2}) = - 4 x$.\nTherefore the correct derivative is $f'(x) = 6 x^{2} - 4 x + 3$.\nFinal answer: $6 x^{2} - 4 x + 3$", "final_answer": "6 x^{2} - 4 x + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "36ca3a68c8971acc", "problem_id": "66d0068f08e7eacf", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 22 red balls and 15 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(37,7).\nFavorable samples: choose k reds and n−k blues: C(22,2)·C(15,5).\nCompute totals: C(37,7) = 10295472, and favorable = 693693.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{21021}{311984}$.\nFinal answer: $\\frac{21021}{311984}$", "final_answer": "\\frac{21021}{311984}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "05e47d85e6636d5b", "problem_id": "66d0068f08e7eacf", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "An urn has 22 red balls and 15 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(37,7).\n3) Favorable samples: choose k reds and n−k blues: C(22,2)·C(15,5).\n4) Compute totals: C(37,7) = 10295472, and favorable = 693693.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{21021}{311984}$", "final_answer": "\\frac{21021}{311984}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4aa6782581511720", "problem_id": "94e7b436be7dc82e", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Two friends split $525 in the ratio 1:2. How much does each receive?", "output": "1) Total ratio parts = 1+2 = 3.\n2) Each part = 525 ÷ 3 = 175.\n3) First share = 1 × 175 = 175.\n4) Second share = 2 × 175 = 350.\nFinal answer: $350$", "final_answer": "350", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fc3e2a5c3787fe1b", "problem_id": "94e7b436be7dc82e", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Two friends split $525 in the ratio 1:2. How much does each receive?", "output": "Total ratio parts = 1+2 = 3.\nEach part = 525 ÷ 3 = 175.\nFirst share = 1 × 175 = 175.\nSecond share = 2 × 175 = 350.\nCheck: 175+350=525.\nFinal answer: $350$", "final_answer": "350", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dae8ae573c4a437d", "problem_id": "3b357082ec16db46", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "An urn has 27 red balls and 28 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(55,7).\nFavorable samples: choose k reds and n−k blues: C(27,2)·C(28,5).\nFinal answer: $\\frac{8424}{49555}$", "final_answer": "\\frac{8424}{49555}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e38cb10727f2e7cc", "problem_id": "3b357082ec16db46", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 27 red balls and 28 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(55,7).\n3) Favorable samples: choose k reds and n−k blues: C(27,2)·C(28,5).\n4) Compute totals: C(55,7) = 202927725, and favorable = 34496280.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{8424}{49555}$", "final_answer": "\\frac{8424}{49555}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4afd64d16e391d88", "problem_id": "b05a48c0ac086cfb", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A cyclist rides at 50 miles per hour for 3/2 hours, then at 28 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 50 × 3/2 = 75 miles.\n2) Second leg distance = 28 × 3/2 = 42 miles.\n3) Total distance = 75 + 42 = 117 miles.\nFinal answer: $117$", "final_answer": "117", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d9e1a5ed7d69d642", "problem_id": "683f51bf6c7a9c8e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{4} - x^{3} - x^{2} - 2 x - 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x) = - 2 x$.\n3) So $f'(x) = 4 x^{3} - 3 x^{2} - 4 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x) = -2$.\nTherefore the correct derivative is $f'(x) = 4 x^{3} - 3 x^{2} - 2 x - 2$.\nFinal answer: $4 x^{3} - 3 x^{2} - 2 x - 2$", "final_answer": "4 x^{3} - 3 x^{2} - 2 x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b90721d86ebdff62", "problem_id": "27b3ab511b08e064", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} - 2 x - 5$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to - 3 x - 5.\n3) Then the limit is -8.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: -7.\nFinal answer: $-7$", "final_answer": "-7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "1b2076ba90e06fc2", "problem_id": "2b5ccd7e9d60b847", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many binary strings of length 11 contain exactly 2 ones with no two ones adjacent?", "output": "Reserve 1 mandatory zeros between the 2 ones.\nThis leaves n−k+1 = 10 valid slots for the ones.\nChoose the slots: C(10,2) = 45.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $45$", "final_answer": "45", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "19b5c2372d14b434", "problem_id": "aa09d628565fb217", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_4=14$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=14 and k=4.\n4) Compute C(17,3) = 680.\nFinal answer: $680$", "final_answer": "680", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7b5ed1463be37d56", "problem_id": "aa09d628565fb217", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_4=14$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=14 and k=4.\nCompute C(17,3) = 680.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $680$", "final_answer": "680", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "90a12fd8df92f0f6", "problem_id": "30d6cf9a97495d00", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{x^{2} + 3}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 3}) = \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} - \\frac{2 x^{4}}{2 x^{4} + 12 x^{2} + 18} + \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 6 x^{2} + 9} + \\frac{6 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} + \\frac{12 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} - \\frac{12 x^{2}}{2 x^{4} + 12 x^{2} + 18} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 6} - \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3} + \\frac{36 x}{2 x^{4} + 12 x^{2} + 18} - \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 3} + \\frac{9 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} + \\frac{18 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} - \\frac{18}{2 x^{4} + 12 x^{2} + 18}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 3}) = - \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3\\right)^{2}} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x^{2} + 3\\right)} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3} + \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 3}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 4 x \\left(- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} + 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x^{2} + 3\\right)^{2}}$.\nFinal answer: $\\frac{- 4 x \\left(- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} + 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x^{2} + 3\\right)^{2}}$", "final_answer": "\\frac{- 4 x \\left(- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} + 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x^{2} + 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b94303ea249274e8", "problem_id": "a6287f0c0734c26f", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{5}$. If it is flipped 21 times, what is the probability of exactly 11 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=21, k=11, p=$\\frac{3}{5}$ and simplify.\nHere C(21,11) = 352716.\nCheck: expression simplifies to $\\frac{63982163202048}{476837158203125}$.\nFinal answer: $\\frac{63982163202048}{476837158203125}$", "final_answer": "\\frac{63982163202048}{476837158203125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "85f93b249000cb24", "problem_id": "9ecfb39a34acea9c", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡0 (mod 7), x≡4 (mod 9). Give the solution modulo 63.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 63.\nCompute the CRT solution to get x ≡ 49 (mod 63).\nCheck: 49 mod 7=0 and mod 9=4.\nFinal answer: $x\\equiv 49\\pmod{63}$", "final_answer": "x\\equiv 49\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "126c083ce6045feb", "problem_id": "9ecfb39a34acea9c", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the congruences: x≡0 (mod 7), x≡4 (mod 9). Give the solution modulo 63.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 63.\n3) Compute the CRT solution to get x ≡ 49 (mod 63).\nFinal answer: $x\\equiv 49\\pmod{63}$", "final_answer": "x\\equiv 49\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8050de9348f3efa8", "problem_id": "fc368635571647dc", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{11 x}{2} + \\frac{9}{4}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$\\frac{11}{2}$, c=$\\frac{9}{4}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{49}{4}$, so √Δ = $\\frac{7}{2}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{1}{2}$.\nx2 = $- \\frac{9}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{1}{2}, - \\frac{9}{4}\\}$", "final_answer": "x \\in \\{- \\frac{1}{2}, - \\frac{9}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7180f9f5b98a0dcc", "problem_id": "fc368635571647dc", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{11 x}{2} + \\frac{9}{4}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$\\frac{11}{2}$, c=$\\frac{9}{4}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{- \\frac{1}{2}, - \\frac{9}{4}\\}$", "final_answer": "x \\in \\{- \\frac{1}{2}, - \\frac{9}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9feb2ef6817107ec", "problem_id": "d5970969212e7c28", "style": "answer_only", "instruction": "Final answer only.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 x - 9$.", "output": "$-15$", "final_answer": "-15", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5bbd5ff3ce5206ef", "problem_id": "d5970969212e7c28", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 x - 9$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $3 x^{2} - 3 x - 2$ and a = -2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $6 x - 3$.\nEvaluate at a: p'(-2) = $-15$.\nCheck: the identity gives $-15$.\nFinal answer: $-15$", "final_answer": "-15", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f20f29a22a460e89", "problem_id": "90e73896315f0f34", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 17 times, what is the probability of exactly 11 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=17, k=11, p=$\\frac{4}{5}$ and simplify.\n4) Here C(17,11) = 12376.\nFinal answer: $\\frac{51908706304}{762939453125}$", "final_answer": "\\frac{51908706304}{762939453125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c7a085c9339c8caf", "problem_id": "3451b6d8ff262c62", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 2 x^{2} \\sin{\\left(x \\right)}) = - 2 x^{2} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - x^{3} \\cos{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 2 x^{2} \\sin{\\left(x \\right)}) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - x^{3} \\cos{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.\nFinal answer: $- x^{3} \\cos{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$", "final_answer": "- x^{3} \\cos{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e708138c9e747fc5", "problem_id": "d682eeff446385a2", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 15 red balls and 13 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(28,11).\nFavorable samples: choose k reds and n−k blues: C(15,2)·C(13,9).\nCompute totals: C(28,11) = 21474180, and favorable = 75075.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{55}{15732}$.\nFinal answer: $\\frac{55}{15732}$", "final_answer": "\\frac{55}{15732}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "93f9670bb12f7e44", "problem_id": "d682eeff446385a2", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 15 red balls and 13 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(28,11).\n3) Favorable samples: choose k reds and n−k blues: C(15,2)·C(13,9).\n4) Compute totals: C(28,11) = 21474180, and favorable = 75075.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{55}{15732}$", "final_answer": "\\frac{55}{15732}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4ccb6e42c541148e", "problem_id": "f83aac52927c79be", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$7 x - 5 y$ = $4$\n$3 x - 2 y$ = $1$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $3$: $21 x - 15 y$ = $12$.\n3) Multiply the second equation by $7$: $21 x - 14 y$ = $7$.\n4) Subtract to eliminate x: $- y$ = $5$.\n5) Solve for y: y = $-5$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, -5\\right)$", "final_answer": "(x,y)=\\left(-3, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2c4645a5a6657331", "problem_id": "f83aac52927c79be", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$7 x - 5 y$ = $4$\n$3 x - 2 y$ = $1$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $21 x - 15 y$ = $12$.\nMultiply the second equation by $7$: $21 x - 14 y$ = $7$.\nSubtract to eliminate x: $- y$ = $5$.\nSolve for y: y = $-5$.\nSubstitute back to find x: x = $-3$.\nCheck: plug (x,y)=(-3,-5) into both equations.\nFinal answer: $(x,y)=\\left(-3, -5\\right)$", "final_answer": "(x,y)=\\left(-3, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "29da4e6b28107463", "problem_id": "c50ffcb8639d21e8", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + \\frac{5 x}{3} - \\frac{8}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{5}{3}$, c=$- \\frac{8}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{121}{9}$, so √Δ = $\\frac{11}{3}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $1$.\n7) x2 = $- \\frac{8}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{1, - \\frac{8}{3}\\}$", "final_answer": "x \\in \\{1, - \\frac{8}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4144a84af01a6558", "problem_id": "c50ffcb8639d21e8", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + \\frac{5 x}{3} - \\frac{8}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{5}{3}$, c=$- \\frac{8}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{121}{9}$, so √Δ = $\\frac{11}{3}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $1$.\nx2 = $- \\frac{8}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{1, - \\frac{8}{3}\\}$", "final_answer": "x \\in \\{1, - \\frac{8}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "63dd02bb7290e7a9", "problem_id": "df9c13e6d5e72633", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 x - \\frac{2 \\sqrt{11} x}{3} + \\frac{20}{9} + \\frac{8 \\sqrt{11}}{9}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$-3 - \\frac{2 \\sqrt{11}}{3}$, c=$\\frac{20}{9} + \\frac{8 \\sqrt{11}}{9}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{4 \\sqrt{11}}{9} + 5$, so √Δ = $\\sqrt{\\frac{4 \\sqrt{11}}{9} + 5}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{11}}{3} + \\frac{\\sqrt{4 \\sqrt{11} + 45}}{6} + \\frac{3}{2}$.\nx2 = $- \\frac{\\sqrt{4 \\sqrt{11} + 45}}{6} + \\frac{\\sqrt{11}}{3} + \\frac{3}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{11}}{3} + \\frac{\\sqrt{4\\cdot \\sqrt{11} + 45}}{6} + \\frac{3}{2}, - \\frac{\\sqrt{4\\cdot \\sqrt{11} + 45}}{6} + \\frac{\\sqrt{11}}{3} + \\frac{3}{2}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{11}}{3} + \\frac{\\sqrt{4\\cdot \\sqrt{11} + 45}}{6} + \\frac{3}{2}, - \\frac{\\sqrt{4\\cdot \\sqrt{11} + 45}}{6} + \\frac{\\sqrt{11}}{3} + \\frac{3}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "84b9849901067fa9", "problem_id": "43199a59849d810a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(3 x^{3} - 3 x^{2} + 2\\right) \\cos{\\left(x - 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 x^{3} \\cos{\\left(x - 3 \\right)}) = - 3 x^{3} \\sin{\\left(x - 3 \\right)}$.\n3) So $f'(x) = - 3 x^{3} \\sin{\\left(x - 3 \\right)} + 3 x^{2} \\sin{\\left(x - 3 \\right)} + x e^{x} - 6 x \\cos{\\left(x - 3 \\right)} + 3 e^{x} - 2 \\sin{\\left(x - 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 x^{3} \\cos{\\left(x - 3 \\right)}) = - 3 x^{3} \\sin{\\left(x - 3 \\right)} + 9 x^{2} \\cos{\\left(x - 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = 3 x \\left(3 x - 2\\right) \\cos{\\left(x - 3 \\right)} + \\left(x + 2\\right) e^{x} - \\left(3 x^{2} \\left(x - 1\\right) + 2\\right) \\sin{\\left(x - 3 \\right)} + e^{x}$.\nFinal answer: $3 x \\left(3 x - 2\\right) \\cos{\\left(x - 3 \\right)} + \\left(x + 2\\right) e^{x} - \\left(3 x^{2} \\left(x - 1\\right) + 2\\right) \\sin{\\left(x - 3 \\right)} + e^{x}$", "final_answer": "3 x \\left(3 x - 2\\right) \\cos{\\left(x - 3 \\right)} + \\left(x + 2\\right) e^{x} - \\left(3 x^{2} \\left(x - 1\\right) + 2\\right) \\sin{\\left(x - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8bfb6751f12af2d1", "problem_id": "4115ba855c0d2f15", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 3$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} - 3$ = $3 \\left(x - 1\\right) \\left(x + 1\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $1$.\n5) From the second factor: x = $-1$.\nFinal answer: $x \\in \\{1, -1\\}$", "final_answer": "x \\in \\{1, -1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ca694413d721f1b9", "problem_id": "92fe8011858adae9", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 3 x^{3} - 6 x^{2} + 21 x - 60$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} - 6 x^{2} + 20 x - 60.\n3) Then the limit is -255.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -258.\nFinal answer: $-258$", "final_answer": "-258", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "343be119ce931f7a", "problem_id": "4f515adb1a211f69", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 25 red balls and 11 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(36,11).\n3) Favorable samples: choose k reds and n−k blues: C(25,2)·C(11,9).\n4) Compute totals: C(36,11) = 600805296, and favorable = 16500.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1375}{50067108}$", "final_answer": "\\frac{1375}{50067108}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c3154c3e7701f0e9", "problem_id": "4f515adb1a211f69", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 25 red balls and 11 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(36,11).\nFavorable samples: choose k reds and n−k blues: C(25,2)·C(11,9).\nCompute totals: C(36,11) = 600805296, and favorable = 16500.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1375}{50067108}$.\nFinal answer: $\\frac{1375}{50067108}$", "final_answer": "\\frac{1375}{50067108}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "885e04f2dc9f8248", "problem_id": "d2266df0b4a0f3d1", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(2 x^{3} - 3 x^{2} - 3 x\\right) \\cos{\\left(3 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 3 x^{2} \\cos{\\left(3 x \\right)}) = 3 x^{2} \\sin{\\left(3 x \\right)}$.\n3) So $f'(x) = - 6 x^{3} \\sin{\\left(3 x \\right)} + 3 x^{2} \\sin{\\left(3 x \\right)} + 6 x^{2} \\cos{\\left(3 x \\right)} + 2 x e^{x} + 9 x \\sin{\\left(3 x \\right)} - 4 e^{x} - 3 \\cos{\\left(3 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 3 x^{2} \\cos{\\left(3 x \\right)}) = 9 x^{2} \\sin{\\left(3 x \\right)} - 6 x \\cos{\\left(3 x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 3 x \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x \\right)} + 2 \\left(x - 3\\right) e^{x} + 3 \\left(2 x^{2} - 2 x - 1\\right) \\cos{\\left(3 x \\right)} + 2 e^{x}$.\nFinal answer: $- 3 x \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x \\right)} + 2 \\left(x - 3\\right) e^{x} + 3 \\left(2 x^{2} - 2 x - 1\\right) \\cos{\\left(3 x \\right)} + 2 e^{x}$", "final_answer": "- 3 x \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x \\right)} + 2 \\left(x - 3\\right) e^{x} + 3 \\left(2 x^{2} - 2 x - 1\\right) \\cos{\\left(3 x \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "85b26d50e7bc77eb", "problem_id": "723ec62673b50dfa", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$7 x + 5 y$ = $-35$\n$3 x + 4 y$ = $-28$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $21 x + 15 y$ = $-105$.\nMultiply the second equation by $7$: $21 x + 28 y$ = $-196$.\nSubtract to eliminate x: $- 13 y$ = $91$.\nSolve for y: y = $-7$.\nSubstitute back to find x: x = $0$.\nCheck: plug (x,y)=(0,-7) into both equations.\nFinal answer: $(x,y)=\\left(0, -7\\right)$", "final_answer": "(x,y)=\\left(0, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1d84df93b9c1bec7", "problem_id": "6a66d1a56ad1f380", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 10 liters of a 37% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 10−x liters is 50% solution.\nSalt equation: 25x + 50(10−x) = 37·10.\n(25-50)x = (37-50)·10 ⇒ x = 26/5.\nThen 10−x = 24/5.\nCheck: 25·26/5 + 50·24/5 = 37·10.\nFinal answer: $26/5 L of 25% and 24/5 L of 50%$", "final_answer": "26/5 L of 25% and 24/5 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "47fbd485c729b5c8", "problem_id": "6a66d1a56ad1f380", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "You want 10 liters of a 37% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 10−x liters is 50% solution.\n2) Salt equation: 25x + 50(10−x) = 37·10.\n3) (25-50)x = (37-50)·10 ⇒ x = 26/5.\n4) Then 10−x = 24/5.\nFinal answer: $26/5 L of 25% and 24/5 L of 50%$", "final_answer": "26/5 L of 25% and 24/5 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0ad5c61d8e612f42", "problem_id": "7248c21ddaad7476", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "You want 19 liters of a 30% salt solution. You have a 15% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 19−x liters is 40% solution.\n2) Salt equation: 15x + 40(19−x) = 30·19.\n3) (15-40)x = (30-40)·19 ⇒ x = 38/5.\n4) Then 19−x = 57/5.\nFinal answer: $38/5 L of 15% and 57/5 L of 40%$", "final_answer": "38/5 L of 15% and 57/5 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b7842251ac741d2d", "problem_id": "7248c21ddaad7476", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 19 liters of a 30% salt solution. You have a 15% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 19−x liters is 40% solution.\nSalt equation: 15x + 40(19−x) = 30·19.\n(15-40)x = (30-40)·19 ⇒ x = 38/5.\nThen 19−x = 57/5.\nCheck: 15·38/5 + 40·57/5 = 30·19.\nFinal answer: $38/5 L of 15% and 57/5 L of 40%$", "final_answer": "38/5 L of 15% and 57/5 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8d5e83bd056c359b", "problem_id": "7e207b8828a53ff3", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$3 x + 3 y$ = $21$\n$- 3 y$ = $-6$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $3$: $- 9 y$ = $-18$.\n4) Subtract to eliminate x: $9 y$ = $18$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, 2\\right)$", "final_answer": "(x,y)=\\left(5, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1524e57f0ed79a74", "problem_id": "4de46a39da1882c5", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(- 2 x^{3} - 3\\right) \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x e^{x}) = x e^{x}$.\n3) So $f'(x) = 4 x^{3} \\sin{\\left(2 x \\right)} - 6 x^{2} \\cos{\\left(2 x \\right)} + x e^{x} + 2 e^{x} + 6 \\sin{\\left(2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x e^{x}) = x e^{x} + e^{x}$.\nTherefore the correct derivative is $f'(x) = - 6 x^{2} \\cos{\\left(2 x \\right)} + \\left(x + 2\\right) e^{x} - 2 \\left(- 2 x^{3} - 3\\right) \\sin{\\left(2 x \\right)} + e^{x}$.\nFinal answer: $- 6 x^{2} \\cos{\\left(2 x \\right)} + \\left(x + 2\\right) e^{x} - 2 \\left(- 2 x^{3} - 3\\right) \\sin{\\left(2 x \\right)} + e^{x}$", "final_answer": "- 6 x^{2} \\cos{\\left(2 x \\right)} + \\left(x + 2\\right) e^{x} - 2 \\left(- 2 x^{3} - 3\\right) \\sin{\\left(2 x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d28c78727197e5df", "problem_id": "c0642024b6190ae5", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "You want 50 liters of a 35% salt solution. You have a 15% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 50−x liters is 70% solution.\nSalt equation: 15x + 70(50−x) = 35·50.\n(15-70)x = (35-70)·50 ⇒ x = 350/11.\nFinal answer: $350/11 L of 15% and 200/11 L of 70%$", "final_answer": "350/11 L of 15% and 200/11 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fb7f4c16dfbf21eb", "problem_id": "c0642024b6190ae5", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "You want 50 liters of a 35% salt solution. You have a 15% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 50−x liters is 70% solution.\n2) Salt equation: 15x + 70(50−x) = 35·50.\n3) (15-70)x = (35-70)·50 ⇒ x = 350/11.\n4) Then 50−x = 200/11.\nFinal answer: $350/11 L of 15% and 200/11 L of 70%$", "final_answer": "350/11 L of 15% and 200/11 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cc6a500cb8bb6353", "problem_id": "5dca2162731b3139", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A store sells a backpack for $705. It is discounted by 25%. You buy 8 backpacks. What is the total cost?", "output": "1) Discounted price = 705 × (1 − 25/100) = 705 × 75/100 = 2115/4.\n2) Subtotal = 8 × 2115/4 = 4230.\nFinal answer: $4230$", "final_answer": "4230", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "806153ebe566ba1a", "problem_id": "5dca2162731b3139", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $705. It is discounted by 25%. You buy 8 backpacks. What is the total cost?", "output": "Discounted price = 705 × (1 − 25/100) = 705 × 75/100 = 2115/4.\nSubtotal = 8 × 2115/4 = 4230.\nCheck: recomputing the formula yields 4230.\nFinal answer: $4230$", "final_answer": "4230", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3d7be344f3d9428c", "problem_id": "a2cbe723ec626207", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 9 x + 4$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$-9$, c=$4$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $49$, so √Δ = $7$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $4$.\n7) x2 = $\\frac{1}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{4, \\frac{1}{2}\\}$", "final_answer": "x \\in \\{4, \\frac{1}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5fd95348ba2b3f11", "problem_id": "a2cbe723ec626207", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 9 x + 4$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$-9$, c=$4$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $49$, so √Δ = $7$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $4$.\nx2 = $\\frac{1}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{4, \\frac{1}{2}\\}$", "final_answer": "x \\in \\{4, \\frac{1}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f0a4d465358d5e03", "problem_id": "e5d248107fd20ff5", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{28 x}{3} + \\frac{16}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$\\frac{28}{3}$, c=$\\frac{16}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{400}{9}$, so √Δ = $\\frac{20}{3}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{2}{3}$.\n7) x2 = $-4$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{2}{3}, -4\\}$", "final_answer": "x \\in \\{- \\frac{2}{3}, -4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "473cf04012c10629", "problem_id": "e5d248107fd20ff5", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{28 x}{3} + \\frac{16}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$\\frac{28}{3}$, c=$\\frac{16}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{400}{9}$, so √Δ = $\\frac{20}{3}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{2}{3}$.\nx2 = $-4$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{2}{3}, -4\\}$", "final_answer": "x \\in \\{- \\frac{2}{3}, -4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cf1e374e6a36e4c3", "problem_id": "5b15246bdec86269", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- 3 x - y$ = $-10$\n$3 x + 2 y$ = $11$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $- 9 x - 3 y$ = $-30$.\nMultiply the second equation by $-3$: $- 9 x - 6 y$ = $-33$.\nSubtract to eliminate x: $3 y$ = $3$.\nSolve for y: y = $1$.\nSubstitute back to find x: x = $3$.\nCheck: plug (x,y)=(3,1) into both equations.\nFinal answer: $(x,y)=\\left(3, 1\\right)$", "final_answer": "(x,y)=\\left(3, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fd2da829fb2b3065", "problem_id": "5b15246bdec86269", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$- 3 x - y$ = $-10$\n$3 x + 2 y$ = $11$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $3$: $- 9 x - 3 y$ = $-30$.\n3) Multiply the second equation by $-3$: $- 9 x - 6 y$ = $-33$.\n4) Subtract to eliminate x: $3 y$ = $3$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, 1\\right)$", "final_answer": "(x,y)=\\left(3, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4c6ca7a1de51f473", "problem_id": "97f3120141571f31", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A disease has prevalence $\\frac{3}{50}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{99}{1250}$.\n3) Compute P(D|+) = $\\frac{85}{132}$.\nFinal answer: $\\frac{85}{132}$", "final_answer": "\\frac{85}{132}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "901a7b6430426282", "problem_id": "97f3120141571f31", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "A disease has prevalence $\\frac{3}{50}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{99}{1250}$.\nCompute P(D|+) = $\\frac{85}{132}$.\nFinal answer: $\\frac{85}{132}$", "final_answer": "\\frac{85}{132}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f6a4e83807c1cec0", "problem_id": "d19acf513dbb0394", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A store sells a backpack for $300. It is discounted by 25%. You buy 5 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 300 × (1 − 25/100) = 300 × 75/100 = 225.\n2) Subtotal = 5 × 225 = 1125.\n3) Total with tax = 1125 × 110/100 = 2475/2.\nFinal answer: $\\frac{2475}{2}$", "final_answer": "\\frac{2475}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "38b76f9c36cad1f1", "problem_id": "d37a8dd6d1d2c67d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{3} - 3 x^{2} - 2 x - 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x) = - 2 x$.\n3) So $f'(x) = - 3 x^{2} - 8 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x) = -2$.\nTherefore the correct derivative is $f'(x) = - 3 x^{2} - 6 x - 2$.\nFinal answer: $- 3 x^{2} - 6 x - 2$", "final_answer": "- 3 x^{2} - 6 x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5438cf778b31d58c", "problem_id": "045fd91fec7f01b6", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 15 red balls and 11 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(26,9).\n3) Favorable samples: choose k reds and n−k blues: C(15,3)·C(11,6).\n4) Compute totals: C(26,9) = 3124550, and favorable = 210210.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{147}{2185}$", "final_answer": "\\frac{147}{2185}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8731762cc8ab085d", "problem_id": "045fd91fec7f01b6", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 15 red balls and 11 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(26,9).\nFavorable samples: choose k reds and n−k blues: C(15,3)·C(11,6).\nCompute totals: C(26,9) = 3124550, and favorable = 210210.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{147}{2185}$.\nFinal answer: $\\frac{147}{2185}$", "final_answer": "\\frac{147}{2185}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e6e254f82b8b0929", "problem_id": "d7e36d94105da57f", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} 3 x^{3} + x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{3} + x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4$.\n2) Evaluate at the bounds: F(\\pi) and F(0).\n3) Subtract: F(\\pi) − F(0) = $\\frac{\\pi \\left(- 24 \\pi - 48 + 4 \\pi^{2} + 9 \\pi^{3}\\right)}{12}$.\n4) Simplify the result.\nFinal answer: $\\frac{\\pi\\cdot \\left(- 24\\cdot \\pi - 48 + 4\\cdot \\pi^{2} + 9\\cdot \\pi^{3}\\right)}{12}$", "final_answer": "\\frac{\\pi\\cdot \\left(- 24\\cdot \\pi - 48 + 4\\cdot \\pi^{2} + 9\\cdot \\pi^{3}\\right)}{12}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3cb71d9d84f90683", "problem_id": "49578745ecf6750f", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - 2 x - 4$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to - 3 x - 4.\n3) Then the limit is -13.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: -10.\nFinal answer: $-10$", "final_answer": "-10", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f19c14a6fc07290f", "problem_id": "127845561a322a32", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{3} + 4 x^{2} + 7 x + 11$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{4} - x^{2} - 3 x - 2$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $8 x^{3} - 2 x - 3$.\nEvaluate at a: p'(2) = $57$.\nCheck: the identity gives $57$.\nFinal answer: $57$", "final_answer": "57", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3689527e932ebfc3", "problem_id": "127845561a322a32", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{3} + 4 x^{2} + 7 x + 11$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $2 x^{4} - x^{2} - 3 x - 2$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $8 x^{3} - 2 x - 3$.\n5) Evaluate at a: p'(2) = $57$.\nFinal answer: $57$", "final_answer": "57", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7489650abbee823e", "problem_id": "401967b13f7f0358", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A cyclist rides at 25 miles per hour for 1 hours, then at 52 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 25 × 1 = 25 miles.\n2) Second leg distance = 52 × 2 = 104 miles.\n3) Total distance = 25 + 104 = 129 miles.\nFinal answer: $129$", "final_answer": "129", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3a5ec3bdeda2247d", "problem_id": "401967b13f7f0358", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 25 miles per hour for 1 hours, then at 52 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 25 × 1 = 25 miles.\nSecond leg distance = 52 × 2 = 104 miles.\nTotal distance = 25 + 104 = 129 miles.\nCheck: adding both legs gives 129.\nFinal answer: $129$", "final_answer": "129", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "05edcef898343e07", "problem_id": "5cb873e2515a6e52", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$6 x - 4 y$ = $18$\n$x - 7 y$ = $22$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $6 x - 4 y$ = $18$.\n3) Multiply the second equation by $6$: $6 x - 42 y$ = $132$.\n4) Subtract to eliminate x: $38 y$ = $-114$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $1$.\nFinal answer: $(x,y)=\\left(1, -3\\right)$", "final_answer": "(x,y)=\\left(1, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "385db183d2212338", "problem_id": "5cb873e2515a6e52", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$6 x - 4 y$ = $18$\n$x - 7 y$ = $22$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $6 x - 4 y$ = $18$.\nMultiply the second equation by $6$: $6 x - 42 y$ = $132$.\nSubtract to eliminate x: $38 y$ = $-114$.\nSolve for y: y = $-3$.\nSubstitute back to find x: x = $1$.\nCheck: plug (x,y)=(1,-3) into both equations.\nFinal answer: $(x,y)=\\left(1, -3\\right)$", "final_answer": "(x,y)=\\left(1, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "79531aa6bd7d693c", "problem_id": "2527ff8a3f2741fb", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 6 red balls and 15 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(21,5).\n3) Favorable samples: choose k reds and n−k blues: C(6,1)·C(15,4).\n4) Compute totals: C(21,5) = 20349, and favorable = 8190.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{130}{323}$", "final_answer": "\\frac{130}{323}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "38f8ff616e8bd5be", "problem_id": "9673fa0311086037", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x^{3} + 5 x^{2} + 11 x + 20$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to x^{3} + 5 x^{2} + 10 x + 20.\n3) Then the limit is 68.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 70.\nFinal answer: $70$", "final_answer": "70", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "15654fba0a604b4e", "problem_id": "a50f95657eb1c781", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the congruences: x≡2 (mod 3), x≡0 (mod 11). Give the solution modulo 33.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 33.\nCompute the CRT solution to get x ≡ 11 (mod 33).\nFinal answer: $x\\equiv 11\\pmod{33}$", "final_answer": "x\\equiv 11\\pmod{33}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "209895e003444f6d", "problem_id": "a50f95657eb1c781", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡2 (mod 3), x≡0 (mod 11). Give the solution modulo 33.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 33.\n3) Compute the CRT solution to get x ≡ 11 (mod 33).\nFinal answer: $x\\equiv 11\\pmod{33}$", "final_answer": "x\\equiv 11\\pmod{33}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "77ed8d51a88c5055", "problem_id": "e360968035bc1178", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡6 (mod 9), x≡11 (mod 13). Give the solution modulo 117.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 117.\n3) Compute the CRT solution to get x ≡ 24 (mod 117).\nFinal answer: $x\\equiv 24\\pmod{117}$", "final_answer": "x\\equiv 24\\pmod{117}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8506d18ddd852895", "problem_id": "e360968035bc1178", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the congruences: x≡6 (mod 9), x≡11 (mod 13). Give the solution modulo 117.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 117.\nCompute the CRT solution to get x ≡ 24 (mod 117).\nFinal answer: $x\\equiv 24\\pmod{117}$", "final_answer": "x\\equiv 24\\pmod{117}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0923ca0bbe4c8b7e", "problem_id": "01c5ab669600b31d", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the multiplicative inverse of 23 modulo 43.", "output": "1) We want x such that 23x ≡ 1 (mod 43).\n2) Use extended Euclid to find s,t with 23s + 43t = 1.\n3) One Bezout identity is: 23·(15) + 43·(-8) = 1.\n4) Reduce s modulo 43: x ≡ 15 (mod 43).\nFinal answer: $x\\equiv 15\\pmod{43}$", "final_answer": "x\\equiv 15\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d9f632b8d24b6872", "problem_id": "01c5ab669600b31d", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 23 modulo 43.", "output": "We want x such that 23x ≡ 1 (mod 43).\nUse extended Euclid to find s,t with 23s + 43t = 1.\nOne Bezout identity is: 23·(15) + 43·(-8) = 1.\nReduce s modulo 43: x ≡ 15 (mod 43).\nCheck: 23·15 mod 43 = 1.\nFinal answer: $x\\equiv 15\\pmod{43}$", "final_answer": "x\\equiv 15\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7b5130a63a1d4d85", "problem_id": "dac9b6e574924d85", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 22 red balls and 12 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(34,9).\n3) Favorable samples: choose k reds and n−k blues: C(22,2)·C(12,7).\n4) Compute totals: C(34,9) = 52451256, and favorable = 182952.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{693}{198679}$", "final_answer": "\\frac{693}{198679}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b4378d2416dcbec4", "problem_id": "dac9b6e574924d85", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 22 red balls and 12 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(34,9).\nFavorable samples: choose k reds and n−k blues: C(22,2)·C(12,7).\nCompute totals: C(34,9) = 52451256, and favorable = 182952.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{693}{198679}$.\nFinal answer: $\\frac{693}{198679}$", "final_answer": "\\frac{693}{198679}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "59953ae4b6114696", "problem_id": "11024044ec7ec250", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{3} + 4 x^{2} + 9 x + 20$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} + 4 x^{2} + 8 x + 20.\n3) Then the limit is 68.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 70.\nFinal answer: $70$", "final_answer": "70", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e164f8c6ba837b42", "problem_id": "2d1a8562ee4b100d", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Worker A finishes a job in 16 hours and Worker B finishes it in 8 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/16 job/hour. B's rate = 1/8 job/hour.\n2) Combined rate = 1/16 + 1/8 = 3/16 job/hour.\n3) Time = 1 ÷ (combined rate) = 16/3 hours.\nFinal answer: $\\frac{16}{3}$", "final_answer": "\\frac{16}{3}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "34f8e106988e79ab", "problem_id": "344f33ecb0617f1c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} 3 x + 14$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to 2 x + 14.\n3) Then the limit is 22.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: 26.\nFinal answer: $26$", "final_answer": "26", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "9cd5ae0f615c019d", "problem_id": "8eaa9b2d0a26909e", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} - 4 x^{4} + 2 x^{3} - 4 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 4 x^{4} + 2 x^{3} - 4 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $- \\frac{\\pi^{5}}{20} - \\frac{\\pi^{3}}{3} + 4 + 3 \\pi$.\n4) Simplify the result.\nFinal answer: $- \\frac{\\pi^{5}}{20} - \\frac{\\pi^{3}}{3} + 4 + 3\\cdot \\pi$", "final_answer": "- \\frac{\\pi^{5}}{20} - \\frac{\\pi^{3}}{3} + 4 + 3\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "276c3c03df8425f7", "problem_id": "e285d2fcda429ab6", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=22$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=22 and k=5.\n4) Compute C(26,4) = 14950.\nFinal answer: $14950$", "final_answer": "14950", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a36028f7ccc5b711", "problem_id": "e285d2fcda429ab6", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=22$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=22 and k=5.\nCompute C(26,4) = 14950.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $14950$", "final_answer": "14950", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "492648852aae4524", "problem_id": "ca37e305e6636670", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{3 x^{2} - 3 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{2 x - 3}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 3}) = \\frac{2 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 3}$.\n3) So $f'(x) = \\frac{4 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{2} - 24 x + 18} + \\frac{8 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{2} - 24 x + 18} + \\frac{12 x^{2}}{8 x^{2} - 24 x + 18} - \\frac{12 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{2} - 24 x + 18} - \\frac{24 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{2} - 24 x + 18} - \\frac{36 x}{8 x^{2} - 24 x + 18} + \\frac{4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{4 x^{2} - 12 x + 9} + \\frac{2 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 3} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 3} - \\frac{2 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 3} + \\frac{9 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{2} - 24 x + 18} + \\frac{18 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{2} - 24 x + 18} + \\frac{26}{8 x^{2} - 24 x + 18} - \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 3}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 3}) = \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 3} + \\frac{2 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x - 3} - \\frac{4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x - 3\\right)^{2}} + \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 3}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 12 x^{2} + 12 x + \\left(2 x - 3\\right) \\left(12 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6\\right) - 4 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8}{2 \\left(2 x - 3\\right)^{2}}$.\nFinal answer: $\\frac{- 12 x^{2} + 12 x + \\left(2 x - 3\\right) \\left(12 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6\\right) - 4 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8}{2 \\left(2 x - 3\\right)^{2}}$", "final_answer": "\\frac{- 12 x^{2} + 12 x + \\left(2 x - 3\\right) \\left(12 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6\\right) - 4 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8}{2 \\left(2 x - 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "4966bfbae914128b", "problem_id": "744b910235c70588", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $-2 + \\frac{3 \\pi}{2} + \\frac{3 \\pi^{2}}{4} + \\frac{3 \\pi^{3}}{8}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $-2 + \\frac{3\\cdot \\pi}{2} + \\frac{3\\cdot \\pi^{2}}{4} + \\frac{3\\cdot \\pi^{3}}{8}$", "final_answer": "-2 + \\frac{3\\cdot \\pi}{2} + \\frac{3\\cdot \\pi^{2}}{4} + \\frac{3\\cdot \\pi^{3}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6ef12c8fc9f9b75b", "problem_id": "b43ffb979041fd9e", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the multiplicative inverse of 20 modulo 37.", "output": "1) We want x such that 20x ≡ 1 (mod 37).\n2) Use extended Euclid to find s,t with 20s + 37t = 1.\n3) One Bezout identity is: 20·(13) + 37·(-7) = 1.\n4) Reduce s modulo 37: x ≡ 13 (mod 37).\nFinal answer: $x\\equiv 13\\pmod{37}$", "final_answer": "x\\equiv 13\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9a9370bd0922af92", "problem_id": "b43ffb979041fd9e", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Find the multiplicative inverse of 20 modulo 37.", "output": "We want x such that 20x ≡ 1 (mod 37).\nUse extended Euclid to find s,t with 20s + 37t = 1.\nOne Bezout identity is: 20·(13) + 37·(-7) = 1.\nFinal answer: $x\\equiv 13\\pmod{37}$", "final_answer": "x\\equiv 13\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "be4e320d94252c8e", "problem_id": "5f1d52bcce33dad5", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $3 x^{2} \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "def00c8c5a0618e9", "problem_id": "5f1d52bcce33dad5", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $3 x^{2} \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ce63d78d5453352b", "problem_id": "ba8dc36f60e26550", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$4 x - 4 y$ = $36$\n$- 6 x + 4 y$ = $-48$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-6$: $- 24 x + 24 y$ = $-216$.\n3) Multiply the second equation by $4$: $- 24 x + 16 y$ = $-192$.\n4) Subtract to eliminate x: $8 y$ = $-24$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $6$.\nFinal answer: $(x,y)=\\left(6, -3\\right)$", "final_answer": "(x,y)=\\left(6, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b90bdf28bd513055", "problem_id": "34640ab62238606d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 2 x - 5$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to x - 5.\n3) Then the limit is -7.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -9.\nFinal answer: $-9$", "final_answer": "-9", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a80c58bd4be1327f", "problem_id": "552ce9f80be21247", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{2} + 3 x + 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 3 x^{2}) = - 6 x^{2}$.\n3) So $f'(x) = 3 - 6 x^{2}$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 3 x^{2}) = - 6 x$.\nTherefore the correct derivative is $f'(x) = 3 \\left(1 - 2 x\\right)$.\nFinal answer: $3 \\left(1 - 2 x\\right)$", "final_answer": "3 \\left(1 - 2 x\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e0eadb7b52cc8a1b", "problem_id": "fb48325ff085c86f", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_4=24$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=24 and k=4.\nCompute C(27,3) = 2925.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $2925$", "final_answer": "2925", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "746ea37d84acde87", "problem_id": "d50a734cc7903740", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_8=30$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_8=14.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=14 and k=8.\n4) Compute C(21,7) = 116280.\nFinal answer: $116280$", "final_answer": "116280", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "af42b46658cae4a4", "problem_id": "d50a734cc7903740", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_8=30$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_8=14.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=14 and k=8.\nFinal answer: $116280$", "final_answer": "116280", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ba280ae45df899e8", "problem_id": "353453472b2d9921", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "You want 53 liters of a 25% salt solution. You have a 20% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 53−x liters is 60% solution.\n2) Salt equation: 20x + 60(53−x) = 25·53.\n3) (20-60)x = (25-60)·53 ⇒ x = 371/8.\n4) Then 53−x = 53/8.\nFinal answer: $371/8 L of 20% and 53/8 L of 60%$", "final_answer": "371/8 L of 20% and 53/8 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1faf723383914e7c", "problem_id": "353453472b2d9921", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "You want 53 liters of a 25% salt solution. You have a 20% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 53−x liters is 60% solution.\nSalt equation: 20x + 60(53−x) = 25·53.\n(20-60)x = (25-60)·53 ⇒ x = 371/8.\nFinal answer: $371/8 L of 20% and 53/8 L of 60%$", "final_answer": "371/8 L of 20% and 53/8 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6d1a89c52d77f12d", "problem_id": "163f47684f213743", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{7}$. If it is flipped 21 times, what is the probability of exactly 12 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=21, k=12, p=$\\frac{4}{7}$ and simplify.\n4) Here C(21,12) = 293930.\nFinal answer: $\\frac{13866187326750720}{79792266297612001}$", "final_answer": "\\frac{13866187326750720}{79792266297612001}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8f3b618d674721e3", "problem_id": "163f47684f213743", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{7}$. If it is flipped 21 times, what is the probability of exactly 12 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=21, k=12, p=$\\frac{4}{7}$ and simplify.\nHere C(21,12) = 293930.\nCheck: expression simplifies to $\\frac{13866187326750720}{79792266297612001}$.\nFinal answer: $\\frac{13866187326750720}{79792266297612001}$", "final_answer": "\\frac{13866187326750720}{79792266297612001}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "01dc23d3e67c4ad9", "problem_id": "1541f66f24e7a6e3", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A cyclist rides at 68 miles per hour for 1/2 hours, then at 53 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 68 × 1/2 = 34 miles.\nSecond leg distance = 53 × 1 = 53 miles.\nTotal distance = 34 + 53 = 87 miles.\nCheck: adding both legs gives 87.\nFinal answer: $87$", "final_answer": "87", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5ee5a9be1015a1c5", "problem_id": "1541f66f24e7a6e3", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A cyclist rides at 68 miles per hour for 1/2 hours, then at 53 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 68 × 1/2 = 34 miles.\n2) Second leg distance = 53 × 1 = 53 miles.\n3) Total distance = 34 + 53 = 87 miles.\nFinal answer: $87$", "final_answer": "87", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5e530f1d8082c591", "problem_id": "29fb21d09a8c08ed", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the congruences: x≡2 (mod 4), x≡4 (mod 9). Give the solution modulo 36.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 36.\n3) Compute the CRT solution to get x ≡ 22 (mod 36).\nFinal answer: $x\\equiv 22\\pmod{36}$", "final_answer": "x\\equiv 22\\pmod{36}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "040276de6f3e8d8b", "problem_id": "29fb21d09a8c08ed", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the congruences: x≡2 (mod 4), x≡4 (mod 9). Give the solution modulo 36.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 36.\nCompute the CRT solution to get x ≡ 22 (mod 36).\nFinal answer: $x\\equiv 22\\pmod{36}$", "final_answer": "x\\equiv 22\\pmod{36}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "01e01a4c836a2928", "problem_id": "de24f34d8714bf30", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A cyclist rides at 23 miles per hour for 3/2 hours, then at 82 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 23 × 3/2 = 69/2 miles.\n2) Second leg distance = 82 × 3 = 246 miles.\n3) Total distance = 69/2 + 246 = 561/2 miles.\nFinal answer: $\\frac{561}{2}$", "final_answer": "\\frac{561}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "989fa7fa858e1e84", "problem_id": "de24f34d8714bf30", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "A cyclist rides at 23 miles per hour for 3/2 hours, then at 82 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 23 × 3/2 = 69/2 miles.\nSecond leg distance = 82 × 3 = 246 miles.\nTotal distance = 69/2 + 246 = 561/2 miles.\nFinal answer: $\\frac{561}{2}$", "final_answer": "\\frac{561}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "05f643d01ac826da", "problem_id": "22c467a302f12234", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_5=41$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_5=26.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=26 and k=5.\n4) Compute C(30,4) = 27405.\nFinal answer: $27405$", "final_answer": "27405", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "188d39f1abb78e81", "problem_id": "22c467a302f12234", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_5=41$?", "output": "$27405$", "final_answer": "27405", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "45e23d7a5a20ff02", "problem_id": "acb2b65d1c016f30", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$- 4 x - 7 y$ = $-81$\n$- 5 y$ = $-35$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $-4$: $20 y$ = $140$.\n4) Subtract to eliminate x: $- 20 y$ = $-140$.\n5) Solve for y: y = $7$.\n6) Substitute back to find x: x = $8$.\nFinal answer: $(x,y)=\\left(8, 7\\right)$", "final_answer": "(x,y)=\\left(8, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "588bf390c7c7ed6e", "problem_id": "acb2b65d1c016f30", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the linear system:\n$- 4 x - 7 y$ = $-81$\n$- 5 y$ = $-35$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $-4$: $20 y$ = $140$.\nFinal answer: $(x,y)=\\left(8, 7\\right)$", "final_answer": "(x,y)=\\left(8, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f961047cab960df9", "problem_id": "de3d1c64a27f4567", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A cyclist rides at 54 miles per hour for 1/2 hours, then at 28 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 54 × 1/2 = 27 miles.\n2) Second leg distance = 28 × 3 = 84 miles.\n3) Total distance = 27 + 84 = 111 miles.\nFinal answer: $111$", "final_answer": "111", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eb6de51b7e8e9477", "problem_id": "de3d1c64a27f4567", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A cyclist rides at 54 miles per hour for 1/2 hours, then at 28 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 54 × 1/2 = 27 miles.\nSecond leg distance = 28 × 3 = 84 miles.\nTotal distance = 27 + 84 = 111 miles.\nCheck: adding both legs gives 111.\nFinal answer: $111$", "final_answer": "111", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1b361f609e1927f8", "problem_id": "1b12cfcf5d6a7505", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 4 x - \\frac{5}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$-4$, c=$- \\frac{5}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $36$, so √Δ = $6$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{5}{2}$.\n7) x2 = $- \\frac{1}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{5}{2}, - \\frac{1}{2}\\}$", "final_answer": "x \\in \\{\\frac{5}{2}, - \\frac{1}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d9ed387e63c7ab89", "problem_id": "1b12cfcf5d6a7505", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 4 x - \\frac{5}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$-4$, c=$- \\frac{5}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $36$, so √Δ = $6$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{5}{2}$.\nx2 = $- \\frac{1}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{5}{2}, - \\frac{1}{2}\\}$", "final_answer": "x \\in \\{\\frac{5}{2}, - \\frac{1}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "77b0492bcb6ac9dd", "problem_id": "f1ed4407003e6691", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "An urn has 6 red balls and 6 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(12,4).\n3) Favorable samples: choose k reds and n−k blues: C(6,2)·C(6,2).\n4) Compute totals: C(12,4) = 495, and favorable = 225.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{5}{11}$", "final_answer": "\\frac{5}{11}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "382208e236a56759", "problem_id": "f1ed4407003e6691", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "An urn has 6 red balls and 6 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(12,4).\nFavorable samples: choose k reds and n−k blues: C(6,2)·C(6,2).\nFinal answer: $\\frac{5}{11}$", "final_answer": "\\frac{5}{11}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0a161f27b20e0c33", "problem_id": "828dfb556b7fd1b0", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $- 9 x^{2} + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 3$.\n2) Slope at x=-1: m = f'(-1) = $-12 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\n3) Point on curve: (-1, f(-1)) = (-1, $\\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 7$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 7 = -12 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}(x − -1).\n6) Optionally expand: y = $- 12 x + \\frac{x}{2 e^{\\frac{1}{2}}} + 2 x \\sin{\\left(2 \\right)} - 5 + \\cos{\\left(2 \\right)} + \\frac{3}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = - 12\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 5 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 12\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 5 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c7866d89fcd0a8c2", "problem_id": "828dfb556b7fd1b0", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=-1$.", "output": "Differentiate: f'(x) = $- 9 x^{2} + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 3$.\nSlope at x=-1: m = f'(-1) = $-12 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nPoint on curve: (-1, f(-1)) = (-1, $\\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 7$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 7 = -12 + \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}(x − -1).\nOptionally expand: y = $- 12 x + \\frac{x}{2 e^{\\frac{1}{2}}} + 2 x \\sin{\\left(2 \\right)} - 5 + \\cos{\\left(2 \\right)} + \\frac{3}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 12\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 5 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = - 12\\cdot x + \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} - 5 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f6fd7397d05a61ab", "problem_id": "4ed39bc30dff5c5a", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A cyclist rides at 50 miles per hour for 2 hours, then at 80 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 50 × 2 = 100 miles.\n2) Second leg distance = 80 × 5/2 = 200 miles.\n3) Total distance = 100 + 200 = 300 miles.\nFinal answer: $300$", "final_answer": "300", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "205482d1133bf7bd", "problem_id": "4ed39bc30dff5c5a", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "A cyclist rides at 50 miles per hour for 2 hours, then at 80 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 50 × 2 = 100 miles.\nSecond leg distance = 80 × 5/2 = 200 miles.\nTotal distance = 100 + 200 = 300 miles.\nFinal answer: $300$", "final_answer": "300", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "42bf8df2cb38aef1", "problem_id": "a2a66f50dbdf6063", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Two friends split $1890 in the ratio 5:9. How much does each receive?", "output": "1) Total ratio parts = 5+9 = 14.\n2) Each part = 1890 ÷ 14 = 135.\n3) First share = 5 × 135 = 675.\n4) Second share = 9 × 135 = 1215.\nFinal answer: $1215$", "final_answer": "1215", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1b3eeb10c4d248c7", "problem_id": "a2a66f50dbdf6063", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $1890 in the ratio 5:9. How much does each receive?", "output": "Total ratio parts = 5+9 = 14.\nEach part = 1890 ÷ 14 = 135.\nFirst share = 5 × 135 = 675.\nSecond share = 9 × 135 = 1215.\nCheck: 675+1215=1890.\nFinal answer: $1215$", "final_answer": "1215", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6aef00e13f332e3c", "problem_id": "5d55738fc8123509", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A cyclist rides at 38 miles per hour for 1 hours, then at 34 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 38 × 1 = 38 miles.\n2) Second leg distance = 34 × 2 = 68 miles.\n3) Total distance = 38 + 68 = 106 miles.\nFinal answer: $106$", "final_answer": "106", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "efec700bef7c8206", "problem_id": "5d55738fc8123509", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A cyclist rides at 38 miles per hour for 1 hours, then at 34 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 38 × 1 = 38 miles.\nSecond leg distance = 34 × 2 = 68 miles.\nTotal distance = 38 + 68 = 106 miles.\nCheck: adding both legs gives 106.\nFinal answer: $106$", "final_answer": "106", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d39ed955a7d90506", "problem_id": "cfa835e6fd77480a", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "You want 28 liters of a 25% salt solution. You have a 15% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 28−x liters is 70% solution.\n2) Salt equation: 15x + 70(28−x) = 25·28.\n3) (15-70)x = (25-70)·28 ⇒ x = 252/11.\n4) Then 28−x = 56/11.\nFinal answer: $252/11 L of 15% and 56/11 L of 70%$", "final_answer": "252/11 L of 15% and 56/11 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b91985b5cef0e429", "problem_id": "cfa835e6fd77480a", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 28 liters of a 25% salt solution. You have a 15% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 28−x liters is 70% solution.\nSalt equation: 15x + 70(28−x) = 25·28.\n(15-70)x = (25-70)·28 ⇒ x = 252/11.\nThen 28−x = 56/11.\nCheck: 15·252/11 + 70·56/11 = 25·28.\nFinal answer: $252/11 L of 15% and 56/11 L of 70%$", "final_answer": "252/11 L of 15% and 56/11 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff6e52d3a311a44d", "problem_id": "fd402060e79fe985", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A store sells a backpack for $83. It is discounted by 40%. You buy 5 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 83 × (1 − 40/100) = 83 × 60/100 = 249/5.\n2) Subtotal = 5 × 249/5 = 249.\n3) Total with tax = 249 × 105/100 = 5229/20.\nFinal answer: $\\frac{5229}{20}$", "final_answer": "\\frac{5229}{20}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "96169ceeee83270e", "problem_id": "fd402060e79fe985", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A store sells a backpack for $83. It is discounted by 40%. You buy 5 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 83 × (1 − 40/100) = 83 × 60/100 = 249/5.\nSubtotal = 5 × 249/5 = 249.\nTotal with tax = 249 × 105/100 = 5229/20.\nCheck: recomputing the formula yields 5229/20.\nFinal answer: $\\frac{5229}{20}$", "final_answer": "\\frac{5229}{20}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "77d783d1e391f05e", "problem_id": "27dedc45114d264b", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡2 (mod 11), x≡0 (mod 8). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 24 (mod 88).\nFinal answer: $x\\equiv 24\\pmod{88}$", "final_answer": "x\\equiv 24\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9a54fb081e4c4369", "problem_id": "4dd4cc37b96485d5", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} + x - 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(2 x^{2}) = 4 x^{2}$.\n3) So $f'(x) = 4 x^{2} + 1$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(2 x^{2}) = 4 x$.\nTherefore the correct derivative is $f'(x) = 4 x + 1$.\nFinal answer: $4 x + 1$", "final_answer": "4 x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "cbef5587ba4e014d", "problem_id": "96b616a30075703a", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=26$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=26 and k=5.\n4) Compute C(30,4) = 27405.\nFinal answer: $27405$", "final_answer": "27405", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "802a624a02161bb5", "problem_id": "96b616a30075703a", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=26$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=26 and k=5.\nCompute C(30,4) = 27405.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $27405$", "final_answer": "27405", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "be4ece2a36e5ddc0", "problem_id": "23bc3d37d5fac0d4", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Two friends split $1710 in the ratio 1:9. How much does each receive?", "output": "1) Total ratio parts = 1+9 = 10.\n2) Each part = 1710 ÷ 10 = 171.\n3) First share = 1 × 171 = 171.\n4) Second share = 9 × 171 = 1539.\nFinal answer: $1539$", "final_answer": "1539", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2864563ab72bbc8a", "problem_id": "23bc3d37d5fac0d4", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Two friends split $1710 in the ratio 1:9. How much does each receive?", "output": "Total ratio parts = 1+9 = 10.\nEach part = 1710 ÷ 10 = 171.\nFirst share = 1 × 171 = 171.\nFinal answer: $1539$", "final_answer": "1539", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3963b506728b179a", "problem_id": "840ae8c796faf587", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the linear system:\n$- 6 x + 5 y$ = $24$\n$5 x - 7 y$ = $-37$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $5$: $- 30 x + 25 y$ = $120$.\n3) Multiply the second equation by $-6$: $- 30 x + 42 y$ = $222$.\n4) Subtract to eliminate x: $- 17 y$ = $-102$.\n5) Solve for y: y = $6$.\n6) Substitute back to find x: x = $1$.\nFinal answer: $(x,y)=\\left(1, 6\\right)$", "final_answer": "(x,y)=\\left(1, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "23deaed3e9b3aa81", "problem_id": "140ac9627555efca", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- \\frac{x^{3}}{2} - x^{2} + x + \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - \\frac{1}{2}}{x + 2}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 2}) = \\frac{2 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 2}$.\n3) So $f'(x) = - \\frac{2 x^{3}}{2 x^{2} + 8 x + 8} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 8 x + 8} + \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + 8 x + 8} - \\frac{8 x^{2}}{2 x^{2} + 8 x + 8} + \\frac{4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 8 x + 8} + \\frac{8 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + 8 x + 8} - \\frac{8 x}{2 x^{2} + 8 x + 8} + \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 8 x + 8} + \\frac{8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + 8 x + 8} + \\frac{5}{2 x^{2} + 8 x + 8} + \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 4 x + 4} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 2} - \\frac{2 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 2} + \\frac{2 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 2}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 2}) = \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 2} + \\frac{2 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 2} - \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x + 2\\right)^{2}}$.\nTherefore the correct derivative is $f'(x) = \\frac{x^{3} + 2 x^{2} - 2 x + \\left(x + 2\\right) \\left(- 3 x^{2} - 4 x + \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2\\right) - 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{2 \\left(x + 2\\right)^{2}}$.\nFinal answer: $\\frac{x^{3} + 2 x^{2} - 2 x + \\left(x + 2\\right) \\left(- 3 x^{2} - 4 x + \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2\\right) - 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{2 \\left(x + 2\\right)^{2}}$", "final_answer": "\\frac{x^{3} + 2 x^{2} - 2 x + \\left(x + 2\\right) \\left(- 3 x^{2} - 4 x + \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2\\right) - 2 \\left(x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{2 \\left(x + 2\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "63695ac1c2069eb2", "problem_id": "b9b783c249a1b483", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 3 x^{3} + x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{3} + x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $\\frac{3 \\pi \\left(- 8 \\pi - 16 + 4 \\pi^{2} + 15 \\pi^{3}\\right)}{4}$.\n4) Simplify the result.\nFinal answer: $\\frac{3\\cdot \\pi\\cdot \\left(- 8\\cdot \\pi - 16 + 4\\cdot \\pi^{2} + 15\\cdot \\pi^{3}\\right)}{4}$", "final_answer": "\\frac{3\\cdot \\pi\\cdot \\left(- 8\\cdot \\pi - 16 + 4\\cdot \\pi^{2} + 15\\cdot \\pi^{3}\\right)}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b170da2d9e5fc35e", "problem_id": "b9b783c249a1b483", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 3 x^{3} + x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{3} + x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $\\frac{3 \\pi \\left(- 8 \\pi - 16 + 4 \\pi^{2} + 15 \\pi^{3}\\right)}{4}$.\nFinal answer: $\\frac{3\\cdot \\pi\\cdot \\left(- 8\\cdot \\pi - 16 + 4\\cdot \\pi^{2} + 15\\cdot \\pi^{3}\\right)}{4}$", "final_answer": "\\frac{3\\cdot \\pi\\cdot \\left(- 8\\cdot \\pi - 16 + 4\\cdot \\pi^{2} + 15\\cdot \\pi^{3}\\right)}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1200b43fa3de3742", "problem_id": "7b6c644c1933885b", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "An urn has 19 red balls and 30 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(49,10).\nFavorable samples: choose k reds and n−k blues: C(19,4)·C(30,6).\nFinal answer: $\\frac{82195425}{293493662}$", "final_answer": "\\frac{82195425}{293493662}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2b66844386660853", "problem_id": "7b6c644c1933885b", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 19 red balls and 30 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(49,10).\nFavorable samples: choose k reds and n−k blues: C(19,4)·C(30,6).\nCompute totals: C(49,10) = 8217822536, and favorable = 2301471900.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{82195425}{293493662}$.\nFinal answer: $\\frac{82195425}{293493662}$", "final_answer": "\\frac{82195425}{293493662}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ccf3ff8ae007ee66", "problem_id": "63874d17c8efcca3", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - \\frac{3 x}{5} - \\frac{12}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$- \\frac{3}{5}$, c=$- \\frac{12}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{1, - \\frac{4}{5}\\}$", "final_answer": "x \\in \\{1, - \\frac{4}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "178b7badbadbf246", "problem_id": "63874d17c8efcca3", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - \\frac{3 x}{5} - \\frac{12}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$- \\frac{3}{5}$, c=$- \\frac{12}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{729}{25}$, so √Δ = $\\frac{27}{5}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $1$.\n7) x2 = $- \\frac{4}{5}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{1, - \\frac{4}{5}\\}$", "final_answer": "x \\in \\{1, - \\frac{4}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d489e512cd2e0029", "problem_id": "2ffd71bcbe668fb3", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} + 2$ at $x=0$.", "output": "1) Differentiate: f'(x) = $4 x + 2 \\cos{\\left(x \\right)} - 2$.\n2) Slope at x=0: m = f'(0) = $0$.\n3) Point on curve: (0, f(0)) = (0, $2$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 2 = 0(x − 0).\n6) Optionally expand: y = $2$.\nFinal answer: $y = 2$", "final_answer": "y = 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3bb8ba3af09b4a44", "problem_id": "2ffd71bcbe668fb3", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{2} - 2 x + 2 \\sin{\\left(x \\right)} + 2$ at $x=0$.", "output": "Differentiate: f'(x) = $4 x + 2 \\cos{\\left(x \\right)} - 2$.\nSlope at x=0: m = f'(0) = $0$.\nPoint on curve: (0, f(0)) = (0, $2$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 2 = 0(x − 0).\nOptionally expand: y = $2$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 2$", "final_answer": "y = 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fe01a33f9d0547dd", "problem_id": "ded7a6e63b753bcc", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{4} - x^{3} + 4 x^{2} + 2 x - 4\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $2 x^{4} - x^{3} + 4 x^{2} + 2 x - 4$.\n3) Compute an antiderivative: F(x) = $\\frac{2 x^{5}}{5} - \\frac{x^{4}}{4} + \\frac{4 x^{3}}{3} + x^{2} - 4 x$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{2\\cdot x^{5}}{5} - \\frac{x^{4}}{4} + \\frac{4\\cdot x^{3}}{3} + x^{2} - 4\\cdot x$", "final_answer": "C + \\frac{2\\cdot x^{5}}{5} - \\frac{x^{4}}{4} + \\frac{4\\cdot x^{3}}{3} + x^{2} - 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9782643b38f839a8", "problem_id": "ded7a6e63b753bcc", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{4} - x^{3} + 4 x^{2} + 2 x - 4\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $2 x^{4} - x^{3} + 4 x^{2} + 2 x - 4$.\nCompute an antiderivative: F(x) = $\\frac{2 x^{5}}{5} - \\frac{x^{4}}{4} + \\frac{4 x^{3}}{3} + x^{2} - 4 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{2\\cdot x^{5}}{5} - \\frac{x^{4}}{4} + \\frac{4\\cdot x^{3}}{3} + x^{2} - 4\\cdot x$", "final_answer": "C + \\frac{2\\cdot x^{5}}{5} - \\frac{x^{4}}{4} + \\frac{4\\cdot x^{3}}{3} + x^{2} - 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "298bf4cd2402cc00", "problem_id": "fd2bcac6548591c0", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\left(x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)}\\right) + 3 \\sin{\\left(x \\right)}) = \\left(x^{2} + x + 3\\right) \\cos{\\left(x \\right)}$.\n3) So $f'(x) = x^{2} \\cos{\\left(x \\right)} + x \\cos{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\left(x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)}\\right) + 3 \\sin{\\left(x \\right)}) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} + 3 \\cos{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$", "final_answer": "x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5822f7c574c36382", "problem_id": "e35ff03e13909c5a", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 11 times, what is the probability of exactly 9 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=9, p=$\\frac{2}{5}$ and simplify.\n4) Here C(11,9) = 55.\nFinal answer: $\\frac{50688}{9765625}$", "final_answer": "\\frac{50688}{9765625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c6f6f8284a652803", "problem_id": "2d814437b0ff9e94", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A store sells a backpack for $63. It is discounted by 10%. You buy 8 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 63 × (1 − 10/100) = 63 × 90/100 = 567/10.\n2) Subtotal = 8 × 567/10 = 2268/5.\n3) Total with tax = 2268/5 × 105/100 = 11907/25.\nFinal answer: $\\frac{11907}{25}$", "final_answer": "\\frac{11907}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ada2bd77c453ba8e", "problem_id": "2d814437b0ff9e94", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $63. It is discounted by 10%. You buy 8 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 63 × (1 − 10/100) = 63 × 90/100 = 567/10.\nSubtotal = 8 × 567/10 = 2268/5.\nTotal with tax = 2268/5 × 105/100 = 11907/25.\nCheck: recomputing the formula yields 11907/25.\nFinal answer: $\\frac{11907}{25}$", "final_answer": "\\frac{11907}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9c2c9c3655ba48f2", "problem_id": "098cc5caec1ba39e", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 40 liters of a 35% salt solution. You have a 25% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 40−x liters is 60% solution.\nSalt equation: 25x + 60(40−x) = 35·40.\n(25-60)x = (35-60)·40 ⇒ x = 200/7.\nThen 40−x = 80/7.\nCheck: 25·200/7 + 60·80/7 = 35·40.\nFinal answer: $200/7 L of 25% and 80/7 L of 60%$", "final_answer": "200/7 L of 25% and 80/7 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bd27ff35ac7e13bd", "problem_id": "098cc5caec1ba39e", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "You want 40 liters of a 35% salt solution. You have a 25% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 40−x liters is 60% solution.\n2) Salt equation: 25x + 60(40−x) = 35·40.\n3) (25-60)x = (35-60)·40 ⇒ x = 200/7.\n4) Then 40−x = 80/7.\nFinal answer: $200/7 L of 25% and 80/7 L of 60%$", "final_answer": "200/7 L of 25% and 80/7 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a1cd790b428030b6", "problem_id": "02949f8acba1e126", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the multiplicative inverse of 6 modulo 23.", "output": "1) We want x such that 6x ≡ 1 (mod 23).\n2) Use extended Euclid to find s,t with 6s + 23t = 1.\n3) One Bezout identity is: 6·(4) + 23·(-1) = 1.\n4) Reduce s modulo 23: x ≡ 4 (mod 23).\nFinal answer: $x\\equiv 4\\pmod{23}$", "final_answer": "x\\equiv 4\\pmod{23}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "df14f5372470c845", "problem_id": "02949f8acba1e126", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 6 modulo 23.", "output": "We want x such that 6x ≡ 1 (mod 23).\nUse extended Euclid to find s,t with 6s + 23t = 1.\nOne Bezout identity is: 6·(4) + 23·(-1) = 1.\nReduce s modulo 23: x ≡ 4 (mod 23).\nCheck: 6·4 mod 23 = 1.\nFinal answer: $x\\equiv 4\\pmod{23}$", "final_answer": "x\\equiv 4\\pmod{23}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "827180acda050d71", "problem_id": "524fd53c8a3be5c7", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} - 3 x^{2} - 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 3$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 3 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 3$.\n2) Slope at x=2: m = f'(2) = $-27 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\n3) Point on curve: (2, f(2)) = (2, $-23 + \\cos{\\left(4 \\right)} + e$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -23 + \\cos{\\left(4 \\right)} + e = -27 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\n6) Optionally expand: y = $- 27 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 31$.\nFinal answer: $y = - 27\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 31$", "final_answer": "y = - 27\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 31", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "06dc994ff9678bdf", "problem_id": "524fd53c8a3be5c7", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} - 3 x^{2} - 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 3$ at $x=2$.", "output": "Differentiate: f'(x) = $- 3 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 3$.\nSlope at x=2: m = f'(2) = $-27 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\nPoint on curve: (2, f(2)) = (2, $-23 + \\cos{\\left(4 \\right)} + e$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -23 + \\cos{\\left(4 \\right)} + e = -27 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\nOptionally expand: y = $- 27 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 31$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 27\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 31$", "final_answer": "y = - 27\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 31", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6fc4ec78e4dc032e", "problem_id": "803c180199805916", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$- 6 x - 8 y$ = $-24$\n$5 x + 6 y$ = $18$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $5$: $- 30 x - 40 y$ = $-120$.\n3) Multiply the second equation by $-6$: $- 30 x - 36 y$ = $-108$.\n4) Subtract to eliminate x: $- 4 y$ = $-12$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, 3\\right)$", "final_answer": "(x,y)=\\left(0, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff8e6a9de689df62", "problem_id": "803c180199805916", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$- 6 x - 8 y$ = $-24$\n$5 x + 6 y$ = $18$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $5$: $- 30 x - 40 y$ = $-120$.\nMultiply the second equation by $-6$: $- 30 x - 36 y$ = $-108$.\nSubtract to eliminate x: $- 4 y$ = $-12$.\nSolve for y: y = $3$.\nSubstitute back to find x: x = $0$.\nCheck: plug (x,y)=(0,3) into both equations.\nFinal answer: $(x,y)=\\left(0, 3\\right)$", "final_answer": "(x,y)=\\left(0, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fad4669d360b555e", "problem_id": "bff714c9c13b7fdc", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} + x^{2} + 2 \\sin{\\left(x \\right)} - 3$ at $x=2$.", "output": "Differentiate: f'(x) = $6 x^{2} + 2 x + 2 \\cos{\\left(x \\right)}$.\nSlope at x=2: m = f'(2) = $2 \\cos{\\left(2 \\right)} + 28$.\nPoint on curve: (2, f(2)) = (2, $2 \\sin{\\left(2 \\right)} + 17$).\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 28\\cdot x - 39 - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 28\\cdot x - 39 - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9bb3193d0e19ccd8", "problem_id": "bff714c9c13b7fdc", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} + x^{2} + 2 \\sin{\\left(x \\right)} - 3$ at $x=2$.", "output": "1) Differentiate: f'(x) = $6 x^{2} + 2 x + 2 \\cos{\\left(x \\right)}$.\n2) Slope at x=2: m = f'(2) = $2 \\cos{\\left(2 \\right)} + 28$.\n3) Point on curve: (2, f(2)) = (2, $2 \\sin{\\left(2 \\right)} + 17$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 2 \\sin{\\left(2 \\right)} + 17 = 2 \\cos{\\left(2 \\right)} + 28(x − 2).\n6) Optionally expand: y = $2 x \\cos{\\left(2 \\right)} + 28 x - 39 - 4 \\cos{\\left(2 \\right)} + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 28\\cdot x - 39 - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 28\\cdot x - 39 - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec9764f1f92a2ff6", "problem_id": "f4a1d70cc436bfdb", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Find the multiplicative inverse of 3 modulo 53.", "output": "We want x such that 3x ≡ 1 (mod 53).\nUse extended Euclid to find s,t with 3s + 53t = 1.\nOne Bezout identity is: 3·(18) + 53·(-1) = 1.\nFinal answer: $x\\equiv 18\\pmod{53}$", "final_answer": "x\\equiv 18\\pmod{53}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "55066b2b90e80926", "problem_id": "f4a1d70cc436bfdb", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the multiplicative inverse of 3 modulo 53.", "output": "We want x such that 3x ≡ 1 (mod 53).\nUse extended Euclid to find s,t with 3s + 53t = 1.\nOne Bezout identity is: 3·(18) + 53·(-1) = 1.\nReduce s modulo 53: x ≡ 18 (mod 53).\nCheck: 3·18 mod 53 = 1.\nFinal answer: $x\\equiv 18\\pmod{53}$", "final_answer": "x\\equiv 18\\pmod{53}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5406fb7d835e7144", "problem_id": "18cef2fd6d765628", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the multiplicative inverse of 5 modulo 31.", "output": "1) We want x such that 5x ≡ 1 (mod 31).\n2) Use extended Euclid to find s,t with 5s + 31t = 1.\n3) One Bezout identity is: 5·(-6) + 31·(1) = 1.\n4) Reduce s modulo 31: x ≡ 25 (mod 31).\nFinal answer: $x\\equiv 25\\pmod{31}$", "final_answer": "x\\equiv 25\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b5d43d35519c873d", "problem_id": "18cef2fd6d765628", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the multiplicative inverse of 5 modulo 31.", "output": "We want x such that 5x ≡ 1 (mod 31).\nUse extended Euclid to find s,t with 5s + 31t = 1.\nOne Bezout identity is: 5·(-6) + 31·(1) = 1.\nReduce s modulo 31: x ≡ 25 (mod 31).\nCheck: 5·25 mod 31 = 1.\nFinal answer: $x\\equiv 25\\pmod{31}$", "final_answer": "x\\equiv 25\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "346d6071195e64a6", "problem_id": "aa4c860b8a6ecbe2", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 36 liters of a 35% salt solution. You have a 15% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 36−x liters is 70% solution.\nSalt equation: 15x + 70(36−x) = 35·36.\n(15-70)x = (35-70)·36 ⇒ x = 252/11.\nThen 36−x = 144/11.\nCheck: 15·252/11 + 70·144/11 = 35·36.\nFinal answer: $252/11 L of 15% and 144/11 L of 70%$", "final_answer": "252/11 L of 15% and 144/11 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d26482c8b245ba7a", "problem_id": "aa4c860b8a6ecbe2", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "You want 36 liters of a 35% salt solution. You have a 15% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 36−x liters is 70% solution.\n2) Salt equation: 15x + 70(36−x) = 35·36.\n3) (15-70)x = (35-70)·36 ⇒ x = 252/11.\n4) Then 36−x = 144/11.\nFinal answer: $252/11 L of 15% and 144/11 L of 70%$", "final_answer": "252/11 L of 15% and 144/11 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cf9d2251ca5cbba2", "problem_id": "99cef67943bc92f5", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$4 x - 4 y$ = $-28$\n$x + 4 y$ = $18$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $4 x - 4 y$ = $-28$.\n3) Multiply the second equation by $4$: $4 x + 16 y$ = $72$.\n4) Subtract to eliminate x: $- 20 y$ = $-100$.\n5) Solve for y: y = $5$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, 5\\right)$", "final_answer": "(x,y)=\\left(-2, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c1f73e090aa013b8", "problem_id": "99cef67943bc92f5", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$4 x - 4 y$ = $-28$\n$x + 4 y$ = $18$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $4 x - 4 y$ = $-28$.\nMultiply the second equation by $4$: $4 x + 16 y$ = $72$.\nSubtract to eliminate x: $- 20 y$ = $-100$.\nSolve for y: y = $5$.\nSubstitute back to find x: x = $-2$.\nCheck: plug (x,y)=(-2,5) into both equations.\nFinal answer: $(x,y)=\\left(-2, 5\\right)$", "final_answer": "(x,y)=\\left(-2, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b2ca619880f1d023", "problem_id": "e11a5b9ccc10cb70", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 12 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=12, k=3, p=$\\frac{1}{2}$ and simplify.\nFinal answer: $\\frac{55}{1024}$", "final_answer": "\\frac{55}{1024}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "623cdd522f87d8bd", "problem_id": "e11a5b9ccc10cb70", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 12 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=12, k=3, p=$\\frac{1}{2}$ and simplify.\nHere C(12,3) = 220.\nCheck: expression simplifies to $\\frac{55}{1024}$.\nFinal answer: $\\frac{55}{1024}$", "final_answer": "\\frac{55}{1024}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2332563f2e7e6f97", "problem_id": "b8715fa3d9a22b4f", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} - 2 x^{2} - x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=0$.", "output": "Differentiate: f'(x) = $9 x^{2} - 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 1$.\nSlope at x=0: m = f'(0) = $- \\frac{1}{2}$.\nPoint on curve: (0, f(0)) = (0, $0$).\nFinal answer: $y = - \\frac{x}{2}$", "final_answer": "y = - \\frac{x}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "910e79afb53b0ed7", "problem_id": "b8715fa3d9a22b4f", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} - 2 x^{2} - x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=0$.", "output": "1) Differentiate: f'(x) = $9 x^{2} - 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 1$.\n2) Slope at x=0: m = f'(0) = $- \\frac{1}{2}$.\n3) Point on curve: (0, f(0)) = (0, $0$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 0 = - \\frac{1}{2}(x − 0).\n6) Optionally expand: y = $- \\frac{x}{2}$.\nFinal answer: $y = - \\frac{x}{2}$", "final_answer": "y = - \\frac{x}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fa7e01fdc9d30b86", "problem_id": "906d658ac1bba00d", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "You want 50 liters of a 42% salt solution. You have a 25% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 50−x liters is 60% solution.\n2) Salt equation: 25x + 60(50−x) = 42·50.\n3) (25-60)x = (42-60)·50 ⇒ x = 180/7.\n4) Then 50−x = 170/7.\nFinal answer: $180/7 L of 25% and 170/7 L of 60%$", "final_answer": "180/7 L of 25% and 170/7 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8e9b4614411601e3", "problem_id": "070b8d8e920c171e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- x^{2} + 3 x - 1\\right) \\cos{\\left(3 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x e^{x}) = 2 x e^{x}$.\n3) So $f'(x) = 3 x^{2} \\sin{\\left(3 x \\right)} + 2 x e^{x} - 9 x \\sin{\\left(3 x \\right)} - 2 x \\cos{\\left(3 x \\right)} + 2 e^{x} + 3 \\sin{\\left(3 x \\right)} + 3 \\cos{\\left(3 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x e^{x}) = 2 x e^{x} + 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = \\left(3 - 2 x\\right) \\cos{\\left(3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 3 \\left(x \\left(3 - x\\right) - 1\\right) \\sin{\\left(3 x \\right)} + 2 e^{x}$.\nFinal answer: $\\left(3 - 2 x\\right) \\cos{\\left(3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 3 \\left(x \\left(3 - x\\right) - 1\\right) \\sin{\\left(3 x \\right)} + 2 e^{x}$", "final_answer": "\\left(3 - 2 x\\right) \\cos{\\left(3 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 3 \\left(x \\left(3 - x\\right) - 1\\right) \\sin{\\left(3 x \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0f28009dd4255d5b", "problem_id": "c5f51839679194f0", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{2} x}{2} + \\frac{x}{3} - \\frac{14}{3} - \\frac{7 \\sqrt{2}}{6}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{1}{3} - \\frac{\\sqrt{2}}{2}$, c=$- \\frac{14}{3} - \\frac{7 \\sqrt{2}}{6}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{13 \\sqrt{2}}{3} + \\frac{347}{18}$, so √Δ = $\\sqrt{\\frac{13 \\sqrt{2}}{3} + \\frac{347}{18}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{1}{6} + \\frac{\\sqrt{2}}{4} + \\frac{\\sqrt{156 \\sqrt{2} + 694}}{12}$.\n7) x2 = $- \\frac{\\sqrt{156 \\sqrt{2} + 694}}{12} - \\frac{1}{6} + \\frac{\\sqrt{2}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{1}{6} + \\frac{\\sqrt{2}}{4} + \\frac{\\sqrt{156\\cdot \\sqrt{2} + 694}}{12}, - \\frac{\\sqrt{156\\cdot \\sqrt{2} + 694}}{12} - \\frac{1}{6} + \\frac{\\sqrt{2}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{1}{6} + \\frac{\\sqrt{2}}{4} + \\frac{\\sqrt{156\\cdot \\sqrt{2} + 694}}{12}, - \\frac{\\sqrt{156\\cdot \\sqrt{2} + 694}}{12} - \\frac{1}{6} + \\frac{\\sqrt{2}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "56bdd2126bb17927", "problem_id": "c5f51839679194f0", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{2} x}{2} + \\frac{x}{3} - \\frac{14}{3} - \\frac{7 \\sqrt{2}}{6}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{1}{3} - \\frac{\\sqrt{2}}{2}$, c=$- \\frac{14}{3} - \\frac{7 \\sqrt{2}}{6}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{13 \\sqrt{2}}{3} + \\frac{347}{18}$, so √Δ = $\\sqrt{\\frac{13 \\sqrt{2}}{3} + \\frac{347}{18}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{1}{6} + \\frac{\\sqrt{2}}{4} + \\frac{\\sqrt{156 \\sqrt{2} + 694}}{12}$.\nx2 = $- \\frac{\\sqrt{156 \\sqrt{2} + 694}}{12} - \\frac{1}{6} + \\frac{\\sqrt{2}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{1}{6} + \\frac{\\sqrt{2}}{4} + \\frac{\\sqrt{156\\cdot \\sqrt{2} + 694}}{12}, - \\frac{\\sqrt{156\\cdot \\sqrt{2} + 694}}{12} - \\frac{1}{6} + \\frac{\\sqrt{2}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{1}{6} + \\frac{\\sqrt{2}}{4} + \\frac{\\sqrt{156\\cdot \\sqrt{2} + 694}}{12}, - \\frac{\\sqrt{156\\cdot \\sqrt{2} + 694}}{12} - \\frac{1}{6} + \\frac{\\sqrt{2}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4abfd5b25d7dad85", "problem_id": "0818c08299023185", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} - 3 x^{2} + 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=1$.", "output": "1) Differentiate: f'(x) = $- 3 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 3$.\n2) Slope at x=1: m = f'(1) = $-6 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}$.\n3) Point on curve: (1, f(1)) = (1, $\\cos{\\left(2 \\right)} + e^{\\frac{1}{2}}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} = -6 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}(x − 1).\n6) Optionally expand: y = $- 6 x - 2 x \\sin{\\left(2 \\right)} + \\frac{x e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2 \\sin{\\left(2 \\right)} + 6$.\nFinal answer: $y = - 6\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 6$", "final_answer": "y = - 6\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "89a36e31a048cfd2", "problem_id": "0818c08299023185", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} - 3 x^{2} + 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=1$.", "output": "Differentiate: f'(x) = $- 3 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 3$.\nSlope at x=1: m = f'(1) = $-6 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}$.\nPoint on curve: (1, f(1)) = (1, $\\cos{\\left(2 \\right)} + e^{\\frac{1}{2}}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − \\cos{\\left(2 \\right)} + e^{\\frac{1}{2}} = -6 - 2 \\sin{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2}(x − 1).\nOptionally expand: y = $- 6 x - 2 x \\sin{\\left(2 \\right)} + \\frac{x e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2 \\sin{\\left(2 \\right)} + 6$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 6\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 6$", "final_answer": "y = - 6\\cdot x - 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + \\frac{x\\cdot e^{\\frac{1}{2}}}{2} + \\cos{\\left(2 \\right)} + \\frac{e^{\\frac{1}{2}}}{2} + 2\\cdot \\sin{\\left(2 \\right)} + 6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dc23d94b5bdb7b0d", "problem_id": "ab52e1e1df339686", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "An urn has 12 red balls and 26 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 6 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(38,12).\nFavorable samples: choose k reds and n−k blues: C(12,6)·C(26,6).\nFinal answer: $\\frac{2532530}{32231847}$", "final_answer": "\\frac{2532530}{32231847}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5c24043a394d5aea", "problem_id": "ab52e1e1df339686", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 12 red balls and 26 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 6 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(38,12).\nFavorable samples: choose k reds and n−k blues: C(12,6)·C(26,6).\nCompute totals: C(38,12) = 2707475148, and favorable = 212732520.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{2532530}{32231847}$.\nFinal answer: $\\frac{2532530}{32231847}$", "final_answer": "\\frac{2532530}{32231847}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "73d0878dd75a3d88", "problem_id": "1cb99ed15fbe73e1", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Two friends split $462 in the ratio 8:6. How much does each receive?", "output": "1) Total ratio parts = 8+6 = 14.\n2) Each part = 462 ÷ 14 = 33.\n3) First share = 8 × 33 = 264.\n4) Second share = 6 × 33 = 198.\nFinal answer: $198$", "final_answer": "198", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9adfba7fbf0d47ee", "problem_id": "0d1ed7df721c833d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(3 x^{2} + x + 1\\right) \\cos{\\left(x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\cos{\\left(x + 3 \\right)}) = - \\sin{\\left(x + 3 \\right)}$.\n3) So $f'(x) = - 3 x^{2} \\sin{\\left(x + 3 \\right)} + 2 x e^{x} - x \\sin{\\left(x + 3 \\right)} + 6 x \\cos{\\left(x + 3 \\right)} - 4 e^{x} - \\sin{\\left(x + 3 \\right)} + \\cos{\\left(x + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\cos{\\left(x + 3 \\right)}) = - \\sin{\\left(x + 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x - 3\\right) e^{x} + \\left(6 x + 1\\right) \\cos{\\left(x + 3 \\right)} - \\left(x \\left(3 x + 1\\right) + 1\\right) \\sin{\\left(x + 3 \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(x - 3\\right) e^{x} + \\left(6 x + 1\\right) \\cos{\\left(x + 3 \\right)} - \\left(x \\left(3 x + 1\\right) + 1\\right) \\sin{\\left(x + 3 \\right)} + 2 e^{x}$", "final_answer": "2 \\left(x - 3\\right) e^{x} + \\left(6 x + 1\\right) \\cos{\\left(x + 3 \\right)} - \\left(x \\left(3 x + 1\\right) + 1\\right) \\sin{\\left(x + 3 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e41e3d2a1e1eb7f8", "problem_id": "1fef6943fe01aaa4", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "An urn has 6 red balls and 10 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(16,8).\n3) Favorable samples: choose k reds and n−k blues: C(6,1)·C(10,7).\n4) Compute totals: C(16,8) = 12870, and favorable = 720.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{8}{143}$", "final_answer": "\\frac{8}{143}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9271497561a1baa3", "problem_id": "1fef6943fe01aaa4", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 6 red balls and 10 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(16,8).\nFavorable samples: choose k reds and n−k blues: C(6,1)·C(10,7).\nCompute totals: C(16,8) = 12870, and favorable = 720.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{8}{143}$.\nFinal answer: $\\frac{8}{143}$", "final_answer": "\\frac{8}{143}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "551538c23d92fb5f", "problem_id": "eb85472489534222", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Worker A finishes a job in 13 hours and Worker B finishes it in 19 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/13 job/hour. B's rate = 1/19 job/hour.\n2) Combined rate = 1/13 + 1/19 = 32/247 job/hour.\n3) Time = 1 ÷ (combined rate) = 247/32 hours.\nFinal answer: $\\frac{247}{32}$", "final_answer": "\\frac{247}{32}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b54c436416207b73", "problem_id": "eb85472489534222", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Worker A finishes a job in 13 hours and Worker B finishes it in 19 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/13 job/hour. B's rate = 1/19 job/hour.\nCombined rate = 1/13 + 1/19 = 32/247 job/hour.\nTime = 1 ÷ (combined rate) = 247/32 hours.\nFinal answer: $\\frac{247}{32}$", "final_answer": "\\frac{247}{32}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "767f5aeb99b7a556", "problem_id": "07fb1717a578f563", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the equation of the tangent line to $y=2 x^{3} - x^{2} + 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)}$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $6 x^{2} - 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 3$.\n2) Slope at x=-1: m = f'(-1) = $\\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)} + 11$.\n3) Point on curve: (-1, f(-1)) = (-1, $-6 + \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -6 + \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} = \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)} + 11(x − -1).\n6) Optionally expand: y = $\\frac{x}{2 e^{\\frac{1}{2}}} + 2 x \\sin{\\left(2 \\right)} + 11 x + \\cos{\\left(2 \\right)} + \\frac{3}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)} + 5$.\nFinal answer: $y = \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + 11\\cdot x + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)} + 5$", "final_answer": "y = \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + 11\\cdot x + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)} + 5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a583cbfc13e4e9a7", "problem_id": "ba579723ea38e816", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} x^{2} + 4 x + 14$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to x^{2} + 3 x + 14.\n3) Then the limit is 32.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: 35.\nFinal answer: $35$", "final_answer": "35", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "08178f9d42fca910", "problem_id": "e294049ac4228959", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the multiplicative inverse of 14 modulo 43.", "output": "We want x such that 14x ≡ 1 (mod 43).\nUse extended Euclid to find s,t with 14s + 43t = 1.\nOne Bezout identity is: 14·(-3) + 43·(1) = 1.\nReduce s modulo 43: x ≡ 40 (mod 43).\nCheck: 14·40 mod 43 = 1.\nFinal answer: $x\\equiv 40\\pmod{43}$", "final_answer": "x\\equiv 40\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "76b56e64f6c8da2e", "problem_id": "e294049ac4228959", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Find the multiplicative inverse of 14 modulo 43.", "output": "1) We want x such that 14x ≡ 1 (mod 43).\n2) Use extended Euclid to find s,t with 14s + 43t = 1.\n3) One Bezout identity is: 14·(-3) + 43·(1) = 1.\n4) Reduce s modulo 43: x ≡ 40 (mod 43).\nFinal answer: $x\\equiv 40\\pmod{43}$", "final_answer": "x\\equiv 40\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dd883a7d899f5f0e", "problem_id": "d7a54611ac930014", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A cyclist rides at 55 miles per hour for 2 hours, then at 41 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 55 × 2 = 110 miles.\n2) Second leg distance = 41 × 5/2 = 205/2 miles.\n3) Total distance = 110 + 205/2 = 425/2 miles.\nFinal answer: $\\frac{425}{2}$", "final_answer": "\\frac{425}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "679f2bfdceaec9a9", "problem_id": "d7a54611ac930014", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A cyclist rides at 55 miles per hour for 2 hours, then at 41 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 55 × 2 = 110 miles.\nSecond leg distance = 41 × 5/2 = 205/2 miles.\nTotal distance = 110 + 205/2 = 425/2 miles.\nCheck: adding both legs gives 425/2.\nFinal answer: $\\frac{425}{2}$", "final_answer": "\\frac{425}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0d63f2a0610a9d0f", "problem_id": "609c482d931e69e9", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the multiplicative inverse of 33 modulo 41.", "output": "1) We want x such that 33x ≡ 1 (mod 41).\n2) Use extended Euclid to find s,t with 33s + 41t = 1.\n3) One Bezout identity is: 33·(5) + 41·(-4) = 1.\n4) Reduce s modulo 41: x ≡ 5 (mod 41).\nFinal answer: $x\\equiv 5\\pmod{41}$", "final_answer": "x\\equiv 5\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4f0099851646ce2f", "problem_id": "609c482d931e69e9", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 33 modulo 41.", "output": "We want x such that 33x ≡ 1 (mod 41).\nUse extended Euclid to find s,t with 33s + 41t = 1.\nOne Bezout identity is: 33·(5) + 41·(-4) = 1.\nReduce s modulo 41: x ≡ 5 (mod 41).\nCheck: 33·5 mod 41 = 1.\nFinal answer: $x\\equiv 5\\pmod{41}$", "final_answer": "x\\equiv 5\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bf7d79602c5d0d18", "problem_id": "1cd62e237f45cfa6", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Worker A finishes a job in 22 hours and Worker B finishes it in 5 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/22 job/hour. B's rate = 1/5 job/hour.\n2) Combined rate = 1/22 + 1/5 = 27/110 job/hour.\n3) Time = 1 ÷ (combined rate) = 110/27 hours.\nFinal answer: $\\frac{110}{27}$", "final_answer": "\\frac{110}{27}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f91f5cb54280a7e0", "problem_id": "98e765fee2d42e08", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} - 4 x + 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x^{2}) = - 4 x^{2}$.\n3) So $f'(x) = - 4 x^{2} - 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x^{2}) = - 4 x$.\nTherefore the correct derivative is $f'(x) = 4 \\left(- x - 1\\right)$.\nFinal answer: $4 \\left(- x - 1\\right)$", "final_answer": "4 \\left(- x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "90506df03f135af6", "problem_id": "1c90adfa6c9c4c91", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the equation of the tangent line to $y=- 4 x^{3} + 4 x^{2} - 2$ at $x=0$.", "output": "1) Differentiate: f'(x) = $- 12 x^{2} + 8 x$.\n2) Slope at x=0: m = f'(0) = $0$.\n3) Point on curve: (0, f(0)) = (0, $-2$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -2 = 0(x − 0).\n6) Optionally expand: y = $-2$.\nFinal answer: $y = -2$", "final_answer": "y = -2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0fc08e5fb12ac99b", "problem_id": "1c90adfa6c9c4c91", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 4 x^{3} + 4 x^{2} - 2$ at $x=0$.", "output": "Differentiate: f'(x) = $- 12 x^{2} + 8 x$.\nSlope at x=0: m = f'(0) = $0$.\nPoint on curve: (0, f(0)) = (0, $-2$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -2 = 0(x − 0).\nOptionally expand: y = $-2$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = -2$", "final_answer": "y = -2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "19184f1bc78fe53c", "problem_id": "3462a47b0c9804b8", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $57. It is discounted by 25%. You buy 7 backpacks. What is the total cost?", "output": "Discounted price = 57 × (1 − 25/100) = 57 × 75/100 = 171/4.\nSubtotal = 7 × 171/4 = 1197/4.\nCheck: recomputing the formula yields 1197/4.\nFinal answer: $\\frac{1197}{4}$", "final_answer": "\\frac{1197}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6bb47d0ef947930c", "problem_id": "3462a47b0c9804b8", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "A store sells a backpack for $57. It is discounted by 25%. You buy 7 backpacks. What is the total cost?", "output": "$\\frac{1197}{4}$", "final_answer": "\\frac{1197}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fc74d3bd7f221497", "problem_id": "2a246ee421db8c36", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- x + 2 y$ = $-2$\n$- 2 x + 3 y$ = $-1$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $2 x - 4 y$ = $4$.\nMultiply the second equation by $-1$: $2 x - 3 y$ = $1$.\nSubtract to eliminate x: $- y$ = $3$.\nSolve for y: y = $-3$.\nSubstitute back to find x: x = $-4$.\nCheck: plug (x,y)=(-4,-3) into both equations.\nFinal answer: $(x,y)=\\left(-4, -3\\right)$", "final_answer": "(x,y)=\\left(-4, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff279f581281144d", "problem_id": "2a246ee421db8c36", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$- x + 2 y$ = $-2$\n$- 2 x + 3 y$ = $-1$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $2 x - 4 y$ = $4$.\n3) Multiply the second equation by $-1$: $2 x - 3 y$ = $1$.\n4) Subtract to eliminate x: $- y$ = $3$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $-4$.\nFinal answer: $(x,y)=\\left(-4, -3\\right)$", "final_answer": "(x,y)=\\left(-4, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "09bc6c8908162bce", "problem_id": "f848801b2fcc751e", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 22 liters of a 35% salt solution. You have a 25% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 22−x liters is 60% solution.\nSalt equation: 25x + 60(22−x) = 35·22.\n(25-60)x = (35-60)·22 ⇒ x = 110/7.\nThen 22−x = 44/7.\nCheck: 25·110/7 + 60·44/7 = 35·22.\nFinal answer: $110/7 L of 25% and 44/7 L of 60%$", "final_answer": "110/7 L of 25% and 44/7 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "75689c3c52820ce4", "problem_id": "dc42cd99772c0e8b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(x^{2} - x + 2\\right) \\cos{\\left(3 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x^{2} \\cos{\\left(3 x \\right)}) = - x^{2} \\sin{\\left(3 x \\right)}$.\n3) So $f'(x) = - x^{2} \\sin{\\left(3 x \\right)} + x e^{x} + 3 x \\sin{\\left(3 x \\right)} - 2 e^{x} - 6 \\sin{\\left(3 x \\right)} - \\cos{\\left(3 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x^{2} \\cos{\\left(3 x \\right)}) = - 3 x^{2} \\sin{\\left(3 x \\right)} + 2 x \\cos{\\left(3 x \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x - 3\\right) e^{x} + \\left(2 x - 1\\right) \\cos{\\left(3 x \\right)} - 3 \\left(x \\left(x - 1\\right) + 2\\right) \\sin{\\left(3 x \\right)} + e^{x}$.\nFinal answer: $\\left(x - 3\\right) e^{x} + \\left(2 x - 1\\right) \\cos{\\left(3 x \\right)} - 3 \\left(x \\left(x - 1\\right) + 2\\right) \\sin{\\left(3 x \\right)} + e^{x}$", "final_answer": "\\left(x - 3\\right) e^{x} + \\left(2 x - 1\\right) \\cos{\\left(3 x \\right)} - 3 \\left(x \\left(x - 1\\right) + 2\\right) \\sin{\\left(3 x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "216ec0c78003e4ad", "problem_id": "4d3359a2db7508fc", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_7=30$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_7=23.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=23 and k=7.\n4) Compute C(29,6) = 475020.\nFinal answer: $475020$", "final_answer": "475020", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8e7ed95c41c48558", "problem_id": "c94653b55cacc193", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} \\sin{\\left(x \\right)} - x^{2} \\sin{\\left(x \\right)} - 3 x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 \\sin{\\left(x \\right)}) = 3 \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 \\sin{\\left(x \\right)}) = 3 \\cos{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$", "final_answer": "- 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "502f55a4220b8101", "problem_id": "9b41eefa73ce62dc", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 7 red balls and 11 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(18,8).\n3) Favorable samples: choose k reds and n−k blues: C(7,2)·C(11,6).\n4) Compute totals: C(18,8) = 43758, and favorable = 9702.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{49}{221}$", "final_answer": "\\frac{49}{221}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "373221f19cb623ff", "problem_id": "9b41eefa73ce62dc", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 7 red balls and 11 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(18,8).\nFavorable samples: choose k reds and n−k blues: C(7,2)·C(11,6).\nCompute totals: C(18,8) = 43758, and favorable = 9702.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{49}{221}$.\nFinal answer: $\\frac{49}{221}$", "final_answer": "\\frac{49}{221}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f2b99be5d295e033", "problem_id": "b1227542c89b22d1", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the equation of the tangent line to $y=x^{3} + x^{2} + 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 3$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $3 x^{2} + 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 3$.\n2) Slope at x=-1: m = f'(-1) = $\\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)} + 4$.\n3) Point on curve: (-1, f(-1)) = (-1, $-6 + \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -6 + \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} = \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)} + 4(x − -1).\n6) Optionally expand: y = $\\frac{x}{2 e^{\\frac{1}{2}}} + 2 x \\sin{\\left(2 \\right)} + 4 x - 2 + \\cos{\\left(2 \\right)} + \\frac{3}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + 4\\cdot x - 2 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + 4\\cdot x - 2 + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7fa5d0b4695f93f8", "problem_id": "e4305ec474fba739", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\cos{\\left(2 x \\right)}) = - \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\cos{\\left(2 x \\right)}) = - 2 \\sin{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.\nFinal answer: $x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$", "final_answer": "x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d50d48db8908819f", "problem_id": "43b7c1442bdc199d", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_4=13$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=13 and k=4.\n4) Compute C(16,3) = 560.\nFinal answer: $560$", "final_answer": "560", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f2b76908591d26c3", "problem_id": "52119ff68cb17c7e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(- 3 x^{3} + x^{2} + 2 x - 2\\right) \\cos{\\left(x - 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x^{2} \\cos{\\left(x - 3 \\right)}) = - x^{2} \\sin{\\left(x - 3 \\right)}$.\n3) So $f'(x) = 3 x^{3} \\sin{\\left(x - 3 \\right)} - x^{2} \\sin{\\left(x - 3 \\right)} - 9 x^{2} \\cos{\\left(x - 3 \\right)} + x e^{x} - 2 x \\sin{\\left(x - 3 \\right)} + 3 e^{x} + 2 \\sin{\\left(x - 3 \\right)} + 2 \\cos{\\left(x - 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x^{2} \\cos{\\left(x - 3 \\right)}) = - x^{2} \\sin{\\left(x - 3 \\right)} + 2 x \\cos{\\left(x - 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 2\\right) e^{x} - \\left(x \\left(x \\left(1 - 3 x\\right) + 2\\right) - 2\\right) \\sin{\\left(x - 3 \\right)} + \\left(- 9 x^{2} + 2 x + 2\\right) \\cos{\\left(x - 3 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 2\\right) e^{x} - \\left(x \\left(x \\left(1 - 3 x\\right) + 2\\right) - 2\\right) \\sin{\\left(x - 3 \\right)} + \\left(- 9 x^{2} + 2 x + 2\\right) \\cos{\\left(x - 3 \\right)} + e^{x}$", "final_answer": "\\left(x + 2\\right) e^{x} - \\left(x \\left(x \\left(1 - 3 x\\right) + 2\\right) - 2\\right) \\sin{\\left(x - 3 \\right)} + \\left(- 9 x^{2} + 2 x + 2\\right) \\cos{\\left(x - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "15145be1e3ea189b", "problem_id": "c4f7d4f095c40579", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$- x - 2 y$ = $10$\n$4 x - 4 y$ = $8$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $- 4 x - 8 y$ = $40$.\n3) Multiply the second equation by $-1$: $- 4 x + 4 y$ = $-8$.\n4) Subtract to eliminate x: $- 12 y$ = $48$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, -4\\right)$", "final_answer": "(x,y)=\\left(-2, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "11148b2b9cb4d371", "problem_id": "c4f7d4f095c40579", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the linear system:\n$- x - 2 y$ = $10$\n$4 x - 4 y$ = $8$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $- 4 x - 8 y$ = $40$.\nMultiply the second equation by $-1$: $- 4 x + 4 y$ = $-8$.\nFinal answer: $(x,y)=\\left(-2, -4\\right)$", "final_answer": "(x,y)=\\left(-2, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "92777fe1e078ed0f", "problem_id": "68d9882999e63aed", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Worker A finishes a job in 9 hours and Worker B finishes it in 4 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/9 job/hour. B's rate = 1/4 job/hour.\n2) Combined rate = 1/9 + 1/4 = 13/36 job/hour.\n3) Time = 1 ÷ (combined rate) = 36/13 hours.\nFinal answer: $\\frac{36}{13}$", "final_answer": "\\frac{36}{13}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d71aa4edaa55238", "problem_id": "68d9882999e63aed", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Worker A finishes a job in 9 hours and Worker B finishes it in 4 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/9 job/hour. B's rate = 1/4 job/hour.\nCombined rate = 1/9 + 1/4 = 13/36 job/hour.\nTime = 1 ÷ (combined rate) = 36/13 hours.\nCheck: (combined rate)×(time) = 13/36×36/13 = 1.\nFinal answer: $\\frac{36}{13}$", "final_answer": "\\frac{36}{13}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b223d3b93d9b2145", "problem_id": "701769d70f54ffaf", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡6 (mod 7), x≡2 (mod 5). Give the solution modulo 35.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 35.\nCompute the CRT solution to get x ≡ 27 (mod 35).\nCheck: 27 mod 7=6 and mod 5=2.\nFinal answer: $x\\equiv 27\\pmod{35}$", "final_answer": "x\\equiv 27\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5424920baa1d8c0b", "problem_id": "701769d70f54ffaf", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡6 (mod 7), x≡2 (mod 5). Give the solution modulo 35.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 35.\n3) Compute the CRT solution to get x ≡ 27 (mod 35).\nFinal answer: $x\\equiv 27\\pmod{35}$", "final_answer": "x\\equiv 27\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3dab6ebaa5e98a4d", "problem_id": "5b5878d9cefb6d3a", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$- 3 x - y$ = $-5$\n$- 4 x - 4 y$ = $4$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $12 x + 4 y$ = $20$.\n3) Multiply the second equation by $-3$: $12 x + 12 y$ = $-12$.\n4) Subtract to eliminate x: $- 8 y$ = $32$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, -4\\right)$", "final_answer": "(x,y)=\\left(3, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bce07f1c8de093d7", "problem_id": "5b5878d9cefb6d3a", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- 3 x - y$ = $-5$\n$- 4 x - 4 y$ = $4$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $12 x + 4 y$ = $20$.\nMultiply the second equation by $-3$: $12 x + 12 y$ = $-12$.\nSubtract to eliminate x: $- 8 y$ = $32$.\nSolve for y: y = $-4$.\nSubstitute back to find x: x = $3$.\nCheck: plug (x,y)=(3,-4) into both equations.\nFinal answer: $(x,y)=\\left(3, -4\\right)$", "final_answer": "(x,y)=\\left(3, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aec536c5b88d1adb", "problem_id": "997e107cfd1f9db7", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 15 x - 42$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} - 15 x - 42$ = $3 \\left(x - 7\\right) \\left(x + 2\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $7$.\n5) From the second factor: x = $-2$.\nFinal answer: $x \\in \\{7, -2\\}$", "final_answer": "x \\in \\{7, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a22c0c5c6eb317e6", "problem_id": "997e107cfd1f9db7", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 15 x - 42$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} - 15 x - 42$ = $3 \\left(x - 7\\right) \\left(x + 2\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $7$.\nFrom the second factor: x = $-2$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{7, -2\\}$", "final_answer": "x \\in \\{7, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aee6bed0ef847026", "problem_id": "c8a9487805a76300", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$- x - 4 y$ = $18$\n$4 y$ = $-20$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $-1$: $- 4 y$ = $20$.\n4) Subtract to eliminate x: $4 y$ = $-20$.\n5) Solve for y: y = $-5$.\n6) Substitute back to find x: x = $2$.\nFinal answer: $(x,y)=\\left(2, -5\\right)$", "final_answer": "(x,y)=\\left(2, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f16c8f58ddf9205a", "problem_id": "dbe34a8b224711d0", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 x^{3} - 4 x^{2} + 11 x - 19$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} - 4 x^{2} + 10 x - 19.\n3) Then the limit is -79.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -81.\nFinal answer: $-81$", "final_answer": "-81", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ce1f5fe595d5127f", "problem_id": "0b334a977a19e1a5", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "An urn has 10 red balls and 17 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(27,10).\n3) Favorable samples: choose k reds and n−k blues: C(10,3)·C(17,7).\n4) Compute totals: C(27,10) = 8436285, and favorable = 2333760.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1088}{3933}$", "final_answer": "\\frac{1088}{3933}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "63359bf0a4cb7be5", "problem_id": "0b334a977a19e1a5", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 10 red balls and 17 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(27,10).\nFavorable samples: choose k reds and n−k blues: C(10,3)·C(17,7).\nCompute totals: C(27,10) = 8436285, and favorable = 2333760.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1088}{3933}$.\nFinal answer: $\\frac{1088}{3933}$", "final_answer": "\\frac{1088}{3933}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9573e379f5867dab", "problem_id": "ca95a07d638d1bd1", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many binary strings of length 24 contain exactly 5 ones with no two ones adjacent?", "output": "1) Reserve 4 mandatory zeros between the 5 ones.\n2) This leaves n−k+1 = 20 valid slots for the ones.\n3) Choose the slots: C(20,5) = 15504.\nFinal answer: $15504$", "final_answer": "15504", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f5a1e549fdf7c0a5", "problem_id": "ca95a07d638d1bd1", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many binary strings of length 24 contain exactly 5 ones with no two ones adjacent?", "output": "Reserve 4 mandatory zeros between the 5 ones.\nThis leaves n−k+1 = 20 valid slots for the ones.\nChoose the slots: C(20,5) = 15504.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $15504$", "final_answer": "15504", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "43044e45993c972a", "problem_id": "c3285721304806bc", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A cyclist rides at 69 miles per hour for 2 hours, then at 61 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 69 × 2 = 138 miles.\n2) Second leg distance = 61 × 2 = 122 miles.\n3) Total distance = 138 + 122 = 260 miles.\nFinal answer: $260$", "final_answer": "260", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "584bc2d6e2445e35", "problem_id": "c3285721304806bc", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A cyclist rides at 69 miles per hour for 2 hours, then at 61 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 69 × 2 = 138 miles.\nSecond leg distance = 61 × 2 = 122 miles.\nTotal distance = 138 + 122 = 260 miles.\nCheck: adding both legs gives 260.\nFinal answer: $260$", "final_answer": "260", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a0da08bcbc0f4c86", "problem_id": "e920a2b80a8880ab", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} - x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3$.\n2) Evaluate at the bounds: F(\\pi) and F(0).\n3) Subtract: F(\\pi) − F(0) = $- \\frac{3 \\pi^{2}}{2} - \\frac{\\pi^{3}}{3} + 4 + 3 \\pi$.\n4) Simplify the result.\nFinal answer: $- \\frac{3\\cdot \\pi^{2}}{2} - \\frac{\\pi^{3}}{3} + 4 + 3\\cdot \\pi$", "final_answer": "- \\frac{3\\cdot \\pi^{2}}{2} - \\frac{\\pi^{3}}{3} + 4 + 3\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ad913878baa92030", "problem_id": "e920a2b80a8880ab", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} - x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3$.\nEvaluate at the bounds: F(\\pi) and F(0).\nSubtract: F(\\pi) − F(0) = $- \\frac{3 \\pi^{2}}{2} - \\frac{\\pi^{3}}{3} + 4 + 3 \\pi$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{3\\cdot \\pi^{2}}{2} - \\frac{\\pi^{3}}{3} + 4 + 3\\cdot \\pi$", "final_answer": "- \\frac{3\\cdot \\pi^{2}}{2} - \\frac{\\pi^{3}}{3} + 4 + 3\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b1df46d542b73076", "problem_id": "27b3ab511b08e064", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} - 2 x - 5$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 2 x^{2} - 3 x - 2$ and a = 1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 4 x - 3$.\n5) Evaluate at a: p'(1) = $-7$.\nFinal answer: $-7$", "final_answer": "-7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a660a93cdf0424ea", "problem_id": "27b3ab511b08e064", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} - 2 x - 5$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 2 x^{2} - 3 x - 2$ and a = 1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 4 x - 3$.\nEvaluate at a: p'(1) = $-7$.\nCheck: the identity gives $-7$.\nFinal answer: $-7$", "final_answer": "-7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5f6823d20bfbacdb", "problem_id": "90825bf152ddb151", "style": "answer_only", "instruction": "Final answer only.", "input": "Solve the linear system:\n$- 7 x + y$ = $-3$\n$- 4 x + 7 y$ = $24$", "output": "$(x,y)=\\left(1, 4\\right)$", "final_answer": "(x,y)=\\left(1, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "371c397ac86c0c4c", "problem_id": "90825bf152ddb151", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$- 7 x + y$ = $-3$\n$- 4 x + 7 y$ = $24$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $28 x - 4 y$ = $12$.\n3) Multiply the second equation by $-7$: $28 x - 49 y$ = $-168$.\n4) Subtract to eliminate x: $45 y$ = $180$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $1$.\nFinal answer: $(x,y)=\\left(1, 4\\right)$", "final_answer": "(x,y)=\\left(1, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ee6d96ca5edfc45", "problem_id": "a493fdf58ab85b0c", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 13 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=13, k=6, p=$\\frac{1}{2}$ and simplify.\n4) Here C(13,6) = 1716.\nFinal answer: $\\frac{429}{2048}$", "final_answer": "\\frac{429}{2048}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "805fa6bf383854f8", "problem_id": "a493fdf58ab85b0c", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 13 times, what is the probability of exactly 6 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=13, k=6, p=$\\frac{1}{2}$ and simplify.\nHere C(13,6) = 1716.\nCheck: expression simplifies to $\\frac{429}{2048}$.\nFinal answer: $\\frac{429}{2048}$", "final_answer": "\\frac{429}{2048}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f33315a794e25a8b", "problem_id": "d336e4864b1c050d", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$x - 4 y$ = $14$\n$2 x - 7 y$ = $25$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $2$: $2 x - 8 y$ = $28$.\nMultiply the second equation by $1$: $2 x - 7 y$ = $25$.\nSubtract to eliminate x: $- y$ = $3$.\nSolve for y: y = $-3$.\nSubstitute back to find x: x = $2$.\nCheck: plug (x,y)=(2,-3) into both equations.\nFinal answer: $(x,y)=\\left(2, -3\\right)$", "final_answer": "(x,y)=\\left(2, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "38f8ff616e8bd5be", "problem_id": "9673fa0311086037", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x^{3} + 5 x^{2} + 11 x + 20$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to x^{3} + 5 x^{2} + 10 x + 20.\n3) Then the limit is 68.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 70.\nFinal answer: $70$", "final_answer": "70", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7f346046f232554a", "problem_id": "ecab1c5401a8527b", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 11 x + 18$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} - 11 x + 18$ = $\\left(x - 9\\right) \\left(x - 2\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $9$.\n5) From the second factor: x = $2$.\nFinal answer: $x \\in \\{9, 2\\}$", "final_answer": "x \\in \\{9, 2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "89869ce3603ef8b2", "problem_id": "ecab1c5401a8527b", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 11 x + 18$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} - 11 x + 18$ = $\\left(x - 9\\right) \\left(x - 2\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $9$.\nFrom the second factor: x = $2$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{9, 2\\}$", "final_answer": "x \\in \\{9, 2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "63b5508366671021", "problem_id": "22da7ead2bd936e6", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} - 3 x^{2} + x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 2$ at $x=2$.", "output": "Differentiate: f'(x) = $- 6 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 1$.\nSlope at x=2: m = f'(2) = $-35 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\nPoint on curve: (2, f(2)) = (2, $-24 + \\cos{\\left(4 \\right)} + e$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -24 + \\cos{\\left(4 \\right)} + e = -35 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\nOptionally expand: y = $- 35 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 46$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 35\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 46$", "final_answer": "y = - 35\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 46", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0a7e7751cb79e0c2", "problem_id": "22da7ead2bd936e6", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} - 3 x^{2} + x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 2$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 6 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 1$.\n2) Slope at x=2: m = f'(2) = $-35 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\n3) Point on curve: (2, f(2)) = (2, $-24 + \\cos{\\left(4 \\right)} + e$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -24 + \\cos{\\left(4 \\right)} + e = -35 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\n6) Optionally expand: y = $- 35 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 46$.\nFinal answer: $y = - 35\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 46$", "final_answer": "y = - 35\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 46", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "12c80cb8c68184e5", "problem_id": "5e9e51100b50237b", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the equation of the tangent line to $y=- 4 x^{3} - 2 x^{2} - 2 x - 3$ at $x=2$.", "output": "Differentiate: f'(x) = $- 12 x^{2} - 4 x - 2$.\nSlope at x=2: m = f'(2) = $-58$.\nPoint on curve: (2, f(2)) = (2, $-47$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -47 = -58(x − 2).\nOptionally expand: y = $69 - 58 x$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 69 - 58\\cdot x$", "final_answer": "y = 69 - 58\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7005ac2d34b43975", "problem_id": "5e9e51100b50237b", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Find the equation of the tangent line to $y=- 4 x^{3} - 2 x^{2} - 2 x - 3$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 12 x^{2} - 4 x - 2$.\n2) Slope at x=2: m = f'(2) = $-58$.\n3) Point on curve: (2, f(2)) = (2, $-47$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -47 = -58(x − 2).\n6) Optionally expand: y = $69 - 58 x$.\nFinal answer: $y = 69 - 58\\cdot x$", "final_answer": "y = 69 - 58\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e846ff87d320e62f", "problem_id": "030ad33f101f2a00", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} + 2 x^{2} + 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $- 3 x^{2} + 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 3$.\n2) Slope at x=-2: m = f'(-2) = $-17 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}$.\n3) Point on curve: (-2, f(-2)) = (-2, $\\cos{\\left(4 \\right)} + e^{-1} + 9$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(4 \\right)} + e^{-1} + 9 = -17 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}(x − -2).\n6) Optionally expand: y = $- 17 x + 2 x \\sin{\\left(4 \\right)} + \\frac{x}{2 e} - 25 + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$.\nFinal answer: $y = - 17\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 25 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$", "final_answer": "y = - 17\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 25 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ba6140faf8e0f897", "problem_id": "030ad33f101f2a00", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} + 2 x^{2} + 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=-2$.", "output": "Differentiate: f'(x) = $- 3 x^{2} + 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 3$.\nSlope at x=-2: m = f'(-2) = $-17 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}$.\nPoint on curve: (-2, f(-2)) = (-2, $\\cos{\\left(4 \\right)} + e^{-1} + 9$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − \\cos{\\left(4 \\right)} + e^{-1} + 9 = -17 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}(x − -2).\nOptionally expand: y = $- 17 x + 2 x \\sin{\\left(4 \\right)} + \\frac{x}{2 e} - 25 + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 17\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 25 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$", "final_answer": "y = - 17\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 25 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "71776964bf14dafc", "problem_id": "c01030582e81ed59", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the equation of the tangent line to $y=- x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 1$ at $x=1$.", "output": "1) Differentiate: f'(x) = $- 2 x + 2 \\cos{\\left(x \\right)} + 2$.\n2) Slope at x=1: m = f'(1) = $2 \\cos{\\left(1 \\right)}$.\n3) Point on curve: (1, f(1)) = (1, $2 \\sin{\\left(1 \\right)}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 2 \\sin{\\left(1 \\right)} = 2 \\cos{\\left(1 \\right)}(x − 1).\n6) Optionally expand: y = $2 x \\cos{\\left(1 \\right)} - 2 \\cos{\\left(1 \\right)} + 2 \\sin{\\left(1 \\right)}$.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} - 2\\cdot \\cos{\\left(1 \\right)} + 2\\cdot \\sin{\\left(1 \\right)}$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(1 \\right)} - 2\\cdot \\cos{\\left(1 \\right)} + 2\\cdot \\sin{\\left(1 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9a097218438ac4cf", "problem_id": "434fb9ff87e5e1ff", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{3} - 3 x^{2} + 2 x + \\left(x^{2} + 3 x + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{x^{2} + 3 x + 7}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) + 7}) = \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3 x + 7}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} - \\frac{2 x^{4}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{6 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{12 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} - \\frac{12 x^{3}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{23 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{46 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} - \\frac{64 x^{2}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{6 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 6 x^{3} + 23 x^{2} + 42 x + 49} + \\frac{42 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{84 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} - \\frac{92 x}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{9 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 6 x^{3} + 23 x^{2} + 42 x + 49} - \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 6 x + 14} + \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3 x + 7} - \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) + 7} + \\frac{49 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{98 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} + \\frac{16}{2 x^{4} + 12 x^{3} + 46 x^{2} + 84 x + 98} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) + 7}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) + 7}) = \\frac{3 x \\left(- 2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(x^{2} + 3 x\\right) + 7\\right)^{2}} + \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(x^{2} + 3 x\\right) + 7\\right)} + \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) + 7} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3 x\\right) + 7}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(- 2 x - 3\\right) \\left(- x^{3} - 3 x^{2} + 2 x + \\left(x \\left(x + 3\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2\\right) + \\left(x \\left(x + 3\\right) + 7\\right) \\left(- 6 x^{2} - 12 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x + 3\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x + 3\\right) + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x + 3\\right) + 7\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(- 2 x - 3\\right) \\left(- x^{3} - 3 x^{2} + 2 x + \\left(x \\left(x + 3\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2\\right) + \\left(x \\left(x + 3\\right) + 7\\right) \\left(- 6 x^{2} - 12 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x + 3\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x + 3\\right) + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x + 3\\right) + 7\\right)^{2}}$", "final_answer": "\\frac{2 \\left(- 2 x - 3\\right) \\left(- x^{3} - 3 x^{2} + 2 x + \\left(x \\left(x + 3\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2\\right) + \\left(x \\left(x + 3\\right) + 7\\right) \\left(- 6 x^{2} - 12 x + 2 \\left(2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x + 3\\right) + 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x + 3\\right) + 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(x + 3\\right) + 7\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "1f944a30b150b9c9", "problem_id": "d746237f1658f738", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$4 x - 4 y$ = $-4$\n$3 x + 4 y$ = $-10$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $12 x - 12 y$ = $-12$.\nMultiply the second equation by $4$: $12 x + 16 y$ = $-40$.\nSubtract to eliminate x: $- 28 y$ = $28$.\nSolve for y: y = $-1$.\nSubstitute back to find x: x = $-2$.\nCheck: plug (x,y)=(-2,-1) into both equations.\nFinal answer: $(x,y)=\\left(-2, -1\\right)$", "final_answer": "(x,y)=\\left(-2, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3b08038e1eccb73d", "problem_id": "2f460b9885f27eca", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x^{2} - 4 x - 5$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} - 5 x - 5.\n3) Then the limit is -19.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -17.\nFinal answer: $-17$", "final_answer": "-17", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6fa23b767ce6c0d5", "problem_id": "8ace5bf22ab7eae3", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{3} - x^{2} - x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1}{x^{2} + 2}$.", "output": "1) Let f(x) = $\\frac{- 3 x^{3} - x^{2} - x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1}{x^{2} + 2}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{- 4 x \\left(- 3 x^{3} - x^{2} - x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) + \\left(x^{2} + 2\\right) \\left(- 18 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4 x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x^{2} + 2\\right)^{2}}$.\nFinal answer: $\\frac{- 4\\cdot x\\cdot \\left(- 3\\cdot x^{3} - x^{2} - x + \\left(x^{2} + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 1\\right) + \\left(x^{2} + 2\\right)\\cdot \\left(- 18\\cdot x^{2} + 4\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 4\\cdot x + \\left(x^{2} + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 2\\right)}{2\\cdot \\left(x^{2} + 2\\right)^{2}}$", "final_answer": "\\frac{- 4\\cdot x\\cdot \\left(- 3\\cdot x^{3} - x^{2} - x + \\left(x^{2} + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 1\\right) + \\left(x^{2} + 2\\right)\\cdot \\left(- 18\\cdot x^{2} + 4\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 4\\cdot x + \\left(x^{2} + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 2\\right)}{2\\cdot \\left(x^{2} + 2\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b3418ff5bdf59f18", "problem_id": "8ace5bf22ab7eae3", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{3} - x^{2} - x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1}{x^{2} + 2}$.", "output": "Let f(x) = $\\frac{- 3 x^{3} - x^{2} - x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1}{x^{2} + 2}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{- 4 x \\left(- 3 x^{3} - x^{2} - x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) + \\left(x^{2} + 2\\right) \\left(- 18 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4 x + \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x^{2} + 2\\right)^{2}}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\frac{- 4\\cdot x\\cdot \\left(- 3\\cdot x^{3} - x^{2} - x + \\left(x^{2} + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 1\\right) + \\left(x^{2} + 2\\right)\\cdot \\left(- 18\\cdot x^{2} + 4\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 4\\cdot x + \\left(x^{2} + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 2\\right)}{2\\cdot \\left(x^{2} + 2\\right)^{2}}$", "final_answer": "\\frac{- 4\\cdot x\\cdot \\left(- 3\\cdot x^{3} - x^{2} - x + \\left(x^{2} + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 1\\right) + \\left(x^{2} + 2\\right)\\cdot \\left(- 18\\cdot x^{2} + 4\\cdot x\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 4\\cdot x + \\left(x^{2} + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 2\\right)}{2\\cdot \\left(x^{2} + 2\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "154c82f4c333e4fe", "problem_id": "aa0660d344fe68be", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}$.\nCompute an antiderivative: F(x) = $\\frac{2 x^{3}}{3} + x^{2} - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{2\\cdot x^{3}}{3} + x^{2} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{2\\cdot x^{3}}{3} + x^{2} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f81b0bb40b6e4187", "problem_id": "aa0660d344fe68be", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)}$.\n3) Compute an antiderivative: F(x) = $\\frac{2 x^{3}}{3} + x^{2} - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{2\\cdot x^{3}}{3} + x^{2} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{2\\cdot x^{3}}{3} + x^{2} - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0df8c588fbbe9557", "problem_id": "c818cbf308226a75", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "You want 49 liters of a 35% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 49−x liters is 50% solution.\n2) Salt equation: 25x + 50(49−x) = 35·49.\n3) (25-50)x = (35-50)·49 ⇒ x = 147/5.\n4) Then 49−x = 98/5.\nFinal answer: $147/5 L of 25% and 98/5 L of 50%$", "final_answer": "147/5 L of 25% and 98/5 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b42eda7186b72e70", "problem_id": "c818cbf308226a75", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "You want 49 liters of a 35% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 49−x liters is 50% solution.\nSalt equation: 25x + 50(49−x) = 35·49.\n(25-50)x = (35-50)·49 ⇒ x = 147/5.\nThen 49−x = 98/5.\nCheck: 25·147/5 + 50·98/5 = 35·49.\nFinal answer: $147/5 L of 25% and 98/5 L of 50%$", "final_answer": "147/5 L of 25% and 98/5 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "206bea1a37cb3450", "problem_id": "f4864bda1183ed5f", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 3), x≡11 (mod 13). Give the solution modulo 39.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 39.\nCompute the CRT solution to get x ≡ 11 (mod 39).\nCheck: 11 mod 3=2 and mod 13=11.\nFinal answer: $x\\equiv 11\\pmod{39}$", "final_answer": "x\\equiv 11\\pmod{39}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "433022b5ef9c5302", "problem_id": "f4864bda1183ed5f", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡2 (mod 3), x≡11 (mod 13). Give the solution modulo 39.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 39.\n3) Compute the CRT solution to get x ≡ 11 (mod 39).\nFinal answer: $x\\equiv 11\\pmod{39}$", "final_answer": "x\\equiv 11\\pmod{39}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7a42993a6bacf265", "problem_id": "87e1d1d1c6222c9e", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(- 2 x^{2} - 2 x\\right) \\cos{\\left(- 3 x^{2} + 2 x + 3 \\right)}$.", "output": "Let f(x) = $2 \\left(x - 3\\right) e^{x} + \\left(- 2 x^{2} - 2 x\\right) \\cos{\\left(- 3 x^{2} + 2 x + 3 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $2 \\left(- 2 x \\left(1 - 3 x\\right) \\left(- x - 1\\right) \\sin{\\left(- 3 x^{2} + 2 x + 3 \\right)} + \\left(- 2 x - 1\\right) \\cos{\\left(- 3 x^{2} + 2 x + 3 \\right)} + \\left(x - 3\\right) e^{x} + e^{x}\\right)$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 4\\cdot x\\cdot \\left(1 - 3\\cdot x\\right)\\cdot \\left(- x - 1\\right)\\cdot \\sin{\\left(- 3\\cdot x^{2} + 2\\cdot x + 3 \\right)} + 2\\cdot \\left(- 2\\cdot x - 1\\right)\\cdot \\cos{\\left(- 3\\cdot x^{2} + 2\\cdot x + 3 \\right)} + 2\\cdot \\left(x - 3\\right)\\cdot e^{x} + 2\\cdot e^{x}$", "final_answer": "- 4\\cdot x\\cdot \\left(1 - 3\\cdot x\\right)\\cdot \\left(- x - 1\\right)\\cdot \\sin{\\left(- 3\\cdot x^{2} + 2\\cdot x + 3 \\right)} + 2\\cdot \\left(- 2\\cdot x - 1\\right)\\cdot \\cos{\\left(- 3\\cdot x^{2} + 2\\cdot x + 3 \\right)} + 2\\cdot \\left(x - 3\\right)\\cdot e^{x} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0a2c4563f2485012", "problem_id": "87e1d1d1c6222c9e", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(- 2 x^{2} - 2 x\\right) \\cos{\\left(- 3 x^{2} + 2 x + 3 \\right)}$.", "output": "1) Let f(x) = $2 \\left(x - 3\\right) e^{x} + \\left(- 2 x^{2} - 2 x\\right) \\cos{\\left(- 3 x^{2} + 2 x + 3 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $2 \\left(- 2 x \\left(1 - 3 x\\right) \\left(- x - 1\\right) \\sin{\\left(- 3 x^{2} + 2 x + 3 \\right)} + \\left(- 2 x - 1\\right) \\cos{\\left(- 3 x^{2} + 2 x + 3 \\right)} + \\left(x - 3\\right) e^{x} + e^{x}\\right)$.\nFinal answer: $- 4\\cdot x\\cdot \\left(1 - 3\\cdot x\\right)\\cdot \\left(- x - 1\\right)\\cdot \\sin{\\left(- 3\\cdot x^{2} + 2\\cdot x + 3 \\right)} + 2\\cdot \\left(- 2\\cdot x - 1\\right)\\cdot \\cos{\\left(- 3\\cdot x^{2} + 2\\cdot x + 3 \\right)} + 2\\cdot \\left(x - 3\\right)\\cdot e^{x} + 2\\cdot e^{x}$", "final_answer": "- 4\\cdot x\\cdot \\left(1 - 3\\cdot x\\right)\\cdot \\left(- x - 1\\right)\\cdot \\sin{\\left(- 3\\cdot x^{2} + 2\\cdot x + 3 \\right)} + 2\\cdot \\left(- 2\\cdot x - 1\\right)\\cdot \\cos{\\left(- 3\\cdot x^{2} + 2\\cdot x + 3 \\right)} + 2\\cdot \\left(x - 3\\right)\\cdot e^{x} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "897f0a45d1821238", "problem_id": "322a8d19c0ec7200", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Solve the linear system:\n$- 4 x + 11 y$ = $-38$\n$9 x + 11 y$ = $-129$", "output": "$(x,y)=\\left(-7, -6\\right)$", "final_answer": "(x,y)=\\left(-7, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4aa6e34823d2b1c8", "problem_id": "322a8d19c0ec7200", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$- 4 x + 11 y$ = $-38$\n$9 x + 11 y$ = $-129$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $9$: $- 36 x + 99 y$ = $-342$.\nMultiply the second equation by $-4$: $- 36 x - 44 y$ = $516$.\nSubtract to eliminate x: $143 y$ = $-858$.\nSolve for y: y = $-6$.\nSubstitute back to find x: x = $-7$.\nCheck: plug (x,y)=(-7,-6) into both equations.\nFinal answer: $(x,y)=\\left(-7, -6\\right)$", "final_answer": "(x,y)=\\left(-7, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3e7fe7c2e4423571", "problem_id": "6a85e47fb642310e", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 16 red balls and 7 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(23,7).\nFavorable samples: choose k reds and n−k blues: C(16,3)·C(7,4).\nCompute totals: C(23,7) = 245157, and favorable = 19600.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{19600}{245157}$.\nFinal answer: $\\frac{19600}{245157}$", "final_answer": "\\frac{19600}{245157}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a42c547303e723e3", "problem_id": "6a85e47fb642310e", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 16 red balls and 7 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(23,7).\n3) Favorable samples: choose k reds and n−k blues: C(16,3)·C(7,4).\n4) Compute totals: C(23,7) = 245157, and favorable = 19600.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{19600}{245157}$", "final_answer": "\\frac{19600}{245157}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "040666855338a978", "problem_id": "6c9e3f9cbf28c365", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$- 2 x - 4 y$ = $12$\n$x - 2 y$ = $14$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $- 2 x - 4 y$ = $12$.\n3) Multiply the second equation by $-2$: $- 2 x + 4 y$ = $-28$.\n4) Subtract to eliminate x: $- 8 y$ = $40$.\n5) Solve for y: y = $-5$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, -5\\right)$", "final_answer": "(x,y)=\\left(4, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8ef836105ed60964", "problem_id": "6c9e3f9cbf28c365", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the linear system:\n$- 2 x - 4 y$ = $12$\n$x - 2 y$ = $14$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $- 2 x - 4 y$ = $12$.\nMultiply the second equation by $-2$: $- 2 x + 4 y$ = $-28$.\nSubtract to eliminate x: $- 8 y$ = $40$.\nSolve for y: y = $-5$.\nSubstitute back to find x: x = $4$.\nCheck: plug (x,y)=(4,-5) into both equations.\nFinal answer: $(x,y)=\\left(4, -5\\right)$", "final_answer": "(x,y)=\\left(4, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "97cda7188f248c60", "problem_id": "ca355cb679c66c1d", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the multiplicative inverse of 16 modulo 53.", "output": "1) We want x such that 16x ≡ 1 (mod 53).\n2) Use extended Euclid to find s,t with 16s + 53t = 1.\n3) One Bezout identity is: 16·(10) + 53·(-3) = 1.\n4) Reduce s modulo 53: x ≡ 10 (mod 53).\nFinal answer: $x\\equiv 10\\pmod{53}$", "final_answer": "x\\equiv 10\\pmod{53}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "89556bb4665c17b5", "problem_id": "ca355cb679c66c1d", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the multiplicative inverse of 16 modulo 53.", "output": "We want x such that 16x ≡ 1 (mod 53).\nUse extended Euclid to find s,t with 16s + 53t = 1.\nOne Bezout identity is: 16·(10) + 53·(-3) = 1.\nReduce s modulo 53: x ≡ 10 (mod 53).\nCheck: 16·10 mod 53 = 1.\nFinal answer: $x\\equiv 10\\pmod{53}$", "final_answer": "x\\equiv 10\\pmod{53}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "383319aef25af944", "problem_id": "ed2ed975f3e47993", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "You want 70 liters of a 45% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 70−x liters is 70% solution.\n2) Salt equation: 25x + 70(70−x) = 45·70.\n3) (25-70)x = (45-70)·70 ⇒ x = 350/9.\n4) Then 70−x = 280/9.\nFinal answer: $350/9 L of 25% and 280/9 L of 70%$", "final_answer": "350/9 L of 25% and 280/9 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f79afa3df921f4fb", "problem_id": "ed2ed975f3e47993", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "You want 70 liters of a 45% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 70−x liters is 70% solution.\nSalt equation: 25x + 70(70−x) = 45·70.\n(25-70)x = (45-70)·70 ⇒ x = 350/9.\nThen 70−x = 280/9.\nCheck: 25·350/9 + 70·280/9 = 45·70.\nFinal answer: $350/9 L of 25% and 280/9 L of 70%$", "final_answer": "350/9 L of 25% and 280/9 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "81396d67e760b575", "problem_id": "97cbb6047e4471c5", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 3 x^{2} + 9 x + 18$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{3} + 3 x^{2} - 1$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $9 x^{2} + 6 x$.\n5) Evaluate at a: p'(2) = $48$.\nFinal answer: $48$", "final_answer": "48", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7a8538bb2c844132", "problem_id": "97cbb6047e4471c5", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 3 x^{2} + 9 x + 18$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $3 x^{3} + 3 x^{2} - 1$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $9 x^{2} + 6 x$.\nEvaluate at a: p'(2) = $48$.\nCheck: the identity gives $48$.\nFinal answer: $48$", "final_answer": "48", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e561a198f9042277", "problem_id": "7253d4677acf777a", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} + 3 x + \\left(3 x - 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{3 x - 7}$.", "output": "1) Let f(x) = $\\frac{- x^{2} + 3 x + \\left(3 x - 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{3 x - 7}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{6 x^{2} - 18 x + \\left(3 x - 7\\right) \\left(- 4 x + \\left(3 x - 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x - 7\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 6 \\left(3 x - 7\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6}{2 \\left(3 x - 7\\right)^{2}}$.\nFinal answer: $\\frac{6\\cdot x^{2} - 18\\cdot x + \\left(3\\cdot x - 7\\right)\\cdot \\left(- 4\\cdot x + \\left(3\\cdot x - 7\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x - 14\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6\\right) - \\left(18\\cdot x - 42\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 6}{2\\cdot \\left(3\\cdot x - 7\\right)^{2}}$", "final_answer": "\\frac{6\\cdot x^{2} - 18\\cdot x + \\left(3\\cdot x - 7\\right)\\cdot \\left(- 4\\cdot x + \\left(3\\cdot x - 7\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x - 14\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 6\\right) - \\left(18\\cdot x - 42\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 6}{2\\cdot \\left(3\\cdot x - 7\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "466e0fef3d84cafc", "problem_id": "83a3b0f783dfd312", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 4 x^{3} + 3 x^{2} - 4 x - 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 4 x^{3}) = - 12 x^{3}$.\n3) So $f'(x) = - 12 x^{3} + 6 x - 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 4 x^{3}) = - 12 x^{2}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- 6 x^{2} + 3 x - 2\\right)$.\nFinal answer: $2 \\left(- 6 x^{2} + 3 x - 2\\right)$", "final_answer": "2 \\left(- 6 x^{2} + 3 x - 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "97459a0faaee80f6", "problem_id": "69d0a8247000fbf3", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=15$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=15 and k=5.\n4) Compute C(19,4) = 3876.\nFinal answer: $3876$", "final_answer": "3876", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e08f35b136211ec7", "problem_id": "69d0a8247000fbf3", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=15$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=15 and k=5.\nCompute C(19,4) = 3876.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $3876$", "final_answer": "3876", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c134b619de4444be", "problem_id": "6304f475937b3a06", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 13 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=13, k=3, p=$\\frac{3}{8}$ and simplify.\nFinal answer: $\\frac{37705078125}{274877906944}$", "final_answer": "\\frac{37705078125}{274877906944}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0a6cc2c66a3f76d6", "problem_id": "6304f475937b3a06", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 13 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=13, k=3, p=$\\frac{3}{8}$ and simplify.\n4) Here C(13,3) = 286.\nFinal answer: $\\frac{37705078125}{274877906944}$", "final_answer": "\\frac{37705078125}{274877906944}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "91055e7ad5f2a40f", "problem_id": "ac21aa6e5f4b63d3", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A cyclist rides at 75 miles per hour for 2 hours, then at 65 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 75 × 2 = 150 miles.\n2) Second leg distance = 65 × 1 = 65 miles.\n3) Total distance = 150 + 65 = 215 miles.\nFinal answer: $215$", "final_answer": "215", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "54ba9a4a73e8f5df", "problem_id": "ac21aa6e5f4b63d3", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A cyclist rides at 75 miles per hour for 2 hours, then at 65 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 75 × 2 = 150 miles.\nSecond leg distance = 65 × 1 = 65 miles.\nTotal distance = 150 + 65 = 215 miles.\nCheck: adding both legs gives 215.\nFinal answer: $215$", "final_answer": "215", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ca880d1f916c2897", "problem_id": "98f16065aac99e0f", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$8 x^{2} - 12 \\sqrt{10} x - 18 x + 27 \\sqrt{10}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$8$, b=$- 12 \\sqrt{10} - 18$, c=$27 \\sqrt{10}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $1764 - 432 \\sqrt{10}$, so √Δ = $\\sqrt{1764 - 432 \\sqrt{10}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{9}{8} + \\frac{3 \\sqrt{49 - 12 \\sqrt{10}}}{8} + \\frac{3 \\sqrt{10}}{4}$.\nx2 = $- \\frac{3 \\sqrt{49 - 12 \\sqrt{10}}}{8} + \\frac{9}{8} + \\frac{3 \\sqrt{10}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{9}{8} + \\frac{3\\cdot \\sqrt{49 - 12\\cdot \\sqrt{10}}}{8} + \\frac{3\\cdot \\sqrt{10}}{4}, - \\frac{3\\cdot \\sqrt{49 - 12\\cdot \\sqrt{10}}}{8} + \\frac{9}{8} + \\frac{3\\cdot \\sqrt{10}}{4}\\}$", "final_answer": "x \\in \\{\\frac{9}{8} + \\frac{3\\cdot \\sqrt{49 - 12\\cdot \\sqrt{10}}}{8} + \\frac{3\\cdot \\sqrt{10}}{4}, - \\frac{3\\cdot \\sqrt{49 - 12\\cdot \\sqrt{10}}}{8} + \\frac{9}{8} + \\frac{3\\cdot \\sqrt{10}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "425a5898ca083979", "problem_id": "98f16065aac99e0f", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$8 x^{2} - 12 \\sqrt{10} x - 18 x + 27 \\sqrt{10}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$8$, b=$- 12 \\sqrt{10} - 18$, c=$27 \\sqrt{10}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $1764 - 432 \\sqrt{10}$, so √Δ = $\\sqrt{1764 - 432 \\sqrt{10}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{9}{8} + \\frac{3 \\sqrt{49 - 12 \\sqrt{10}}}{8} + \\frac{3 \\sqrt{10}}{4}$.\n7) x2 = $- \\frac{3 \\sqrt{49 - 12 \\sqrt{10}}}{8} + \\frac{9}{8} + \\frac{3 \\sqrt{10}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{9}{8} + \\frac{3\\cdot \\sqrt{49 - 12\\cdot \\sqrt{10}}}{8} + \\frac{3\\cdot \\sqrt{10}}{4}, - \\frac{3\\cdot \\sqrt{49 - 12\\cdot \\sqrt{10}}}{8} + \\frac{9}{8} + \\frac{3\\cdot \\sqrt{10}}{4}\\}$", "final_answer": "x \\in \\{\\frac{9}{8} + \\frac{3\\cdot \\sqrt{49 - 12\\cdot \\sqrt{10}}}{8} + \\frac{3\\cdot \\sqrt{10}}{4}, - \\frac{3\\cdot \\sqrt{49 - 12\\cdot \\sqrt{10}}}{8} + \\frac{9}{8} + \\frac{3\\cdot \\sqrt{10}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7e54ed1bc4236fa4", "problem_id": "77f11d81ae150b6c", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{13 x}{5} + \\frac{8}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{13}{5}$, c=$\\frac{8}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{9}{25}$, so √Δ = $\\frac{3}{5}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{8}{5}$.\nx2 = $1$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{8}{5}, 1\\}$", "final_answer": "x \\in \\{\\frac{8}{5}, 1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c70426359c4d76b4", "problem_id": "77f11d81ae150b6c", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{13 x}{5} + \\frac{8}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{13}{5}$, c=$\\frac{8}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{9}{25}$, so √Δ = $\\frac{3}{5}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{8}{5}$.\n7) x2 = $1$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{8}{5}, 1\\}$", "final_answer": "x \\in \\{\\frac{8}{5}, 1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "042791dedbe3ab45", "problem_id": "5afac4e642394115", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the multiplicative inverse of 20 modulo 31.", "output": "1) We want x such that 20x ≡ 1 (mod 31).\n2) Use extended Euclid to find s,t with 20s + 31t = 1.\n3) One Bezout identity is: 20·(14) + 31·(-9) = 1.\n4) Reduce s modulo 31: x ≡ 14 (mod 31).\nFinal answer: $x\\equiv 14\\pmod{31}$", "final_answer": "x\\equiv 14\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1162e5a8562a8401", "problem_id": "955f37ef1895c469", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- x^{3} + x^{2} - x + 1\\right) \\cos{\\left(x + 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x \\cos{\\left(x + 2 \\right)}) = x \\sin{\\left(x + 2 \\right)}$.\n3) So $f'(x) = x^{3} \\sin{\\left(x + 2 \\right)} - x^{2} \\sin{\\left(x + 2 \\right)} - 3 x^{2} \\cos{\\left(x + 2 \\right)} + 2 x e^{x} + x \\sin{\\left(x + 2 \\right)} + 2 x \\cos{\\left(x + 2 \\right)} + 4 e^{x} - \\sin{\\left(x + 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x \\cos{\\left(x + 2 \\right)}) = x \\sin{\\left(x + 2 \\right)} - \\cos{\\left(x + 2 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x + 1\\right) e^{x} - \\left(x \\left(x \\left(1 - x\\right) - 1\\right) + 1\\right) \\sin{\\left(x + 2 \\right)} + \\left(- 3 x^{2} + 2 x - 1\\right) \\cos{\\left(x + 2 \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(x + 1\\right) e^{x} - \\left(x \\left(x \\left(1 - x\\right) - 1\\right) + 1\\right) \\sin{\\left(x + 2 \\right)} + \\left(- 3 x^{2} + 2 x - 1\\right) \\cos{\\left(x + 2 \\right)} + 2 e^{x}$", "final_answer": "2 \\left(x + 1\\right) e^{x} - \\left(x \\left(x \\left(1 - x\\right) - 1\\right) + 1\\right) \\sin{\\left(x + 2 \\right)} + \\left(- 3 x^{2} + 2 x - 1\\right) \\cos{\\left(x + 2 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "476b2024ba5e4b3d", "problem_id": "417ed5ef80a0ce66", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Two friends split $1128 in the ratio 6:6. How much does each receive?", "output": "1) Total ratio parts = 6+6 = 12.\n2) Each part = 1128 ÷ 12 = 94.\n3) First share = 6 × 94 = 564.\n4) Second share = 6 × 94 = 564.\nFinal answer: $564$", "final_answer": "564", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e43bc2c9fd55456d", "problem_id": "417ed5ef80a0ce66", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Two friends split $1128 in the ratio 6:6. How much does each receive?", "output": "Total ratio parts = 6+6 = 12.\nEach part = 1128 ÷ 12 = 94.\nFirst share = 6 × 94 = 564.\nSecond share = 6 × 94 = 564.\nCheck: 564+564=1128.\nFinal answer: $564$", "final_answer": "564", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "71fa8b379df6f979", "problem_id": "86d981d33637d333", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 22 times, what is the probability of exactly 13 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=22, k=13, p=$\\frac{4}{5}$ and simplify.\n4) Here C(22,13) = 497420.\nFinal answer: $\\frac{6676258226176}{476837158203125}$", "final_answer": "\\frac{6676258226176}{476837158203125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b0f579322000627", "problem_id": "86d981d33637d333", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 22 times, what is the probability of exactly 13 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=22, k=13, p=$\\frac{4}{5}$ and simplify.\nHere C(22,13) = 497420.\nCheck: expression simplifies to $\\frac{6676258226176}{476837158203125}$.\nFinal answer: $\\frac{6676258226176}{476837158203125}$", "final_answer": "\\frac{6676258226176}{476837158203125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "04ea6dd38a201983", "problem_id": "5c4d171cb85a355c", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - 3 x^{3} + 9 x^{2} - 15 x + 31$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 3 x^{4} + 3 x^{3} + 3 x^{2} + x$ and a = -2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 12 x^{3} + 9 x^{2} + 6 x + 1$.\n5) Evaluate at a: p'(-2) = $121$.\nFinal answer: $121$", "final_answer": "121", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8b1ba2156463cbf2", "problem_id": "5c4d171cb85a355c", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - 3 x^{3} + 9 x^{2} - 15 x + 31$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 3 x^{4} + 3 x^{3} + 3 x^{2} + x$ and a = -2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 12 x^{3} + 9 x^{2} + 6 x + 1$.\nEvaluate at a: p'(-2) = $121$.\nCheck: the identity gives $121$.\nFinal answer: $121$", "final_answer": "121", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff751bfe8fe60235", "problem_id": "a81dc39c7bdfecc9", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 18 red balls and 22 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(40,8).\n3) Favorable samples: choose k reds and n−k blues: C(18,2)·C(22,6).\n4) Compute totals: C(40,8) = 76904685, and favorable = 11415789.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{357}{2405}$", "final_answer": "\\frac{357}{2405}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "617ad58fd740ece4", "problem_id": "a81dc39c7bdfecc9", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 18 red balls and 22 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(40,8).\nFavorable samples: choose k reds and n−k blues: C(18,2)·C(22,6).\nCompute totals: C(40,8) = 76904685, and favorable = 11415789.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{357}{2405}$.\nFinal answer: $\\frac{357}{2405}$", "final_answer": "\\frac{357}{2405}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f8bda5bd98695239", "problem_id": "920cfc8c868816a4", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A cyclist rides at 80 miles per hour for 2 hours, then at 27 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 80 × 2 = 160 miles.\n2) Second leg distance = 27 × 1 = 27 miles.\n3) Total distance = 160 + 27 = 187 miles.\nFinal answer: $187$", "final_answer": "187", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dc2d45e126c4661e", "problem_id": "920cfc8c868816a4", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "A cyclist rides at 80 miles per hour for 2 hours, then at 27 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 80 × 2 = 160 miles.\nSecond leg distance = 27 × 1 = 27 miles.\nTotal distance = 160 + 27 = 187 miles.\nFinal answer: $187$", "final_answer": "187", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2303ffcd3aeb352b", "problem_id": "7c9c8118092ec7c9", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A cyclist rides at 33 miles per hour for 1 hours, then at 30 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 33 × 1 = 33 miles.\n2) Second leg distance = 30 × 2 = 60 miles.\n3) Total distance = 33 + 60 = 93 miles.\nFinal answer: $93$", "final_answer": "93", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9d2e9649b40a9e8e", "problem_id": "7c9c8118092ec7c9", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A cyclist rides at 33 miles per hour for 1 hours, then at 30 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 33 × 1 = 33 miles.\nSecond leg distance = 30 × 2 = 60 miles.\nTotal distance = 33 + 60 = 93 miles.\nCheck: adding both legs gives 93.\nFinal answer: $93$", "final_answer": "93", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e03a205168ca6d8d", "problem_id": "2f4e6d66fd1cb08e", "style": "answer_only", "instruction": "Final answer only.", "input": "You want 21 liters of a 30% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "84/5 L of 25% and 21/5 L of 50%", "final_answer": "84/5 L of 25% and 21/5 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3338d6cbdb3ea388", "problem_id": "2f4e6d66fd1cb08e", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "You want 21 liters of a 30% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 21−x liters is 50% solution.\nSalt equation: 25x + 50(21−x) = 30·21.\n(25-50)x = (30-50)·21 ⇒ x = 84/5.\nFinal answer: $84/5 L of 25% and 21/5 L of 50%$", "final_answer": "84/5 L of 25% and 21/5 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0c217c8a19159712", "problem_id": "0a1be025f503c38a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{2} + x + 3$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{2} + 3.\n3) Then the limit is 11.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 13.\nFinal answer: $13$", "final_answer": "13", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0706dbe68b7272bf", "problem_id": "b7845101aca07d1c", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Two friends split $1143 in the ratio 6:3. How much does each receive?", "output": "1) Total ratio parts = 6+3 = 9.\n2) Each part = 1143 ÷ 9 = 127.\n3) First share = 6 × 127 = 762.\n4) Second share = 3 × 127 = 381.\nFinal answer: $381$", "final_answer": "381", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c7e75a11ea9171a3", "problem_id": "2266af0b0dafe10f", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} 2 x^{3} - 9 x^{2} + 33 x - 129$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} - 9 x^{2} + 32 x - 129.\n3) Then the limit is -529.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: -533.\nFinal answer: $-533$", "final_answer": "-533", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3ab5405fa643b166", "problem_id": "3bd33c59fe1be88f", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 2 x^{2} - 2 x + \\left(- 3 x^{2} + x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{- 3 x^{2} + x + 2}$.", "output": "1) Let f(x) = $\\frac{- 2 x^{2} - 2 x + \\left(- 3 x^{2} + x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{- 3 x^{2} + x + 2}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{2 \\left(6 x - 1\\right) \\left(- 2 x^{2} - 2 x + \\left(- 3 x^{2} + x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(- 3 x^{2} + x + 2\\right) \\left(- 8 x + 2 \\left(1 - 6 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(- 3 x^{2} + x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(- 3 x^{2} + x + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 4\\right)}{2 \\left(- 3 x^{2} + x + 2\\right)^{2}}$.\nFinal answer: $\\frac{\\left(12\\cdot x - 2\\right)\\cdot \\left(- 2\\cdot x^{2} - 2\\cdot x + \\left(- 3\\cdot x^{2} + x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 3\\right) + \\left(- 3\\cdot x^{2} + x + 2\\right)\\cdot \\left(- 8\\cdot x + \\left(2 - 12\\cdot x\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(- 6\\cdot x^{2} + 2\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + \\left(- 3\\cdot x^{2} + x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 4\\right)}{2\\cdot \\left(- 3\\cdot x^{2} + x + 2\\right)^{2}}$", "final_answer": "\\frac{\\left(12\\cdot x - 2\\right)\\cdot \\left(- 2\\cdot x^{2} - 2\\cdot x + \\left(- 3\\cdot x^{2} + x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 3\\right) + \\left(- 3\\cdot x^{2} + x + 2\\right)\\cdot \\left(- 8\\cdot x + \\left(2 - 12\\cdot x\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(- 6\\cdot x^{2} + 2\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + \\left(- 3\\cdot x^{2} + x + 2\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 4\\right)}{2\\cdot \\left(- 3\\cdot x^{2} + x + 2\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "de6dec86114495ae", "problem_id": "1b0242d6b27e0901", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "An urn has 7 red balls and 11 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(18,4).\n3) Favorable samples: choose k reds and n−k blues: C(7,2)·C(11,2).\n4) Compute totals: C(18,4) = 3060, and favorable = 1155.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{77}{204}$", "final_answer": "\\frac{77}{204}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "be4c9dae00bb2a7b", "problem_id": "1b0242d6b27e0901", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 7 red balls and 11 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(18,4).\nFavorable samples: choose k reds and n−k blues: C(7,2)·C(11,2).\nCompute totals: C(18,4) = 3060, and favorable = 1155.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{77}{204}$.\nFinal answer: $\\frac{77}{204}$", "final_answer": "\\frac{77}{204}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8f0985e31ee1998b", "problem_id": "3c656865e3fcf88d", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{27 x}{10} - \\frac{7}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$\\frac{27}{10}$, c=$- \\frac{7}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{1849}{100}$, so √Δ = $\\frac{43}{10}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{2}{5}$.\n7) x2 = $- \\frac{7}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{2}{5}, - \\frac{7}{4}\\}$", "final_answer": "x \\in \\{\\frac{2}{5}, - \\frac{7}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "af75e3ef5c2b88f6", "problem_id": "755a7e23efb2939a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} - x^{2} + 4 x - 16$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} + 3 x - 16.\n3) Then the limit is -44.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: -48.\nFinal answer: $-48$", "final_answer": "-48", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "026808c637854988", "problem_id": "e4cb5250cce8ce42", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Find the multiplicative inverse of 3 modulo 11.", "output": "We want x such that 3x ≡ 1 (mod 11).\nUse extended Euclid to find s,t with 3s + 11t = 1.\nOne Bezout identity is: 3·(4) + 11·(-1) = 1.\nFinal answer: $x\\equiv 4\\pmod{11}$", "final_answer": "x\\equiv 4\\pmod{11}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72ff32162e2a43c8", "problem_id": "e4cb5250cce8ce42", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the multiplicative inverse of 3 modulo 11.", "output": "1) We want x such that 3x ≡ 1 (mod 11).\n2) Use extended Euclid to find s,t with 3s + 11t = 1.\n3) One Bezout identity is: 3·(4) + 11·(-1) = 1.\n4) Reduce s modulo 11: x ≡ 4 (mod 11).\nFinal answer: $x\\equiv 4\\pmod{11}$", "final_answer": "x\\equiv 4\\pmod{11}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f33b6d02d9485d50", "problem_id": "5a0603aebf4db81b", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 3 x^{3} - 4 x^{2} + x - 1$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{4} - x^{3} - 3 x^{2} + 3$ and a = -1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $12 x^{3} - 3 x^{2} - 6 x$.\n5) Evaluate at a: p'(-1) = $-9$.\nFinal answer: $-9$", "final_answer": "-9", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "94b0b63d43552c42", "problem_id": "5a0603aebf4db81b", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 3 x^{3} - 4 x^{2} + x - 1$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $3 x^{4} - x^{3} - 3 x^{2} + 3$ and a = -1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $12 x^{3} - 3 x^{2} - 6 x$.\nEvaluate at a: p'(-1) = $-9$.\nCheck: the identity gives $-9$.\nFinal answer: $-9$", "final_answer": "-9", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0567b03ae6ae9217", "problem_id": "685864062d14069f", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "You want 18 liters of a 35% salt solution. You have a 15% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 18−x liters is 70% solution.\n2) Salt equation: 15x + 70(18−x) = 35·18.\n3) (15-70)x = (35-70)·18 ⇒ x = 126/11.\n4) Then 18−x = 72/11.\nFinal answer: $126/11 L of 15% and 72/11 L of 70%$", "final_answer": "126/11 L of 15% and 72/11 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d8dd941998b482e", "problem_id": "685864062d14069f", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "You want 18 liters of a 35% salt solution. You have a 15% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 18−x liters is 70% solution.\nSalt equation: 15x + 70(18−x) = 35·18.\n(15-70)x = (35-70)·18 ⇒ x = 126/11.\nFinal answer: $126/11 L of 15% and 72/11 L of 70%$", "final_answer": "126/11 L of 15% and 72/11 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c6fa280aabb2c832", "problem_id": "1270bf1b738bde8c", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{17 x}{3} + \\frac{7}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$- \\frac{17}{3}$, c=$\\frac{7}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{121}{9}$, so √Δ = $\\frac{11}{3}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{7}{3}$.\n7) x2 = $\\frac{1}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{7}{3}, \\frac{1}{2}\\}$", "final_answer": "x \\in \\{\\frac{7}{3}, \\frac{1}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "17ea7058a26d4dc2", "problem_id": "1270bf1b738bde8c", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{17 x}{3} + \\frac{7}{3}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$- \\frac{17}{3}$, c=$\\frac{7}{3}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{7}{3}, \\frac{1}{2}\\}$", "final_answer": "x \\in \\{\\frac{7}{3}, \\frac{1}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6f359826e6eb85ca", "problem_id": "d2dd3866e4577e2c", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A cyclist rides at 38 miles per hour for 1/2 hours, then at 20 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 38 × 1/2 = 19 miles.\n2) Second leg distance = 20 × 2 = 40 miles.\n3) Total distance = 19 + 40 = 59 miles.\nFinal answer: $59$", "final_answer": "59", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e5c21cbb50aa5dcb", "problem_id": "d2dd3866e4577e2c", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A cyclist rides at 38 miles per hour for 1/2 hours, then at 20 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 38 × 1/2 = 19 miles.\nSecond leg distance = 20 × 2 = 40 miles.\nTotal distance = 19 + 40 = 59 miles.\nFinal answer: $59$", "final_answer": "59", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dd0338008c61afd1", "problem_id": "b55ad2a0708c7c22", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "An urn has 11 red balls and 20 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(31,8).\n3) Favorable samples: choose k reds and n−k blues: C(11,3)·C(20,5).\n4) Compute totals: C(31,8) = 7888725, and favorable = 2558160.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{56848}{175305}$", "final_answer": "\\frac{56848}{175305}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cf4f655c1f3dbaf1", "problem_id": "b55ad2a0708c7c22", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 11 red balls and 20 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(31,8).\nFavorable samples: choose k reds and n−k blues: C(11,3)·C(20,5).\nCompute totals: C(31,8) = 7888725, and favorable = 2558160.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{56848}{175305}$.\nFinal answer: $\\frac{56848}{175305}$", "final_answer": "\\frac{56848}{175305}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f9af8653d946ea48", "problem_id": "3809b6f49cc4eac8", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡5 (mod 7), x≡6 (mod 9). Give the solution modulo 63.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 63.\n3) Compute the CRT solution to get x ≡ 33 (mod 63).\nFinal answer: $x\\equiv 33\\pmod{63}$", "final_answer": "x\\equiv 33\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "12de8b15ee4e1b09", "problem_id": "3809b6f49cc4eac8", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡5 (mod 7), x≡6 (mod 9). Give the solution modulo 63.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 63.\nCompute the CRT solution to get x ≡ 33 (mod 63).\nCheck: 33 mod 7=5 and mod 9=6.\nFinal answer: $x\\equiv 33\\pmod{63}$", "final_answer": "x\\equiv 33\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "32e696aa68808a13", "problem_id": "d520b8d0b8b25235", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{7} x}{2} - \\frac{9 x}{10} - \\frac{7}{10} + \\frac{21 \\sqrt{7}}{10}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{3 \\sqrt{7}}{2} - \\frac{9}{10}$, c=$- \\frac{7}{10} + \\frac{21 \\sqrt{7}}{10}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{484}{25} - \\frac{57 \\sqrt{7}}{10}$, so √Δ = $\\sqrt{\\frac{484}{25} - \\frac{57 \\sqrt{7}}{10}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{9}{20} + \\frac{\\sqrt{1936 - 570 \\sqrt{7}}}{20} + \\frac{3 \\sqrt{7}}{4}$.\nx2 = $- \\frac{\\sqrt{1936 - 570 \\sqrt{7}}}{20} + \\frac{9}{20} + \\frac{3 \\sqrt{7}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{9}{20} + \\frac{\\sqrt{1936 - 570\\cdot \\sqrt{7}}}{20} + \\frac{3\\cdot \\sqrt{7}}{4}, - \\frac{\\sqrt{1936 - 570\\cdot \\sqrt{7}}}{20} + \\frac{9}{20} + \\frac{3\\cdot \\sqrt{7}}{4}\\}$", "final_answer": "x \\in \\{\\frac{9}{20} + \\frac{\\sqrt{1936 - 570\\cdot \\sqrt{7}}}{20} + \\frac{3\\cdot \\sqrt{7}}{4}, - \\frac{\\sqrt{1936 - 570\\cdot \\sqrt{7}}}{20} + \\frac{9}{20} + \\frac{3\\cdot \\sqrt{7}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7fc1025163653c3e", "problem_id": "d520b8d0b8b25235", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{7} x}{2} - \\frac{9 x}{10} - \\frac{7}{10} + \\frac{21 \\sqrt{7}}{10}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{3 \\sqrt{7}}{2} - \\frac{9}{10}$, c=$- \\frac{7}{10} + \\frac{21 \\sqrt{7}}{10}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{9}{20} + \\frac{\\sqrt{1936 - 570\\cdot \\sqrt{7}}}{20} + \\frac{3\\cdot \\sqrt{7}}{4}, - \\frac{\\sqrt{1936 - 570\\cdot \\sqrt{7}}}{20} + \\frac{9}{20} + \\frac{3\\cdot \\sqrt{7}}{4}\\}$", "final_answer": "x \\in \\{\\frac{9}{20} + \\frac{\\sqrt{1936 - 570\\cdot \\sqrt{7}}}{20} + \\frac{3\\cdot \\sqrt{7}}{4}, - \\frac{\\sqrt{1936 - 570\\cdot \\sqrt{7}}}{20} + \\frac{9}{20} + \\frac{3\\cdot \\sqrt{7}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "81a120630493367f", "problem_id": "a0a028757c5d03a7", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Two friends split $2475 in the ratio 6:9. How much does each receive?", "output": "1) Total ratio parts = 6+9 = 15.\n2) Each part = 2475 ÷ 15 = 165.\n3) First share = 6 × 165 = 990.\n4) Second share = 9 × 165 = 1485.\nFinal answer: $1485$", "final_answer": "1485", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "afb40da0d5b79ec3", "problem_id": "a0a028757c5d03a7", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $2475 in the ratio 6:9. How much does each receive?", "output": "Total ratio parts = 6+9 = 15.\nEach part = 2475 ÷ 15 = 165.\nFirst share = 6 × 165 = 990.\nSecond share = 9 × 165 = 1485.\nCheck: 990+1485=2475.\nFinal answer: $1485$", "final_answer": "1485", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1cd75f708b0d8b93", "problem_id": "7431162434492cba", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} - 3 x^{3} + x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{3} + x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\nEvaluate at the bounds: F(\\pi) and F(- \\pi).\nSubtract: F(\\pi) − F(- \\pi) = $\\frac{2 \\pi \\left(-6 + \\pi^{2}\\right)}{3}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(-6 + \\pi^{2}\\right)}{3}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(-6 + \\pi^{2}\\right)}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bf2b175196868140", "problem_id": "7431162434492cba", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} - 3 x^{3} + x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{3} + x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\pi).\n3) Subtract: F(\\pi) − F(- \\pi) = $\\frac{2 \\pi \\left(-6 + \\pi^{2}\\right)}{3}$.\n4) Simplify the result.\nFinal answer: $\\frac{2\\cdot \\pi\\cdot \\left(-6 + \\pi^{2}\\right)}{3}$", "final_answer": "\\frac{2\\cdot \\pi\\cdot \\left(-6 + \\pi^{2}\\right)}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0b04257e65b96271", "problem_id": "660f83bc358400a6", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{3} - 3 x^{2} - 2$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-2) = 0.\n3) d/dx(- 3 x^{2}) = - 6 x.\n4) d/dx(4 x^{3}) = 12 x^{2}.\n5) Combine: f'(x) = 6 x \\left(2 x - 1\\right).\nFinal answer: $6\\cdot x\\cdot \\left(2\\cdot x - 1\\right)$", "final_answer": "6\\cdot x\\cdot \\left(2\\cdot x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c05e7b06bb729098", "problem_id": "660f83bc358400a6", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{3} - 3 x^{2} - 2$.", "output": "Differentiate term-by-term:\nd/dx(-2) = 0.\nd/dx(- 3 x^{2}) = - 6 x.\nd/dx(4 x^{3}) = 12 x^{2}.\nCombine: f'(x) = 6 x \\left(2 x - 1\\right).\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $6\\cdot x\\cdot \\left(2\\cdot x - 1\\right)$", "final_answer": "6\\cdot x\\cdot \\left(2\\cdot x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7db8b350d0b9af84", "problem_id": "e336008ab7883bd8", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Two friends split $660 in the ratio 4:7. How much does each receive?", "output": "1) Total ratio parts = 4+7 = 11.\n2) Each part = 660 ÷ 11 = 60.\n3) First share = 4 × 60 = 240.\n4) Second share = 7 × 60 = 420.\nFinal answer: $420$", "final_answer": "420", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c68c34dcad9c20dd", "problem_id": "e336008ab7883bd8", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $660 in the ratio 4:7. How much does each receive?", "output": "Total ratio parts = 4+7 = 11.\nEach part = 660 ÷ 11 = 60.\nFirst share = 4 × 60 = 240.\nSecond share = 7 × 60 = 420.\nCheck: 240+420=660.\nFinal answer: $420$", "final_answer": "420", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "70d03710f55396c3", "problem_id": "62bd1e780e5012f8", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡6 (mod 9), x≡4 (mod 8). Give the solution modulo 72.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 72.\n3) Compute the CRT solution to get x ≡ 60 (mod 72).\nFinal answer: $x\\equiv 60\\pmod{72}$", "final_answer": "x\\equiv 60\\pmod{72}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "093a5cb92d92c313", "problem_id": "62bd1e780e5012f8", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the congruences: x≡6 (mod 9), x≡4 (mod 8). Give the solution modulo 72.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 72.\nCompute the CRT solution to get x ≡ 60 (mod 72).\nFinal answer: $x\\equiv 60\\pmod{72}$", "final_answer": "x\\equiv 60\\pmod{72}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f16c8f58ddf9205a", "problem_id": "dbe34a8b224711d0", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 x^{3} - 4 x^{2} + 11 x - 19$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} - 4 x^{2} + 10 x - 19.\n3) Then the limit is -79.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -81.\nFinal answer: $-81$", "final_answer": "-81", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "63156c922a57f6ce", "problem_id": "5d04b4ea4ae6f4b1", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 10 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=5, p=$\\frac{1}{6}$ and simplify.\nHere C(10,5) = 252.\nCheck: expression simplifies to $\\frac{21875}{1679616}$.\nFinal answer: $\\frac{21875}{1679616}$", "final_answer": "\\frac{21875}{1679616}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3b79cae92251ad8b", "problem_id": "0660057be7541f3d", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the equation of the tangent line to $y=x^{3} - 3 x^{2} + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=2$.", "output": "1) Differentiate: f'(x) = $3 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)}$.\n2) Slope at x=2: m = f'(2) = $\\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\n3) Point on curve: (2, f(2)) = (2, $-6 + \\cos{\\left(4 \\right)} + e$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -6 + \\cos{\\left(4 \\right)} + e = \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\n6) Optionally expand: y = $\\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} - 6 + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$.\nFinal answer: $y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} - 6 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$", "final_answer": "y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} - 6 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3ff581d59e814f86", "problem_id": "0660057be7541f3d", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the equation of the tangent line to $y=x^{3} - 3 x^{2} + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=2$.", "output": "Differentiate: f'(x) = $3 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)}$.\nSlope at x=2: m = f'(2) = $\\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\nPoint on curve: (2, f(2)) = (2, $-6 + \\cos{\\left(4 \\right)} + e$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -6 + \\cos{\\left(4 \\right)} + e = \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\nOptionally expand: y = $\\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} - 6 + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} - 6 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$", "final_answer": "y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} - 6 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cfbaaa20d2246bfc", "problem_id": "b1991bb4ad98f6ce", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{13} x}{2} + \\frac{25 x}{6} - \\frac{5 \\sqrt{13}}{6} + \\frac{25}{6}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{25}{6} - \\frac{\\sqrt{13}}{2}$, c=$\\frac{25}{6} - \\frac{5 \\sqrt{13}}{6}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{71}{18} - \\frac{5 \\sqrt{13}}{6}$, so √Δ = $\\sqrt{\\frac{71}{18} - \\frac{5 \\sqrt{13}}{6}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{25}{12} + \\frac{\\sqrt{142 - 30 \\sqrt{13}}}{12} + \\frac{\\sqrt{13}}{4}$.\n7) x2 = $- \\frac{25}{12} - \\frac{\\sqrt{142 - 30 \\sqrt{13}}}{12} + \\frac{\\sqrt{13}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{25}{12} + \\frac{\\sqrt{142 - 30\\cdot \\sqrt{13}}}{12} + \\frac{\\sqrt{13}}{4}, - \\frac{25}{12} - \\frac{\\sqrt{142 - 30\\cdot \\sqrt{13}}}{12} + \\frac{\\sqrt{13}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{25}{12} + \\frac{\\sqrt{142 - 30\\cdot \\sqrt{13}}}{12} + \\frac{\\sqrt{13}}{4}, - \\frac{25}{12} - \\frac{\\sqrt{142 - 30\\cdot \\sqrt{13}}}{12} + \\frac{\\sqrt{13}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "08d6c3c44640ec3f", "problem_id": "b1991bb4ad98f6ce", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{13} x}{2} + \\frac{25 x}{6} - \\frac{5 \\sqrt{13}}{6} + \\frac{25}{6}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{25}{6} - \\frac{\\sqrt{13}}{2}$, c=$\\frac{25}{6} - \\frac{5 \\sqrt{13}}{6}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{- \\frac{25}{12} + \\frac{\\sqrt{142 - 30\\cdot \\sqrt{13}}}{12} + \\frac{\\sqrt{13}}{4}, - \\frac{25}{12} - \\frac{\\sqrt{142 - 30\\cdot \\sqrt{13}}}{12} + \\frac{\\sqrt{13}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{25}{12} + \\frac{\\sqrt{142 - 30\\cdot \\sqrt{13}}}{12} + \\frac{\\sqrt{13}}{4}, - \\frac{25}{12} - \\frac{\\sqrt{142 - 30\\cdot \\sqrt{13}}}{12} + \\frac{\\sqrt{13}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b03cd89140418d1e", "problem_id": "ce51192a114a8a2c", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 14 times, what is the probability of exactly 8 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=14, k=8, p=$\\frac{1}{10}$ and simplify.\n4) Here C(14,8) = 3003.\nFinal answer: $\\frac{1595917323}{100000000000000}$", "final_answer": "\\frac{1595917323}{100000000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cb2f3efaa631b740", "problem_id": "ce51192a114a8a2c", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 14 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=14, k=8, p=$\\frac{1}{10}$ and simplify.\nHere C(14,8) = 3003.\nCheck: expression simplifies to $\\frac{1595917323}{100000000000000}$.\nFinal answer: $\\frac{1595917323}{100000000000000}$", "final_answer": "\\frac{1595917323}{100000000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "94de97ec490a49d7", "problem_id": "0164c41f7047fa8b", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$- x + 2 y$ = $2$\n$2 x + 4 y$ = $20$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $- 2 x + 4 y$ = $4$.\n3) Multiply the second equation by $-1$: $- 2 x - 4 y$ = $-20$.\n4) Subtract to eliminate x: $8 y$ = $24$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, 3\\right)$", "final_answer": "(x,y)=\\left(4, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bef6d0ced5d595d5", "problem_id": "0164c41f7047fa8b", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the linear system:\n$- x + 2 y$ = $2$\n$2 x + 4 y$ = $20$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $2$: $- 2 x + 4 y$ = $4$.\nMultiply the second equation by $-1$: $- 2 x - 4 y$ = $-20$.\nFinal answer: $(x,y)=\\left(4, 3\\right)$", "final_answer": "(x,y)=\\left(4, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f9241d35205f0afd", "problem_id": "4164dedb99c7086f", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Two friends split $1104 in the ratio 5:3. How much does each receive?", "output": "1) Total ratio parts = 5+3 = 8.\n2) Each part = 1104 ÷ 8 = 138.\n3) First share = 5 × 138 = 690.\n4) Second share = 3 × 138 = 414.\nFinal answer: $414$", "final_answer": "414", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "658f61f0cfe1a75c", "problem_id": "4164dedb99c7086f", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $1104 in the ratio 5:3. How much does each receive?", "output": "Total ratio parts = 5+3 = 8.\nEach part = 1104 ÷ 8 = 138.\nFirst share = 5 × 138 = 690.\nSecond share = 3 × 138 = 414.\nCheck: 690+414=1104.\nFinal answer: $414$", "final_answer": "414", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7c98121ba2df2c74", "problem_id": "59f5c893a8232453", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} - x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $- 3 \\pi^{3} - 3 \\pi^{2} - 4$.\n4) Simplify the result.\nFinal answer: $- 3\\cdot \\pi^{3} - 3\\cdot \\pi^{2} - 4$", "final_answer": "- 3\\cdot \\pi^{3} - 3\\cdot \\pi^{2} - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d6d6e1397d68bf0", "problem_id": "59f5c893a8232453", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} - x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{2} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $- 3 \\pi^{3} - 3 \\pi^{2} - 4$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- 3\\cdot \\pi^{3} - 3\\cdot \\pi^{2} - 4$", "final_answer": "- 3\\cdot \\pi^{3} - 3\\cdot \\pi^{2} - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "32299974a31216a0", "problem_id": "276c94ad624a95fe", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x^{3} - 4 x^{2} - 7 x - 12$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - x^{3} - 4 x^{2} - 8 x - 12.\n3) Then the limit is -52.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -50.\nFinal answer: $-50$", "final_answer": "-50", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6a077c90c957b79d", "problem_id": "7916337ec3a2e9ef", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} - x^{2} + 2 \\sin{\\left(x \\right)} + 2$ at $x=2$.", "output": "1) Differentiate: f'(x) = $6 x^{2} - 2 x + 2 \\cos{\\left(x \\right)}$.\n2) Slope at x=2: m = f'(2) = $2 \\cos{\\left(2 \\right)} + 20$.\n3) Point on curve: (2, f(2)) = (2, $2 \\sin{\\left(2 \\right)} + 14$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 2 \\sin{\\left(2 \\right)} + 14 = 2 \\cos{\\left(2 \\right)} + 20(x − 2).\n6) Optionally expand: y = $2 x \\cos{\\left(2 \\right)} + 20 x - 26 - 4 \\cos{\\left(2 \\right)} + 2 \\sin{\\left(2 \\right)}$.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 20\\cdot x - 26 - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)}$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 20\\cdot x - 26 - 4\\cdot \\cos{\\left(2 \\right)} + 2\\cdot \\sin{\\left(2 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b0075a0134dbbc91", "problem_id": "26535aaad4aaf8dc", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 21 red balls and 24 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(45,9).\nFavorable samples: choose k reds and n−k blues: C(21,3)·C(24,6).\nCompute totals: C(45,9) = 886163135, and favorable = 179012680.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{171304}{848003}$.\nFinal answer: $\\frac{171304}{848003}$", "final_answer": "\\frac{171304}{848003}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c47bec61245df39a", "problem_id": "26535aaad4aaf8dc", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 21 red balls and 24 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(45,9).\n3) Favorable samples: choose k reds and n−k blues: C(21,3)·C(24,6).\n4) Compute totals: C(45,9) = 886163135, and favorable = 179012680.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{171304}{848003}$", "final_answer": "\\frac{171304}{848003}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "91234c9225693e4a", "problem_id": "0695b8eff042c92d", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} 3 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\n3) Subtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{3 \\pi}{2} + \\frac{\\pi^{2}}{8} + 2 + \\frac{\\pi^{3}}{8}$.\n4) Simplify the result.\nFinal answer: $- \\frac{3\\cdot \\pi}{2} + \\frac{\\pi^{2}}{8} + 2 + \\frac{\\pi^{3}}{8}$", "final_answer": "- \\frac{3\\cdot \\pi}{2} + \\frac{\\pi^{2}}{8} + 2 + \\frac{\\pi^{3}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "30b913592f2766e4", "problem_id": "0695b8eff042c92d", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} 3 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\nSubtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{3 \\pi}{2} + \\frac{\\pi^{2}}{8} + 2 + \\frac{\\pi^{3}}{8}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{3\\cdot \\pi}{2} + \\frac{\\pi^{2}}{8} + 2 + \\frac{\\pi^{3}}{8}$", "final_answer": "- \\frac{3\\cdot \\pi}{2} + \\frac{\\pi^{2}}{8} + 2 + \\frac{\\pi^{3}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ffc5c382483ec856", "problem_id": "9d208654e1e7a33b", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$- 3 x - 4 y$ = $55$\n$- x + y$ = $2$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $3 x + 4 y$ = $-55$.\nMultiply the second equation by $-3$: $3 x - 3 y$ = $-6$.\nSubtract to eliminate x: $7 y$ = $-49$.\nSolve for y: y = $-7$.\nSubstitute back to find x: x = $-9$.\nCheck: plug (x,y)=(-9,-7) into both equations.\nFinal answer: $(x,y)=\\left(-9, -7\\right)$", "final_answer": "(x,y)=\\left(-9, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d5fcc72ec7554fec", "problem_id": "9d208654e1e7a33b", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$- 3 x - 4 y$ = $55$\n$- x + y$ = $2$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $3 x + 4 y$ = $-55$.\n3) Multiply the second equation by $-3$: $3 x - 3 y$ = $-6$.\n4) Subtract to eliminate x: $7 y$ = $-49$.\n5) Solve for y: y = $-7$.\n6) Substitute back to find x: x = $-9$.\nFinal answer: $(x,y)=\\left(-9, -7\\right)$", "final_answer": "(x,y)=\\left(-9, -7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3b08038e1eccb73d", "problem_id": "2f460b9885f27eca", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x^{2} - 4 x - 5$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} - 5 x - 5.\n3) Then the limit is -19.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -17.\nFinal answer: $-17$", "final_answer": "-17", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "dbd455f89de54cff", "problem_id": "7939d1b46e07d88d", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$x - 2 y$ = $0$\n$- 3 x + 4 y$ = $-4$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $- 3 x + 6 y$ = $0$.\n3) Multiply the second equation by $1$: $- 3 x + 4 y$ = $-4$.\n4) Subtract to eliminate x: $2 y$ = $4$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, 2\\right)$", "final_answer": "(x,y)=\\left(4, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "96cf051c73b7fb18", "problem_id": "39755f7adb260959", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$- 2 x - 2 y$ = $-16$\n$- 7 x + 5 y$ = $-44$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-7$: $14 x + 14 y$ = $112$.\n3) Multiply the second equation by $-2$: $14 x - 10 y$ = $88$.\n4) Subtract to eliminate x: $24 y$ = $24$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $7$.\nFinal answer: $(x,y)=\\left(7, 1\\right)$", "final_answer": "(x,y)=\\left(7, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5469cdaf7cbe53f5", "problem_id": "39755f7adb260959", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the linear system:\n$- 2 x - 2 y$ = $-16$\n$- 7 x + 5 y$ = $-44$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-7$: $14 x + 14 y$ = $112$.\nMultiply the second equation by $-2$: $14 x - 10 y$ = $88$.\nFinal answer: $(x,y)=\\left(7, 1\\right)$", "final_answer": "(x,y)=\\left(7, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec1a47dafa23f257", "problem_id": "4de46a39da1882c5", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(- 2 x^{3} - 3\\right) \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $\\left(x + 2\\right) e^{x} + \\left(- 2 x^{3} - 3\\right) \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- 6 x^{2} \\cos{\\left(2 x \\right)} + \\left(x + 2\\right) e^{x} - 2 \\left(- 2 x^{3} - 3\\right) \\sin{\\left(2 x \\right)} + e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 6\\cdot x^{2}\\cdot \\cos{\\left(2\\cdot x \\right)} + \\left(x + 2\\right)\\cdot e^{x} - \\left(- 4\\cdot x^{3} - 6\\right)\\cdot \\sin{\\left(2\\cdot x \\right)} + e^{x}$", "final_answer": "- 6\\cdot x^{2}\\cdot \\cos{\\left(2\\cdot x \\right)} + \\left(x + 2\\right)\\cdot e^{x} - \\left(- 4\\cdot x^{3} - 6\\right)\\cdot \\sin{\\left(2\\cdot x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f463d8d29a5748ac", "problem_id": "4de46a39da1882c5", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(- 2 x^{3} - 3\\right) \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $\\left(x + 2\\right) e^{x} + \\left(- 2 x^{3} - 3\\right) \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- 6 x^{2} \\cos{\\left(2 x \\right)} + \\left(x + 2\\right) e^{x} - 2 \\left(- 2 x^{3} - 3\\right) \\sin{\\left(2 x \\right)} + e^{x}$.\nFinal answer: $- 6\\cdot x^{2}\\cdot \\cos{\\left(2\\cdot x \\right)} + \\left(x + 2\\right)\\cdot e^{x} - \\left(- 4\\cdot x^{3} - 6\\right)\\cdot \\sin{\\left(2\\cdot x \\right)} + e^{x}$", "final_answer": "- 6\\cdot x^{2}\\cdot \\cos{\\left(2\\cdot x \\right)} + \\left(x + 2\\right)\\cdot e^{x} - \\left(- 4\\cdot x^{3} - 6\\right)\\cdot \\sin{\\left(2\\cdot x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bc364755ba0a5555", "problem_id": "6bd9d4e120536af9", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Two friends split $350 in the ratio 4:3. How much does each receive?", "output": "1) Total ratio parts = 4+3 = 7.\n2) Each part = 350 ÷ 7 = 50.\n3) First share = 4 × 50 = 200.\n4) Second share = 3 × 50 = 150.\nFinal answer: $150$", "final_answer": "150", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f589f34dc6a75987", "problem_id": "6bd9d4e120536af9", "style": "answer_only", "instruction": "Final answer only.", "input": "Two friends split $350 in the ratio 4:3. How much does each receive?", "output": "$150$", "final_answer": "150", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f8f094407fbf81c6", "problem_id": "ad32b22caaf5c830", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$x - y$ = $7$\n$- 7 x + 3 y$ = $-13$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-7$: $- 7 x + 7 y$ = $-49$.\n3) Multiply the second equation by $1$: $- 7 x + 3 y$ = $-13$.\n4) Subtract to eliminate x: $4 y$ = $-36$.\n5) Solve for y: y = $-9$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, -9\\right)$", "final_answer": "(x,y)=\\left(-2, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1e2fe6d96e069a82", "problem_id": "ad32b22caaf5c830", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$x - y$ = $7$\n$- 7 x + 3 y$ = $-13$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-7$: $- 7 x + 7 y$ = $-49$.\nMultiply the second equation by $1$: $- 7 x + 3 y$ = $-13$.\nSubtract to eliminate x: $4 y$ = $-36$.\nSolve for y: y = $-9$.\nSubstitute back to find x: x = $-2$.\nCheck: plug (x,y)=(-2,-9) into both equations.\nFinal answer: $(x,y)=\\left(-2, -9\\right)$", "final_answer": "(x,y)=\\left(-2, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e2580834c57b00f", "problem_id": "e9329e89f2bc78af", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 19 red balls and 14 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(33,9).\nFavorable samples: choose k reds and n−k blues: C(19,2)·C(14,7).\nCompute totals: C(33,9) = 38567100, and favorable = 586872.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{342}{22475}$.\nFinal answer: $\\frac{342}{22475}$", "final_answer": "\\frac{342}{22475}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "86e86dfc9c36111b", "problem_id": "e9329e89f2bc78af", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 19 red balls and 14 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(33,9).\n3) Favorable samples: choose k reds and n−k blues: C(19,2)·C(14,7).\n4) Compute totals: C(33,9) = 38567100, and favorable = 586872.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{342}{22475}$", "final_answer": "\\frac{342}{22475}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "20960dd5b134e90d", "problem_id": "52377b7868c8768b", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - x^{2} + 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=2$.", "output": "Differentiate: f'(x) = $- 9 x^{2} - 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 3$.\nSlope at x=2: m = f'(2) = $-37 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\nPoint on curve: (2, f(2)) = (2, $-23 + \\cos{\\left(4 \\right)} + e$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -23 + \\cos{\\left(4 \\right)} + e = -37 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\nOptionally expand: y = $- 37 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 51$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 37\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 51$", "final_answer": "y = - 37\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 51", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "521a0a59f922504c", "problem_id": "52377b7868c8768b", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - x^{2} + 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 1$ at $x=2$.", "output": "Differentiate: f'(x) = $- 9 x^{2} - 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 3$.\nSlope at x=2: m = f'(2) = $-37 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\nPoint on curve: (2, f(2)) = (2, $-23 + \\cos{\\left(4 \\right)} + e$).\nFinal answer: $y = - 37\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 51$", "final_answer": "y = - 37\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 51", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1d6843301b7b84a7", "problem_id": "ec76aaf9d0f1d6a1", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Worker A finishes a job in 16 hours and Worker B finishes it in 4 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/16 job/hour. B's rate = 1/4 job/hour.\n2) Combined rate = 1/16 + 1/4 = 5/16 job/hour.\n3) Time = 1 ÷ (combined rate) = 16/5 hours.\nFinal answer: $\\frac{16}{5}$", "final_answer": "\\frac{16}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ae45c0aa815b326", "problem_id": "ec76aaf9d0f1d6a1", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Worker A finishes a job in 16 hours and Worker B finishes it in 4 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/16 job/hour. B's rate = 1/4 job/hour.\nCombined rate = 1/16 + 1/4 = 5/16 job/hour.\nTime = 1 ÷ (combined rate) = 16/5 hours.\nFinal answer: $\\frac{16}{5}$", "final_answer": "\\frac{16}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "837e663924005bd7", "problem_id": "c850d75b6ee3ace8", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=38$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=38 and k=10.\nCompute C(47,9) = 1362649145.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $1362649145$", "final_answer": "1362649145", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "001524f6a6db45f1", "problem_id": "c850d75b6ee3ace8", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=38$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=38 and k=10.\n4) Compute C(47,9) = 1362649145.\nFinal answer: $1362649145$", "final_answer": "1362649145", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e657b4ab49849752", "problem_id": "844160d486ff7dfa", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} - x^{3} + 2 x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{3} + 2 x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\nSubtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{\\pi^{4}}{64} + 2 + \\frac{\\pi^{3}}{12} + \\pi$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{\\pi^{4}}{64} + 2 + \\frac{\\pi^{3}}{12} + \\pi$", "final_answer": "- \\frac{\\pi^{4}}{64} + 2 + \\frac{\\pi^{3}}{12} + \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "466c045989c381d8", "problem_id": "844160d486ff7dfa", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} - x^{3} + 2 x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{3} + 2 x^{2} + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\n3) Subtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{\\pi^{4}}{64} + 2 + \\frac{\\pi^{3}}{12} + \\pi$.\n4) Simplify the result.\nFinal answer: $- \\frac{\\pi^{4}}{64} + 2 + \\frac{\\pi^{3}}{12} + \\pi$", "final_answer": "- \\frac{\\pi^{4}}{64} + 2 + \\frac{\\pi^{3}}{12} + \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2eaef04e70129c41", "problem_id": "ec520a3d58a37ed8", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3$.\n3) Compute an antiderivative: F(x) = $\\frac{x^{3}}{3} + \\frac{3 x^{2}}{2} - 3 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e153bffe0e37527a", "problem_id": "ec520a3d58a37ed8", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 3$.\nCompute an antiderivative: F(x) = $\\frac{x^{3}}{3} + \\frac{3 x^{2}}{2} - 3 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} - 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "512a8957af42f9a6", "problem_id": "e3000c6b13e01f35", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(- x^{3} - 3 x^{2} + x - 2\\right) \\cos{\\left(3 x - 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x^{3} \\cos{\\left(3 x - 1 \\right)}) = x^{3} \\sin{\\left(3 x - 1 \\right)}$.\n3) So $f'(x) = x^{3} \\sin{\\left(3 x - 1 \\right)} + 9 x^{2} \\sin{\\left(3 x - 1 \\right)} + 2 x e^{x} - 3 x \\sin{\\left(3 x - 1 \\right)} - 6 x \\cos{\\left(3 x - 1 \\right)} + 6 e^{x} + 6 \\sin{\\left(3 x - 1 \\right)} + \\cos{\\left(3 x - 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x^{3} \\cos{\\left(3 x - 1 \\right)}) = 3 x^{3} \\sin{\\left(3 x - 1 \\right)} - 3 x^{2} \\cos{\\left(3 x - 1 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x + 2\\right) e^{x} - 3 \\left(x \\left(x \\left(- x - 3\\right) + 1\\right) - 2\\right) \\sin{\\left(3 x - 1 \\right)} + \\left(- 3 x^{2} - 6 x + 1\\right) \\cos{\\left(3 x - 1 \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(x + 2\\right) e^{x} - 3 \\left(x \\left(x \\left(- x - 3\\right) + 1\\right) - 2\\right) \\sin{\\left(3 x - 1 \\right)} + \\left(- 3 x^{2} - 6 x + 1\\right) \\cos{\\left(3 x - 1 \\right)} + 2 e^{x}$", "final_answer": "2 \\left(x + 2\\right) e^{x} - 3 \\left(x \\left(x \\left(- x - 3\\right) + 1\\right) - 2\\right) \\sin{\\left(3 x - 1 \\right)} + \\left(- 3 x^{2} - 6 x + 1\\right) \\cos{\\left(3 x - 1 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0fb62d970ef23727", "problem_id": "e208c79d4c482a59", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡2 (mod 11). Give the solution modulo 33.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 33.\nCompute the CRT solution to get x ≡ 13 (mod 33).\nFinal answer: $x\\equiv 13\\pmod{33}$", "final_answer": "x\\equiv 13\\pmod{33}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6985de516a261215", "problem_id": "32d8d20afd531b76", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{3} + x^{2} - \\frac{x}{2} + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - \\frac{1}{2}}{x - 4}$.", "output": "1) Let f(x) = $\\frac{x^{3} + x^{2} - \\frac{x}{2} + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - \\frac{1}{2}}{x - 4}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{- 2 x^{3} - 2 x^{2} + x + \\left(x - 4\\right) \\left(6 x^{2} + 4 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{2 \\left(x - 4\\right)^{2}}$.\nFinal answer: $\\frac{- 2\\cdot x^{3} - 2\\cdot x^{2} + x + \\left(x - 4\\right)\\cdot \\left(6\\cdot x^{2} + 4\\cdot x + \\left(x - 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x - 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 1\\right) - \\left(2\\cdot x - 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 1}{2\\cdot \\left(x - 4\\right)^{2}}$", "final_answer": "\\frac{- 2\\cdot x^{3} - 2\\cdot x^{2} + x + \\left(x - 4\\right)\\cdot \\left(6\\cdot x^{2} + 4\\cdot x + \\left(x - 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x - 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 1\\right) - \\left(2\\cdot x - 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 1}{2\\cdot \\left(x - 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4ce190720ae80912", "problem_id": "32d8d20afd531b76", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{3} + x^{2} - \\frac{x}{2} + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - \\frac{1}{2}}{x - 4}$.", "output": "Let f(x) = $\\frac{x^{3} + x^{2} - \\frac{x}{2} + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - \\frac{1}{2}}{x - 4}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{- 2 x^{3} - 2 x^{2} + x + \\left(x - 4\\right) \\left(6 x^{2} + 4 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{2 \\left(x - 4\\right)^{2}}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\frac{- 2\\cdot x^{3} - 2\\cdot x^{2} + x + \\left(x - 4\\right)\\cdot \\left(6\\cdot x^{2} + 4\\cdot x + \\left(x - 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x - 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 1\\right) - \\left(2\\cdot x - 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 1}{2\\cdot \\left(x - 4\\right)^{2}}$", "final_answer": "\\frac{- 2\\cdot x^{3} - 2\\cdot x^{2} + x + \\left(x - 4\\right)\\cdot \\left(6\\cdot x^{2} + 4\\cdot x + \\left(x - 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x - 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 2\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 1\\right) - \\left(2\\cdot x - 8\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 1}{2\\cdot \\left(x - 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bf77d1e289722d47", "problem_id": "83a3b0f783dfd312", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 4 x^{3} + 3 x^{2} - 4 x - 1$.", "output": "Differentiate term-by-term:\nd/dx(-1) = 0.\nd/dx(- 4 x) = -4.\nd/dx(- 4 x^{3}) = - 12 x^{2}.\nd/dx(3 x^{2}) = 6 x.\nCombine: f'(x) = 2 \\left(- 6 x^{2} + 3 x - 2\\right).\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 12\\cdot x^{2} + 6\\cdot x - 4$", "final_answer": "- 12\\cdot x^{2} + 6\\cdot x - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d875adc49fd6e646", "problem_id": "83a3b0f783dfd312", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 4 x^{3} + 3 x^{2} - 4 x - 1$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-1) = 0.\n3) d/dx(- 4 x) = -4.\n4) d/dx(- 4 x^{3}) = - 12 x^{2}.\n5) d/dx(3 x^{2}) = 6 x.\n6) Combine: f'(x) = 2 \\left(- 6 x^{2} + 3 x - 2\\right).\nFinal answer: $- 12\\cdot x^{2} + 6\\cdot x - 4$", "final_answer": "- 12\\cdot x^{2} + 6\\cdot x - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b842f3b6f4200224", "problem_id": "ddbac8b99a18096b", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Worker A finishes a job in 18 hours and Worker B finishes it in 19 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/18 job/hour. B's rate = 1/19 job/hour.\n2) Combined rate = 1/18 + 1/19 = 37/342 job/hour.\n3) Time = 1 ÷ (combined rate) = 342/37 hours.\nFinal answer: $\\frac{342}{37}$", "final_answer": "\\frac{342}{37}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fd84726cd03ad3f0", "problem_id": "ddbac8b99a18096b", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Worker A finishes a job in 18 hours and Worker B finishes it in 19 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/18 job/hour. B's rate = 1/19 job/hour.\nCombined rate = 1/18 + 1/19 = 37/342 job/hour.\nTime = 1 ÷ (combined rate) = 342/37 hours.\nCheck: (combined rate)×(time) = 37/342×342/37 = 1.\nFinal answer: $\\frac{342}{37}$", "final_answer": "\\frac{342}{37}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eb8d197e479301d7", "problem_id": "8b1a348fa710b7f2", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} - 2 x + \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{2 x - 5}$.", "output": "1) Let f(x) = $\\frac{- x^{2} - 2 x + \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{2 x - 5}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{4 x^{2} + 8 x + \\left(2 x - 5\\right) \\left(- 4 x + \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4\\right) - 4 \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8}{2 \\left(2 x - 5\\right)^{2}}$.\nFinal answer: $\\frac{4\\cdot x^{2} + 8\\cdot x + \\left(2\\cdot x - 5\\right)\\cdot \\left(- 4\\cdot x + \\left(2\\cdot x - 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x - 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 4\\right) - \\left(8\\cdot x - 20\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 8}{2\\cdot \\left(2\\cdot x - 5\\right)^{2}}$", "final_answer": "\\frac{4\\cdot x^{2} + 8\\cdot x + \\left(2\\cdot x - 5\\right)\\cdot \\left(- 4\\cdot x + \\left(2\\cdot x - 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x - 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 4\\right) - \\left(8\\cdot x - 20\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 8}{2\\cdot \\left(2\\cdot x - 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4e76c9787a851486", "problem_id": "8b1a348fa710b7f2", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} - 2 x + \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{2 x - 5}$.", "output": "Let f(x) = $\\frac{- x^{2} - 2 x + \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{2 x - 5}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{4 x^{2} + 8 x + \\left(2 x - 5\\right) \\left(- 4 x + \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 4\\right) - 4 \\left(2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 8}{2 \\left(2 x - 5\\right)^{2}}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\frac{4\\cdot x^{2} + 8\\cdot x + \\left(2\\cdot x - 5\\right)\\cdot \\left(- 4\\cdot x + \\left(2\\cdot x - 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x - 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 4\\right) - \\left(8\\cdot x - 20\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 8}{2\\cdot \\left(2\\cdot x - 5\\right)^{2}}$", "final_answer": "\\frac{4\\cdot x^{2} + 8\\cdot x + \\left(2\\cdot x - 5\\right)\\cdot \\left(- 4\\cdot x + \\left(2\\cdot x - 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x - 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 4\\right) - \\left(8\\cdot x - 20\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 8}{2\\cdot \\left(2\\cdot x - 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3bbf1fa235467d43", "problem_id": "d26079fb8633df7a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "You want 70 liters of a 45% salt solution. You have a 15% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 70−x liters is 50% solution.\nSalt equation: 15x + 50(70−x) = 45·70.\n(15-50)x = (45-50)·70 ⇒ x = 10.\nThen 70−x = 60.\nCheck: 15·10 + 50·60 = 45·70.\nFinal answer: $10 L of 15% and 60 L of 50%$", "final_answer": "10 L of 15% and 60 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f543c8f79a19177b", "problem_id": "d26079fb8633df7a", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "You want 70 liters of a 45% salt solution. You have a 15% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 70−x liters is 50% solution.\n2) Salt equation: 15x + 50(70−x) = 45·70.\n3) (15-50)x = (45-50)·70 ⇒ x = 10.\n4) Then 70−x = 60.\nFinal answer: $10 L of 15% and 60 L of 50%$", "final_answer": "10 L of 15% and 60 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b26be5b65192556f", "problem_id": "fc153c18530ce403", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} x^{3} - x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $x^{3} - x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $\\frac{3 \\pi \\left(- 4 \\pi^{2} - 6 \\pi - 8 + 5 \\pi^{3}\\right)}{4}$.\n4) Simplify the result.\nFinal answer: $\\frac{3\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} - 6\\cdot \\pi - 8 + 5\\cdot \\pi^{3}\\right)}{4}$", "final_answer": "\\frac{3\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} - 6\\cdot \\pi - 8 + 5\\cdot \\pi^{3}\\right)}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d354da3c73a9f3a0", "problem_id": "fc153c18530ce403", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} x^{3} - x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $x^{3} - x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $\\frac{3 \\pi \\left(- 4 \\pi^{2} - 6 \\pi - 8 + 5 \\pi^{3}\\right)}{4}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{3\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} - 6\\cdot \\pi - 8 + 5\\cdot \\pi^{3}\\right)}{4}$", "final_answer": "\\frac{3\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{2} - 6\\cdot \\pi - 8 + 5\\cdot \\pi^{3}\\right)}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3e602f710fc564c3", "problem_id": "fb1e3471e7c8c796", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the linear system:\n$12 x - 11 y$ = $183$\n$7 x + 2 y$ = $31$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $7$: $84 x - 77 y$ = $1281$.\nMultiply the second equation by $12$: $84 x + 24 y$ = $372$.\nFinal answer: $(x,y)=\\left(7, -9\\right)$", "final_answer": "(x,y)=\\left(7, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eb27cef60bbf1f93", "problem_id": "fb1e3471e7c8c796", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$12 x - 11 y$ = $183$\n$7 x + 2 y$ = $31$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $7$: $84 x - 77 y$ = $1281$.\n3) Multiply the second equation by $12$: $84 x + 24 y$ = $372$.\n4) Subtract to eliminate x: $- 101 y$ = $909$.\n5) Solve for y: y = $-9$.\n6) Substitute back to find x: x = $7$.\nFinal answer: $(x,y)=\\left(7, -9\\right)$", "final_answer": "(x,y)=\\left(7, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dbfa2c43748e323e", "problem_id": "0b6d4ff83ea19918", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{1}{7}$. If it is flipped 11 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=3, p=$\\frac{1}{7}$ and simplify.\n4) Here C(11,3) = 165.\nFinal answer: $\\frac{277136640}{1977326743}$", "final_answer": "\\frac{277136640}{1977326743}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b5a26ea53df792f8", "problem_id": "0b6d4ff83ea19918", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{7}$. If it is flipped 11 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=11, k=3, p=$\\frac{1}{7}$ and simplify.\nFinal answer: $\\frac{277136640}{1977326743}$", "final_answer": "\\frac{277136640}{1977326743}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "442fa4d079bcb51c", "problem_id": "e3e1a2aac03ec873", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_9=40$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_9=31.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=31 and k=9.\nFinal answer: $61523748$", "final_answer": "61523748", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a0d3573066bb27ec", "problem_id": "e3e1a2aac03ec873", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_9=40$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_9=31.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=31 and k=9.\nCompute C(39,8) = 61523748.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $61523748$", "final_answer": "61523748", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "629065a44cd89371", "problem_id": "cb014a9ffa0465e3", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 15 red balls and 6 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(21,4).\nFavorable samples: choose k reds and n−k blues: C(15,2)·C(6,2).\nCompute totals: C(21,4) = 5985, and favorable = 1575.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{5}{19}$.\nFinal answer: $\\frac{5}{19}$", "final_answer": "\\frac{5}{19}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f0330bcfe3bfd356", "problem_id": "cb014a9ffa0465e3", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 15 red balls and 6 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(21,4).\n3) Favorable samples: choose k reds and n−k blues: C(15,2)·C(6,2).\n4) Compute totals: C(21,4) = 5985, and favorable = 1575.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{5}{19}$", "final_answer": "\\frac{5}{19}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c90fdfb5a08c11bd", "problem_id": "fb6e80da31c9c103", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 4 x^{2} - x - 4\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 4 x^{2} - x - 4$.\n3) Compute an antiderivative: F(x) = $- \\frac{4 x^{3}}{3} - \\frac{x^{2}}{2} - 4 x$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{4\\cdot x^{3}}{3} - \\frac{x^{2}}{2} - 4\\cdot x$", "final_answer": "C - \\frac{4\\cdot x^{3}}{3} - \\frac{x^{2}}{2} - 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "76cf55c9ada5f91b", "problem_id": "fb6e80da31c9c103", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 4 x^{2} - x - 4\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 4 x^{2} - x - 4$.\nCompute an antiderivative: F(x) = $- \\frac{4 x^{3}}{3} - \\frac{x^{2}}{2} - 4 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - \\frac{4\\cdot x^{3}}{3} - \\frac{x^{2}}{2} - 4\\cdot x$", "final_answer": "C - \\frac{4\\cdot x^{3}}{3} - \\frac{x^{2}}{2} - 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a4391c1100006278", "problem_id": "74b96f4090a07e4a", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 21 times, what is the probability of exactly 18 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=21, k=18, p=$\\frac{1}{6}$ and simplify.\n4) Here C(21,18) = 1330.\nFinal answer: $\\frac{83125}{10968475320188928}$", "final_answer": "\\frac{83125}{10968475320188928}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f67585c6ff39d1cf", "problem_id": "74b96f4090a07e4a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 21 times, what is the probability of exactly 18 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=21, k=18, p=$\\frac{1}{6}$ and simplify.\nHere C(21,18) = 1330.\nCheck: expression simplifies to $\\frac{83125}{10968475320188928}$.\nFinal answer: $\\frac{83125}{10968475320188928}$", "final_answer": "\\frac{83125}{10968475320188928}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1ecc898f1994d177", "problem_id": "7bd4a540e16dacb3", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡7 (mod 8), x≡5 (mod 11). Give the solution modulo 88.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 88.\nCompute the CRT solution to get x ≡ 71 (mod 88).\nCheck: 71 mod 8=7 and mod 11=5.\nFinal answer: $x\\equiv 71\\pmod{88}$", "final_answer": "x\\equiv 71\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2eea107028c639ba", "problem_id": "7bd4a540e16dacb3", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡7 (mod 8), x≡5 (mod 11). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 71 (mod 88).\nFinal answer: $x\\equiv 71\\pmod{88}$", "final_answer": "x\\equiv 71\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "305a9316c34c8bf5", "problem_id": "189ea9dd6588a425", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Worker A finishes a job in 8 hours and Worker B finishes it in 21 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/8 job/hour. B's rate = 1/21 job/hour.\n2) Combined rate = 1/8 + 1/21 = 29/168 job/hour.\n3) Time = 1 ÷ (combined rate) = 168/29 hours.\nFinal answer: $\\frac{168}{29}$", "final_answer": "\\frac{168}{29}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "50d1532bca09c292", "problem_id": "189ea9dd6588a425", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Worker A finishes a job in 8 hours and Worker B finishes it in 21 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/8 job/hour. B's rate = 1/21 job/hour.\nCombined rate = 1/8 + 1/21 = 29/168 job/hour.\nTime = 1 ÷ (combined rate) = 168/29 hours.\nCheck: (combined rate)×(time) = 29/168×168/29 = 1.\nFinal answer: $\\frac{168}{29}$", "final_answer": "\\frac{168}{29}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "098e39124cb3d7ce", "problem_id": "1663119665a7aba5", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{4} + 2 x^{3} + 4 x^{2} - 3$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-3) = 0.\n3) d/dx(2 x^{3}) = 6 x^{2}.\n4) d/dx(2 x^{4}) = 8 x^{3}.\n5) d/dx(4 x^{2}) = 8 x.\n6) Combine: f'(x) = 2 x \\left(4 x^{2} + 3 x + 4\\right).\nFinal answer: $2\\cdot x\\cdot \\left(4\\cdot x^{2} + 3\\cdot x + 4\\right)$", "final_answer": "2\\cdot x\\cdot \\left(4\\cdot x^{2} + 3\\cdot x + 4\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f0190f27373d2c9f", "problem_id": "1663119665a7aba5", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{4} + 2 x^{3} + 4 x^{2} - 3$.", "output": "$2\\cdot x\\cdot \\left(4\\cdot x^{2} + 3\\cdot x + 4\\right)$", "final_answer": "2\\cdot x\\cdot \\left(4\\cdot x^{2} + 3\\cdot x + 4\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d95bb63ac5efa205", "problem_id": "4bc282ec563d07d7", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=23$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=23 and k=5.\nCompute C(27,4) = 17550.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $17550$", "final_answer": "17550", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec768e1461d3d235", "problem_id": "4bc282ec563d07d7", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=23$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=23 and k=5.\n4) Compute C(27,4) = 17550.\nFinal answer: $17550$", "final_answer": "17550", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bd194ece37709ddc", "problem_id": "5ef9a085dea11d2d", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} 2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\pi).\n3) Subtract: F(\\pi) − F(- \\pi) = $6 \\pi + \\frac{4 \\pi^{3}}{3}$.\n4) Simplify the result.\nFinal answer: $6\\cdot \\pi + \\frac{4\\cdot \\pi^{3}}{3}$", "final_answer": "6\\cdot \\pi + \\frac{4\\cdot \\pi^{3}}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7d0f1bff84eef991", "problem_id": "5ef9a085dea11d2d", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} 2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3$.\nEvaluate at the bounds: F(\\pi) and F(- \\pi).\nSubtract: F(\\pi) − F(- \\pi) = $6 \\pi + \\frac{4 \\pi^{3}}{3}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $6\\cdot \\pi + \\frac{4\\cdot \\pi^{3}}{3}$", "final_answer": "6\\cdot \\pi + \\frac{4\\cdot \\pi^{3}}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ed2cd90076c97b1f", "problem_id": "cda0d48c7532f72b", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$- 4 x$ = $16$\n$2 x + 3 y$ = $-23$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $- 8 x$ = $32$.\n3) Multiply the second equation by $-4$: $- 8 x - 12 y$ = $92$.\n4) Subtract to eliminate x: $12 y$ = $-60$.\n5) Solve for y: y = $-5$.\n6) Substitute back to find x: x = $-4$.\nFinal answer: $(x,y)=\\left(-4, -5\\right)$", "final_answer": "(x,y)=\\left(-4, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "77484ce4872c3718", "problem_id": "cda0d48c7532f72b", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the linear system:\n$- 4 x$ = $16$\n$2 x + 3 y$ = $-23$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $2$: $- 8 x$ = $32$.\nMultiply the second equation by $-4$: $- 8 x - 12 y$ = $92$.\nFinal answer: $(x,y)=\\left(-4, -5\\right)$", "final_answer": "(x,y)=\\left(-4, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "47b66f8c86336307", "problem_id": "35adf5a8eee38d06", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} - 2 x^{3} - x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{3} - x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\nEvaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{15 \\pi^{4}}{32} - \\frac{3 \\pi^{3}}{8} - 3 \\pi - \\frac{3 \\pi^{2}}{8} + 2$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{15\\cdot \\pi^{4}}{32} - \\frac{3\\cdot \\pi^{3}}{8} - 3\\cdot \\pi - \\frac{3\\cdot \\pi^{2}}{8} + 2$", "final_answer": "- \\frac{15\\cdot \\pi^{4}}{32} - \\frac{3\\cdot \\pi^{3}}{8} - 3\\cdot \\pi - \\frac{3\\cdot \\pi^{2}}{8} + 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d14e97d156af1ea1", "problem_id": "35adf5a8eee38d06", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} - 2 x^{3} - x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{3} - x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 2$.\nEvaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{15 \\pi^{4}}{32} - \\frac{3 \\pi^{3}}{8} - 3 \\pi - \\frac{3 \\pi^{2}}{8} + 2$.\nFinal answer: $- \\frac{15\\cdot \\pi^{4}}{32} - \\frac{3\\cdot \\pi^{3}}{8} - 3\\cdot \\pi - \\frac{3\\cdot \\pi^{2}}{8} + 2$", "final_answer": "- \\frac{15\\cdot \\pi^{4}}{32} - \\frac{3\\cdot \\pi^{3}}{8} - 3\\cdot \\pi - \\frac{3\\cdot \\pi^{2}}{8} + 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d480b476c01e7bb9", "problem_id": "8647dc3a64100f15", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 19 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=19, k=6, p=$\\frac{2}{7}$ and simplify.\n4) Here C(19,6) = 27132.\nFinal answer: $\\frac{302812500000000}{1628413597910449}$", "final_answer": "\\frac{302812500000000}{1628413597910449}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "37be484e7ee77f99", "problem_id": "c70b1fa0bce0719e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} 2 - x$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to 2 - 2 x.\n3) Then the limit is 10.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: 6.\nFinal answer: $6$", "final_answer": "6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "86d10932990c3449", "problem_id": "3387a198be5e670a", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡1 (mod 9), x≡3 (mod 13). Give the solution modulo 117.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 117.\n3) Compute the CRT solution to get x ≡ 55 (mod 117).\nFinal answer: $x\\equiv 55\\pmod{117}$", "final_answer": "x\\equiv 55\\pmod{117}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "03949f7aa9e1d234", "problem_id": "3387a198be5e670a", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡1 (mod 9), x≡3 (mod 13). Give the solution modulo 117.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 117.\nCompute the CRT solution to get x ≡ 55 (mod 117).\nCheck: 55 mod 9=1 and mod 13=3.\nFinal answer: $x\\equiv 55\\pmod{117}$", "final_answer": "x\\equiv 55\\pmod{117}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c3fcc9d682380e2b", "problem_id": "26e9853dcb83fb74", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{4} - 2 x^{3} - x^{2} + x - 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x) = x$.\n3) So $f'(x) = 12 x^{3} - 6 x^{2} - x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x) = 1$.\nTherefore the correct derivative is $f'(x) = 12 x^{3} - 6 x^{2} - 2 x + 1$.\nFinal answer: $12 x^{3} - 6 x^{2} - 2 x + 1$", "final_answer": "12 x^{3} - 6 x^{2} - 2 x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "36d4367aeacbc963", "problem_id": "c4aa75c753ae9dfe", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x^{2} - 5 x - 13$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{2} - 6 x - 13.\n3) Then the limit is -33.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -31.\nFinal answer: $-31$", "final_answer": "-31", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "1497e8194ff5d43d", "problem_id": "9cc3163a1358061a", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 8), x≡6 (mod 7). Give the solution modulo 56.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 56.\nCompute the CRT solution to get x ≡ 34 (mod 56).\nCheck: 34 mod 8=2 and mod 7=6.\nFinal answer: $x\\equiv 34\\pmod{56}$", "final_answer": "x\\equiv 34\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eb94440e168876d1", "problem_id": "9cc3163a1358061a", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡2 (mod 8), x≡6 (mod 7). Give the solution modulo 56.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 56.\n3) Compute the CRT solution to get x ≡ 34 (mod 56).\nFinal answer: $x\\equiv 34\\pmod{56}$", "final_answer": "x\\equiv 34\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1f61d15b55d56342", "problem_id": "99a732ca6ece72c0", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "An urn has 24 red balls and 18 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(42,7).\n3) Favorable samples: choose k reds and n−k blues: C(24,3)·C(18,4).\n4) Compute totals: C(42,7) = 26978328, and favorable = 6193440.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{86020}{374699}$", "final_answer": "\\frac{86020}{374699}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a252f3b0aa0b3cb", "problem_id": "d6334c18f690852d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} \\cos{\\left(x^{2} + 3 x + 1 \\right)} + 2 \\left(x + 1\\right) e^{x}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x e^{x}) = 2 x e^{x}$.\n3) So $f'(x) = - 6 x^{3} \\sin{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} - 9 x^{2} \\sin{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 2 x e^{x} + 6 x \\cos{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 2 e^{x}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x e^{x}) = 2 x e^{x} + 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = - 3 x^{2} \\left(2 x + 3\\right) \\sin{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 6 x \\cos{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 2 \\left(x + 1\\right) e^{x} + 2 e^{x}$.\nFinal answer: $- 3 x^{2} \\left(2 x + 3\\right) \\sin{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 6 x \\cos{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 2 \\left(x + 1\\right) e^{x} + 2 e^{x}$", "final_answer": "- 3 x^{2} \\left(2 x + 3\\right) \\sin{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 6 x \\cos{\\left(\\left(x^{2} + 3 x\\right) + 1 \\right)} + 2 \\left(x + 1\\right) e^{x} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0d0496763f63f37f", "problem_id": "b2fb39bb5b5d5827", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 12 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=5, p=$\\frac{3}{7}$ and simplify.\n4) Here C(12,5) = 792.\nFinal answer: $\\frac{3153199104}{13841287201}$", "final_answer": "\\frac{3153199104}{13841287201}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d89e27e20a97fd84", "problem_id": "b2fb39bb5b5d5827", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 12 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=12, k=5, p=$\\frac{3}{7}$ and simplify.\nFinal answer: $\\frac{3153199104}{13841287201}$", "final_answer": "\\frac{3153199104}{13841287201}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72918001b9c0d129", "problem_id": "6752fd1a8d0f215d", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 18 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=18, k=5, p=$\\frac{1}{4}$ and simplify.\nHere C(18,5) = 8568.\nCheck: expression simplifies to $\\frac{1707519933}{8589934592}$.\nFinal answer: $\\frac{1707519933}{8589934592}$", "final_answer": "\\frac{1707519933}{8589934592}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "64ef954aefb2b4db", "problem_id": "6752fd1a8d0f215d", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 18 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=18, k=5, p=$\\frac{1}{4}$ and simplify.\n4) Here C(18,5) = 8568.\nFinal answer: $\\frac{1707519933}{8589934592}$", "final_answer": "\\frac{1707519933}{8589934592}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "32a67812e651327f", "problem_id": "9324b99eb1983050", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(3 x^{2} - 2 x - 3\\right) \\cos{\\left(2 x^{2} + x \\right)}$.", "output": "1) Let f(x) = $\\left(x + 2\\right) e^{x} + \\left(3 x^{2} - 2 x - 3\\right) \\cos{\\left(2 x^{2} + x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\left(x + 2\\right) e^{x} + 2 \\left(3 x - 1\\right) \\cos{\\left(2 x^{2} + x \\right)} - \\left(4 x + 1\\right) \\left(3 x^{2} - 2 x - 3\\right) \\sin{\\left(2 x^{2} + x \\right)} + e^{x}$.\nFinal answer: $\\left(x + 2\\right)\\cdot e^{x} - \\left(4\\cdot x + 1\\right)\\cdot \\left(3\\cdot x^{2} - 2\\cdot x - 3\\right)\\cdot \\sin{\\left(2\\cdot x^{2} + x \\right)} + \\left(6\\cdot x - 2\\right)\\cdot \\cos{\\left(2\\cdot x^{2} + x \\right)} + e^{x}$", "final_answer": "\\left(x + 2\\right)\\cdot e^{x} - \\left(4\\cdot x + 1\\right)\\cdot \\left(3\\cdot x^{2} - 2\\cdot x - 3\\right)\\cdot \\sin{\\left(2\\cdot x^{2} + x \\right)} + \\left(6\\cdot x - 2\\right)\\cdot \\cos{\\left(2\\cdot x^{2} + x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "22c44e0ec1869452", "problem_id": "eb3dc200753b87c5", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 2 x - 2$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to x - 2.\n3) Then the limit is -3.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: -4.\nFinal answer: $-4$", "final_answer": "-4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e0ab7bc851462c0a", "problem_id": "27081d9e89fd835b", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A cyclist rides at 57 miles per hour for 1 hours, then at 72 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 57 × 1 = 57 miles.\n2) Second leg distance = 72 × 5/2 = 180 miles.\n3) Total distance = 57 + 180 = 237 miles.\nFinal answer: $237$", "final_answer": "237", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fb3939f5320f34bd", "problem_id": "27081d9e89fd835b", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 57 miles per hour for 1 hours, then at 72 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 57 × 1 = 57 miles.\nSecond leg distance = 72 × 5/2 = 180 miles.\nTotal distance = 57 + 180 = 237 miles.\nCheck: adding both legs gives 237.\nFinal answer: $237$", "final_answer": "237", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "891917d6002d99e8", "problem_id": "6044218390e9c04a", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 8 red balls and 14 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(22,5).\nFavorable samples: choose k reds and n−k blues: C(8,1)·C(14,4).\nCompute totals: C(22,5) = 26334, and favorable = 8008.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{52}{171}$.\nFinal answer: $\\frac{52}{171}$", "final_answer": "\\frac{52}{171}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e2a41871dada666d", "problem_id": "6044218390e9c04a", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "An urn has 8 red balls and 14 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(22,5).\n3) Favorable samples: choose k reds and n−k blues: C(8,1)·C(14,4).\n4) Compute totals: C(22,5) = 26334, and favorable = 8008.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{52}{171}$", "final_answer": "\\frac{52}{171}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a172b9be7ac6aea8", "problem_id": "7b653f824f1c4771", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "An urn has 9 red balls and 14 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(23,4).\n3) Favorable samples: choose k reds and n−k blues: C(9,2)·C(14,2).\n4) Compute totals: C(23,4) = 8855, and favorable = 3276.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{468}{1265}$", "final_answer": "\\frac{468}{1265}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9b992a14644d3426", "problem_id": "7b653f824f1c4771", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 9 red balls and 14 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(23,4).\nFavorable samples: choose k reds and n−k blues: C(9,2)·C(14,2).\nCompute totals: C(23,4) = 8855, and favorable = 3276.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{468}{1265}$.\nFinal answer: $\\frac{468}{1265}$", "final_answer": "\\frac{468}{1265}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c828b2a719021faa", "problem_id": "431878a2c0c8fbc0", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A store sells a backpack for $890. It is discounted by 5%. You buy 7 backpacks. What is the total cost?", "output": "1) Discounted price = 890 × (1 − 5/100) = 890 × 95/100 = 28*2**(152/281)*3**(66/281)*5**(210/281)*7**(227/281).\n2) Subtotal = 7 × 28*2**(152/281)*3**(66/281)*5**(210/281)*7**(227/281) = 196*2**(152/281)*3**(66/281)*5**(210/281)*7**(227/281).\nFinal answer: $196\\cdot 2^{\\frac{152}{281}}\\cdot 3^{\\frac{66}{281}}\\cdot 5^{\\frac{210}{281}}\\cdot 7^{\\frac{227}{281}}$", "final_answer": "196\\cdot 2^{\\frac{152}{281}}\\cdot 3^{\\frac{66}{281}}\\cdot 5^{\\frac{210}{281}}\\cdot 7^{\\frac{227}{281}}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7bfa2dd44c3a9962", "problem_id": "431878a2c0c8fbc0", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A store sells a backpack for $890. It is discounted by 5%. You buy 7 backpacks. What is the total cost?", "output": "Discounted price = 890 × (1 − 5/100) = 890 × 95/100 = 28*2**(152/281)*3**(66/281)*5**(210/281)*7**(227/281).\nSubtotal = 7 × 28*2**(152/281)*3**(66/281)*5**(210/281)*7**(227/281) = 196*2**(152/281)*3**(66/281)*5**(210/281)*7**(227/281).\nCheck: recomputing the formula yields 196*2**(152/281)*3**(66/281)*5**(210/281)*7**(227/281).\nFinal answer: $196\\cdot 2^{\\frac{152}{281}}\\cdot 3^{\\frac{66}{281}}\\cdot 5^{\\frac{210}{281}}\\cdot 7^{\\frac{227}{281}}$", "final_answer": "196\\cdot 2^{\\frac{152}{281}}\\cdot 3^{\\frac{66}{281}}\\cdot 5^{\\frac{210}{281}}\\cdot 7^{\\frac{227}{281}}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7274c676bb250f4c", "problem_id": "b27429be0e70f212", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - \\frac{35 x}{4} + \\frac{25}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$- \\frac{35}{4}$, c=$\\frac{25}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{25}{16}$, so √Δ = $\\frac{5}{4}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{5}{3}$.\n7) x2 = $\\frac{5}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{5}{3}, \\frac{5}{4}\\}$", "final_answer": "x \\in \\{\\frac{5}{3}, \\frac{5}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "af2c5de5af0a4f0e", "problem_id": "b27429be0e70f212", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - \\frac{35 x}{4} + \\frac{25}{4}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$- \\frac{35}{4}$, c=$\\frac{25}{4}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{25}{16}$, so √Δ = $\\frac{5}{4}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{5}{3}$.\nx2 = $\\frac{5}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{5}{3}, \\frac{5}{4}\\}$", "final_answer": "x \\in \\{\\frac{5}{3}, \\frac{5}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "291ada32baafcf7f", "problem_id": "f6ac966a463560d8", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 28 red balls and 19 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(47,10).\n3) Favorable samples: choose k reds and n−k blues: C(28,3)·C(19,7).\n4) Compute totals: C(47,10) = 5178066751, and favorable = 165071088.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{668304}{20963833}$", "final_answer": "\\frac{668304}{20963833}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "58249cadf3dd7490", "problem_id": "f6ac966a463560d8", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 28 red balls and 19 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(47,10).\nFavorable samples: choose k reds and n−k blues: C(28,3)·C(19,7).\nCompute totals: C(47,10) = 5178066751, and favorable = 165071088.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{668304}{20963833}$.\nFinal answer: $\\frac{668304}{20963833}$", "final_answer": "\\frac{668304}{20963833}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "508ed4a41ff1eb5f", "problem_id": "badc1efc537ac953", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $294 in the ratio 3:3. How much does each receive?", "output": "1) Total ratio parts = 3+3 = 6.\n2) Each part = 294 ÷ 6 = 49.\n3) First share = 3 × 49 = 147.\n4) Second share = 3 × 49 = 147.\nFinal answer: $147$", "final_answer": "147", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "636c782467e73612", "problem_id": "6b0e3c2fd06ac29e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 3 x^{2} \\sin{\\left(x \\right)}) = - 3 x^{2} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 3 x^{2} \\cos{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 3 x^{2} \\sin{\\left(x \\right)}) = - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.\nFinal answer: $- 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$", "final_answer": "- 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "372ee476a26a3c60", "problem_id": "9a6e02b92a9edd0c", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 12 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=2, p=$\\frac{2}{3}$ and simplify.\n4) Here C(12,2) = 66.\nFinal answer: $\\frac{88}{177147}$", "final_answer": "\\frac{88}{177147}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c27b0accdef38773", "problem_id": "9a6e02b92a9edd0c", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 12 times, what is the probability of exactly 2 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=12, k=2, p=$\\frac{2}{3}$ and simplify.\nHere C(12,2) = 66.\nCheck: expression simplifies to $\\frac{88}{177147}$.\nFinal answer: $\\frac{88}{177147}$", "final_answer": "\\frac{88}{177147}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "887225e59d01170e", "problem_id": "a103649bbf101192", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 8 red balls and 9 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(17,6).\nFavorable samples: choose k reds and n−k blues: C(8,3)·C(9,3).\nCompute totals: C(17,6) = 12376, and favorable = 4704.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{84}{221}$.\nFinal answer: $\\frac{84}{221}$", "final_answer": "\\frac{84}{221}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d62c5dcc8f23b95c", "problem_id": "a103649bbf101192", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 8 red balls and 9 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(17,6).\n3) Favorable samples: choose k reds and n−k blues: C(8,3)·C(9,3).\n4) Compute totals: C(17,6) = 12376, and favorable = 4704.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{84}{221}$", "final_answer": "\\frac{84}{221}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3afc60c60142c55f", "problem_id": "81acc0307cd9c85e", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} 3 x^{3} - 2 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{3} - 2 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1$.\n2) Evaluate at the bounds: F(\\pi) and F(0).\n3) Subtract: F(\\pi) − F(0) = $- \\frac{2 \\pi^{3}}{3} - \\frac{3 \\pi^{2}}{2} - \\pi + 4 + \\frac{3 \\pi^{4}}{4}$.\n4) Simplify the result.\nFinal answer: $- \\frac{2\\cdot \\pi^{3}}{3} - \\frac{3\\cdot \\pi^{2}}{2} - \\pi + 4 + \\frac{3\\cdot \\pi^{4}}{4}$", "final_answer": "- \\frac{2\\cdot \\pi^{3}}{3} - \\frac{3\\cdot \\pi^{2}}{2} - \\pi + 4 + \\frac{3\\cdot \\pi^{4}}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "03a76345b4bed607", "problem_id": "81acc0307cd9c85e", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} 3 x^{3} - 2 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{3} - 2 x^{2} - 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 1$.\nEvaluate at the bounds: F(\\pi) and F(0).\nSubtract: F(\\pi) − F(0) = $- \\frac{2 \\pi^{3}}{3} - \\frac{3 \\pi^{2}}{2} - \\pi + 4 + \\frac{3 \\pi^{4}}{4}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{2\\cdot \\pi^{3}}{3} - \\frac{3\\cdot \\pi^{2}}{2} - \\pi + 4 + \\frac{3\\cdot \\pi^{4}}{4}$", "final_answer": "- \\frac{2\\cdot \\pi^{3}}{3} - \\frac{3\\cdot \\pi^{2}}{2} - \\pi + 4 + \\frac{3\\cdot \\pi^{4}}{4}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ee6aebbc5d927d4b", "problem_id": "994510e05a010aa0", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 18 times, what is the probability of exactly 16 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=18, k=16, p=$\\frac{1}{2}$ and simplify.\n4) Here C(18,16) = 153.\nFinal answer: $\\frac{153}{262144}$", "final_answer": "\\frac{153}{262144}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "524392c562ffbd44", "problem_id": "994510e05a010aa0", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 18 times, what is the probability of exactly 16 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=18, k=16, p=$\\frac{1}{2}$ and simplify.\nFinal answer: $\\frac{153}{262144}$", "final_answer": "\\frac{153}{262144}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "776236dd16e96e02", "problem_id": "69aca37b034b0f3f", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 20 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=20, k=6, p=$\\frac{1}{6}$ and simplify.\n4) Here C(20,6) = 38760.\nFinal answer: $\\frac{9857177734375}{152339935002624}$", "final_answer": "\\frac{9857177734375}{152339935002624}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "10b9df6cc21d5143", "problem_id": "69aca37b034b0f3f", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 20 times, what is the probability of exactly 6 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=20, k=6, p=$\\frac{1}{6}$ and simplify.\nFinal answer: $\\frac{9857177734375}{152339935002624}$", "final_answer": "\\frac{9857177734375}{152339935002624}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e7d9669cc471c347", "problem_id": "32f03e858bc2b527", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{6} x}{2} + 3 x - \\frac{\\sqrt{6}}{2} + 2$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$3 - \\frac{\\sqrt{6}}{2}$, c=$2 - \\frac{\\sqrt{6}}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{5}{2} - \\sqrt{6}$, so √Δ = $\\sqrt{\\frac{5}{2} - \\sqrt{6}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{3}{2} + \\frac{\\sqrt{10 - 4 \\sqrt{6}}}{4} + \\frac{\\sqrt{6}}{4}$.\nx2 = $- \\frac{3}{2} - \\frac{\\sqrt{10 - 4 \\sqrt{6}}}{4} + \\frac{\\sqrt{6}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{3}{2} + \\frac{\\sqrt{10 - 4\\cdot \\sqrt{6}}}{4} + \\frac{\\sqrt{6}}{4}, - \\frac{3}{2} - \\frac{\\sqrt{10 - 4\\cdot \\sqrt{6}}}{4} + \\frac{\\sqrt{6}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{3}{2} + \\frac{\\sqrt{10 - 4\\cdot \\sqrt{6}}}{4} + \\frac{\\sqrt{6}}{4}, - \\frac{3}{2} - \\frac{\\sqrt{10 - 4\\cdot \\sqrt{6}}}{4} + \\frac{\\sqrt{6}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bf6bd719110da050", "problem_id": "32f03e858bc2b527", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{6} x}{2} + 3 x - \\frac{\\sqrt{6}}{2} + 2$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$3 - \\frac{\\sqrt{6}}{2}$, c=$2 - \\frac{\\sqrt{6}}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{5}{2} - \\sqrt{6}$, so √Δ = $\\sqrt{\\frac{5}{2} - \\sqrt{6}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{3}{2} + \\frac{\\sqrt{10 - 4 \\sqrt{6}}}{4} + \\frac{\\sqrt{6}}{4}$.\n7) x2 = $- \\frac{3}{2} - \\frac{\\sqrt{10 - 4 \\sqrt{6}}}{4} + \\frac{\\sqrt{6}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{3}{2} + \\frac{\\sqrt{10 - 4\\cdot \\sqrt{6}}}{4} + \\frac{\\sqrt{6}}{4}, - \\frac{3}{2} - \\frac{\\sqrt{10 - 4\\cdot \\sqrt{6}}}{4} + \\frac{\\sqrt{6}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{3}{2} + \\frac{\\sqrt{10 - 4\\cdot \\sqrt{6}}}{4} + \\frac{\\sqrt{6}}{4}, - \\frac{3}{2} - \\frac{\\sqrt{10 - 4\\cdot \\sqrt{6}}}{4} + \\frac{\\sqrt{6}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cb1562961cb1b897", "problem_id": "21460b1ea079a484", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_9=39$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=39 and k=9.\n4) Compute C(47,8) = 314457495.\nFinal answer: $314457495$", "final_answer": "314457495", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "83d83f57366df1c1", "problem_id": "21460b1ea079a484", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_9=39$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=39 and k=9.\nCompute C(47,8) = 314457495.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $314457495$", "final_answer": "314457495", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7efa4dac962c6765", "problem_id": "5aac33b0caf13a11", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 21 x - 24$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} - 21 x - 24$ = $3 \\left(x - 8\\right) \\left(x + 1\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $8$.\n5) From the second factor: x = $-1$.\nFinal answer: $x \\in \\{8, -1\\}$", "final_answer": "x \\in \\{8, -1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "73e47f0b9b0ae7fd", "problem_id": "5aac33b0caf13a11", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 21 x - 24$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} - 21 x - 24$ = $3 \\left(x - 8\\right) \\left(x + 1\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFinal answer: $x \\in \\{8, -1\\}$", "final_answer": "x \\in \\{8, -1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b76d4effefc8326d", "problem_id": "737116416f2e47ee", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A disease has prevalence $\\frac{2}{25}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{59}{500}$.\n3) Compute P(D|+) = $\\frac{36}{59}$.\nFinal answer: $\\frac{36}{59}$", "final_answer": "\\frac{36}{59}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f7d5c846c1eb2b5", "problem_id": "737116416f2e47ee", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A disease has prevalence $\\frac{2}{25}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{19}{20}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{59}{500}$.\nCompute P(D|+) = $\\frac{36}{59}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{36}{59}$", "final_answer": "\\frac{36}{59}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c4e8b7c7d3d5a5c8", "problem_id": "2305c87209fbbead", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 19 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=19, k=5, p=$\\frac{4}{5}$ and simplify.\n4) Here C(19,5) = 11628.\nFinal answer: $\\frac{11907072}{19073486328125}$", "final_answer": "\\frac{11907072}{19073486328125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e8c5d0d3990a069e", "problem_id": "2305c87209fbbead", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 19 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=19, k=5, p=$\\frac{4}{5}$ and simplify.\nHere C(19,5) = 11628.\nCheck: expression simplifies to $\\frac{11907072}{19073486328125}$.\nFinal answer: $\\frac{11907072}{19073486328125}$", "final_answer": "\\frac{11907072}{19073486328125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "686d75144d498704", "problem_id": "40380a61c83835b6", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$8 x + 4 y$ = $-4$\n$10 x$ = $10$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $10$: $80 x + 40 y$ = $-40$.\n3) Multiply the second equation by $8$: $80 x$ = $80$.\n4) Subtract to eliminate x: $40 y$ = $-120$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $1$.\nFinal answer: $(x,y)=\\left(1, -3\\right)$", "final_answer": "(x,y)=\\left(1, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b2369e271ac2cdfa", "problem_id": "40380a61c83835b6", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$8 x + 4 y$ = $-4$\n$10 x$ = $10$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $10$: $80 x + 40 y$ = $-40$.\nMultiply the second equation by $8$: $80 x$ = $80$.\nSubtract to eliminate x: $40 y$ = $-120$.\nSolve for y: y = $-3$.\nSubstitute back to find x: x = $1$.\nCheck: plug (x,y)=(1,-3) into both equations.\nFinal answer: $(x,y)=\\left(1, -3\\right)$", "final_answer": "(x,y)=\\left(1, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d6ed29823b2a43f2", "problem_id": "6a0b059a094ec977", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 10 x + 12$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 10 x + 12$ = $2 \\left(x - 3\\right) \\left(x - 2\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $3$.\n5) From the second factor: x = $2$.\nFinal answer: $x \\in \\{3, 2\\}$", "final_answer": "x \\in \\{3, 2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "abe2818f0df7c05c", "problem_id": "6a0b059a094ec977", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 10 x + 12$ = 0.", "output": "$x \\in \\{3, 2\\}$", "final_answer": "x \\in \\{3, 2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d42c6099b2912ea5", "problem_id": "a47696d65a799548", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A store sells a backpack for $340. It is discounted by 20%. You buy 4 backpacks. What is the total cost?", "output": "1) Discounted price = 340 × (1 − 20/100) = 340 × 80/100 = 272.\n2) Subtotal = 4 × 272 = 1088.\nFinal answer: $1088$", "final_answer": "1088", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4c79db24a4647ed3", "problem_id": "c2beab8a129c5d2e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} 3 x + 9$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to 2 x + 9.\n3) Then the limit is 15.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: 18.\nFinal answer: $18$", "final_answer": "18", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e9a42bf911cfb799", "problem_id": "7efa7d436beb2ee8", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 17 times, what is the probability of exactly 9 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=17, k=9, p=$\\frac{1}{10}$ and simplify.\n4) Here C(17,9) = 24310.\nFinal answer: $\\frac{104646578751}{10000000000000000}$", "final_answer": "\\frac{104646578751}{10000000000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3e5f4b0909d037b9", "problem_id": "d723cd3fbf671a71", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{3} + 2 x^{2} - x + \\left(2 x^{2} + x - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{2 x^{2} + x - 6}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x^{2} + x\\right) - 6}) = - \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + x - 6}$.\n3) So $f'(x) = \\frac{4 x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} + \\frac{8 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} - \\frac{4 x^{4}}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} + \\frac{4 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} + \\frac{8 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} - \\frac{4 x^{3}}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} - \\frac{23 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} - \\frac{46 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} + \\frac{44 x^{2}}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} - \\frac{12 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} - \\frac{24 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} - \\frac{24 x}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} - \\frac{24 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{4 x^{4} + 4 x^{3} - 23 x^{2} - 12 x + 36} + \\frac{36 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} + \\frac{72 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} + \\frac{18}{8 x^{4} + 8 x^{3} - 46 x^{2} - 24 x + 72} - \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{4 x^{4} + 4 x^{3} - 23 x^{2} - 12 x + 36} - \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + x - 6} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x^{2} + x\\right) - 6} + \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(2 x^{2} + x\\right) - 6}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x^{2} + x\\right) - 6}) = - \\frac{6 \\left(- 4 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(2 x^{2} + x\\right) - 6\\right)^{2}} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x^{2} + x\\right) - 6} - \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(2 x^{2} + x\\right) - 6}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(- 4 x - 1\\right) \\left(- x^{3} + 2 x^{2} - x + \\left(x \\left(2 x + 1\\right) - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x \\left(2 x + 1\\right) - 6\\right) \\left(- 6 x^{2} + 8 x + 2 \\left(4 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 x + 1\\right) - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 x + 1\\right) - 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x \\left(2 x + 1\\right) - 6\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(- 4 x - 1\\right) \\left(- x^{3} + 2 x^{2} - x + \\left(x \\left(2 x + 1\\right) - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x \\left(2 x + 1\\right) - 6\\right) \\left(- 6 x^{2} + 8 x + 2 \\left(4 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 x + 1\\right) - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 x + 1\\right) - 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x \\left(2 x + 1\\right) - 6\\right)^{2}}$", "final_answer": "\\frac{2 \\left(- 4 x - 1\\right) \\left(- x^{3} + 2 x^{2} - x + \\left(x \\left(2 x + 1\\right) - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x \\left(2 x + 1\\right) - 6\\right) \\left(- 6 x^{2} + 8 x + 2 \\left(4 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 x + 1\\right) - 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 x + 1\\right) - 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x \\left(2 x + 1\\right) - 6\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f207317e9fbdea6f", "problem_id": "a0a0008722a0b394", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 18 times, what is the probability of exactly 9 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=18, k=9, p=$\\frac{4}{5}$ and simplify.\nHere C(18,9) = 48620.\nCheck: expression simplifies to $\\frac{2549088256}{762939453125}$.\nFinal answer: $\\frac{2549088256}{762939453125}$", "final_answer": "\\frac{2549088256}{762939453125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b7f7e975bda077e6", "problem_id": "a0a0008722a0b394", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 18 times, what is the probability of exactly 9 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=18, k=9, p=$\\frac{4}{5}$ and simplify.\n4) Here C(18,9) = 48620.\nFinal answer: $\\frac{2549088256}{762939453125}$", "final_answer": "\\frac{2549088256}{762939453125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "860ce16a6f06f424", "problem_id": "26e9853dcb83fb74", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{4} - 2 x^{3} - x^{2} + x - 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- x^{2}) = - 2 x^{2}$.\n3) So $f'(x) = 12 x^{3} - 8 x^{2} + 1$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- x^{2}) = - 2 x$.\nTherefore the correct derivative is $f'(x) = 12 x^{3} - 6 x^{2} - 2 x + 1$.\nFinal answer: $12 x^{3} - 6 x^{2} - 2 x + 1$", "final_answer": "12 x^{3} - 6 x^{2} - 2 x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8a3dba68db9bae8a", "problem_id": "2b9a91cab911086c", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 26 red balls and 23 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(49,10).\n3) Favorable samples: choose k reds and n−k blues: C(26,3)·C(23,7).\n4) Compute totals: C(49,10) = 8217822536, and favorable = 637408200.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{314925}{4060189}$", "final_answer": "\\frac{314925}{4060189}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "436d75c4cbab914d", "problem_id": "31c6b01ead2b29df", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$4 x - 6 y$ = $-34$\n$8 x + 2 y$ = $-54$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $8$: $32 x - 48 y$ = $-272$.\n3) Multiply the second equation by $4$: $32 x + 8 y$ = $-216$.\n4) Subtract to eliminate x: $- 56 y$ = $-56$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $-7$.\nFinal answer: $(x,y)=\\left(-7, 1\\right)$", "final_answer": "(x,y)=\\left(-7, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9f92cbb219da3d4a", "problem_id": "d9a0f61d269f57de", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(3 x^{3} - x^{2} - x - 3\\right) \\cos{\\left(2 x - 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 3 \\cos{\\left(2 x - 3 \\right)}) = 3 \\sin{\\left(2 x - 3 \\right)}$.\n3) So $f'(x) = - 6 x^{3} \\sin{\\left(2 x - 3 \\right)} + 2 x^{2} \\sin{\\left(2 x - 3 \\right)} + 9 x^{2} \\cos{\\left(2 x - 3 \\right)} + 2 x e^{x} + 2 x \\sin{\\left(2 x - 3 \\right)} - 2 x \\cos{\\left(2 x - 3 \\right)} + 6 e^{x} + 3 \\sin{\\left(2 x - 3 \\right)} - \\cos{\\left(2 x - 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 3 \\cos{\\left(2 x - 3 \\right)}) = 6 \\sin{\\left(2 x - 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x + 2\\right) e^{x} - 2 \\left(x \\left(x \\left(3 x - 1\\right) - 1\\right) - 3\\right) \\sin{\\left(2 x - 3 \\right)} + \\left(9 x^{2} - 2 x - 1\\right) \\cos{\\left(2 x - 3 \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(x + 2\\right) e^{x} - 2 \\left(x \\left(x \\left(3 x - 1\\right) - 1\\right) - 3\\right) \\sin{\\left(2 x - 3 \\right)} + \\left(9 x^{2} - 2 x - 1\\right) \\cos{\\left(2 x - 3 \\right)} + 2 e^{x}$", "final_answer": "2 \\left(x + 2\\right) e^{x} - 2 \\left(x \\left(x \\left(3 x - 1\\right) - 1\\right) - 3\\right) \\sin{\\left(2 x - 3 \\right)} + \\left(9 x^{2} - 2 x - 1\\right) \\cos{\\left(2 x - 3 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f37e4b546d7e1f26", "problem_id": "aca94374356aad13", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 7 red balls and 10 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(17,4).\n3) Favorable samples: choose k reds and n−k blues: C(7,1)·C(10,3).\n4) Compute totals: C(17,4) = 2380, and favorable = 840.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{6}{17}$", "final_answer": "\\frac{6}{17}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dd98baae82a0e229", "problem_id": "aca94374356aad13", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 7 red balls and 10 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(17,4).\nFavorable samples: choose k reds and n−k blues: C(7,1)·C(10,3).\nCompute totals: C(17,4) = 2380, and favorable = 840.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{6}{17}$.\nFinal answer: $\\frac{6}{17}$", "final_answer": "\\frac{6}{17}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "edaf675d92ede571", "problem_id": "f3fc9dcea72fd323", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$- 8 x + 12 y$ = $24$\n$12 x - 8 y$ = $-56$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $12$: $- 96 x + 144 y$ = $288$.\n3) Multiply the second equation by $-8$: $- 96 x + 64 y$ = $448$.\n4) Subtract to eliminate x: $80 y$ = $-160$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $-6$.\nFinal answer: $(x,y)=\\left(-6, -2\\right)$", "final_answer": "(x,y)=\\left(-6, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a3c6b14340ff0832", "problem_id": "f3fc9dcea72fd323", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$- 8 x + 12 y$ = $24$\n$12 x - 8 y$ = $-56$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $12$: $- 96 x + 144 y$ = $288$.\nMultiply the second equation by $-8$: $- 96 x + 64 y$ = $448$.\nSubtract to eliminate x: $80 y$ = $-160$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $-6$.\nCheck: plug (x,y)=(-6,-2) into both equations.\nFinal answer: $(x,y)=\\left(-6, -2\\right)$", "final_answer": "(x,y)=\\left(-6, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b92a30a0829b4f2a", "problem_id": "196bab3ed92a7ce2", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$x + 8 y$ = $46$\n$- 6 x + 7 y$ = $-1$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-6$: $- 6 x - 48 y$ = $-276$.\n3) Multiply the second equation by $1$: $- 6 x + 7 y$ = $-1$.\n4) Subtract to eliminate x: $- 55 y$ = $-275$.\n5) Solve for y: y = $5$.\n6) Substitute back to find x: x = $6$.\nFinal answer: $(x,y)=\\left(6, 5\\right)$", "final_answer": "(x,y)=\\left(6, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "184c614a2ad6e063", "problem_id": "196bab3ed92a7ce2", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$x + 8 y$ = $46$\n$- 6 x + 7 y$ = $-1$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-6$: $- 6 x - 48 y$ = $-276$.\nMultiply the second equation by $1$: $- 6 x + 7 y$ = $-1$.\nSubtract to eliminate x: $- 55 y$ = $-275$.\nSolve for y: y = $5$.\nSubstitute back to find x: x = $6$.\nCheck: plug (x,y)=(6,5) into both equations.\nFinal answer: $(x,y)=\\left(6, 5\\right)$", "final_answer": "(x,y)=\\left(6, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5094467f1f070e63", "problem_id": "d63f003c95220905", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Worker A finishes a job in 12 hours and Worker B finishes it in 9 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/12 job/hour. B's rate = 1/9 job/hour.\n2) Combined rate = 1/12 + 1/9 = 7/36 job/hour.\n3) Time = 1 ÷ (combined rate) = 36/7 hours.\nFinal answer: $\\frac{36}{7}$", "final_answer": "\\frac{36}{7}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9c1381dcac5a1e19", "problem_id": "8a6217f9111d785d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 - 3 x$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 3 - 4 x.\n3) Then the limit is 11.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 9.\nFinal answer: $9$", "final_answer": "9", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "fbb6f7ab639c5ac6", "problem_id": "773884c3d52bc6f8", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_8=22$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=22 and k=8.\n4) Compute C(29,7) = 1560780.\nFinal answer: $1560780$", "final_answer": "1560780", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "41fbf4c70e3454a5", "problem_id": "773884c3d52bc6f8", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_8=22$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=22 and k=8.\nCompute C(29,7) = 1560780.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $1560780$", "final_answer": "1560780", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3b83a714ded48454", "problem_id": "c91a94a342c9dab0", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many binary strings of length 21 contain exactly 4 ones with no two ones adjacent?", "output": "Reserve 3 mandatory zeros between the 4 ones.\nThis leaves n−k+1 = 18 valid slots for the ones.\nChoose the slots: C(18,4) = 3060.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $3060$", "final_answer": "3060", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f2d2eda309171c8d", "problem_id": "c91a94a342c9dab0", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many binary strings of length 21 contain exactly 4 ones with no two ones adjacent?", "output": "1) Reserve 3 mandatory zeros between the 4 ones.\n2) This leaves n−k+1 = 18 valid slots for the ones.\n3) Choose the slots: C(18,4) = 3060.\nFinal answer: $3060$", "final_answer": "3060", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d3524ae22911f3d5", "problem_id": "97b2382ffc83e1b2", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the multiplicative inverse of 23 modulo 29.", "output": "1) We want x such that 23x ≡ 1 (mod 29).\n2) Use extended Euclid to find s,t with 23s + 29t = 1.\n3) One Bezout identity is: 23·(-5) + 29·(4) = 1.\n4) Reduce s modulo 29: x ≡ 24 (mod 29).\nFinal answer: $x\\equiv 24\\pmod{29}$", "final_answer": "x\\equiv 24\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "209fae24e160b580", "problem_id": "97b2382ffc83e1b2", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the multiplicative inverse of 23 modulo 29.", "output": "We want x such that 23x ≡ 1 (mod 29).\nUse extended Euclid to find s,t with 23s + 29t = 1.\nOne Bezout identity is: 23·(-5) + 29·(4) = 1.\nReduce s modulo 29: x ≡ 24 (mod 29).\nCheck: 23·24 mod 29 = 1.\nFinal answer: $x\\equiv 24\\pmod{29}$", "final_answer": "x\\equiv 24\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c7ddeb9b4988ed47", "problem_id": "202a61ad39538f6a", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$- 2 y$ = $8$\n$3 x + 4 y$ = $-22$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $3$: $- 6 y$ = $24$.\n3) Multiply the second equation by $0$: $0$ = $0$.\n4) Subtract to eliminate x: $- 6 y$ = $24$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, -4\\right)$", "final_answer": "(x,y)=\\left(-2, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "daa126d59f335918", "problem_id": "202a61ad39538f6a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$- 2 y$ = $8$\n$3 x + 4 y$ = $-22$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $- 6 y$ = $24$.\nMultiply the second equation by $0$: $0$ = $0$.\nSubtract to eliminate x: $- 6 y$ = $24$.\nSolve for y: y = $-4$.\nSubstitute back to find x: x = $-2$.\nCheck: plug (x,y)=(-2,-4) into both equations.\nFinal answer: $(x,y)=\\left(-2, -4\\right)$", "final_answer": "(x,y)=\\left(-2, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4f9445d8d7b4cf5a", "problem_id": "35346f4f38d0b788", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "You want 52 liters of a 35% salt solution. You have a 30% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 52−x liters is 40% solution.\n2) Salt equation: 30x + 40(52−x) = 35·52.\n3) (30-40)x = (35-40)·52 ⇒ x = 26.\n4) Then 52−x = 26.\nFinal answer: $26 L of 30% and 26 L of 40%$", "final_answer": "26 L of 30% and 26 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7a5e5c5ba69f53fe", "problem_id": "9e84074b80b7df79", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(- x^{2} - 2 x - 1\\right) \\cos{\\left(x^{2} + x - 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x^{2} \\cos{\\left(\\left(x^{2} + x\\right) - 1 \\right)}) = x^{2} \\sin{\\left(x^{2} + x - 1 \\right)}$.\n3) So $f'(x) = 5 x^{2} \\sin{\\left(x^{2} + x - 1 \\right)} + x e^{x} + 4 x \\sin{\\left(x^{2} + x - 1 \\right)} + 3 e^{x} + \\sin{\\left(x^{2} + x - 1 \\right)} - 2 \\cos{\\left(\\left(x^{2} + x\\right) - 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x^{2} \\cos{\\left(\\left(x^{2} + x\\right) - 1 \\right)}) = x^{2} \\left(2 x + 1\\right) \\sin{\\left(x^{2} + x - 1 \\right)} - 2 x \\cos{\\left(\\left(x^{2} + x\\right) - 1 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- x - 1\\right) \\cos{\\left(\\left(x^{2} + x\\right) - 1 \\right)} + \\left(x + 2\\right) e^{x} - \\left(2 x + 1\\right) \\left(x \\left(- x - 2\\right) - 1\\right) \\sin{\\left(x^{2} + x - 1 \\right)} + e^{x}$.\nFinal answer: $2 \\left(- x - 1\\right) \\cos{\\left(\\left(x^{2} + x\\right) - 1 \\right)} + \\left(x + 2\\right) e^{x} - \\left(2 x + 1\\right) \\left(x \\left(- x - 2\\right) - 1\\right) \\sin{\\left(x^{2} + x - 1 \\right)} + e^{x}$", "final_answer": "2 \\left(- x - 1\\right) \\cos{\\left(\\left(x^{2} + x\\right) - 1 \\right)} + \\left(x + 2\\right) e^{x} - \\left(2 x + 1\\right) \\left(x \\left(- x - 2\\right) - 1\\right) \\sin{\\left(x^{2} + x - 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5adedf6e308fea39", "problem_id": "c159ec006ad391f6", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{2} x + \\frac{5 x}{6} - \\frac{\\sqrt{2}}{2} + \\frac{1}{6}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{5}{6} - \\sqrt{2}$, c=$\\frac{1}{6} - \\frac{\\sqrt{2}}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{\\sqrt{2}}{3} + \\frac{73}{36}$, so √Δ = $\\sqrt{\\frac{\\sqrt{2}}{3} + \\frac{73}{36}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{5}{12} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{12 \\sqrt{2} + 73}}{12}$.\n7) x2 = $- \\frac{\\sqrt{12 \\sqrt{2} + 73}}{12} - \\frac{5}{12} + \\frac{\\sqrt{2}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{5}{12} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{12\\cdot \\sqrt{2} + 73}}{12}, - \\frac{\\sqrt{12\\cdot \\sqrt{2} + 73}}{12} - \\frac{5}{12} + \\frac{\\sqrt{2}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{5}{12} + \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{12\\cdot \\sqrt{2} + 73}}{12}, - \\frac{\\sqrt{12\\cdot \\sqrt{2} + 73}}{12} - \\frac{5}{12} + \\frac{\\sqrt{2}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "122dc14d85969570", "problem_id": "4c1989da639ec4c7", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - x^{2} - 5 x - 17$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} - 6 x - 17.\n3) Then the limit is -57.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: -53.\nFinal answer: $-53$", "final_answer": "-53", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ee711a92ac702d5b", "problem_id": "0e3d556dc8cc042e", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_5=29$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_5=14.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=14 and k=5.\n4) Compute C(18,4) = 3060.\nFinal answer: $3060$", "final_answer": "3060", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0857436a64f8926b", "problem_id": "7d441d4fa9f7e32e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 2 x^{3} + 2 x^{2} + x - 1$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{3} + 2 x^{2} - 1.\n3) Then the limit is 3.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: 2.\nFinal answer: $2$", "final_answer": "2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "135e0d0aa40a831a", "problem_id": "47b8789b6e4ef509", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 12 red balls and 25 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(37,10).\n3) Favorable samples: choose k reds and n−k blues: C(12,1)·C(25,9).\n4) Compute totals: C(37,10) = 348330136, and favorable = 24515700.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{32775}{465682}$", "final_answer": "\\frac{32775}{465682}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d339c6b3bea682ee", "problem_id": "47b8789b6e4ef509", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "An urn has 12 red balls and 25 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(37,10).\nFavorable samples: choose k reds and n−k blues: C(12,1)·C(25,9).\nFinal answer: $\\frac{32775}{465682}$", "final_answer": "\\frac{32775}{465682}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9290628de67c28b6", "problem_id": "a3343dda6c927aaf", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} - x + \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{x + 6}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 6}) = \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 6}$.\n3) So $f'(x) = \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 24 x + 72} + \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + 24 x + 72} - \\frac{2 x^{2}}{2 x^{2} + 24 x + 72} + \\frac{12 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 24 x + 72} + \\frac{24 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + 24 x + 72} - \\frac{24 x}{2 x^{2} + 24 x + 72} + \\frac{36 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 24 x + 72} + \\frac{72 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + 24 x + 72} - \\frac{6}{2 x^{2} + 24 x + 72} + \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 12 x + 36} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 6} - \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 6} + \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 6}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 6}) = \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x + 6} + \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x + 6} - \\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x + 6\\right)^{2}}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 x^{2} + 2 x + \\left(x + 6\\right) \\left(- 4 x + \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 2 \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6}{2 \\left(x + 6\\right)^{2}}$.\nFinal answer: $\\frac{2 x^{2} + 2 x + \\left(x + 6\\right) \\left(- 4 x + \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 2 \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6}{2 \\left(x + 6\\right)^{2}}$", "final_answer": "\\frac{2 x^{2} + 2 x + \\left(x + 6\\right) \\left(- 4 x + \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 2 \\left(x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6}{2 \\left(x + 6\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d7d2eccdab21ab14", "problem_id": "494d6f591876aa50", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 18 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=18, k=8, p=$\\frac{1}{2}$ and simplify.\nHere C(18,8) = 43758.\nCheck: expression simplifies to $\\frac{21879}{131072}$.\nFinal answer: $\\frac{21879}{131072}$", "final_answer": "\\frac{21879}{131072}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d19be37b0a974f8b", "problem_id": "d3314825fd1a2cd9", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{13} x + \\frac{21 x}{4} - \\frac{9}{2} + \\frac{9 \\sqrt{13}}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{21}{4} - 3 \\sqrt{13}$, c=$- \\frac{9}{2} + \\frac{9 \\sqrt{13}}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{2601}{16} - \\frac{81 \\sqrt{13}}{2}$, so √Δ = $\\sqrt{\\frac{2601}{16} - \\frac{81 \\sqrt{13}}{2}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{21}{8} + \\frac{3 \\sqrt{289 - 72 \\sqrt{13}}}{8} + \\frac{3 \\sqrt{13}}{2}$.\n7) x2 = $- \\frac{21}{8} - \\frac{3 \\sqrt{289 - 72 \\sqrt{13}}}{8} + \\frac{3 \\sqrt{13}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{21}{8} + \\frac{3\\cdot \\sqrt{289 - 72\\cdot \\sqrt{13}}}{8} + \\frac{3\\cdot \\sqrt{13}}{2}, - \\frac{21}{8} - \\frac{3\\cdot \\sqrt{289 - 72\\cdot \\sqrt{13}}}{8} + \\frac{3\\cdot \\sqrt{13}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{21}{8} + \\frac{3\\cdot \\sqrt{289 - 72\\cdot \\sqrt{13}}}{8} + \\frac{3\\cdot \\sqrt{13}}{2}, - \\frac{21}{8} - \\frac{3\\cdot \\sqrt{289 - 72\\cdot \\sqrt{13}}}{8} + \\frac{3\\cdot \\sqrt{13}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "433cd5ede27d58a6", "problem_id": "d3314825fd1a2cd9", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{13} x + \\frac{21 x}{4} - \\frac{9}{2} + \\frac{9 \\sqrt{13}}{4}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{21}{4} - 3 \\sqrt{13}$, c=$- \\frac{9}{2} + \\frac{9 \\sqrt{13}}{4}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{- \\frac{21}{8} + \\frac{3\\cdot \\sqrt{289 - 72\\cdot \\sqrt{13}}}{8} + \\frac{3\\cdot \\sqrt{13}}{2}, - \\frac{21}{8} - \\frac{3\\cdot \\sqrt{289 - 72\\cdot \\sqrt{13}}}{8} + \\frac{3\\cdot \\sqrt{13}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{21}{8} + \\frac{3\\cdot \\sqrt{289 - 72\\cdot \\sqrt{13}}}{8} + \\frac{3\\cdot \\sqrt{13}}{2}, - \\frac{21}{8} - \\frac{3\\cdot \\sqrt{289 - 72\\cdot \\sqrt{13}}}{8} + \\frac{3\\cdot \\sqrt{13}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3cb71d9d84f90683", "problem_id": "49578745ecf6750f", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - 2 x - 4$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to - 3 x - 4.\n3) Then the limit is -13.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: -10.\nFinal answer: $-10$", "final_answer": "-10", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "05bd5635be5bb3c9", "problem_id": "0398c0b744502010", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 2 x^{3} - 3 x^{2} + 8 x - 27$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} - 3 x^{2} + 7 x - 27.\n3) Then the limit is -129.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -132.\nFinal answer: $-132$", "final_answer": "-132", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "662986627ab22b2a", "problem_id": "c0b4cfac1ac9929b", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡2 (mod 3), x≡3 (mod 5). Give the solution modulo 15.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 15.\nCompute the CRT solution to get x ≡ 8 (mod 15).\nCheck: 8 mod 3=2 and mod 5=3.\nFinal answer: $x\\equiv 8\\pmod{15}$", "final_answer": "x\\equiv 8\\pmod{15}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4ab9587da6ca85be", "problem_id": "c0b4cfac1ac9929b", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡2 (mod 3), x≡3 (mod 5). Give the solution modulo 15.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 15.\n3) Compute the CRT solution to get x ≡ 8 (mod 15).\nFinal answer: $x\\equiv 8\\pmod{15}$", "final_answer": "x\\equiv 8\\pmod{15}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ae55d3b7fd78857a", "problem_id": "c22f4339baa0bece", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Two friends split $570 in the ratio 8:7. How much does each receive?", "output": "1) Total ratio parts = 8+7 = 15.\n2) Each part = 570 ÷ 15 = 38.\n3) First share = 8 × 38 = 304.\n4) Second share = 7 × 38 = 266.\nFinal answer: $266$", "final_answer": "266", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "654e39058d7fb965", "problem_id": "c22f4339baa0bece", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Two friends split $570 in the ratio 8:7. How much does each receive?", "output": "Total ratio parts = 8+7 = 15.\nEach part = 570 ÷ 15 = 38.\nFirst share = 8 × 38 = 304.\nFinal answer: $266$", "final_answer": "266", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "831a701f97dedaf4", "problem_id": "ba523c36a6f416c8", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $150. It is discounted by 5%. You buy 6 backpacks. What is the total cost?", "output": "Discounted price = 150 × (1 − 5/100) = 150 × 95/100 = 285/2.\nSubtotal = 6 × 285/2 = 855.\nCheck: recomputing the formula yields 855.\nFinal answer: $855$", "final_answer": "855", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6ec077e4e40ee0e0", "problem_id": "ba523c36a6f416c8", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A store sells a backpack for $150. It is discounted by 5%. You buy 6 backpacks. What is the total cost?", "output": "1) Discounted price = 150 × (1 − 5/100) = 150 × 95/100 = 285/2.\n2) Subtotal = 6 × 285/2 = 855.\nFinal answer: $855$", "final_answer": "855", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e6f9c1d0266b5fb4", "problem_id": "5561792674930470", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 20 red balls and 22 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(42,12).\n3) Favorable samples: choose k reds and n−k blues: C(20,1)·C(22,11).\n4) Compute totals: C(42,12) = 11058116888, and favorable = 14108640.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{60}{47027}$", "final_answer": "\\frac{60}{47027}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1bf2bf050f0f808c", "problem_id": "5561792674930470", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 20 red balls and 22 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(42,12).\nFavorable samples: choose k reds and n−k blues: C(20,1)·C(22,11).\nCompute totals: C(42,12) = 11058116888, and favorable = 14108640.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{60}{47027}$.\nFinal answer: $\\frac{60}{47027}$", "final_answer": "\\frac{60}{47027}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e2158a998332ff9", "problem_id": "0046ad9f60a9a4fd", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 2 x^{3} - x^{2} - 2 x - 1\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 2 x^{3} - x^{2} - 2 x - 1$.\n3) Compute an antiderivative: F(x) = $- \\frac{x^{4}}{2} - \\frac{x^{3}}{3} - x^{2} - x$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{x^{4}}{2} - \\frac{x^{3}}{3} - x^{2} - x$", "final_answer": "C - \\frac{x^{4}}{2} - \\frac{x^{3}}{3} - x^{2} - x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6e50c0e18ca6eb9e", "problem_id": "0046ad9f60a9a4fd", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 2 x^{3} - x^{2} - 2 x - 1\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 2 x^{3} - x^{2} - 2 x - 1$.\nCompute an antiderivative: F(x) = $- \\frac{x^{4}}{2} - \\frac{x^{3}}{3} - x^{2} - x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - \\frac{x^{4}}{2} - \\frac{x^{3}}{3} - x^{2} - x$", "final_answer": "C - \\frac{x^{4}}{2} - \\frac{x^{3}}{3} - x^{2} - x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7e8770a457ee2189", "problem_id": "7516466545b610c0", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Two friends split $2175 in the ratio 6:9. How much does each receive?", "output": "1) Total ratio parts = 6+9 = 15.\n2) Each part = 2175 ÷ 15 = 145.\n3) First share = 6 × 145 = 870.\n4) Second share = 9 × 145 = 1305.\nFinal answer: $1305$", "final_answer": "1305", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "598f2444dc5c4083", "problem_id": "7516466545b610c0", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $2175 in the ratio 6:9. How much does each receive?", "output": "Total ratio parts = 6+9 = 15.\nEach part = 2175 ÷ 15 = 145.\nFirst share = 6 × 145 = 870.\nSecond share = 9 × 145 = 1305.\nCheck: 870+1305=2175.\nFinal answer: $1305$", "final_answer": "1305", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2d1416a47e1c2c02", "problem_id": "5cc3365597e1151c", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A cyclist rides at 87 miles per hour for 3/2 hours, then at 81 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 87 × 3/2 = 261/2 miles.\n2) Second leg distance = 81 × 5/2 = 405/2 miles.\n3) Total distance = 261/2 + 405/2 = 333 miles.\nFinal answer: $333$", "final_answer": "333", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c0dd2c764041105b", "problem_id": "5cc3365597e1151c", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 87 miles per hour for 3/2 hours, then at 81 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 87 × 3/2 = 261/2 miles.\nSecond leg distance = 81 × 5/2 = 405/2 miles.\nTotal distance = 261/2 + 405/2 = 333 miles.\nCheck: adding both legs gives 333.\nFinal answer: $333$", "final_answer": "333", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c6b06d18f6583a88", "problem_id": "80cd2d9d70c1bc48", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{3} \\sin{\\left(x \\right)} - x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 \\cos{\\left(2 x \\right)}) = - 3 \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = 2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 3 \\sin{\\left(2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 \\cos{\\left(2 x \\right)}) = - 6 \\sin{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)}$.\nFinal answer: $2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)}$", "final_answer": "2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2d6791d0e965b201", "problem_id": "4d274c6d3171eb5b", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{10}$. If it is flipped 11 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=5, p=$\\frac{3}{10}$ and simplify.\n4) Here C(11,5) = 462.\nFinal answer: $\\frac{6603991317}{50000000000}$", "final_answer": "\\frac{6603991317}{50000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a0451a682f3b766d", "problem_id": "128fc9fc4707633f", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 12 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=6, p=$\\frac{1}{5}$ and simplify.\n4) Here C(12,6) = 924.\nFinal answer: $\\frac{3784704}{244140625}$", "final_answer": "\\frac{3784704}{244140625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "69e00f1933630113", "problem_id": "128fc9fc4707633f", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 12 times, what is the probability of exactly 6 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=12, k=6, p=$\\frac{1}{5}$ and simplify.\nHere C(12,6) = 924.\nCheck: expression simplifies to $\\frac{3784704}{244140625}$.\nFinal answer: $\\frac{3784704}{244140625}$", "final_answer": "\\frac{3784704}{244140625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4c05600ae8b05b63", "problem_id": "f749a85a1e0ade84", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} - 2 x^{2} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=-2$.", "output": "Differentiate: f'(x) = $- 3 x^{2} - 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\nSlope at x=-2: m = f'(-2) = $-6 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}$.\nPoint on curve: (-2, f(-2)) = (-2, $\\cos{\\left(4 \\right)} + e^{-1} + 2$).\nFinal answer: $y = - 6\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 10 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$", "final_answer": "y = - 6\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 10 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "62a46da7f2165bdf", "problem_id": "f749a85a1e0ade84", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} - 2 x^{2} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $- 3 x^{2} - 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\n2) Slope at x=-2: m = f'(-2) = $-6 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}$.\n3) Point on curve: (-2, f(-2)) = (-2, $\\cos{\\left(4 \\right)} + e^{-1} + 2$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(4 \\right)} + e^{-1} + 2 = -6 + 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e}(x − -2).\n6) Optionally expand: y = $- 6 x + 2 x \\sin{\\left(4 \\right)} + \\frac{x}{2 e} - 10 + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$.\nFinal answer: $y = - 6\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 10 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}$", "final_answer": "y = - 6\\cdot x + 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} - 10 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4c1c3ee3291487bc", "problem_id": "028547e130990c10", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $840 in the ratio 7:8. How much does each receive?", "output": "1) Total ratio parts = 7+8 = 15.\n2) Each part = 840 ÷ 15 = 56.\n3) First share = 7 × 56 = 392.\n4) Second share = 8 × 56 = 448.\nFinal answer: $448$", "final_answer": "448", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5824947c9ea08e60", "problem_id": "028547e130990c10", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Two friends split $840 in the ratio 7:8. How much does each receive?", "output": "Total ratio parts = 7+8 = 15.\nEach part = 840 ÷ 15 = 56.\nFirst share = 7 × 56 = 392.\nSecond share = 8 × 56 = 448.\nCheck: 392+448=840.\nFinal answer: $448$", "final_answer": "448", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c661a234d08a8549", "problem_id": "c2e03d97321c7633", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 14 x + 45$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} + 14 x + 45$ = $\\left(x + 5\\right) \\left(x + 9\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-5$.\n5) From the second factor: x = $-9$.\nFinal answer: $x \\in \\{-5, -9\\}$", "final_answer": "x \\in \\{-5, -9\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72138a9dd05b6e1c", "problem_id": "399ece830e72a8be", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - 2 x^{3} - 5 x^{2} - 21 x - 83$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{3} - 5 x^{2} - 22 x - 83.\n3) Then the limit is -379.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: -375.\nFinal answer: $-375$", "final_answer": "-375", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5191a6f23804fb60", "problem_id": "9a20538facc25392", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 22 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=22, k=6, p=$\\frac{1}{8}$ and simplify.\n4) Here C(22,6) = 74613.\nFinal answer: $\\frac{2479608648589639413}{73786976294838206464}$", "final_answer": "\\frac{2479608648589639413}{73786976294838206464}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6cac21c3174c02a5", "problem_id": "9a20538facc25392", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 22 times, what is the probability of exactly 6 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=22, k=6, p=$\\frac{1}{8}$ and simplify.\nHere C(22,6) = 74613.\nCheck: expression simplifies to $\\frac{2479608648589639413}{73786976294838206464}$.\nFinal answer: $\\frac{2479608648589639413}{73786976294838206464}$", "final_answer": "\\frac{2479608648589639413}{73786976294838206464}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9e4cfc6f9f585641", "problem_id": "f94bb98cea397d4b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{3} \\sin{\\left(x \\right)} + x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 x^{3} \\sin{\\left(x \\right)}) = 3 x^{3} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = 3 x^{3} \\cos{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 x^{3} \\sin{\\left(x \\right)}) = 3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$", "final_answer": "3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "08b33ec85b25828b", "problem_id": "6902e069b8e6d1b4", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A cyclist rides at 56 miles per hour for 3/2 hours, then at 79 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 56 × 3/2 = 84 miles.\n2) Second leg distance = 79 × 2 = 158 miles.\n3) Total distance = 84 + 158 = 242 miles.\nFinal answer: $242$", "final_answer": "242", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "33f54f6a7648340b", "problem_id": "6c8c72584c3e57f5", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 17 times, what is the probability of exactly 2 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=17, k=2, p=$\\frac{3}{8}$ and simplify.\nHere C(17,2) = 136.\nCheck: expression simplifies to $\\frac{4669189453125}{281474976710656}$.\nFinal answer: $\\frac{4669189453125}{281474976710656}$", "final_answer": "\\frac{4669189453125}{281474976710656}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c5acf63d8c21eeb0", "problem_id": "6c8c72584c3e57f5", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 17 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=17, k=2, p=$\\frac{3}{8}$ and simplify.\n4) Here C(17,2) = 136.\nFinal answer: $\\frac{4669189453125}{281474976710656}$", "final_answer": "\\frac{4669189453125}{281474976710656}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ade57414b24b2538", "problem_id": "25cc0852ff399e79", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$- 4 x - 4 y$ = $4$\n$- 4 x$ = $12$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $16 x + 16 y$ = $-16$.\n3) Multiply the second equation by $-4$: $16 x$ = $-48$.\n4) Subtract to eliminate x: $16 y$ = $32$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, 2\\right)$", "final_answer": "(x,y)=\\left(-3, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "49d7651b1e10a544", "problem_id": "25cc0852ff399e79", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the linear system:\n$- 4 x - 4 y$ = $4$\n$- 4 x$ = $12$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $16 x + 16 y$ = $-16$.\nMultiply the second equation by $-4$: $16 x$ = $-48$.\nFinal answer: $(x,y)=\\left(-3, 2\\right)$", "final_answer": "(x,y)=\\left(-3, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8c7f78d24b8ee872", "problem_id": "c291d5d1311b3a8e", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the multiplicative inverse of 37 modulo 53.", "output": "1) We want x such that 37x ≡ 1 (mod 53).\n2) Use extended Euclid to find s,t with 37s + 53t = 1.\n3) One Bezout identity is: 37·(-10) + 53·(7) = 1.\n4) Reduce s modulo 53: x ≡ 43 (mod 53).\nFinal answer: $x\\equiv 43\\pmod{53}$", "final_answer": "x\\equiv 43\\pmod{53}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec9ff43afc123a0b", "problem_id": "3ef96dfabadc66d8", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 2 x^{3} + 5 x^{2} - 6 x + 8$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{3} + 5 x^{2} - 7 x + 8.\n3) Then the limit is 22.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: 21.\nFinal answer: $21$", "final_answer": "21", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6ccb75795180213c", "problem_id": "5bb29d35260bb53d", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Find the multiplicative inverse of 38 modulo 43.", "output": "1) We want x such that 38x ≡ 1 (mod 43).\n2) Use extended Euclid to find s,t with 38s + 43t = 1.\n3) One Bezout identity is: 38·(17) + 43·(-15) = 1.\n4) Reduce s modulo 43: x ≡ 17 (mod 43).\nFinal answer: $x\\equiv 17\\pmod{43}$", "final_answer": "x\\equiv 17\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "53c0cdb03769c429", "problem_id": "5bb29d35260bb53d", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 38 modulo 43.", "output": "We want x such that 38x ≡ 1 (mod 43).\nUse extended Euclid to find s,t with 38s + 43t = 1.\nOne Bezout identity is: 38·(17) + 43·(-15) = 1.\nReduce s modulo 43: x ≡ 17 (mod 43).\nCheck: 38·17 mod 43 = 1.\nFinal answer: $x\\equiv 17\\pmod{43}$", "final_answer": "x\\equiv 17\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "85a0826b3d82aa6f", "problem_id": "03e5f510f9fac443", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} 3 x^{3} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{3} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $0$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $0$", "final_answer": "0", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5547a1af271c4416", "problem_id": "03e5f510f9fac443", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\frac{\\pi}{2}} 3 x^{3} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{3} - 2 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\frac{\\pi}{2}) = $0$.\n4) Simplify the result.\nFinal answer: $0$", "final_answer": "0", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bbd6448d3477add5", "problem_id": "1c7b9c297ecb2161", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 13 red balls and 15 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(28,7).\n3) Favorable samples: choose k reds and n−k blues: C(13,1)·C(15,6).\n4) Compute totals: C(28,7) = 1184040, and favorable = 65065.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{91}{1656}$", "final_answer": "\\frac{91}{1656}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e47f13143c5831f5", "problem_id": "1c7b9c297ecb2161", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 13 red balls and 15 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(28,7).\nFavorable samples: choose k reds and n−k blues: C(13,1)·C(15,6).\nCompute totals: C(28,7) = 1184040, and favorable = 65065.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{91}{1656}$.\nFinal answer: $\\frac{91}{1656}$", "final_answer": "\\frac{91}{1656}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9af52ff94ae733e7", "problem_id": "c2e7c72614c1be7e", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$- 2 x - 8 y$ = $-8$\n$3 x + 3 y$ = $3$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $- 6 x - 24 y$ = $-24$.\nMultiply the second equation by $-2$: $- 6 x - 6 y$ = $-6$.\nSubtract to eliminate x: $- 18 y$ = $-18$.\nSolve for y: y = $1$.\nSubstitute back to find x: x = $0$.\nCheck: plug (x,y)=(0,1) into both equations.\nFinal answer: $(x,y)=\\left(0, 1\\right)$", "final_answer": "(x,y)=\\left(0, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5dd581f77707fc20", "problem_id": "c2e7c72614c1be7e", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$- 2 x - 8 y$ = $-8$\n$3 x + 3 y$ = $3$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $3$: $- 6 x - 24 y$ = $-24$.\n3) Multiply the second equation by $-2$: $- 6 x - 6 y$ = $-6$.\n4) Subtract to eliminate x: $- 18 y$ = $-18$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, 1\\right)$", "final_answer": "(x,y)=\\left(0, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bdc68a2b3cd1566a", "problem_id": "bf66467d1229cdad", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{11} x}{2} + \\frac{11 x}{6} - \\frac{2 \\sqrt{11}}{3} + \\frac{2}{3}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{11}{6} - \\frac{\\sqrt{11}}{2}$, c=$\\frac{2}{3} - \\frac{2 \\sqrt{11}}{3}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{5 \\sqrt{11}}{6} + \\frac{31}{9}$, so √Δ = $\\sqrt{\\frac{5 \\sqrt{11}}{6} + \\frac{31}{9}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{11}{12} + \\frac{\\sqrt{11}}{4} + \\frac{\\sqrt{30 \\sqrt{11} + 124}}{12}$.\n7) x2 = $- \\frac{\\sqrt{30 \\sqrt{11} + 124}}{12} - \\frac{11}{12} + \\frac{\\sqrt{11}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{11}{12} + \\frac{\\sqrt{11}}{4} + \\frac{\\sqrt{30\\cdot \\sqrt{11} + 124}}{12}, - \\frac{\\sqrt{30\\cdot \\sqrt{11} + 124}}{12} - \\frac{11}{12} + \\frac{\\sqrt{11}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{11}{12} + \\frac{\\sqrt{11}}{4} + \\frac{\\sqrt{30\\cdot \\sqrt{11} + 124}}{12}, - \\frac{\\sqrt{30\\cdot \\sqrt{11} + 124}}{12} - \\frac{11}{12} + \\frac{\\sqrt{11}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ff3e2d14958746b", "problem_id": "8eb129c8ffc74457", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 x^{2} - 5 x + 8$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{2} - 6 x + 8.\n3) Then the limit is 32.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 30.\nFinal answer: $30$", "final_answer": "30", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "98191602ccf32b02", "problem_id": "208ef6ba63abd874", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{3} - 2 x^{2} + 3 x - 4$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-4) = 0.\n3) d/dx(- 2 x^{2}) = - 4 x.\n4) d/dx(2 x^{3}) = 6 x^{2}.\n5) d/dx(3 x) = 3.\n6) Combine: f'(x) = 6 x^{2} - 4 x + 3.\nFinal answer: $6\\cdot x^{2} - 4\\cdot x + 3$", "final_answer": "6\\cdot x^{2} - 4\\cdot x + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "21426e242d1fb6c9", "problem_id": "208ef6ba63abd874", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{3} - 2 x^{2} + 3 x - 4$.", "output": "Differentiate term-by-term:\nd/dx(-4) = 0.\nd/dx(- 2 x^{2}) = - 4 x.\nd/dx(2 x^{3}) = 6 x^{2}.\nd/dx(3 x) = 3.\nCombine: f'(x) = 6 x^{2} - 4 x + 3.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $6\\cdot x^{2} - 4\\cdot x + 3$", "final_answer": "6\\cdot x^{2} - 4\\cdot x + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "02c071cd3c9660af", "problem_id": "4e985425e3dde495", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the multiplicative inverse of 11 modulo 37.", "output": "1) We want x such that 11x ≡ 1 (mod 37).\n2) Use extended Euclid to find s,t with 11s + 37t = 1.\n3) One Bezout identity is: 11·(-10) + 37·(3) = 1.\n4) Reduce s modulo 37: x ≡ 27 (mod 37).\nFinal answer: $x\\equiv 27\\pmod{37}$", "final_answer": "x\\equiv 27\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "626dd515a50e417f", "problem_id": "4e985425e3dde495", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the multiplicative inverse of 11 modulo 37.", "output": "We want x such that 11x ≡ 1 (mod 37).\nUse extended Euclid to find s,t with 11s + 37t = 1.\nOne Bezout identity is: 11·(-10) + 37·(3) = 1.\nReduce s modulo 37: x ≡ 27 (mod 37).\nCheck: 11·27 mod 37 = 1.\nFinal answer: $x\\equiv 27\\pmod{37}$", "final_answer": "x\\equiv 27\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0133a6fe4f708e41", "problem_id": "5c3ae3b54d6b14fd", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $2160 in the ratio 4:8. How much does each receive?", "output": "1) Total ratio parts = 4+8 = 12.\n2) Each part = 2160 ÷ 12 = 180.\n3) First share = 4 × 180 = 720.\n4) Second share = 8 × 180 = 1440.\nFinal answer: $1440$", "final_answer": "1440", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d190aa992ddde1d", "problem_id": "5c3ae3b54d6b14fd", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Two friends split $2160 in the ratio 4:8. How much does each receive?", "output": "Total ratio parts = 4+8 = 12.\nEach part = 2160 ÷ 12 = 180.\nFirst share = 4 × 180 = 720.\nFinal answer: $1440$", "final_answer": "1440", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "999c2a8056d2e5ff", "problem_id": "c1b2be4760e27590", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 18 x + 16$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 18 x + 16$ = $2 \\left(x - 8\\right) \\left(x - 1\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $8$.\n5) From the second factor: x = $1$.\nFinal answer: $x \\in \\{8, 1\\}$", "final_answer": "x \\in \\{8, 1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b87541d40b3dcb95", "problem_id": "c1b2be4760e27590", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 18 x + 16$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} - 18 x + 16$ = $2 \\left(x - 8\\right) \\left(x - 1\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $8$.\nFrom the second factor: x = $1$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{8, 1\\}$", "final_answer": "x \\in \\{8, 1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "585d531aab9d69cb", "problem_id": "3ecb21c9d55ee786", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 2 x^{3} + 3 x^{2} + 2 x + \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{3 x - 5}$.", "output": "1) Let f(x) = $\\frac{- 2 x^{3} + 3 x^{2} + 2 x + \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{3 x - 5}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{12 x^{3} - 18 x^{2} - 12 x + \\left(3 x - 5\\right) \\left(- 12 x^{2} + 12 x + \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 6 \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 18}{2 \\left(3 x - 5\\right)^{2}}$.\nFinal answer: $\\frac{12\\cdot x^{3} - 18\\cdot x^{2} - 12\\cdot x + \\left(3\\cdot x - 5\\right)\\cdot \\left(- 12\\cdot x^{2} + 12\\cdot x + \\left(3\\cdot x - 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x - 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 4\\right) - \\left(18\\cdot x - 30\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 18}{2\\cdot \\left(3\\cdot x - 5\\right)^{2}}$", "final_answer": "\\frac{12\\cdot x^{3} - 18\\cdot x^{2} - 12\\cdot x + \\left(3\\cdot x - 5\\right)\\cdot \\left(- 12\\cdot x^{2} + 12\\cdot x + \\left(3\\cdot x - 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x - 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 4\\right) - \\left(18\\cdot x - 30\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 18}{2\\cdot \\left(3\\cdot x - 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c1f99f1d4331553a", "problem_id": "3ecb21c9d55ee786", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 2 x^{3} + 3 x^{2} + 2 x + \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{3 x - 5}$.", "output": "Let f(x) = $\\frac{- 2 x^{3} + 3 x^{2} + 2 x + \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{3 x - 5}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{12 x^{3} - 18 x^{2} - 12 x + \\left(3 x - 5\\right) \\left(- 12 x^{2} + 12 x + \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 6 \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 18}{2 \\left(3 x - 5\\right)^{2}}$.\nFinal answer: $\\frac{12\\cdot x^{3} - 18\\cdot x^{2} - 12\\cdot x + \\left(3\\cdot x - 5\\right)\\cdot \\left(- 12\\cdot x^{2} + 12\\cdot x + \\left(3\\cdot x - 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x - 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 4\\right) - \\left(18\\cdot x - 30\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 18}{2\\cdot \\left(3\\cdot x - 5\\right)^{2}}$", "final_answer": "\\frac{12\\cdot x^{3} - 18\\cdot x^{2} - 12\\cdot x + \\left(3\\cdot x - 5\\right)\\cdot \\left(- 12\\cdot x^{2} + 12\\cdot x + \\left(3\\cdot x - 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x - 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 6\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 4\\right) - \\left(18\\cdot x - 30\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 18}{2\\cdot \\left(3\\cdot x - 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7692b2cd4dbbc6f1", "problem_id": "0c80630854bd7d88", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "You want 46 liters of a 30% salt solution. You have a 15% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 46−x liters is 50% solution.\n2) Salt equation: 15x + 50(46−x) = 30·46.\n3) (15-50)x = (30-50)·46 ⇒ x = 184/7.\n4) Then 46−x = 138/7.\nFinal answer: $184/7 L of 15% and 138/7 L of 50%$", "final_answer": "184/7 L of 15% and 138/7 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b301f9e97a25066b", "problem_id": "0c80630854bd7d88", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "You want 46 liters of a 30% salt solution. You have a 15% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 46−x liters is 50% solution.\nSalt equation: 15x + 50(46−x) = 30·46.\n(15-50)x = (30-50)·46 ⇒ x = 184/7.\nThen 46−x = 138/7.\nCheck: 15·184/7 + 50·138/7 = 30·46.\nFinal answer: $184/7 L of 15% and 138/7 L of 50%$", "final_answer": "184/7 L of 15% and 138/7 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "78565e1df2fe3602", "problem_id": "bebf569ca9053e10", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A store sells a backpack for $66. It is discounted by 35%. You buy 5 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 66 × (1 − 35/100) = 66 × 65/100 = 429/10.\nSubtotal = 5 × 429/10 = 429/2.\nTotal with tax = 429/2 × 108/100 = 11583/50.\nCheck: recomputing the formula yields 11583/50.\nFinal answer: $\\frac{11583}{50}$", "final_answer": "\\frac{11583}{50}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a227f7f502a08f90", "problem_id": "bebf569ca9053e10", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A store sells a backpack for $66. It is discounted by 35%. You buy 5 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 66 × (1 − 35/100) = 66 × 65/100 = 429/10.\n2) Subtotal = 5 × 429/10 = 429/2.\n3) Total with tax = 429/2 × 108/100 = 11583/50.\nFinal answer: $\\frac{11583}{50}$", "final_answer": "\\frac{11583}{50}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f52c9e79c0ab21de", "problem_id": "682f63d7c68e490b", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{10}$. If it is flipped 10 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=4, p=$\\frac{3}{10}$ and simplify.\n4) Here C(10,4) = 210.\nFinal answer: $\\frac{200120949}{1000000000}$", "final_answer": "\\frac{200120949}{1000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "32d74a4c34f22880", "problem_id": "682f63d7c68e490b", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{10}$. If it is flipped 10 times, what is the probability of exactly 4 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=4, p=$\\frac{3}{10}$ and simplify.\nHere C(10,4) = 210.\nCheck: expression simplifies to $\\frac{200120949}{1000000000}$.\nFinal answer: $\\frac{200120949}{1000000000}$", "final_answer": "\\frac{200120949}{1000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d3fb554229fdd336", "problem_id": "7dd6dbde5c8c79a3", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} - x^{4} + 4 x^{3} - x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{4} + 4 x^{3} - x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\n3) Subtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{\\pi^{2}}{2} - \\frac{\\pi^{5}}{160} - \\frac{\\pi^{3}}{24} + \\frac{\\pi}{2} + 5 + \\frac{\\pi^{4}}{16}$.\n4) Simplify the result.\nFinal answer: $- \\frac{\\pi^{2}}{2} - \\frac{\\pi^{5}}{160} - \\frac{\\pi^{3}}{24} + \\frac{\\pi}{2} + 5 + \\frac{\\pi^{4}}{16}$", "final_answer": "- \\frac{\\pi^{2}}{2} - \\frac{\\pi^{5}}{160} - \\frac{\\pi^{3}}{24} + \\frac{\\pi}{2} + 5 + \\frac{\\pi^{4}}{16}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eb2b2edf775c4ac1", "problem_id": "7dd6dbde5c8c79a3", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} - x^{4} + 4 x^{3} - x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{4} + 4 x^{3} - x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\nSubtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{\\pi^{2}}{2} - \\frac{\\pi^{5}}{160} - \\frac{\\pi^{3}}{24} + \\frac{\\pi}{2} + 5 + \\frac{\\pi^{4}}{16}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{\\pi^{2}}{2} - \\frac{\\pi^{5}}{160} - \\frac{\\pi^{3}}{24} + \\frac{\\pi}{2} + 5 + \\frac{\\pi^{4}}{16}$", "final_answer": "- \\frac{\\pi^{2}}{2} - \\frac{\\pi^{5}}{160} - \\frac{\\pi^{3}}{24} + \\frac{\\pi}{2} + 5 + \\frac{\\pi^{4}}{16}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "62cacc4c63f88028", "problem_id": "ec6c5605384fe308", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{7} x}{2} + 4 x - \\frac{3 \\sqrt{7}}{2} + 3$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$4 - \\frac{3 \\sqrt{7}}{2}$, c=$3 - \\frac{3 \\sqrt{7}}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{79}{4} - 6 \\sqrt{7}$, so √Δ = $\\sqrt{\\frac{79}{4} - 6 \\sqrt{7}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $-2 + \\frac{\\sqrt{79 - 24 \\sqrt{7}}}{4} + \\frac{3 \\sqrt{7}}{4}$.\n7) x2 = $-2 - \\frac{\\sqrt{79 - 24 \\sqrt{7}}}{4} + \\frac{3 \\sqrt{7}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{-2 + \\frac{\\sqrt{79 - 24\\cdot \\sqrt{7}}}{4} + \\frac{3\\cdot \\sqrt{7}}{4}, -2 - \\frac{\\sqrt{79 - 24\\cdot \\sqrt{7}}}{4} + \\frac{3\\cdot \\sqrt{7}}{4}\\}$", "final_answer": "x \\in \\{-2 + \\frac{\\sqrt{79 - 24\\cdot \\sqrt{7}}}{4} + \\frac{3\\cdot \\sqrt{7}}{4}, -2 - \\frac{\\sqrt{79 - 24\\cdot \\sqrt{7}}}{4} + \\frac{3\\cdot \\sqrt{7}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "90a12fd8df92f0f6", "problem_id": "30d6cf9a97495d00", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{x^{2} + 3}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 3}) = \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} - \\frac{2 x^{4}}{2 x^{4} + 12 x^{2} + 18} + \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} + 6 x^{2} + 9} + \\frac{6 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} + \\frac{12 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} - \\frac{12 x^{2}}{2 x^{4} + 12 x^{2} + 18} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} + 6} - \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3} + \\frac{36 x}{2 x^{4} + 12 x^{2} + 18} - \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 3} + \\frac{9 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} + \\frac{18 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} + 12 x^{2} + 18} - \\frac{18}{2 x^{4} + 12 x^{2} + 18}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 3}) = - \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} + 3\\right)^{2}} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x^{2} + 3\\right)} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} + 3} + \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} + 3}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 4 x \\left(- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} + 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x^{2} + 3\\right)^{2}}$.\nFinal answer: $\\frac{- 4 x \\left(- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} + 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x^{2} + 3\\right)^{2}}$", "final_answer": "\\frac{- 4 x \\left(- x^{3} + 2 x^{2} - 3 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x^{2} + 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8 x + \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x^{2} + 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "aec17a9c7efff9a3", "problem_id": "efc53486cbfa23d6", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_8=37$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=37 and k=8.\nFinal answer: $38320568$", "final_answer": "38320568", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e6cca19b9f2b5545", "problem_id": "efc53486cbfa23d6", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_8=37$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=37 and k=8.\n4) Compute C(44,7) = 38320568.\nFinal answer: $38320568$", "final_answer": "38320568", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ad2ac0b7c302dfc6", "problem_id": "76334532bacb640d", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the multiplicative inverse of 6 modulo 41.", "output": "1) We want x such that 6x ≡ 1 (mod 41).\n2) Use extended Euclid to find s,t with 6s + 41t = 1.\n3) One Bezout identity is: 6·(7) + 41·(-1) = 1.\n4) Reduce s modulo 41: x ≡ 7 (mod 41).\nFinal answer: $x\\equiv 7\\pmod{41}$", "final_answer": "x\\equiv 7\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8d66e40a2d5f1ec1", "problem_id": "76334532bacb640d", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Find the multiplicative inverse of 6 modulo 41.", "output": "We want x such that 6x ≡ 1 (mod 41).\nUse extended Euclid to find s,t with 6s + 41t = 1.\nOne Bezout identity is: 6·(7) + 41·(-1) = 1.\nFinal answer: $x\\equiv 7\\pmod{41}$", "final_answer": "x\\equiv 7\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5366e3dbfe59a220", "problem_id": "7061fd3d8523c4fb", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 17 times, what is the probability of exactly 14 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=17, k=14, p=$\\frac{2}{3}$ and simplify.\nHere C(17,14) = 680.\nCheck: expression simplifies to $\\frac{11141120}{129140163}$.\nFinal answer: $\\frac{11141120}{129140163}$", "final_answer": "\\frac{11141120}{129140163}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2cec0dbf28aba78f", "problem_id": "7061fd3d8523c4fb", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 17 times, what is the probability of exactly 14 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=17, k=14, p=$\\frac{2}{3}$ and simplify.\n4) Here C(17,14) = 680.\nFinal answer: $\\frac{11141120}{129140163}$", "final_answer": "\\frac{11141120}{129140163}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "55248cfb10c521e3", "problem_id": "83b5ec0d86704c54", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡4 (mod 11), x≡7 (mod 13). Give the solution modulo 143.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 143.\nCompute the CRT solution to get x ≡ 59 (mod 143).\nCheck: 59 mod 11=4 and mod 13=7.\nFinal answer: $x\\equiv 59\\pmod{143}$", "final_answer": "x\\equiv 59\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9b30f10a7f6dd247", "problem_id": "83b5ec0d86704c54", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the congruences: x≡4 (mod 11), x≡7 (mod 13). Give the solution modulo 143.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 143.\nCompute the CRT solution to get x ≡ 59 (mod 143).\nFinal answer: $x\\equiv 59\\pmod{143}$", "final_answer": "x\\equiv 59\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6ded2c63604a891c", "problem_id": "ef4e8854daeebb14", "style": "answer_only", "instruction": "Final answer only.", "input": "An urn has 22 red balls and 26 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "$\\frac{28600}{745749}$", "final_answer": "\\frac{28600}{745749}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ab75567d63282be9", "problem_id": "ef4e8854daeebb14", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 22 red balls and 26 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(48,8).\n3) Favorable samples: choose k reds and n−k blues: C(22,1)·C(26,7).\n4) Compute totals: C(48,8) = 377348994, and favorable = 14471600.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{28600}{745749}$", "final_answer": "\\frac{28600}{745749}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6be109f0cec163f7", "problem_id": "2f61de34f6f11791", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A store sells a backpack for $173. It is discounted by 25%. You buy 3 backpacks. What is the total cost?", "output": "Discounted price = 173 × (1 − 25/100) = 173 × 75/100 = 519/4.\nSubtotal = 3 × 519/4 = 1557/4.\nCheck: recomputing the formula yields 1557/4.\nFinal answer: $\\frac{1557}{4}$", "final_answer": "\\frac{1557}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1ad71f660778655a", "problem_id": "2f61de34f6f11791", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A store sells a backpack for $173. It is discounted by 25%. You buy 3 backpacks. What is the total cost?", "output": "1) Discounted price = 173 × (1 − 25/100) = 173 × 75/100 = 519/4.\n2) Subtotal = 3 × 519/4 = 1557/4.\nFinal answer: $\\frac{1557}{4}$", "final_answer": "\\frac{1557}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4a97a1bb8f999423", "problem_id": "c9050a82e4daab6e", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A cyclist rides at 42 miles per hour for 1 hours, then at 85 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 42 × 1 = 42 miles.\nSecond leg distance = 85 × 1 = 85 miles.\nTotal distance = 42 + 85 = 127 miles.\nFinal answer: $127$", "final_answer": "127", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a6e0603a055d255a", "problem_id": "c9050a82e4daab6e", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A cyclist rides at 42 miles per hour for 1 hours, then at 85 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 42 × 1 = 42 miles.\n2) Second leg distance = 85 × 1 = 85 miles.\n3) Total distance = 42 + 85 = 127 miles.\nFinal answer: $127$", "final_answer": "127", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "09b0ed7514ffd40c", "problem_id": "21fb702622360ea1", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_8=37$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_8=29.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=29 and k=8.\n4) Compute C(36,7) = 8347680.\nFinal answer: $8347680$", "final_answer": "8347680", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6bf0531d1fb7dba6", "problem_id": "21fb702622360ea1", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_8=37$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_8=29.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=29 and k=8.\nCompute C(36,7) = 8347680.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $8347680$", "final_answer": "8347680", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1f12962c8998d61e", "problem_id": "b84b61995772c8cd", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{7} x + 2 x - 4 \\sqrt{7} - 8$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$2 - \\sqrt{7}$, c=$- 4 \\sqrt{7} - 8$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $12 \\sqrt{7} + 43$, so √Δ = $\\sqrt{12 \\sqrt{7} + 43}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $-1 + \\frac{\\sqrt{7}}{2} + \\frac{\\sqrt{12 \\sqrt{7} + 43}}{2}$.\n7) x2 = $- \\frac{\\sqrt{12 \\sqrt{7} + 43}}{2} - 1 + \\frac{\\sqrt{7}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{-1 + \\frac{\\sqrt{7}}{2} + \\frac{\\sqrt{12\\cdot \\sqrt{7} + 43}}{2}, - \\frac{\\sqrt{12\\cdot \\sqrt{7} + 43}}{2} - 1 + \\frac{\\sqrt{7}}{2}\\}$", "final_answer": "x \\in \\{-1 + \\frac{\\sqrt{7}}{2} + \\frac{\\sqrt{12\\cdot \\sqrt{7} + 43}}{2}, - \\frac{\\sqrt{12\\cdot \\sqrt{7} + 43}}{2} - 1 + \\frac{\\sqrt{7}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e2b11af34310f0d5", "problem_id": "b84b61995772c8cd", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{7} x + 2 x - 4 \\sqrt{7} - 8$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$2 - \\sqrt{7}$, c=$- 4 \\sqrt{7} - 8$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $12 \\sqrt{7} + 43$, so √Δ = $\\sqrt{12 \\sqrt{7} + 43}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $-1 + \\frac{\\sqrt{7}}{2} + \\frac{\\sqrt{12 \\sqrt{7} + 43}}{2}$.\nx2 = $- \\frac{\\sqrt{12 \\sqrt{7} + 43}}{2} - 1 + \\frac{\\sqrt{7}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-1 + \\frac{\\sqrt{7}}{2} + \\frac{\\sqrt{12\\cdot \\sqrt{7} + 43}}{2}, - \\frac{\\sqrt{12\\cdot \\sqrt{7} + 43}}{2} - 1 + \\frac{\\sqrt{7}}{2}\\}$", "final_answer": "x \\in \\{-1 + \\frac{\\sqrt{7}}{2} + \\frac{\\sqrt{12\\cdot \\sqrt{7} + 43}}{2}, - \\frac{\\sqrt{12\\cdot \\sqrt{7} + 43}}{2} - 1 + \\frac{\\sqrt{7}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e3891e11c3edf32c", "problem_id": "91cfdc0643a29493", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - 2 x^{3} + x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{3} + x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{255 \\pi^{4}}{32} - \\frac{15 \\pi^{2}}{4} - 1 + 10 \\pi + \\frac{65 \\pi^{3}}{24}$.\n4) Simplify the result.\nFinal answer: $- \\frac{255\\cdot \\pi^{4}}{32} - \\frac{15\\cdot \\pi^{2}}{4} - 1 + 10\\cdot \\pi + \\frac{65\\cdot \\pi^{3}}{24}$", "final_answer": "- \\frac{255\\cdot \\pi^{4}}{32} - \\frac{15\\cdot \\pi^{2}}{4} - 1 + 10\\cdot \\pi + \\frac{65\\cdot \\pi^{3}}{24}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "096b5f4349b90c7c", "problem_id": "91cfdc0643a29493", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - 2 x^{3} + x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{3} + x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 4$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{255 \\pi^{4}}{32} - \\frac{15 \\pi^{2}}{4} - 1 + 10 \\pi + \\frac{65 \\pi^{3}}{24}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{255\\cdot \\pi^{4}}{32} - \\frac{15\\cdot \\pi^{2}}{4} - 1 + 10\\cdot \\pi + \\frac{65\\cdot \\pi^{3}}{24}$", "final_answer": "- \\frac{255\\cdot \\pi^{4}}{32} - \\frac{15\\cdot \\pi^{2}}{4} - 1 + 10\\cdot \\pi + \\frac{65\\cdot \\pi^{3}}{24}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "525587691e5e9f01", "problem_id": "0b7e777937ae674a", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 29 miles per hour for 1 hours, then at 52 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 29 × 1 = 29 miles.\nSecond leg distance = 52 × 3/2 = 78 miles.\nTotal distance = 29 + 78 = 107 miles.\nCheck: adding both legs gives 107.\nFinal answer: $107$", "final_answer": "107", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b4e6b9e59cef00f7", "problem_id": "0b7e777937ae674a", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "A cyclist rides at 29 miles per hour for 1 hours, then at 52 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 29 × 1 = 29 miles.\nSecond leg distance = 52 × 3/2 = 78 miles.\nTotal distance = 29 + 78 = 107 miles.\nFinal answer: $107$", "final_answer": "107", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e65a6acc2c021350", "problem_id": "a8351154e9b8e95e", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "An urn has 15 red balls and 15 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(30,4).\nFavorable samples: choose k reds and n−k blues: C(15,2)·C(15,2).\nFinal answer: $\\frac{35}{87}$", "final_answer": "\\frac{35}{87}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ecb203256dd34812", "problem_id": "a8351154e9b8e95e", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 15 red balls and 15 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(30,4).\nFavorable samples: choose k reds and n−k blues: C(15,2)·C(15,2).\nCompute totals: C(30,4) = 27405, and favorable = 11025.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{35}{87}$.\nFinal answer: $\\frac{35}{87}$", "final_answer": "\\frac{35}{87}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a55faf0a1491bd43", "problem_id": "b477cc9156a2b9b9", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 4 x^{4} - 2 x^{3} - 4 x^{2} - x + 2\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 4 x^{4} - 2 x^{3} - 4 x^{2} - x + 2$.\n3) Compute an antiderivative: F(x) = $- \\frac{4 x^{5}}{5} - \\frac{x^{4}}{2} - \\frac{4 x^{3}}{3} - \\frac{x^{2}}{2} + 2 x$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{4\\cdot x^{5}}{5} - \\frac{x^{4}}{2} - \\frac{4\\cdot x^{3}}{3} - \\frac{x^{2}}{2} + 2\\cdot x$", "final_answer": "C - \\frac{4\\cdot x^{5}}{5} - \\frac{x^{4}}{2} - \\frac{4\\cdot x^{3}}{3} - \\frac{x^{2}}{2} + 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4c55fb5391614cd2", "problem_id": "b477cc9156a2b9b9", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 4 x^{4} - 2 x^{3} - 4 x^{2} - x + 2\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $- 4 x^{4} - 2 x^{3} - 4 x^{2} - x + 2$.\nCompute an antiderivative: F(x) = $- \\frac{4 x^{5}}{5} - \\frac{x^{4}}{2} - \\frac{4 x^{3}}{3} - \\frac{x^{2}}{2} + 2 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C - \\frac{4\\cdot x^{5}}{5} - \\frac{x^{4}}{2} - \\frac{4\\cdot x^{3}}{3} - \\frac{x^{2}}{2} + 2\\cdot x$", "final_answer": "C - \\frac{4\\cdot x^{5}}{5} - \\frac{x^{4}}{2} - \\frac{4\\cdot x^{3}}{3} - \\frac{x^{2}}{2} + 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b93ba16817fbd82e", "problem_id": "f43c537ed24e8a94", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A store sells a backpack for $940. It is discounted by 15%. You buy 12 backpacks. What is the total cost?", "output": "1) Discounted price = 940 × (1 − 15/100) = 940 × 85/100 = 799.\n2) Subtotal = 12 × 799 = 9588.\nFinal answer: $9588$", "final_answer": "9588", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0690f1a139a81aa7", "problem_id": "a6d73985fdff9722", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(2 x^{3} - x^{2} - 3 x - 1\\right) \\cos{\\left(2 x - 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x e^{x}) = x e^{x}$.\n3) So $f'(x) = - 4 x^{3} \\sin{\\left(2 x - 2 \\right)} + 2 x^{2} \\sin{\\left(2 x - 2 \\right)} + 6 x^{2} \\cos{\\left(2 x - 2 \\right)} + x e^{x} + 6 x \\sin{\\left(2 x - 2 \\right)} - 2 x \\cos{\\left(2 x - 2 \\right)} + e^{x} + 2 \\sin{\\left(2 x - 2 \\right)} - 3 \\cos{\\left(2 x - 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x e^{x}) = x e^{x} + e^{x}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 1\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) - 3\\right) - 1\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(6 x^{2} - 2 x - 3\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 1\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) - 3\\right) - 1\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(6 x^{2} - 2 x - 3\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}$", "final_answer": "\\left(x + 1\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) - 3\\right) - 1\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(6 x^{2} - 2 x - 3\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "620c3fa1be20e58b", "problem_id": "f7005da7c43d50c0", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A cyclist rides at 60 miles per hour for 3/2 hours, then at 71 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 60 × 3/2 = 90 miles.\nSecond leg distance = 71 × 2 = 142 miles.\nTotal distance = 90 + 142 = 232 miles.\nFinal answer: $232$", "final_answer": "232", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "444253659b8df4ac", "problem_id": "f7005da7c43d50c0", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A cyclist rides at 60 miles per hour for 3/2 hours, then at 71 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 60 × 3/2 = 90 miles.\n2) Second leg distance = 71 × 2 = 142 miles.\n3) Total distance = 90 + 142 = 232 miles.\nFinal answer: $232$", "final_answer": "232", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c678ed794f531b66", "problem_id": "275f565facfc66fb", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 17 red balls and 11 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(28,11).\nFavorable samples: choose k reds and n−k blues: C(17,3)·C(11,8).\nCompute totals: C(28,11) = 21474180, and favorable = 112200.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1870}{357903}$.\nFinal answer: $\\frac{1870}{357903}$", "final_answer": "\\frac{1870}{357903}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "85937987d7953ba8", "problem_id": "275f565facfc66fb", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 17 red balls and 11 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(28,11).\n3) Favorable samples: choose k reds and n−k blues: C(17,3)·C(11,8).\n4) Compute totals: C(28,11) = 21474180, and favorable = 112200.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1870}{357903}$", "final_answer": "\\frac{1870}{357903}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "30e62e75c54e266b", "problem_id": "617d067a530118cd", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} 4 x^{3} + x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $4 x^{3} + x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{15 \\pi^{4}}{16} - 6 \\pi + 5 + \\frac{3 \\pi^{3}}{8} + \\frac{3 \\pi^{2}}{2}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{15\\cdot \\pi^{4}}{16} - 6\\cdot \\pi + 5 + \\frac{3\\cdot \\pi^{3}}{8} + \\frac{3\\cdot \\pi^{2}}{2}$", "final_answer": "- \\frac{15\\cdot \\pi^{4}}{16} - 6\\cdot \\pi + 5 + \\frac{3\\cdot \\pi^{3}}{8} + \\frac{3\\cdot \\pi^{2}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "677256fff5f840f0", "problem_id": "617d067a530118cd", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} 4 x^{3} + x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $4 x^{3} + x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 4$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{15 \\pi^{4}}{16} - 6 \\pi + 5 + \\frac{3 \\pi^{3}}{8} + \\frac{3 \\pi^{2}}{2}$.\n4) Simplify the result.\nFinal answer: $- \\frac{15\\cdot \\pi^{4}}{16} - 6\\cdot \\pi + 5 + \\frac{3\\cdot \\pi^{3}}{8} + \\frac{3\\cdot \\pi^{2}}{2}$", "final_answer": "- \\frac{15\\cdot \\pi^{4}}{16} - 6\\cdot \\pi + 5 + \\frac{3\\cdot \\pi^{3}}{8} + \\frac{3\\cdot \\pi^{2}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fc88b43d310ecdf6", "problem_id": "4f855f9d0337c84a", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "You want 29 liters of a 35% salt solution. You have a 30% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 29−x liters is 70% solution.\n2) Salt equation: 30x + 70(29−x) = 35·29.\n3) (30-70)x = (35-70)·29 ⇒ x = 203/8.\n4) Then 29−x = 29/8.\nFinal answer: $203/8 L of 30% and 29/8 L of 70%$", "final_answer": "203/8 L of 30% and 29/8 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ea0ea68e6a1dbb40", "problem_id": "4f855f9d0337c84a", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "You want 29 liters of a 35% salt solution. You have a 30% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 29−x liters is 70% solution.\nSalt equation: 30x + 70(29−x) = 35·29.\n(30-70)x = (35-70)·29 ⇒ x = 203/8.\nFinal answer: $203/8 L of 30% and 29/8 L of 70%$", "final_answer": "203/8 L of 30% and 29/8 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "55e6e3d8cc90b4fd", "problem_id": "ee8ca69cc5ede5ab", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} x + 1$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{2} + 3 x$ and a = -2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $2 x + 3$.\n5) Evaluate at a: p'(-2) = $-1$.\nFinal answer: $-1$", "final_answer": "-1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5bacac342079e0f2", "problem_id": "ee8ca69cc5ede5ab", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} x + 1$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $x^{2} + 3 x$ and a = -2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $2 x + 3$.\nEvaluate at a: p'(-2) = $-1$.\nCheck: the identity gives $-1$.\nFinal answer: $-1$", "final_answer": "-1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d239fb8667305cf0", "problem_id": "440b487fd488d85d", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$2 x + 4 y$ = $-10$\n$3 x + 2 y$ = $-11$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $6 x + 12 y$ = $-30$.\nMultiply the second equation by $2$: $6 x + 4 y$ = $-22$.\nSubtract to eliminate x: $8 y$ = $-8$.\nSolve for y: y = $-1$.\nSubstitute back to find x: x = $-3$.\nCheck: plug (x,y)=(-3,-1) into both equations.\nFinal answer: $(x,y)=\\left(-3, -1\\right)$", "final_answer": "(x,y)=\\left(-3, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "012a362e8c0d70df", "problem_id": "440b487fd488d85d", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$2 x + 4 y$ = $-10$\n$3 x + 2 y$ = $-11$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $3$: $6 x + 12 y$ = $-30$.\n3) Multiply the second equation by $2$: $6 x + 4 y$ = $-22$.\n4) Subtract to eliminate x: $8 y$ = $-8$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, -1\\right)$", "final_answer": "(x,y)=\\left(-3, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "99f391aa7baa93b6", "problem_id": "e27ea56d13b98ebb", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$- 3 y$ = $0$\n$- x + 3 y$ = $5$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $3 y$ = $0$.\n3) Multiply the second equation by $0$: $0$ = $0$.\n4) Subtract to eliminate x: $3 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, 0\\right)$", "final_answer": "(x,y)=\\left(-5, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ce30fbda66bcbb9", "problem_id": "15a878ae69b5201a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(3 x^{3} - 3 x^{2} + 3 x\\right) \\cos{\\left(3 x + 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 6 e^{x}) = - 6 e^{x}$.\n3) So $f'(x) = - 9 x^{3} \\sin{\\left(3 x + 2 \\right)} + 9 x^{2} \\sin{\\left(3 x + 2 \\right)} + 9 x^{2} \\cos{\\left(3 x + 2 \\right)} + 2 x e^{x} - 9 x \\sin{\\left(3 x + 2 \\right)} - 6 x \\cos{\\left(3 x + 2 \\right)} - 4 e^{x} + 3 \\cos{\\left(3 x + 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 6 e^{x}) = - 6 e^{x}$.\nTherefore the correct derivative is $f'(x) = - 9 x \\left(x \\left(x - 1\\right) + 1\\right) \\sin{\\left(3 x + 2 \\right)} + 2 \\left(x - 3\\right) e^{x} + 3 \\left(3 x^{2} - 2 x + 1\\right) \\cos{\\left(3 x + 2 \\right)} + 2 e^{x}$.\nFinal answer: $- 9 x \\left(x \\left(x - 1\\right) + 1\\right) \\sin{\\left(3 x + 2 \\right)} + 2 \\left(x - 3\\right) e^{x} + 3 \\left(3 x^{2} - 2 x + 1\\right) \\cos{\\left(3 x + 2 \\right)} + 2 e^{x}$", "final_answer": "- 9 x \\left(x \\left(x - 1\\right) + 1\\right) \\sin{\\left(3 x + 2 \\right)} + 2 \\left(x - 3\\right) e^{x} + 3 \\left(3 x^{2} - 2 x + 1\\right) \\cos{\\left(3 x + 2 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "23a78f6222128748", "problem_id": "9c1fdb32a2eba309", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 11 red balls and 10 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(21,9).\nFavorable samples: choose k reds and n−k blues: C(11,4)·C(10,5).\nCompute totals: C(21,9) = 293930, and favorable = 83160.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1188}{4199}$.\nFinal answer: $\\frac{1188}{4199}$", "final_answer": "\\frac{1188}{4199}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "00c3982043fd0664", "problem_id": "9c1fdb32a2eba309", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 11 red balls and 10 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(21,9).\n3) Favorable samples: choose k reds and n−k blues: C(11,4)·C(10,5).\n4) Compute totals: C(21,9) = 293930, and favorable = 83160.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1188}{4199}$", "final_answer": "\\frac{1188}{4199}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "da43fe264133d8d2", "problem_id": "53a2a5610a87ab23", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{10} x + \\frac{29 x}{5} - \\frac{12 \\sqrt{10}}{5} + 4$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{29}{5} - 3 \\sqrt{10}$, c=$4 - \\frac{12 \\sqrt{10}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{2691}{25} - \\frac{126 \\sqrt{10}}{5}$, so √Δ = $\\sqrt{\\frac{2691}{25} - \\frac{126 \\sqrt{10}}{5}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{29}{10} + \\frac{3 \\sqrt{299 - 70 \\sqrt{10}}}{10} + \\frac{3 \\sqrt{10}}{2}$.\n7) x2 = $- \\frac{29}{10} - \\frac{3 \\sqrt{299 - 70 \\sqrt{10}}}{10} + \\frac{3 \\sqrt{10}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{29}{10} + \\frac{3\\cdot \\sqrt{299 - 70\\cdot \\sqrt{10}}}{10} + \\frac{3\\cdot \\sqrt{10}}{2}, - \\frac{29}{10} - \\frac{3\\cdot \\sqrt{299 - 70\\cdot \\sqrt{10}}}{10} + \\frac{3\\cdot \\sqrt{10}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{29}{10} + \\frac{3\\cdot \\sqrt{299 - 70\\cdot \\sqrt{10}}}{10} + \\frac{3\\cdot \\sqrt{10}}{2}, - \\frac{29}{10} - \\frac{3\\cdot \\sqrt{299 - 70\\cdot \\sqrt{10}}}{10} + \\frac{3\\cdot \\sqrt{10}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "75a441cf84363da8", "problem_id": "53a2a5610a87ab23", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{10} x + \\frac{29 x}{5} - \\frac{12 \\sqrt{10}}{5} + 4$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{29}{5} - 3 \\sqrt{10}$, c=$4 - \\frac{12 \\sqrt{10}}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{- \\frac{29}{10} + \\frac{3\\cdot \\sqrt{299 - 70\\cdot \\sqrt{10}}}{10} + \\frac{3\\cdot \\sqrt{10}}{2}, - \\frac{29}{10} - \\frac{3\\cdot \\sqrt{299 - 70\\cdot \\sqrt{10}}}{10} + \\frac{3\\cdot \\sqrt{10}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{29}{10} + \\frac{3\\cdot \\sqrt{299 - 70\\cdot \\sqrt{10}}}{10} + \\frac{3\\cdot \\sqrt{10}}{2}, - \\frac{29}{10} - \\frac{3\\cdot \\sqrt{299 - 70\\cdot \\sqrt{10}}}{10} + \\frac{3\\cdot \\sqrt{10}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "944f5c34068508de", "problem_id": "e660d1583dc731ed", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A store sells a backpack for $790. It is discounted by 20%. You buy 13 backpacks. What is the total cost?", "output": "1) Discounted price = 790 × (1 − 20/100) = 790 × 80/100 = 632.\n2) Subtotal = 13 × 632 = 8216.\nFinal answer: $8216$", "final_answer": "8216", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f96bd6e8e7bd3757", "problem_id": "e660d1583dc731ed", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $790. It is discounted by 20%. You buy 13 backpacks. What is the total cost?", "output": "Discounted price = 790 × (1 − 20/100) = 790 × 80/100 = 632.\nSubtotal = 13 × 632 = 8216.\nCheck: recomputing the formula yields 8216.\nFinal answer: $8216$", "final_answer": "8216", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b029b9d806dc08bb", "problem_id": "f9dcf7da32fa90f6", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "You want 36 liters of a 30% salt solution. You have a 15% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 36−x liters is 40% solution.\n2) Salt equation: 15x + 40(36−x) = 30·36.\n3) (15-40)x = (30-40)·36 ⇒ x = 72/5.\n4) Then 36−x = 108/5.\nFinal answer: $72/5 L of 15% and 108/5 L of 40%$", "final_answer": "72/5 L of 15% and 108/5 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e02d489d96ddcde6", "problem_id": "f9dcf7da32fa90f6", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "You want 36 liters of a 30% salt solution. You have a 15% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 36−x liters is 40% solution.\nSalt equation: 15x + 40(36−x) = 30·36.\n(15-40)x = (30-40)·36 ⇒ x = 72/5.\nThen 36−x = 108/5.\nCheck: 15·72/5 + 40·108/5 = 30·36.\nFinal answer: $72/5 L of 15% and 108/5 L of 40%$", "final_answer": "72/5 L of 15% and 108/5 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "828f26e93c576811", "problem_id": "04bd3777b43fc71e", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the multiplicative inverse of 25 modulo 43.", "output": "1) We want x such that 25x ≡ 1 (mod 43).\n2) Use extended Euclid to find s,t with 25s + 43t = 1.\n3) One Bezout identity is: 25·(-12) + 43·(7) = 1.\n4) Reduce s modulo 43: x ≡ 31 (mod 43).\nFinal answer: $x\\equiv 31\\pmod{43}$", "final_answer": "x\\equiv 31\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "86aad4daf83d3f86", "problem_id": "04bd3777b43fc71e", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Find the multiplicative inverse of 25 modulo 43.", "output": "We want x such that 25x ≡ 1 (mod 43).\nUse extended Euclid to find s,t with 25s + 43t = 1.\nOne Bezout identity is: 25·(-12) + 43·(7) = 1.\nFinal answer: $x\\equiv 31\\pmod{43}$", "final_answer": "x\\equiv 31\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d2a006bbef6d95bf", "problem_id": "ba04bd6423854d8d", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_9=43$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=43 and k=9.\n4) Compute C(51,8) = 636763050.\nFinal answer: $636763050$", "final_answer": "636763050", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "56c68874ff85a7c6", "problem_id": "ba04bd6423854d8d", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_9=43$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=43 and k=9.\nCompute C(51,8) = 636763050.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $636763050$", "final_answer": "636763050", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7db4e4a6e7f3f675", "problem_id": "f2d39b6479b1229c", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Two friends split $432 in the ratio 7:5. How much does each receive?", "output": "1) Total ratio parts = 7+5 = 12.\n2) Each part = 432 ÷ 12 = 36.\n3) First share = 7 × 36 = 252.\n4) Second share = 5 × 36 = 180.\nFinal answer: $180$", "final_answer": "180", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b7057184e60273b5", "problem_id": "4de46a39da1882c5", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(- 2 x^{3} - 3\\right) \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 2 x^{3} \\cos{\\left(2 x \\right)}) = 2 x^{3} \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = 2 x^{3} \\sin{\\left(2 x \\right)} + x e^{x} + 3 e^{x} + 6 \\sin{\\left(2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 2 x^{3} \\cos{\\left(2 x \\right)}) = 4 x^{3} \\sin{\\left(2 x \\right)} - 6 x^{2} \\cos{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 6 x^{2} \\cos{\\left(2 x \\right)} + \\left(x + 2\\right) e^{x} - 2 \\left(- 2 x^{3} - 3\\right) \\sin{\\left(2 x \\right)} + e^{x}$.\nFinal answer: $- 6 x^{2} \\cos{\\left(2 x \\right)} + \\left(x + 2\\right) e^{x} - 2 \\left(- 2 x^{3} - 3\\right) \\sin{\\left(2 x \\right)} + e^{x}$", "final_answer": "- 6 x^{2} \\cos{\\left(2 x \\right)} + \\left(x + 2\\right) e^{x} - 2 \\left(- 2 x^{3} - 3\\right) \\sin{\\left(2 x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5749ba68121865c1", "problem_id": "f6ce0abaae05971b", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$x - 3 y$ = $-3$\n$x + 4 y$ = $11$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $x - 3 y$ = $-3$.\nMultiply the second equation by $1$: $x + 4 y$ = $11$.\nSubtract to eliminate x: $- 7 y$ = $-14$.\nSolve for y: y = $2$.\nSubstitute back to find x: x = $3$.\nCheck: plug (x,y)=(3,2) into both equations.\nFinal answer: $(x,y)=\\left(3, 2\\right)$", "final_answer": "(x,y)=\\left(3, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a40640d523825cf2", "problem_id": "f6ce0abaae05971b", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$x - 3 y$ = $-3$\n$x + 4 y$ = $11$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $x - 3 y$ = $-3$.\n3) Multiply the second equation by $1$: $x + 4 y$ = $11$.\n4) Subtract to eliminate x: $- 7 y$ = $-14$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, 2\\right)$", "final_answer": "(x,y)=\\left(3, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3a80101e731fcbac", "problem_id": "d425f2e063466925", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 2 x^{3} - 3 x^{2} + 8 x - 23$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} - 3 x^{2} + 7 x - 23.\n3) Then the limit is -125.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -128.\nFinal answer: $-128$", "final_answer": "-128", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "98f9cf18dde91b8a", "problem_id": "ac75772f6f7cde79", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Two friends split $100 in the ratio 7:3. How much does each receive?", "output": "Total ratio parts = 7+3 = 10.\nEach part = 100 ÷ 10 = 10.\nFirst share = 7 × 10 = 70.\nSecond share = 3 × 10 = 30.\nCheck: 70+30=100.\nFinal answer: $30$", "final_answer": "30", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cbe334f3c5312501", "problem_id": "ce0ad96fcc9eded7", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "You want 60 liters of a 35% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 60−x liters is 70% solution.\n2) Salt equation: 25x + 70(60−x) = 35·60.\n3) (25-70)x = (35-70)·60 ⇒ x = 140/3.\n4) Then 60−x = 40/3.\nFinal answer: $140/3 L of 25% and 40/3 L of 70%$", "final_answer": "140/3 L of 25% and 40/3 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f93909e06753109d", "problem_id": "ce0ad96fcc9eded7", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "You want 60 liters of a 35% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "140/3 L of 25% and 40/3 L of 70%", "final_answer": "140/3 L of 25% and 40/3 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0023095c1341ce28", "problem_id": "c8c89f57372d5fb0", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} 4 x^{4} - x^{3} - 4 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $4 x^{4} - x^{3} - 4 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\n2) Evaluate at the bounds: F(\\pi) and F(0).\n3) Subtract: F(\\pi) − F(0) = $\\frac{\\pi \\left(- 80 \\pi^{2} - 15 \\pi^{3} - 60 \\pi + 60 + 48 \\pi^{4}\\right)}{60}$.\n4) Simplify the result.\nFinal answer: $\\frac{\\pi\\cdot \\left(- 80\\cdot \\pi^{2} - 15\\cdot \\pi^{3} - 60\\cdot \\pi + 60 + 48\\cdot \\pi^{4}\\right)}{60}$", "final_answer": "\\frac{\\pi\\cdot \\left(- 80\\cdot \\pi^{2} - 15\\cdot \\pi^{3} - 60\\cdot \\pi + 60 + 48\\cdot \\pi^{4}\\right)}{60}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2bea2fb71b1233d1", "problem_id": "c8c89f57372d5fb0", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} 4 x^{4} - x^{3} - 4 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $4 x^{4} - x^{3} - 4 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\nEvaluate at the bounds: F(\\pi) and F(0).\nSubtract: F(\\pi) − F(0) = $\\frac{\\pi \\left(- 80 \\pi^{2} - 15 \\pi^{3} - 60 \\pi + 60 + 48 \\pi^{4}\\right)}{60}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{\\pi\\cdot \\left(- 80\\cdot \\pi^{2} - 15\\cdot \\pi^{3} - 60\\cdot \\pi + 60 + 48\\cdot \\pi^{4}\\right)}{60}$", "final_answer": "\\frac{\\pi\\cdot \\left(- 80\\cdot \\pi^{2} - 15\\cdot \\pi^{3} - 60\\cdot \\pi + 60 + 48\\cdot \\pi^{4}\\right)}{60}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9a9387482fc5bbca", "problem_id": "61d0e79a7ae8f5ea", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Two friends split $112 in the ratio 6:1. How much does each receive?", "output": "1) Total ratio parts = 6+1 = 7.\n2) Each part = 112 ÷ 7 = 16.\n3) First share = 6 × 16 = 96.\n4) Second share = 1 × 16 = 16.\nFinal answer: $16$", "final_answer": "16", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f62e23f77f605c70", "problem_id": "61d0e79a7ae8f5ea", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Two friends split $112 in the ratio 6:1. How much does each receive?", "output": "$16$", "final_answer": "16", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "089071e2679d7df7", "problem_id": "a8e9db1eb212ba4f", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 21 times, what is the probability of exactly 10 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=21, k=10, p=$\\frac{2}{5}$ and simplify.\n4) Here C(21,10) = 352716.\nFinal answer: $\\frac{63982163202048}{476837158203125}$", "final_answer": "\\frac{63982163202048}{476837158203125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b3e7a03b93e5ad09", "problem_id": "a8e9db1eb212ba4f", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 21 times, what is the probability of exactly 10 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=21, k=10, p=$\\frac{2}{5}$ and simplify.\nHere C(21,10) = 352716.\nCheck: expression simplifies to $\\frac{63982163202048}{476837158203125}$.\nFinal answer: $\\frac{63982163202048}{476837158203125}$", "final_answer": "\\frac{63982163202048}{476837158203125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "10e35b4bf43a8ff3", "problem_id": "e3c8360006f99e77", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "An urn has 15 red balls and 16 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(31,5).\n3) Favorable samples: choose k reds and n−k blues: C(15,1)·C(16,4).\n4) Compute totals: C(31,5) = 169911, and favorable = 27300.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1300}{8091}$", "final_answer": "\\frac{1300}{8091}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "53fe3bb8b4306aee", "problem_id": "e3c8360006f99e77", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 15 red balls and 16 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(31,5).\nFavorable samples: choose k reds and n−k blues: C(15,1)·C(16,4).\nCompute totals: C(31,5) = 169911, and favorable = 27300.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1300}{8091}$.\nFinal answer: $\\frac{1300}{8091}$", "final_answer": "\\frac{1300}{8091}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "00337820f2734243", "problem_id": "e1aab988c701c594", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Find the multiplicative inverse of 3 modulo 23.", "output": "1) We want x such that 3x ≡ 1 (mod 23).\n2) Use extended Euclid to find s,t with 3s + 23t = 1.\n3) One Bezout identity is: 3·(8) + 23·(-1) = 1.\n4) Reduce s modulo 23: x ≡ 8 (mod 23).\nFinal answer: $x\\equiv 8\\pmod{23}$", "final_answer": "x\\equiv 8\\pmod{23}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f57f5952d9fce4fa", "problem_id": "c0fbe97de884fdee", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} + 4 x + 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x^{3}) = 3 x^{3}$.\n3) So $f'(x) = 3 x^{3} + 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x^{3}) = 3 x^{2}$.\nTherefore the correct derivative is $f'(x) = 3 x^{2} + 4$.\nFinal answer: $3 x^{2} + 4$", "final_answer": "3 x^{2} + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "414e61a73b10f5cb", "problem_id": "b19688c68c9016d1", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A store sells a backpack for $235. It is discounted by 30%. You buy 5 backpacks. What is the total cost?", "output": "1) Discounted price = 235 × (1 − 30/100) = 235 × 70/100 = 329/2.\n2) Subtotal = 5 × 329/2 = 1645/2.\nFinal answer: $\\frac{1645}{2}$", "final_answer": "\\frac{1645}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "167b636b2cb3581b", "problem_id": "b19688c68c9016d1", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "A store sells a backpack for $235. It is discounted by 30%. You buy 5 backpacks. What is the total cost?", "output": "Discounted price = 235 × (1 − 30/100) = 235 × 70/100 = 329/2.\nSubtotal = 5 × 329/2 = 1645/2.\nFinal answer: $\\frac{1645}{2}$", "final_answer": "\\frac{1645}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "018ff0f41cea3fe9", "problem_id": "9142f68d8ed3bcd5", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A cyclist rides at 39 miles per hour for 3/2 hours, then at 66 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 39 × 3/2 = 117/2 miles.\nSecond leg distance = 66 × 3 = 198 miles.\nTotal distance = 117/2 + 198 = 513/2 miles.\nCheck: adding both legs gives 513/2.\nFinal answer: $\\frac{513}{2}$", "final_answer": "\\frac{513}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cb9ad1bf90934d87", "problem_id": "9142f68d8ed3bcd5", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A cyclist rides at 39 miles per hour for 3/2 hours, then at 66 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 39 × 3/2 = 117/2 miles.\n2) Second leg distance = 66 × 3 = 198 miles.\n3) Total distance = 117/2 + 198 = 513/2 miles.\nFinal answer: $\\frac{513}{2}$", "final_answer": "\\frac{513}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "254885fe6ffb2dc1", "problem_id": "6ccf58285436a907", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 3$.\n3) Compute an antiderivative: F(x) = $\\frac{x^{3}}{3} + \\frac{3 x^{2}}{2} + 3 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} + 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} + 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "45f25f81c7f28ac5", "problem_id": "6ccf58285436a907", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} + 3$.\nCompute an antiderivative: F(x) = $\\frac{x^{3}}{3} + \\frac{3 x^{2}}{2} + 3 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $C + \\frac{x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} + 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} + 3\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "56b2e33efd5fd8f9", "problem_id": "4273006f371dbbcc", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 \\sqrt{5} x - \\frac{7 x}{4} - \\frac{27 \\sqrt{5}}{4} - 9$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- 3 \\sqrt{5} - \\frac{7}{4}$, c=$- \\frac{27 \\sqrt{5}}{4} - 9$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{75 \\sqrt{5}}{2} + \\frac{1345}{16}$, so √Δ = $\\sqrt{\\frac{75 \\sqrt{5}}{2} + \\frac{1345}{16}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{7}{8} + \\frac{3 \\sqrt{5}}{2} + \\frac{\\sqrt{600 \\sqrt{5} + 1345}}{8}$.\n7) x2 = $- \\frac{\\sqrt{600 \\sqrt{5} + 1345}}{8} + \\frac{7}{8} + \\frac{3 \\sqrt{5}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{7}{8} + \\frac{3\\cdot \\sqrt{5}}{2} + \\frac{\\sqrt{600\\cdot \\sqrt{5} + 1345}}{8}, - \\frac{\\sqrt{600\\cdot \\sqrt{5} + 1345}}{8} + \\frac{7}{8} + \\frac{3\\cdot \\sqrt{5}}{2}\\}$", "final_answer": "x \\in \\{\\frac{7}{8} + \\frac{3\\cdot \\sqrt{5}}{2} + \\frac{\\sqrt{600\\cdot \\sqrt{5} + 1345}}{8}, - \\frac{\\sqrt{600\\cdot \\sqrt{5} + 1345}}{8} + \\frac{7}{8} + \\frac{3\\cdot \\sqrt{5}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8141a159c8e575b9", "problem_id": "fce51105bf7cf726", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{3} \\sin{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 x^{3} \\sin{\\left(x \\right)}) = 3 x^{3} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = 3 x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 x^{3} \\sin{\\left(x \\right)}) = 3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$", "final_answer": "3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a68fc1353b132046", "problem_id": "8764d400a6d9db13", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} 3 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $-2 + \\frac{3 \\pi^{2}}{8} + \\frac{3 \\pi}{2} + \\frac{9 \\pi^{3}}{8}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $-2 + \\frac{3\\cdot \\pi^{2}}{8} + \\frac{3\\cdot \\pi}{2} + \\frac{9\\cdot \\pi^{3}}{8}$", "final_answer": "-2 + \\frac{3\\cdot \\pi^{2}}{8} + \\frac{3\\cdot \\pi}{2} + \\frac{9\\cdot \\pi^{3}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "886bc8597c4243bf", "problem_id": "8764d400a6d9db13", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} 3 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 1$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $-2 + \\frac{3 \\pi^{2}}{8} + \\frac{3 \\pi}{2} + \\frac{9 \\pi^{3}}{8}$.\n4) Simplify the result.\nFinal answer: $-2 + \\frac{3\\cdot \\pi^{2}}{8} + \\frac{3\\cdot \\pi}{2} + \\frac{9\\cdot \\pi^{3}}{8}$", "final_answer": "-2 + \\frac{3\\cdot \\pi^{2}}{8} + \\frac{3\\cdot \\pi}{2} + \\frac{9\\cdot \\pi^{3}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cd58c0541dfae9d7", "problem_id": "bc23fbf179aa2b37", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡0 (mod 3), x≡3 (mod 5). Give the solution modulo 15.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 15.\n3) Compute the CRT solution to get x ≡ 3 (mod 15).\nFinal answer: $x\\equiv 3\\pmod{15}$", "final_answer": "x\\equiv 3\\pmod{15}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "abfee99bc0e61d96", "problem_id": "bc23fbf179aa2b37", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡0 (mod 3), x≡3 (mod 5). Give the solution modulo 15.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 15.\nCompute the CRT solution to get x ≡ 3 (mod 15).\nCheck: 3 mod 3=0 and mod 5=3.\nFinal answer: $x\\equiv 3\\pmod{15}$", "final_answer": "x\\equiv 3\\pmod{15}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a8f4f3d5513e29a", "problem_id": "cd624df101551d61", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} + x^{2} + x + 2 \\sin{\\left(x \\right)}$ at $x=-2$.", "output": "Differentiate: f'(x) = $6 x^{2} + 2 x + 2 \\cos{\\left(x \\right)} + 1$.\nSlope at x=-2: m = f'(-2) = $2 \\cos{\\left(2 \\right)} + 21$.\nPoint on curve: (-2, f(-2)) = (-2, $-14 - 2 \\sin{\\left(2 \\right)}$).\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 21\\cdot x - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)} + 28$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 21\\cdot x - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)} + 28", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2267cd4f285e8edd", "problem_id": "cd624df101551d61", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} + x^{2} + x + 2 \\sin{\\left(x \\right)}$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $6 x^{2} + 2 x + 2 \\cos{\\left(x \\right)} + 1$.\n2) Slope at x=-2: m = f'(-2) = $2 \\cos{\\left(2 \\right)} + 21$.\n3) Point on curve: (-2, f(-2)) = (-2, $-14 - 2 \\sin{\\left(2 \\right)}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -14 - 2 \\sin{\\left(2 \\right)} = 2 \\cos{\\left(2 \\right)} + 21(x − -2).\n6) Optionally expand: y = $2 x \\cos{\\left(2 \\right)} + 21 x - 2 \\sin{\\left(2 \\right)} + 4 \\cos{\\left(2 \\right)} + 28$.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 21\\cdot x - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)} + 28$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 21\\cdot x - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)} + 28", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "75b3661e041fec12", "problem_id": "a8d8abd1efa0269b", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many binary strings of length 25 contain exactly 3 ones with no two ones adjacent?", "output": "1) Reserve 2 mandatory zeros between the 3 ones.\n2) This leaves n−k+1 = 23 valid slots for the ones.\n3) Choose the slots: C(23,3) = 1771.\nFinal answer: $1771$", "final_answer": "1771", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a4e9429f1e850eb", "problem_id": "a8d8abd1efa0269b", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "How many binary strings of length 25 contain exactly 3 ones with no two ones adjacent?", "output": "Reserve 2 mandatory zeros between the 3 ones.\nThis leaves n−k+1 = 23 valid slots for the ones.\nChoose the slots: C(23,3) = 1771.\nFinal answer: $1771$", "final_answer": "1771", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d31982816dc39ead", "problem_id": "f175f8884f0e5adc", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} x^{3} - 2 x^{2} + 6 x - 24$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{4} + 2 x^{3} - 2 x^{2} + 3$ and a = -4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $4 x^{3} + 6 x^{2} - 4 x$.\n5) Evaluate at a: p'(-4) = $-144$.\nFinal answer: $-144$", "final_answer": "-144", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "45f5e0ac83db0d46", "problem_id": "aaa146a38fe8ea5e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{2} + 2 x - 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(4 x^{2}) = 8 x^{2}$.\n3) So $f'(x) = 8 x^{2} + 2$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(4 x^{2}) = 8 x$.\nTherefore the correct derivative is $f'(x) = 2 \\left(4 x + 1\\right)$.\nFinal answer: $2 \\left(4 x + 1\\right)$", "final_answer": "2 \\left(4 x + 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6bf46452444b9841", "problem_id": "8892f8e39d81465c", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} + 2 x^{2} - 4 x + 1$.", "output": "1) Differentiate term-by-term:\n2) d/dx(1) = 0.\n3) d/dx(- 4 x) = -4.\n4) d/dx(- 2 x^{3}) = - 6 x^{2}.\n5) d/dx(2 x^{2}) = 4 x.\n6) Combine: f'(x) = 2 \\left(- 3 x^{2} + 2 x - 2\\right).\nFinal answer: $- 6\\cdot x^{2} + 4\\cdot x - 4$", "final_answer": "- 6\\cdot x^{2} + 4\\cdot x - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fd8818b275f0fff2", "problem_id": "8892f8e39d81465c", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} + 2 x^{2} - 4 x + 1$.", "output": "Differentiate term-by-term:\nd/dx(1) = 0.\nd/dx(- 4 x) = -4.\nd/dx(- 2 x^{3}) = - 6 x^{2}.\nd/dx(2 x^{2}) = 4 x.\nCombine: f'(x) = 2 \\left(- 3 x^{2} + 2 x - 2\\right).\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 6\\cdot x^{2} + 4\\cdot x - 4$", "final_answer": "- 6\\cdot x^{2} + 4\\cdot x - 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b56c8b74efc0a4f2", "problem_id": "1cff8c98aaaecdef", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡6 (mod 11), x≡3 (mod 9). Give the solution modulo 99.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 99.\n3) Compute the CRT solution to get x ≡ 39 (mod 99).\nFinal answer: $x\\equiv 39\\pmod{99}$", "final_answer": "x\\equiv 39\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0671983db05d4745", "problem_id": "1cff8c98aaaecdef", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡6 (mod 11), x≡3 (mod 9). Give the solution modulo 99.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 99.\nCompute the CRT solution to get x ≡ 39 (mod 99).\nCheck: 39 mod 11=6 and mod 9=3.\nFinal answer: $x\\equiv 39\\pmod{99}$", "final_answer": "x\\equiv 39\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2e55c599283da6ae", "problem_id": "2bd4fe86b18f8af4", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the linear system:\n$x + 4 y$ = $-1$\n$- 4 x + 3 y$ = $4$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $- 4 x - 16 y$ = $4$.\nMultiply the second equation by $1$: $- 4 x + 3 y$ = $4$.\nFinal answer: $(x,y)=\\left(-1, 0\\right)$", "final_answer": "(x,y)=\\left(-1, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "49e59bf4ad7bc445", "problem_id": "2bd4fe86b18f8af4", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$x + 4 y$ = $-1$\n$- 4 x + 3 y$ = $4$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $- 4 x - 16 y$ = $4$.\nMultiply the second equation by $1$: $- 4 x + 3 y$ = $4$.\nSubtract to eliminate x: $- 19 y$ = $0$.\nSolve for y: y = $0$.\nSubstitute back to find x: x = $-1$.\nCheck: plug (x,y)=(-1,0) into both equations.\nFinal answer: $(x,y)=\\left(-1, 0\\right)$", "final_answer": "(x,y)=\\left(-1, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0f4e4a19af80fb21", "problem_id": "4a6e18f86adedd24", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{3 x}{2} - \\frac{7}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$\\frac{3}{2}$, c=$- \\frac{7}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{121}{4}$, so √Δ = $\\frac{11}{2}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $1$.\n7) x2 = $- \\frac{7}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{1, - \\frac{7}{4}\\}$", "final_answer": "x \\in \\{1, - \\frac{7}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d260e17af75d8a14", "problem_id": "4a6e18f86adedd24", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{3 x}{2} - \\frac{7}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$\\frac{3}{2}$, c=$- \\frac{7}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{1, - \\frac{7}{4}\\}$", "final_answer": "x \\in \\{1, - \\frac{7}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5dcb8cf3250d11ea", "problem_id": "1b0b18b7722fe111", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡0 (mod 4), x≡7 (mod 11). Give the solution modulo 44.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 44.\n3) Compute the CRT solution to get x ≡ 40 (mod 44).\nFinal answer: $x\\equiv 40\\pmod{44}$", "final_answer": "x\\equiv 40\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "00278099c79167a5", "problem_id": "1b0b18b7722fe111", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡0 (mod 4), x≡7 (mod 11). Give the solution modulo 44.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 44.\nCompute the CRT solution to get x ≡ 40 (mod 44).\nCheck: 40 mod 4=0 and mod 11=7.\nFinal answer: $x\\equiv 40\\pmod{44}$", "final_answer": "x\\equiv 40\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9024edddf13f87ae", "problem_id": "c119b76b10942817", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 72 miles per hour for 1 hours, then at 34 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 72 × 1 = 72 miles.\nSecond leg distance = 34 × 2 = 68 miles.\nTotal distance = 72 + 68 = 140 miles.\nCheck: adding both legs gives 140.\nFinal answer: $140$", "final_answer": "140", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "06dc234e53625d94", "problem_id": "c119b76b10942817", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A cyclist rides at 72 miles per hour for 1 hours, then at 34 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 72 × 1 = 72 miles.\n2) Second leg distance = 34 × 2 = 68 miles.\n3) Total distance = 72 + 68 = 140 miles.\nFinal answer: $140$", "final_answer": "140", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "207ec421d8cc8e1a", "problem_id": "b2874b8922c3d6a1", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Two friends split $660 in the ratio 7:5. How much does each receive?", "output": "Total ratio parts = 7+5 = 12.\nEach part = 660 ÷ 12 = 55.\nFirst share = 7 × 55 = 385.\nFinal answer: $275$", "final_answer": "275", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e54d39b58e06cc93", "problem_id": "b2874b8922c3d6a1", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $660 in the ratio 7:5. How much does each receive?", "output": "Total ratio parts = 7+5 = 12.\nEach part = 660 ÷ 12 = 55.\nFirst share = 7 × 55 = 385.\nSecond share = 5 × 55 = 275.\nCheck: 385+275=660.\nFinal answer: $275$", "final_answer": "275", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f732c4d198117d66", "problem_id": "e6e0397376d11b49", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 20 times, what is the probability of exactly 11 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=20, k=11, p=$\\frac{2}{5}$ and simplify.\n4) Here C(20,11) = 167960.\nFinal answer: $\\frac{1354119856128}{19073486328125}$", "final_answer": "\\frac{1354119856128}{19073486328125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2b2cd3150e1f4d8f", "problem_id": "e6e0397376d11b49", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 20 times, what is the probability of exactly 11 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=20, k=11, p=$\\frac{2}{5}$ and simplify.\nHere C(20,11) = 167960.\nCheck: expression simplifies to $\\frac{1354119856128}{19073486328125}$.\nFinal answer: $\\frac{1354119856128}{19073486328125}$", "final_answer": "\\frac{1354119856128}{19073486328125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cb62bab9a51c6ea6", "problem_id": "e2ca3e9ebeba9ab8", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡0 (mod 3), x≡5 (mod 11). Give the solution modulo 33.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 33.\n3) Compute the CRT solution to get x ≡ 27 (mod 33).\nFinal answer: $x\\equiv 27\\pmod{33}$", "final_answer": "x\\equiv 27\\pmod{33}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a12bb2e205c30afb", "problem_id": "e2ca3e9ebeba9ab8", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡0 (mod 3), x≡5 (mod 11). Give the solution modulo 33.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 33.\nCompute the CRT solution to get x ≡ 27 (mod 33).\nCheck: 27 mod 3=0 and mod 11=5.\nFinal answer: $x\\equiv 27\\pmod{33}$", "final_answer": "x\\equiv 27\\pmod{33}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bb9b221805513bda", "problem_id": "9f2875bdc893df03", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$- 3 x + 6 y$ = $0$\n$7 x - y$ = $-39$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $7$: $- 21 x + 42 y$ = $0$.\nMultiply the second equation by $-3$: $- 21 x + 3 y$ = $117$.\nSubtract to eliminate x: $39 y$ = $-117$.\nSolve for y: y = $-3$.\nSubstitute back to find x: x = $-6$.\nCheck: plug (x,y)=(-6,-3) into both equations.\nFinal answer: $(x,y)=\\left(-6, -3\\right)$", "final_answer": "(x,y)=\\left(-6, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6fca112dd07514c3", "problem_id": "9f2875bdc893df03", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$- 3 x + 6 y$ = $0$\n$7 x - y$ = $-39$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $7$: $- 21 x + 42 y$ = $0$.\n3) Multiply the second equation by $-3$: $- 21 x + 3 y$ = $117$.\n4) Subtract to eliminate x: $39 y$ = $-117$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $-6$.\nFinal answer: $(x,y)=\\left(-6, -3\\right)$", "final_answer": "(x,y)=\\left(-6, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0447a0fb45e08862", "problem_id": "ff43d83db520542d", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 16 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=16, k=4, p=$\\frac{1}{2}$ and simplify.\n4) Here C(16,4) = 1820.\nFinal answer: $\\frac{455}{16384}$", "final_answer": "\\frac{455}{16384}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3d890bf90e27672e", "problem_id": "ff43d83db520542d", "style": "answer_only", "instruction": "Final answer only.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 16 times, what is the probability of exactly 4 heads?", "output": "$\\frac{455}{16384}$", "final_answer": "\\frac{455}{16384}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ea8a1ab1aac78d5", "problem_id": "43b5c70114b1b9b4", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A disease has prevalence $\\frac{1}{100}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{191}{5000}$.\nCompute P(D|+) = $\\frac{85}{382}$.\nCheck: value is between 0 and 1 and increases with prevalence/sensitivity.\nFinal answer: $\\frac{85}{382}$", "final_answer": "\\frac{85}{382}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b5b6faad1e6a5fe0", "problem_id": "43b5c70114b1b9b4", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A disease has prevalence $\\frac{1}{100}$. A test has sensitivity $\\frac{17}{20}$ and specificity $\\frac{97}{100}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{191}{5000}$.\n3) Compute P(D|+) = $\\frac{85}{382}$.\nFinal answer: $\\frac{85}{382}$", "final_answer": "\\frac{85}{382}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b0ef311738573f2b", "problem_id": "c0733be60b977022", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} 2 x^{2} + 5 x + 19$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{3} - 3 x^{2} - x + 1$ and a = 4.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $6 x^{2} - 6 x - 1$.\nEvaluate at a: p'(4) = $71$.\nCheck: the identity gives $71$.\nFinal answer: $71$", "final_answer": "71", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1337923ddc5baf0a", "problem_id": "c0733be60b977022", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} 2 x^{2} + 5 x + 19$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $2 x^{3} - 3 x^{2} - x + 1$ and a = 4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $6 x^{2} - 6 x - 1$.\n5) Evaluate at a: p'(4) = $71$.\nFinal answer: $71$", "final_answer": "71", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9f7bf041c709ca72", "problem_id": "ca59cb48a9bf2e3a", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A store sells a backpack for $375. It is discounted by 30%. You buy 5 backpacks. After the discount, a sales tax of 10% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 375 × (1 − 30/100) = 375 × 70/100 = 525/2.\n2) Subtotal = 5 × 525/2 = 2625/2.\n3) Total with tax = 2625/2 × 110/100 = 5775/4.\nFinal answer: $\\frac{5775}{4}$", "final_answer": "\\frac{5775}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "155617e06777d376", "problem_id": "61597971a8e26afc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - x^{2} - 2 x - 3$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} - 3 x - 3.\n3) Then the limit is -21.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: -18.\nFinal answer: $-18$", "final_answer": "-18", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "484b441a62ce424f", "problem_id": "874c3243e712351a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡2 (mod 3), x≡0 (mod 5). Give the solution modulo 15.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 15.\nCompute the CRT solution to get x ≡ 5 (mod 15).\nCheck: 5 mod 3=2 and mod 5=0.\nFinal answer: $x\\equiv 5\\pmod{15}$", "final_answer": "x\\equiv 5\\pmod{15}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b63ee3d3edb31108", "problem_id": "874c3243e712351a", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡2 (mod 3), x≡0 (mod 5). Give the solution modulo 15.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 15.\n3) Compute the CRT solution to get x ≡ 5 (mod 15).\nFinal answer: $x\\equiv 5\\pmod{15}$", "final_answer": "x\\equiv 5\\pmod{15}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ba7a8a0c54a1079e", "problem_id": "c6955659fd449324", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{10}$. If it is flipped 19 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=19, k=2, p=$\\frac{3}{10}$ and simplify.\n4) Here C(19,2) = 171.\nFinal answer: $\\frac{358018361026311573}{10000000000000000000}$", "final_answer": "\\frac{358018361026311573}{10000000000000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aa268c96b8c8b398", "problem_id": "c6955659fd449324", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{10}$. If it is flipped 19 times, what is the probability of exactly 2 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=19, k=2, p=$\\frac{3}{10}$ and simplify.\nHere C(19,2) = 171.\nCheck: expression simplifies to $\\frac{358018361026311573}{10000000000000000000}$.\nFinal answer: $\\frac{358018361026311573}{10000000000000000000}$", "final_answer": "\\frac{358018361026311573}{10000000000000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f146fad0f716c76f", "problem_id": "743bb328f0e89900", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - 2 x^{2} + 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 3$ at $x=2$.", "output": "Differentiate: f'(x) = $- 9 x^{2} - 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 2$.\nSlope at x=2: m = f'(2) = $-42 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\nPoint on curve: (2, f(2)) = (2, $-25 + \\cos{\\left(4 \\right)} + e$).\nFinal answer: $y = - 42\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 59$", "final_answer": "y = - 42\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 59", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f9ba8b7d94f1c03a", "problem_id": "743bb328f0e89900", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{3} - 2 x^{2} + 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 3$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 9 x^{2} - 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 2$.\n2) Slope at x=2: m = f'(2) = $-42 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\n3) Point on curve: (2, f(2)) = (2, $-25 + \\cos{\\left(4 \\right)} + e$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -25 + \\cos{\\left(4 \\right)} + e = -42 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\n6) Optionally expand: y = $- 42 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 59$.\nFinal answer: $y = - 42\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 59$", "final_answer": "y = - 42\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 59", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "967df8016604f65e", "problem_id": "8488057c8841e322", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A cyclist rides at 37 miles per hour for 1 hours, then at 62 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 37 × 1 = 37 miles.\nSecond leg distance = 62 × 1 = 62 miles.\nTotal distance = 37 + 62 = 99 miles.\nCheck: adding both legs gives 99.\nFinal answer: $99$", "final_answer": "99", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aacab8abfdda467f", "problem_id": "8488057c8841e322", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A cyclist rides at 37 miles per hour for 1 hours, then at 62 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 37 × 1 = 37 miles.\n2) Second leg distance = 62 × 1 = 62 miles.\n3) Total distance = 37 + 62 = 99 miles.\nFinal answer: $99$", "final_answer": "99", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5f039f461fe0afa5", "problem_id": "690aa4c1005708bb", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "You want 69 liters of a 25% salt solution. You have a 10% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 69−x liters is 60% solution.\n2) Salt equation: 10x + 60(69−x) = 25·69.\n3) (10-60)x = (25-60)·69 ⇒ x = 483/10.\n4) Then 69−x = 207/10.\nFinal answer: $483/10 L of 10% and 207/10 L of 60%$", "final_answer": "483/10 L of 10% and 207/10 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e477d2cb07e8f06f", "problem_id": "690aa4c1005708bb", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "You want 69 liters of a 25% salt solution. You have a 10% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 69−x liters is 60% solution.\nSalt equation: 10x + 60(69−x) = 25·69.\n(10-60)x = (25-60)·69 ⇒ x = 483/10.\nFinal answer: $483/10 L of 10% and 207/10 L of 60%$", "final_answer": "483/10 L of 10% and 207/10 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "012fb8684c1f182b", "problem_id": "8348edd404a24880", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 3 x^{3} \\sin{\\left(x \\right)}) = - 3 x^{3} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 3 x^{3} \\cos{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 3 x^{3} \\sin{\\left(x \\right)}) = - 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $- 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$", "final_answer": "- 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c8c0d5523c664e0a", "problem_id": "7693e74f918b38b7", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 11 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=11, k=5, p=$\\frac{1}{10}$ and simplify.\nHere C(11,5) = 462.\nCheck: expression simplifies to $\\frac{122762871}{50000000000}$.\nFinal answer: $\\frac{122762871}{50000000000}$", "final_answer": "\\frac{122762871}{50000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "010926035bb9730a", "problem_id": "b5c4198fa02bf2e8", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡0 (mod 9), x≡12 (mod 13). Give the solution modulo 117.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 117.\nCompute the CRT solution to get x ≡ 90 (mod 117).\nCheck: 90 mod 9=0 and mod 13=12.\nFinal answer: $x\\equiv 90\\pmod{117}$", "final_answer": "x\\equiv 90\\pmod{117}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a5e204b5b3fdee44", "problem_id": "b5c4198fa02bf2e8", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the congruences: x≡0 (mod 9), x≡12 (mod 13). Give the solution modulo 117.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 117.\n3) Compute the CRT solution to get x ≡ 90 (mod 117).\nFinal answer: $x\\equiv 90\\pmod{117}$", "final_answer": "x\\equiv 90\\pmod{117}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "87531d1762d89df0", "problem_id": "ba7cb9a9481a959b", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the multiplicative inverse of 37 modulo 59.", "output": "1) We want x such that 37x ≡ 1 (mod 59).\n2) Use extended Euclid to find s,t with 37s + 59t = 1.\n3) One Bezout identity is: 37·(8) + 59·(-5) = 1.\n4) Reduce s modulo 59: x ≡ 8 (mod 59).\nFinal answer: $x\\equiv 8\\pmod{59}$", "final_answer": "x\\equiv 8\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8c052f32c262edb9", "problem_id": "ba7cb9a9481a959b", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Find the multiplicative inverse of 37 modulo 59.", "output": "We want x such that 37x ≡ 1 (mod 59).\nUse extended Euclid to find s,t with 37s + 59t = 1.\nOne Bezout identity is: 37·(8) + 59·(-5) = 1.\nFinal answer: $x\\equiv 8\\pmod{59}$", "final_answer": "x\\equiv 8\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d4d910a10ae19de2", "problem_id": "27588f2b3c7712a1", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡3 (mod 7), x≡1 (mod 5). Give the solution modulo 35.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 35.\n3) Compute the CRT solution to get x ≡ 31 (mod 35).\nFinal answer: $x\\equiv 31\\pmod{35}$", "final_answer": "x\\equiv 31\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "abda2a9a91a8750a", "problem_id": "27588f2b3c7712a1", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡3 (mod 7), x≡1 (mod 5). Give the solution modulo 35.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 35.\nCompute the CRT solution to get x ≡ 31 (mod 35).\nCheck: 31 mod 7=3 and mod 5=1.\nFinal answer: $x\\equiv 31\\pmod{35}$", "final_answer": "x\\equiv 31\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "59035007f0cf2c11", "problem_id": "26a4abfa9c85fd1e", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the linear system:\n$4 x - 3 y$ = $-26$\n$2 x - 4 y$ = $-18$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $8 x - 6 y$ = $-52$.\n3) Multiply the second equation by $4$: $8 x - 16 y$ = $-72$.\n4) Subtract to eliminate x: $10 y$ = $20$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, 2\\right)$", "final_answer": "(x,y)=\\left(-5, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bb41462a49071e0b", "problem_id": "26a4abfa9c85fd1e", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$4 x - 3 y$ = $-26$\n$2 x - 4 y$ = $-18$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $2$: $8 x - 6 y$ = $-52$.\nMultiply the second equation by $4$: $8 x - 16 y$ = $-72$.\nSubtract to eliminate x: $10 y$ = $20$.\nSolve for y: y = $2$.\nSubstitute back to find x: x = $-5$.\nCheck: plug (x,y)=(-5,2) into both equations.\nFinal answer: $(x,y)=\\left(-5, 2\\right)$", "final_answer": "(x,y)=\\left(-5, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "067a3b17edd87595", "problem_id": "ee77aff34f948a1d", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡1 (mod 5), x≡5 (mod 11). Give the solution modulo 55.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 55.\n3) Compute the CRT solution to get x ≡ 16 (mod 55).\nFinal answer: $x\\equiv 16\\pmod{55}$", "final_answer": "x\\equiv 16\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5c6b353a2daf60e9", "problem_id": "ee77aff34f948a1d", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the congruences: x≡1 (mod 5), x≡5 (mod 11). Give the solution modulo 55.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 55.\nCompute the CRT solution to get x ≡ 16 (mod 55).\nFinal answer: $x\\equiv 16\\pmod{55}$", "final_answer": "x\\equiv 16\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "00f7d70d5f28645c", "problem_id": "2a95124f16e00297", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$- 3 x - y$ = $0$\n$- 2 x - 2 y$ = $0$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $6 x + 2 y$ = $0$.\n3) Multiply the second equation by $-3$: $6 x + 6 y$ = $0$.\n4) Subtract to eliminate x: $- 4 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, 0\\right)$", "final_answer": "(x,y)=\\left(0, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "562dc8bc23a9197b", "problem_id": "2a95124f16e00297", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- 3 x - y$ = $0$\n$- 2 x - 2 y$ = $0$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $6 x + 2 y$ = $0$.\nMultiply the second equation by $-3$: $6 x + 6 y$ = $0$.\nSubtract to eliminate x: $- 4 y$ = $0$.\nSolve for y: y = $0$.\nSubstitute back to find x: x = $0$.\nCheck: plug (x,y)=(0,0) into both equations.\nFinal answer: $(x,y)=\\left(0, 0\\right)$", "final_answer": "(x,y)=\\left(0, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f0f2fa4d1b8d57a5", "problem_id": "be7c28f7bcacefb9", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the linear system:\n$8 x - 9 y$ = $73$\n$- 11 x - 6 y$ = $65$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-11$: $- 88 x + 99 y$ = $-803$.\nMultiply the second equation by $8$: $- 88 x - 48 y$ = $520$.\nFinal answer: $(x,y)=\\left(-1, -9\\right)$", "final_answer": "(x,y)=\\left(-1, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5e757bbdcce714c5", "problem_id": "be7c28f7bcacefb9", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$8 x - 9 y$ = $73$\n$- 11 x - 6 y$ = $65$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-11$: $- 88 x + 99 y$ = $-803$.\n3) Multiply the second equation by $8$: $- 88 x - 48 y$ = $520$.\n4) Subtract to eliminate x: $147 y$ = $-1323$.\n5) Solve for y: y = $-9$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, -9\\right)$", "final_answer": "(x,y)=\\left(-1, -9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a27d3b6d6c5f38e9", "problem_id": "46dda323ae847558", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_8=43$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=43 and k=8.\n4) Compute C(50,7) = 99884400.\nFinal answer: $99884400$", "final_answer": "99884400", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c789decbaafd225d", "problem_id": "4166a418f2c3a4dd", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 \\cos{\\left(2 x \\right)}) = - 3 \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 \\cos{\\left(2 x \\right)}) = - 6 \\sin{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$", "final_answer": "x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3e13a26c6d8ea70f", "problem_id": "8bd6eb09bc478a31", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=- x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $- x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- x^{3} \\cos{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.\nFinal answer: $- x^{3}\\cdot \\cos{\\left(x \\right)} - 3\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} + 3\\cdot x\\cdot \\cos{\\left(x \\right)} + 3\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} + \\cos{\\left(x \\right)}$", "final_answer": "- x^{3}\\cdot \\cos{\\left(x \\right)} - 3\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} + 3\\cdot x\\cdot \\cos{\\left(x \\right)} + 3\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} + \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c66628f9b9fb5d93", "problem_id": "8bd6eb09bc478a31", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $- x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- x^{3} \\cos{\\left(x \\right)} - 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- x^{3}\\cdot \\cos{\\left(x \\right)} - 3\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} + 3\\cdot x\\cdot \\cos{\\left(x \\right)} + 3\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} + \\cos{\\left(x \\right)}$", "final_answer": "- x^{3}\\cdot \\cos{\\left(x \\right)} - 3\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} + 3\\cdot x\\cdot \\cos{\\left(x \\right)} + 3\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} + \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c2cfa3b4aed9ed48", "problem_id": "f73d7603e044faa6", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $2 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $2 \\left(x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}\\right)$.\nFinal answer: $2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 4\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 4\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c29fb4ba169286cb", "problem_id": "f73d7603e044faa6", "style": "answer_only", "instruction": "Final answer only.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.", "output": "$2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 4\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 4\\cdot x\\cdot \\sin{\\left(x \\right)} + 2\\cdot x\\cdot \\cos{\\left(x \\right)} + 2\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4da2c07b64fdc880", "problem_id": "7245174508ae5267", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$2 x + y$ = $2$\n$3 x - 2 y$ = $3$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $6 x + 3 y$ = $6$.\nMultiply the second equation by $2$: $6 x - 4 y$ = $6$.\nSubtract to eliminate x: $7 y$ = $0$.\nSolve for y: y = $0$.\nSubstitute back to find x: x = $1$.\nCheck: plug (x,y)=(1,0) into both equations.\nFinal answer: $(x,y)=\\left(1, 0\\right)$", "final_answer": "(x,y)=\\left(1, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "30b4c26364536c0c", "problem_id": "7245174508ae5267", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$2 x + y$ = $2$\n$3 x - 2 y$ = $3$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $3$: $6 x + 3 y$ = $6$.\n3) Multiply the second equation by $2$: $6 x - 4 y$ = $6$.\n4) Subtract to eliminate x: $7 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $1$.\nFinal answer: $(x,y)=\\left(1, 0\\right)$", "final_answer": "(x,y)=\\left(1, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "217a887b6f317223", "problem_id": "08feabeedede9f89", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "You want 32 liters of a 30% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 32−x liters is 50% solution.\n2) Salt equation: 25x + 50(32−x) = 30·32.\n3) (25-50)x = (30-50)·32 ⇒ x = 128/5.\n4) Then 32−x = 32/5.\nFinal answer: $128/5 L of 25% and 32/5 L of 50%$", "final_answer": "128/5 L of 25% and 32/5 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "97d8be834a2693c8", "problem_id": "08feabeedede9f89", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "You want 32 liters of a 30% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 32−x liters is 50% solution.\nSalt equation: 25x + 50(32−x) = 30·32.\n(25-50)x = (30-50)·32 ⇒ x = 128/5.\nFinal answer: $128/5 L of 25% and 32/5 L of 50%$", "final_answer": "128/5 L of 25% and 32/5 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ed78b20608cd999f", "problem_id": "5681d0b8adffddd6", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 6 x - \\frac{\\sqrt{6} x}{2} + \\frac{3 \\sqrt{6}}{2} + 9$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$-6 - \\frac{\\sqrt{6}}{2}$, c=$\\frac{3 \\sqrt{6}}{2} + 9$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{3}{2}$, so √Δ = $\\frac{\\sqrt{6}}{2}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{6}}{2} + 3$.\nx2 = $3$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{6}}{2} + 3, 3\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{6}}{2} + 3, 3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2ae49f88682487ed", "problem_id": "5681d0b8adffddd6", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 6 x - \\frac{\\sqrt{6} x}{2} + \\frac{3 \\sqrt{6}}{2} + 9$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$-6 - \\frac{\\sqrt{6}}{2}$, c=$\\frac{3 \\sqrt{6}}{2} + 9$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{\\sqrt{6}}{2} + 3, 3\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{6}}{2} + 3, 3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cb9e2c53c2ff28e7", "problem_id": "2279a91ce0d860a3", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the multiplicative inverse of 7 modulo 37.", "output": "1) We want x such that 7x ≡ 1 (mod 37).\n2) Use extended Euclid to find s,t with 7s + 37t = 1.\n3) One Bezout identity is: 7·(16) + 37·(-3) = 1.\n4) Reduce s modulo 37: x ≡ 16 (mod 37).\nFinal answer: $x\\equiv 16\\pmod{37}$", "final_answer": "x\\equiv 16\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "94735180bacae741", "problem_id": "2279a91ce0d860a3", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the multiplicative inverse of 7 modulo 37.", "output": "We want x such that 7x ≡ 1 (mod 37).\nUse extended Euclid to find s,t with 7s + 37t = 1.\nOne Bezout identity is: 7·(16) + 37·(-3) = 1.\nReduce s modulo 37: x ≡ 16 (mod 37).\nCheck: 7·16 mod 37 = 1.\nFinal answer: $x\\equiv 16\\pmod{37}$", "final_answer": "x\\equiv 16\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8b10fc9ea00ab661", "problem_id": "e4bdbe5d6712457b", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{3 x^{2} - x + \\left(- 3 x^{2} + 2 x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{- 3 x^{2} + 2 x + 5}$.", "output": "1) Let f(x) = $\\frac{3 x^{2} - x + \\left(- 3 x^{2} + 2 x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{- 3 x^{2} + 2 x + 5}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{4 \\left(3 x - 1\\right) \\left(3 x^{2} - x + \\left(- 3 x^{2} + 2 x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(- 3 x^{2} + 2 x + 5\\right) \\left(12 x + 4 \\left(1 - 3 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(- 3 x^{2} + 2 x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(- 3 x^{2} + 2 x + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(- 3 x^{2} + 2 x + 5\\right)^{2}}$.\nFinal answer: $\\frac{\\left(12\\cdot x - 4\\right)\\cdot \\left(3\\cdot x^{2} - x + \\left(- 3\\cdot x^{2} + 2\\cdot x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 3\\right) + \\left(- 3\\cdot x^{2} + 2\\cdot x + 5\\right)\\cdot \\left(12\\cdot x + \\left(4 - 12\\cdot x\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(- 6\\cdot x^{2} + 4\\cdot x + 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + \\left(- 3\\cdot x^{2} + 2\\cdot x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right)}{2\\cdot \\left(- 3\\cdot x^{2} + 2\\cdot x + 5\\right)^{2}}$", "final_answer": "\\frac{\\left(12\\cdot x - 4\\right)\\cdot \\left(3\\cdot x^{2} - x + \\left(- 3\\cdot x^{2} + 2\\cdot x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 3\\right) + \\left(- 3\\cdot x^{2} + 2\\cdot x + 5\\right)\\cdot \\left(12\\cdot x + \\left(4 - 12\\cdot x\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(- 6\\cdot x^{2} + 4\\cdot x + 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + \\left(- 3\\cdot x^{2} + 2\\cdot x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right)}{2\\cdot \\left(- 3\\cdot x^{2} + 2\\cdot x + 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9161b99ed45d58b1", "problem_id": "e4bdbe5d6712457b", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{3 x^{2} - x + \\left(- 3 x^{2} + 2 x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{- 3 x^{2} + 2 x + 5}$.", "output": "Let f(x) = $\\frac{3 x^{2} - x + \\left(- 3 x^{2} + 2 x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{- 3 x^{2} + 2 x + 5}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{4 \\left(3 x - 1\\right) \\left(3 x^{2} - x + \\left(- 3 x^{2} + 2 x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(- 3 x^{2} + 2 x + 5\\right) \\left(12 x + 4 \\left(1 - 3 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(- 3 x^{2} + 2 x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(- 3 x^{2} + 2 x + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(- 3 x^{2} + 2 x + 5\\right)^{2}}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\frac{\\left(12\\cdot x - 4\\right)\\cdot \\left(3\\cdot x^{2} - x + \\left(- 3\\cdot x^{2} + 2\\cdot x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 3\\right) + \\left(- 3\\cdot x^{2} + 2\\cdot x + 5\\right)\\cdot \\left(12\\cdot x + \\left(4 - 12\\cdot x\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(- 6\\cdot x^{2} + 4\\cdot x + 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + \\left(- 3\\cdot x^{2} + 2\\cdot x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right)}{2\\cdot \\left(- 3\\cdot x^{2} + 2\\cdot x + 5\\right)^{2}}$", "final_answer": "\\frac{\\left(12\\cdot x - 4\\right)\\cdot \\left(3\\cdot x^{2} - x + \\left(- 3\\cdot x^{2} + 2\\cdot x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 3\\right) + \\left(- 3\\cdot x^{2} + 2\\cdot x + 5\\right)\\cdot \\left(12\\cdot x + \\left(4 - 12\\cdot x\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(- 6\\cdot x^{2} + 4\\cdot x + 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + \\left(- 3\\cdot x^{2} + 2\\cdot x + 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right)}{2\\cdot \\left(- 3\\cdot x^{2} + 2\\cdot x + 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5ded2b335e66f720", "problem_id": "2d6573cee998aca0", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A cyclist rides at 89 miles per hour for 1 hours, then at 50 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 89 × 1 = 89 miles.\n2) Second leg distance = 50 × 3/2 = 75 miles.\n3) Total distance = 89 + 75 = 164 miles.\nFinal answer: $164$", "final_answer": "164", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "32aefed655219bef", "problem_id": "f4bb53485da72686", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{4} - 3 x^{2} + x - 4$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(4 x^{4}) = 16 x^{4}$.\n3) So $f'(x) = 16 x^{4} - 6 x + 1$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(4 x^{4}) = 16 x^{3}$.\nTherefore the correct derivative is $f'(x) = 16 x^{3} - 6 x + 1$.\nFinal answer: $16 x^{3} - 6 x + 1$", "final_answer": "16 x^{3} - 6 x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c400565d9c0148b1", "problem_id": "d282ef9a592e92cc", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many binary strings of length 18 contain exactly 8 ones with no two ones adjacent?", "output": "1) Reserve 7 mandatory zeros between the 8 ones.\n2) This leaves n−k+1 = 11 valid slots for the ones.\n3) Choose the slots: C(11,8) = 165.\nFinal answer: $165$", "final_answer": "165", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3b88ec79fbfaca94", "problem_id": "d282ef9a592e92cc", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many binary strings of length 18 contain exactly 8 ones with no two ones adjacent?", "output": "Reserve 7 mandatory zeros between the 8 ones.\nThis leaves n−k+1 = 11 valid slots for the ones.\nChoose the slots: C(11,8) = 165.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $165$", "final_answer": "165", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "592752658feae7ba", "problem_id": "d8cc166d19e77d96", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x^{3} - 5 x^{2} - 8 x - 14$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 2 x^{4} - x^{3} + 2 x^{2} + 2 x - 2$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 8 x^{3} - 3 x^{2} + 4 x + 2$.\n5) Evaluate at a: p'(2) = $-66$.\nFinal answer: $-66$", "final_answer": "-66", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "75ef00e7f44dbc38", "problem_id": "d8cc166d19e77d96", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x^{3} - 5 x^{2} - 8 x - 14$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 2 x^{4} - x^{3} + 2 x^{2} + 2 x - 2$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 8 x^{3} - 3 x^{2} + 4 x + 2$.\nEvaluate at a: p'(2) = $-66$.\nCheck: the identity gives $-66$.\nFinal answer: $-66$", "final_answer": "-66", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "373154ec2a40d179", "problem_id": "af630a815e14be2a", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $- 2 x^{3} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - x\\cdot \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - x\\cdot \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5e41b0c53f709a90", "problem_id": "af630a815e14be2a", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $- 2 x^{3} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - x\\cdot \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 2\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 6\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - x\\cdot \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3d2a0533bb8e9e52", "problem_id": "f01c85f0a368c490", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Two friends split $40 in the ratio 5:3. How much does each receive?", "output": "Total ratio parts = 5+3 = 8.\nEach part = 40 ÷ 8 = 5.\nFirst share = 5 × 5 = 25.\nSecond share = 3 × 5 = 15.\nCheck: 25+15=40.\nFinal answer: $15$", "final_answer": "15", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4a0044c4c0f41d4a", "problem_id": "f01c85f0a368c490", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Two friends split $40 in the ratio 5:3. How much does each receive?", "output": "1) Total ratio parts = 5+3 = 8.\n2) Each part = 40 ÷ 8 = 5.\n3) First share = 5 × 5 = 25.\n4) Second share = 3 × 5 = 15.\nFinal answer: $15$", "final_answer": "15", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ecc54ce58550353b", "problem_id": "a59c4676fc82d15e", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 10 x + 12$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} + 10 x + 12$ = $2 \\left(x + 2\\right) \\left(x + 3\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-2$.\n5) From the second factor: x = $-3$.\nFinal answer: $x \\in \\{-2, -3\\}$", "final_answer": "x \\in \\{-2, -3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b7a2c4997dd7fc8e", "problem_id": "76ce63b064bca52e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{3 x^{2} + \\left(x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{x^{2} - x + 5}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 5}) = \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 5}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} - \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} - \\frac{4 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} + \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 11 x^{2} - 10 x + 25} + \\frac{11 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} + \\frac{22 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} - \\frac{6 x^{2}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 11 x^{2} - 10 x + 25} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 2 x + 10} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 5} - \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 5} - \\frac{10 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} - \\frac{20 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} + \\frac{72 x}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} - \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 5} + \\frac{25 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} + \\frac{50 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50} - \\frac{6}{2 x^{4} - 4 x^{3} + 22 x^{2} - 20 x + 50}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 5}) = \\frac{x^{2} \\left(1 - 2 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(x^{2} - x\\right) + 5\\right)^{2}} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(x^{2} - x\\right) + 5\\right)} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 5} + \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 5}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(1 - 2 x\\right) \\left(3 x^{2} + \\left(x \\left(x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x \\left(x - 1\\right) + 5\\right) \\left(12 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)}\\right)}{2 \\left(x \\left(x - 1\\right) + 5\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(1 - 2 x\\right) \\left(3 x^{2} + \\left(x \\left(x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x \\left(x - 1\\right) + 5\\right) \\left(12 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)}\\right)}{2 \\left(x \\left(x - 1\\right) + 5\\right)^{2}}$", "final_answer": "\\frac{2 \\left(1 - 2 x\\right) \\left(3 x^{2} + \\left(x \\left(x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x \\left(x - 1\\right) + 5\\right) \\left(12 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)}\\right)}{2 \\left(x \\left(x - 1\\right) + 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "48d9f80e875d3b09", "problem_id": "a8ec2c76674ca8e5", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{4} - 3 x^{3} + x^{2} + 2 x + 4\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $2 x^{4} - 3 x^{3} + x^{2} + 2 x + 4$.\n3) Compute an antiderivative: F(x) = $\\frac{2 x^{5}}{5} - \\frac{3 x^{4}}{4} + \\frac{x^{3}}{3} + x^{2} + 4 x$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{2\\cdot x^{5}}{5} - \\frac{3\\cdot x^{4}}{4} + \\frac{x^{3}}{3} + x^{2} + 4\\cdot x$", "final_answer": "C + \\frac{2\\cdot x^{5}}{5} - \\frac{3\\cdot x^{4}}{4} + \\frac{x^{3}}{3} + x^{2} + 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c7beb87e61d9182e", "problem_id": "a8ec2c76674ca8e5", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int 2 x^{4} - 3 x^{3} + x^{2} + 2 x + 4\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $2 x^{4} - 3 x^{3} + x^{2} + 2 x + 4$.\nCompute an antiderivative: F(x) = $\\frac{2 x^{5}}{5} - \\frac{3 x^{4}}{4} + \\frac{x^{3}}{3} + x^{2} + 4 x$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{2\\cdot x^{5}}{5} - \\frac{3\\cdot x^{4}}{4} + \\frac{x^{3}}{3} + x^{2} + 4\\cdot x$", "final_answer": "C + \\frac{2\\cdot x^{5}}{5} - \\frac{3\\cdot x^{4}}{4} + \\frac{x^{3}}{3} + x^{2} + 4\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bf8fd6ac3c96bd72", "problem_id": "cacd398852168896", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the linear system:\n$- 2 x + 7 y$ = $-51$\n$- x - 5 y$ = $17$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $2 x - 7 y$ = $51$.\n3) Multiply the second equation by $-2$: $2 x + 10 y$ = $-34$.\n4) Subtract to eliminate x: $- 17 y$ = $85$.\n5) Solve for y: y = $-5$.\n6) Substitute back to find x: x = $8$.\nFinal answer: $(x,y)=\\left(8, -5\\right)$", "final_answer": "(x,y)=\\left(8, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6128ed0586b4bf66", "problem_id": "cacd398852168896", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$- 2 x + 7 y$ = $-51$\n$- x - 5 y$ = $17$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $2 x - 7 y$ = $51$.\nMultiply the second equation by $-2$: $2 x + 10 y$ = $-34$.\nSubtract to eliminate x: $- 17 y$ = $85$.\nSolve for y: y = $-5$.\nSubstitute back to find x: x = $8$.\nCheck: plug (x,y)=(8,-5) into both equations.\nFinal answer: $(x,y)=\\left(8, -5\\right)$", "final_answer": "(x,y)=\\left(8, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "544da9467ae58acf", "problem_id": "dd6258205c447f24", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "You want 62 liters of a 30% salt solution. You have a 20% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 62−x liters is 40% solution.\n2) Salt equation: 20x + 40(62−x) = 30·62.\n3) (20-40)x = (30-40)·62 ⇒ x = 31.\n4) Then 62−x = 31.\nFinal answer: $31 L of 20% and 31 L of 40%$", "final_answer": "31 L of 20% and 31 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e50ad8c2927c4b48", "problem_id": "dd6258205c447f24", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "You want 62 liters of a 30% salt solution. You have a 20% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 62−x liters is 40% solution.\nSalt equation: 20x + 40(62−x) = 30·62.\n(20-40)x = (30-40)·62 ⇒ x = 31.\nFinal answer: $31 L of 20% and 31 L of 40%$", "final_answer": "31 L of 20% and 31 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f3f72e7eeb1bc17a", "problem_id": "2704b0ab9709d89a", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=20$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=20 and k=10.\n4) Compute C(29,9) = 10015005.\nFinal answer: $10015005$", "final_answer": "10015005", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "68dfc283f8bc4f24", "problem_id": "2704b0ab9709d89a", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=20$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=20 and k=10.\nFinal answer: $10015005$", "final_answer": "10015005", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0058ea1f3a524f18", "problem_id": "5690e32a2671f4bc", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "A store sells a backpack for $520. It is discounted by 10%. You buy 12 backpacks. What is the total cost?", "output": "Discounted price = 520 × (1 − 10/100) = 520 × 90/100 = 468.\nSubtotal = 12 × 468 = 5616.\nFinal answer: $5616$", "final_answer": "5616", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "23c28044de2e9d99", "problem_id": "5690e32a2671f4bc", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A store sells a backpack for $520. It is discounted by 10%. You buy 12 backpacks. What is the total cost?", "output": "1) Discounted price = 520 × (1 − 10/100) = 520 × 90/100 = 468.\n2) Subtotal = 12 × 468 = 5616.\nFinal answer: $5616$", "final_answer": "5616", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "155617e06777d376", "problem_id": "61597971a8e26afc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - x^{2} - 2 x - 3$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} - 3 x - 3.\n3) Then the limit is -21.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: -18.\nFinal answer: $-18$", "final_answer": "-18", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "efcf400cba95e83e", "problem_id": "36acee256c6ad13a", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the multiplicative inverse of 8 modulo 37.", "output": "1) We want x such that 8x ≡ 1 (mod 37).\n2) Use extended Euclid to find s,t with 8s + 37t = 1.\n3) One Bezout identity is: 8·(14) + 37·(-3) = 1.\n4) Reduce s modulo 37: x ≡ 14 (mod 37).\nFinal answer: $x\\equiv 14\\pmod{37}$", "final_answer": "x\\equiv 14\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7b956a98a7fa5622", "problem_id": "427276eeb7caafad", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 6 red balls and 12 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(18,8).\n3) Favorable samples: choose k reds and n−k blues: C(6,3)·C(12,5).\n4) Compute totals: C(18,8) = 43758, and favorable = 15840.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{80}{221}$", "final_answer": "\\frac{80}{221}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "34a998426d3f9132", "problem_id": "427276eeb7caafad", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 6 red balls and 12 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(18,8).\nFavorable samples: choose k reds and n−k blues: C(6,3)·C(12,5).\nCompute totals: C(18,8) = 43758, and favorable = 15840.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{80}{221}$.\nFinal answer: $\\frac{80}{221}$", "final_answer": "\\frac{80}{221}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "244bb46f34b872fd", "problem_id": "79b7e091f5c7cfef", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{2 x^{3} - x^{2} + 2 x + \\left(3 x^{2} - 2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{3 x^{2} - 2 x + 3}$.", "output": "1) Let f(x) = $\\frac{2 x^{3} - x^{2} + 2 x + \\left(3 x^{2} - 2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{3 x^{2} - 2 x + 3}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{4 \\left(1 - 3 x\\right) \\left(2 x^{3} - x^{2} + 2 x + \\left(3 x^{2} - 2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(3 x^{2} - 2 x + 3\\right) \\left(12 x^{2} - 4 x + 4 \\left(3 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(3 x^{2} - 2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x^{2} - 2 x + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(3 x^{2} - 2 x + 3\\right)^{2}}$.\nFinal answer: $\\frac{\\left(4 - 12\\cdot x\\right)\\cdot \\left(2\\cdot x^{3} - x^{2} + 2\\cdot x + \\left(3\\cdot x^{2} - 2\\cdot x + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right) + \\left(3\\cdot x^{2} - 2\\cdot x + 3\\right)\\cdot \\left(12\\cdot x^{2} - 4\\cdot x + \\left(12\\cdot x - 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(3\\cdot x^{2} - 2\\cdot x + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x^{2} - 4\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\right)}{2\\cdot \\left(3\\cdot x^{2} - 2\\cdot x + 3\\right)^{2}}$", "final_answer": "\\frac{\\left(4 - 12\\cdot x\\right)\\cdot \\left(2\\cdot x^{3} - x^{2} + 2\\cdot x + \\left(3\\cdot x^{2} - 2\\cdot x + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right) + \\left(3\\cdot x^{2} - 2\\cdot x + 3\\right)\\cdot \\left(12\\cdot x^{2} - 4\\cdot x + \\left(12\\cdot x - 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(3\\cdot x^{2} - 2\\cdot x + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x^{2} - 4\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\right)}{2\\cdot \\left(3\\cdot x^{2} - 2\\cdot x + 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a2ae7075869730bd", "problem_id": "79b7e091f5c7cfef", "style": "answer_only", "instruction": "Final answer only.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{2 x^{3} - x^{2} + 2 x + \\left(3 x^{2} - 2 x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{3 x^{2} - 2 x + 3}$.", "output": "$\\frac{\\left(4 - 12\\cdot x\\right)\\cdot \\left(2\\cdot x^{3} - x^{2} + 2\\cdot x + \\left(3\\cdot x^{2} - 2\\cdot x + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right) + \\left(3\\cdot x^{2} - 2\\cdot x + 3\\right)\\cdot \\left(12\\cdot x^{2} - 4\\cdot x + \\left(12\\cdot x - 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(3\\cdot x^{2} - 2\\cdot x + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x^{2} - 4\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\right)}{2\\cdot \\left(3\\cdot x^{2} - 2\\cdot x + 3\\right)^{2}}$", "final_answer": "\\frac{\\left(4 - 12\\cdot x\\right)\\cdot \\left(2\\cdot x^{3} - x^{2} + 2\\cdot x + \\left(3\\cdot x^{2} - 2\\cdot x + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 2\\right) + \\left(3\\cdot x^{2} - 2\\cdot x + 3\\right)\\cdot \\left(12\\cdot x^{2} - 4\\cdot x + \\left(12\\cdot x - 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(3\\cdot x^{2} - 2\\cdot x + 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(6\\cdot x^{2} - 4\\cdot x + 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\right)}{2\\cdot \\left(3\\cdot x^{2} - 2\\cdot x + 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a0b5704bf0880430", "problem_id": "fcbcf76b06eda86d", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} 3 x^{3} - x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{3} - x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{9 \\pi}{2} - \\frac{3 \\pi^{3}}{8} + 2 + \\frac{9 \\pi^{2}}{8} + \\frac{45 \\pi^{4}}{64}$.\n4) Simplify the result.\nFinal answer: $- \\frac{9\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{3}}{8} + 2 + \\frac{9\\cdot \\pi^{2}}{8} + \\frac{45\\cdot \\pi^{4}}{64}$", "final_answer": "- \\frac{9\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{3}}{8} + 2 + \\frac{9\\cdot \\pi^{2}}{8} + \\frac{45\\cdot \\pi^{4}}{64}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "af67926cdf434097", "problem_id": "fcbcf76b06eda86d", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} 3 x^{3} - x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{3} - x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} - 3$.\nEvaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{9 \\pi}{2} - \\frac{3 \\pi^{3}}{8} + 2 + \\frac{9 \\pi^{2}}{8} + \\frac{45 \\pi^{4}}{64}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{9\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{3}}{8} + 2 + \\frac{9\\cdot \\pi^{2}}{8} + \\frac{45\\cdot \\pi^{4}}{64}$", "final_answer": "- \\frac{9\\cdot \\pi}{2} - \\frac{3\\cdot \\pi^{3}}{8} + 2 + \\frac{9\\cdot \\pi^{2}}{8} + \\frac{45\\cdot \\pi^{4}}{64}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9c19a24b959da13f", "problem_id": "4f92bc61595ad3e9", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Find the multiplicative inverse of 29 modulo 41.", "output": "We want x such that 29x ≡ 1 (mod 41).\nUse extended Euclid to find s,t with 29s + 41t = 1.\nOne Bezout identity is: 29·(17) + 41·(-12) = 1.\nFinal answer: $x\\equiv 17\\pmod{41}$", "final_answer": "x\\equiv 17\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b36855b1c4afa4fa", "problem_id": "4f92bc61595ad3e9", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Find the multiplicative inverse of 29 modulo 41.", "output": "1) We want x such that 29x ≡ 1 (mod 41).\n2) Use extended Euclid to find s,t with 29s + 41t = 1.\n3) One Bezout identity is: 29·(17) + 41·(-12) = 1.\n4) Reduce s modulo 41: x ≡ 17 (mod 41).\nFinal answer: $x\\equiv 17\\pmod{41}$", "final_answer": "x\\equiv 17\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d737a9d963ec8c32", "problem_id": "d8d35b6d9871079c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- \\frac{3 x^{2}}{2} - x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1}{x - 4}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}) = - \\frac{4 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4}$.\n3) So $f'(x) = \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{3 x^{2}}{2 x^{2} - 16 x + 32} - \\frac{8 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{16 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{24 x}{2 x^{2} - 16 x + 32} + \\frac{16 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{32 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{10}{2 x^{2} - 16 x + 32} - \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 8 x + 16} - \\frac{4 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} + \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4} + \\frac{4 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}) = - \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4} - \\frac{4 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} + \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x - 4\\right)^{2}}$.\nTherefore the correct derivative is $f'(x) = \\frac{3 x^{2} + 2 x + \\left(x - 4\\right) \\left(- 6 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{2 \\left(x - 4\\right)^{2}}$.\nFinal answer: $\\frac{3 x^{2} + 2 x + \\left(x - 4\\right) \\left(- 6 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{2 \\left(x - 4\\right)^{2}}$", "final_answer": "\\frac{3 x^{2} + 2 x + \\left(x - 4\\right) \\left(- 6 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2}{2 \\left(x - 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a7aef6197803cce0", "problem_id": "8f728f41141675cb", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_7=23$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_7=9.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=9 and k=7.\nCompute C(15,6) = 5005.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $5005$", "final_answer": "5005", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "03d6e452477b3648", "problem_id": "9f3d3b50575d169f", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 64$ = 0.", "output": "$x \\in \\{8, -8\\}$", "final_answer": "x \\in \\{8, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bc451e2e56be0070", "problem_id": "9f3d3b50575d169f", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 64$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} - 64$ = $\\left(x - 8\\right) \\left(x + 8\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $8$.\n5) From the second factor: x = $-8$.\nFinal answer: $x \\in \\{8, -8\\}$", "final_answer": "x \\in \\{8, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b1d42e9ed275c743", "problem_id": "bdb90e46c2e423d8", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the linear system:\n$- x + 4 y$ = $-16$\n$- 3 x + 4 y$ = $-16$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $3 x - 12 y$ = $48$.\n3) Multiply the second equation by $-1$: $3 x - 4 y$ = $16$.\n4) Subtract to eliminate x: $- 8 y$ = $32$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, -4\\right)$", "final_answer": "(x,y)=\\left(0, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cdb4fd749c28a958", "problem_id": "bdb90e46c2e423d8", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the linear system:\n$- x + 4 y$ = $-16$\n$- 3 x + 4 y$ = $-16$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $3 x - 12 y$ = $48$.\nMultiply the second equation by $-1$: $3 x - 4 y$ = $16$.\nFinal answer: $(x,y)=\\left(0, -4\\right)$", "final_answer": "(x,y)=\\left(0, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "51666a11382071f1", "problem_id": "3df20312eccf949e", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "An urn has 7 red balls and 9 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(16,8).\n3) Favorable samples: choose k reds and n−k blues: C(7,1)·C(9,7).\n4) Compute totals: C(16,8) = 12870, and favorable = 252.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{14}{715}$", "final_answer": "\\frac{14}{715}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a3a944a2d6a2d1e", "problem_id": "3df20312eccf949e", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 7 red balls and 9 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(16,8).\nFavorable samples: choose k reds and n−k blues: C(7,1)·C(9,7).\nCompute totals: C(16,8) = 12870, and favorable = 252.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{14}{715}$.\nFinal answer: $\\frac{14}{715}$", "final_answer": "\\frac{14}{715}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "177f08842aed766d", "problem_id": "60d05b0401bfdee7", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_6=28$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_6=16.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=16 and k=6.\n4) Compute C(21,5) = 20349.\nFinal answer: $20349$", "final_answer": "20349", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0423d7cbb5e29e3a", "problem_id": "60d05b0401bfdee7", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_6=28$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_6=16.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=16 and k=6.\nCompute C(21,5) = 20349.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $20349$", "final_answer": "20349", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "49b84e5ca6206f52", "problem_id": "154d1b3dbc9faea2", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A cyclist rides at 31 miles per hour for 3/2 hours, then at 86 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 31 × 3/2 = 93/2 miles.\nSecond leg distance = 86 × 1 = 86 miles.\nTotal distance = 93/2 + 86 = 265/2 miles.\nCheck: adding both legs gives 265/2.\nFinal answer: $\\frac{265}{2}$", "final_answer": "\\frac{265}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cd0014b24b414998", "problem_id": "154d1b3dbc9faea2", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A cyclist rides at 31 miles per hour for 3/2 hours, then at 86 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 31 × 3/2 = 93/2 miles.\n2) Second leg distance = 86 × 1 = 86 miles.\n3) Total distance = 93/2 + 86 = 265/2 miles.\nFinal answer: $\\frac{265}{2}$", "final_answer": "\\frac{265}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "18b06e89dc6ae3e6", "problem_id": "451299f9d1f2a582", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "You want 28 liters of a 40% salt solution. You have a 30% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 28−x liters is 50% solution.\n2) Salt equation: 30x + 50(28−x) = 40·28.\n3) (30-50)x = (40-50)·28 ⇒ x = 14.\n4) Then 28−x = 14.\nFinal answer: $14 L of 30% and 14 L of 50%$", "final_answer": "14 L of 30% and 14 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "077a741ea5e4486a", "problem_id": "451299f9d1f2a582", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "You want 28 liters of a 40% salt solution. You have a 30% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 28−x liters is 50% solution.\nSalt equation: 30x + 50(28−x) = 40·28.\n(30-50)x = (40-50)·28 ⇒ x = 14.\nFinal answer: $14 L of 30% and 14 L of 50%$", "final_answer": "14 L of 30% and 14 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c49434d2b4e2fdcd", "problem_id": "e1d1af62f1d213ef", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 3 x - 10$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $x^{2} + 3 x - 10$ = $\\left(x - 2\\right) \\left(x + 5\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $2$.\n5) From the second factor: x = $-5$.\nFinal answer: $x \\in \\{2, -5\\}$", "final_answer": "x \\in \\{2, -5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7c4e2ac453e53c40", "problem_id": "e1d1af62f1d213ef", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} + 3 x - 10$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $x^{2} + 3 x - 10$ = $\\left(x - 2\\right) \\left(x + 5\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $2$.\nFrom the second factor: x = $-5$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{2, -5\\}$", "final_answer": "x \\in \\{2, -5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bd2a92cced9c202d", "problem_id": "53b6a4668c06466a", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A cyclist rides at 40 miles per hour for 3/2 hours, then at 80 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 40 × 3/2 = 60 miles.\n2) Second leg distance = 80 × 2 = 160 miles.\n3) Total distance = 60 + 160 = 220 miles.\nFinal answer: $220$", "final_answer": "220", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "46a8049a133d8107", "problem_id": "53b6a4668c06466a", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A cyclist rides at 40 miles per hour for 3/2 hours, then at 80 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 40 × 3/2 = 60 miles.\nSecond leg distance = 80 × 2 = 160 miles.\nTotal distance = 60 + 160 = 220 miles.\nFinal answer: $220$", "final_answer": "220", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c76052d3445898d4", "problem_id": "0fb1316309a46063", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the congruences: x≡0 (mod 11), x≡1 (mod 13). Give the solution modulo 143.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 143.\n3) Compute the CRT solution to get x ≡ 66 (mod 143).\nFinal answer: $x\\equiv 66\\pmod{143}$", "final_answer": "x\\equiv 66\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1b2ffc777db8ee9e", "problem_id": "0fb1316309a46063", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡0 (mod 11), x≡1 (mod 13). Give the solution modulo 143.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 143.\nCompute the CRT solution to get x ≡ 66 (mod 143).\nCheck: 66 mod 11=0 and mod 13=1.\nFinal answer: $x\\equiv 66\\pmod{143}$", "final_answer": "x\\equiv 66\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72d439be5665ecc3", "problem_id": "c89fe4c25c35b29d", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡6 (mod 8), x≡2 (mod 5). Give the solution modulo 40.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 40.\n3) Compute the CRT solution to get x ≡ 22 (mod 40).\nFinal answer: $x\\equiv 22\\pmod{40}$", "final_answer": "x\\equiv 22\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a257e71b7986c8b", "problem_id": "c89fe4c25c35b29d", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡6 (mod 8), x≡2 (mod 5). Give the solution modulo 40.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 40.\nCompute the CRT solution to get x ≡ 22 (mod 40).\nCheck: 22 mod 8=6 and mod 5=2.\nFinal answer: $x\\equiv 22\\pmod{40}$", "final_answer": "x\\equiv 22\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "04a090c961075635", "problem_id": "23d5fdbca5f46cb0", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 12 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=4, p=$\\frac{1}{10}$ and simplify.\n4) Here C(12,4) = 495.\nFinal answer: $\\frac{4261625379}{200000000000}$", "final_answer": "\\frac{4261625379}{200000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f3bbf77b1f9d38b", "problem_id": "23d5fdbca5f46cb0", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 12 times, what is the probability of exactly 4 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=12, k=4, p=$\\frac{1}{10}$ and simplify.\nHere C(12,4) = 495.\nCheck: expression simplifies to $\\frac{4261625379}{200000000000}$.\nFinal answer: $\\frac{4261625379}{200000000000}$", "final_answer": "\\frac{4261625379}{200000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4f32a28211aa2f14", "problem_id": "9ed6b62a250ba400", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the congruences: x≡4 (mod 9), x≡3 (mod 13). Give the solution modulo 117.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 117.\nCompute the CRT solution to get x ≡ 94 (mod 117).\nFinal answer: $x\\equiv 94\\pmod{117}$", "final_answer": "x\\equiv 94\\pmod{117}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e7a304c6c4bb55a4", "problem_id": "9ed6b62a250ba400", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡4 (mod 9), x≡3 (mod 13). Give the solution modulo 117.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 117.\nCompute the CRT solution to get x ≡ 94 (mod 117).\nCheck: 94 mod 9=4 and mod 13=3.\nFinal answer: $x\\equiv 94\\pmod{117}$", "final_answer": "x\\equiv 94\\pmod{117}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e8aec572218e217", "problem_id": "5435b6d5436695ed", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡1 (mod 3), x≡6 (mod 10). Give the solution modulo 30.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 30.\n3) Compute the CRT solution to get x ≡ 16 (mod 30).\nFinal answer: $x\\equiv 16\\pmod{30}$", "final_answer": "x\\equiv 16\\pmod{30}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0c681cce47036680", "problem_id": "5435b6d5436695ed", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡6 (mod 10). Give the solution modulo 30.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 30.\nCompute the CRT solution to get x ≡ 16 (mod 30).\nCheck: 16 mod 3=1 and mod 10=6.\nFinal answer: $x\\equiv 16\\pmod{30}$", "final_answer": "x\\equiv 16\\pmod{30}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "136198ed1c5e8031", "problem_id": "31883ab0d3fb94c8", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A disease has prevalence $\\frac{3}{50}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\nP(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{37}{250}$.\nCompute P(D|+) = $\\frac{27}{74}$.\nFinal answer: $\\frac{27}{74}$", "final_answer": "\\frac{27}{74}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f84d92fdb5dc6c65", "problem_id": "31883ab0d3fb94c8", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A disease has prevalence $\\frac{3}{50}$. A test has sensitivity $\\frac{9}{10}$ and specificity $\\frac{9}{10}$. If a person tests positive, what is P(disease | positive)?", "output": "1) Use Bayes' theorem: P(D|+)=P(+|D)P(D)/P(+).\n2) P(+)=P(+|D)P(D)+P(+|¬D)P(¬D) = $\\frac{37}{250}$.\n3) Compute P(D|+) = $\\frac{27}{74}$.\nFinal answer: $\\frac{27}{74}$", "final_answer": "\\frac{27}{74}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "bayes_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1155b41a0959d2b6", "problem_id": "85e6d6fa8471c590", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡0 (mod 7), x≡8 (mod 11). Give the solution modulo 77.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 77.\n3) Compute the CRT solution to get x ≡ 63 (mod 77).\nFinal answer: $x\\equiv 63\\pmod{77}$", "final_answer": "x\\equiv 63\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "51b4ceccacc7b38f", "problem_id": "85e6d6fa8471c590", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡0 (mod 7), x≡8 (mod 11). Give the solution modulo 77.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 77.\nCompute the CRT solution to get x ≡ 63 (mod 77).\nCheck: 63 mod 7=0 and mod 11=8.\nFinal answer: $x\\equiv 63\\pmod{77}$", "final_answer": "x\\equiv 63\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec568c2aeb8d40c1", "problem_id": "f28d8495ee34b534", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 14 times, what is the probability of exactly 6 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=14, k=6, p=$\\frac{4}{5}$ and simplify.\nHere C(14,6) = 3003.\nCheck: expression simplifies to $\\frac{12300288}{6103515625}$.\nFinal answer: $\\frac{12300288}{6103515625}$", "final_answer": "\\frac{12300288}{6103515625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d9d4674ac492c72d", "problem_id": "f28d8495ee34b534", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 14 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=14, k=6, p=$\\frac{4}{5}$ and simplify.\n4) Here C(14,6) = 3003.\nFinal answer: $\\frac{12300288}{6103515625}$", "final_answer": "\\frac{12300288}{6103515625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "57d5e6e5c8194173", "problem_id": "04a70d94b6a91ca3", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "An urn has 13 red balls and 13 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(26,4).\n3) Favorable samples: choose k reds and n−k blues: C(13,2)·C(13,2).\n4) Compute totals: C(26,4) = 14950, and favorable = 6084.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{234}{575}$", "final_answer": "\\frac{234}{575}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "34593ceea0df44a1", "problem_id": "782934a1915a228b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} + 4 x - 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x^{2}) = - 4 x^{2}$.\n3) So $f'(x) = 4 - 4 x^{2}$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x^{2}) = - 4 x$.\nTherefore the correct derivative is $f'(x) = 4 \\left(1 - x\\right)$.\nFinal answer: $4 \\left(1 - x\\right)$", "final_answer": "4 \\left(1 - x\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e9f4cb90127e04db", "problem_id": "388ac049ddbffde1", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡3 (mod 9), x≡5 (mod 13). Give the solution modulo 117.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 117.\n3) Compute the CRT solution to get x ≡ 57 (mod 117).\nFinal answer: $x\\equiv 57\\pmod{117}$", "final_answer": "x\\equiv 57\\pmod{117}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bb55e268246f913f", "problem_id": "388ac049ddbffde1", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡3 (mod 9), x≡5 (mod 13). Give the solution modulo 117.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 117.\nCompute the CRT solution to get x ≡ 57 (mod 117).\nCheck: 57 mod 9=3 and mod 13=5.\nFinal answer: $x\\equiv 57\\pmod{117}$", "final_answer": "x\\equiv 57\\pmod{117}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b6b9752bcbfa608c", "problem_id": "6981b9f715d574cd", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Worker A finishes a job in 13 hours and Worker B finishes it in 21 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/13 job/hour. B's rate = 1/21 job/hour.\n2) Combined rate = 1/13 + 1/21 = 34/273 job/hour.\n3) Time = 1 ÷ (combined rate) = 273/34 hours.\nFinal answer: $\\frac{273}{34}$", "final_answer": "\\frac{273}{34}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "32abb776946c711d", "problem_id": "6981b9f715d574cd", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Worker A finishes a job in 13 hours and Worker B finishes it in 21 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/13 job/hour. B's rate = 1/21 job/hour.\nCombined rate = 1/13 + 1/21 = 34/273 job/hour.\nTime = 1 ÷ (combined rate) = 273/34 hours.\nCheck: (combined rate)×(time) = 34/273×273/34 = 1.\nFinal answer: $\\frac{273}{34}$", "final_answer": "\\frac{273}{34}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5f62995b9a1aa3b5", "problem_id": "c4a8002e9b0eeba3", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} 3 x^{3} - x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{3} - x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{65 \\pi^{3}}{24} - 2 + \\frac{15 \\pi^{2}}{8} + \\frac{765 \\pi^{4}}{64}$.\n4) Simplify the result.\nFinal answer: $- \\frac{65\\cdot \\pi^{3}}{24} - 2 + \\frac{15\\cdot \\pi^{2}}{8} + \\frac{765\\cdot \\pi^{4}}{64}$", "final_answer": "- \\frac{65\\cdot \\pi^{3}}{24} - 2 + \\frac{15\\cdot \\pi^{2}}{8} + \\frac{765\\cdot \\pi^{4}}{64}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "33bacace55c5c34d", "problem_id": "c4a8002e9b0eeba3", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} 3 x^{3} - x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{3} - x^{2} + x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{65 \\pi^{3}}{24} - 2 + \\frac{15 \\pi^{2}}{8} + \\frac{765 \\pi^{4}}{64}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{65\\cdot \\pi^{3}}{24} - 2 + \\frac{15\\cdot \\pi^{2}}{8} + \\frac{765\\cdot \\pi^{4}}{64}$", "final_answer": "- \\frac{65\\cdot \\pi^{3}}{24} - 2 + \\frac{15\\cdot \\pi^{2}}{8} + \\frac{765\\cdot \\pi^{4}}{64}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2ff525b691f04bbb", "problem_id": "2644a31d0bd00c42", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_8=41$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_8=17.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=17 and k=8.\n4) Compute C(24,7) = 346104.\nFinal answer: $346104$", "final_answer": "346104", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0414d6e945d45887", "problem_id": "2644a31d0bd00c42", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_8=41$?", "output": "Let y_i=x_i-3, then y_i≥0 and y_1+…+y_8=17.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=17 and k=8.\nFinal answer: $346104$", "final_answer": "346104", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2f234f33037f5f59", "problem_id": "9de1635d43b49d02", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the multiplicative inverse of 4 modulo 41.", "output": "1) We want x such that 4x ≡ 1 (mod 41).\n2) Use extended Euclid to find s,t with 4s + 41t = 1.\n3) One Bezout identity is: 4·(-10) + 41·(1) = 1.\n4) Reduce s modulo 41: x ≡ 31 (mod 41).\nFinal answer: $x\\equiv 31\\pmod{41}$", "final_answer": "x\\equiv 31\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a1e0612702262713", "problem_id": "9de1635d43b49d02", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Find the multiplicative inverse of 4 modulo 41.", "output": "We want x such that 4x ≡ 1 (mod 41).\nUse extended Euclid to find s,t with 4s + 41t = 1.\nOne Bezout identity is: 4·(-10) + 41·(1) = 1.\nFinal answer: $x\\equiv 31\\pmod{41}$", "final_answer": "x\\equiv 31\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0529b24b8305cfae", "problem_id": "b328ac9baf80fb00", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "An urn has 19 red balls and 24 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(43,10).\nFavorable samples: choose k reds and n−k blues: C(19,2)·C(24,8).\nFinal answer: $\\frac{389367}{5936021}$", "final_answer": "\\frac{389367}{5936021}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a5d5551b6961a5ad", "problem_id": "b328ac9baf80fb00", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 19 red balls and 24 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(43,10).\n3) Favorable samples: choose k reds and n−k blues: C(19,2)·C(24,8).\n4) Compute totals: C(43,10) = 1917334783, and favorable = 125765541.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{389367}{5936021}$", "final_answer": "\\frac{389367}{5936021}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e7ffe5881305fdc3", "problem_id": "3fd8bfe654b680a6", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 22 times, what is the probability of exactly 11 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=22, k=11, p=$\\frac{4}{5}$ and simplify.\nFinal answer: $\\frac{2958796259328}{2384185791015625}$", "final_answer": "\\frac{2958796259328}{2384185791015625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "53ba470e1643e597", "problem_id": "3fd8bfe654b680a6", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 22 times, what is the probability of exactly 11 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=22, k=11, p=$\\frac{4}{5}$ and simplify.\n4) Here C(22,11) = 705432.\nFinal answer: $\\frac{2958796259328}{2384185791015625}$", "final_answer": "\\frac{2958796259328}{2384185791015625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f81fa8444bbf245c", "problem_id": "9dedae09fa481a33", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{5}$. If it is flipped 9 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=9, k=5, p=$\\frac{3}{5}$ and simplify.\nFinal answer: $\\frac{489888}{1953125}$", "final_answer": "\\frac{489888}{1953125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b3f84c02bc6a1f9b", "problem_id": "9dedae09fa481a33", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{5}$. If it is flipped 9 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=9, k=5, p=$\\frac{3}{5}$ and simplify.\n4) Here C(9,5) = 126.\nFinal answer: $\\frac{489888}{1953125}$", "final_answer": "\\frac{489888}{1953125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6462063d56e65cdc", "problem_id": "2a62d90afcb39674", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A store sells a backpack for $495. It is discounted by 20%. You buy 9 backpacks. What is the total cost?", "output": "1) Discounted price = 495 × (1 − 20/100) = 495 × 80/100 = 396.\n2) Subtotal = 9 × 396 = 3564.\nFinal answer: $3564$", "final_answer": "3564", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a0bf4fa952d2b405", "problem_id": "2a62d90afcb39674", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A store sells a backpack for $495. It is discounted by 20%. You buy 9 backpacks. What is the total cost?", "output": "Discounted price = 495 × (1 − 20/100) = 495 × 80/100 = 396.\nSubtotal = 9 × 396 = 3564.\nCheck: recomputing the formula yields 3564.\nFinal answer: $3564$", "final_answer": "3564", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a4f67812f9b79cfd", "problem_id": "05ab454044f0dc1e", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{3} + x^{2} + 4 x + 1$.", "output": "1) Differentiate term-by-term:\n2) d/dx(1) = 0.\n3) d/dx(x^{2}) = 2 x.\n4) d/dx(2 x^{3}) = 6 x^{2}.\n5) d/dx(4 x) = 4.\n6) Combine: f'(x) = 2 \\left(3 x^{2} + x + 2\\right).\nFinal answer: $6\\cdot x^{2} + 2\\cdot x + 4$", "final_answer": "6\\cdot x^{2} + 2\\cdot x + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aec136a46bfa796b", "problem_id": "05ab454044f0dc1e", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{3} + x^{2} + 4 x + 1$.", "output": "Differentiate term-by-term:\nd/dx(1) = 0.\nd/dx(x^{2}) = 2 x.\nd/dx(2 x^{3}) = 6 x^{2}.\nd/dx(4 x) = 4.\nCombine: f'(x) = 2 \\left(3 x^{2} + x + 2\\right).\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $6\\cdot x^{2} + 2\\cdot x + 4$", "final_answer": "6\\cdot x^{2} + 2\\cdot x + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2ab3901472f5cab4", "problem_id": "57c0e39aa121c117", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡4 (mod 5), x≡4 (mod 7). Give the solution modulo 35.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 35.\n3) Compute the CRT solution to get x ≡ 4 (mod 35).\nFinal answer: $x\\equiv 4\\pmod{35}$", "final_answer": "x\\equiv 4\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f73f02f5b1bce956", "problem_id": "57c0e39aa121c117", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡4 (mod 5), x≡4 (mod 7). Give the solution modulo 35.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 35.\nCompute the CRT solution to get x ≡ 4 (mod 35).\nCheck: 4 mod 5=4 and mod 7=4.\nFinal answer: $x\\equiv 4\\pmod{35}$", "final_answer": "x\\equiv 4\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fdce12f0642a142d", "problem_id": "8435a336b183864c", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "You want 42 liters of a 35% salt solution. You have a 10% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 42−x liters is 40% solution.\nSalt equation: 10x + 40(42−x) = 35·42.\n(10-40)x = (35-40)·42 ⇒ x = 7.\nThen 42−x = 35.\nCheck: 10·7 + 40·35 = 35·42.\nFinal answer: $7 L of 10% and 35 L of 40%$", "final_answer": "7 L of 10% and 35 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "669bfca662b2dba3", "problem_id": "8435a336b183864c", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "You want 42 liters of a 35% salt solution. You have a 10% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 42−x liters is 40% solution.\n2) Salt equation: 10x + 40(42−x) = 35·42.\n3) (10-40)x = (35-40)·42 ⇒ x = 7.\n4) Then 42−x = 35.\nFinal answer: $7 L of 10% and 35 L of 40%$", "final_answer": "7 L of 10% and 35 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5f54acdbf6b8580d", "problem_id": "d0caa8abfaf83e96", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A store sells a backpack for $475. It is discounted by 35%. You buy 11 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 475 × (1 − 35/100) = 475 × 65/100 = 1235/4.\n2) Subtotal = 11 × 1235/4 = 13585/4.\n3) Total with tax = 13585/4 × 107/100 = 290719/80.\nFinal answer: $\\frac{290719}{80}$", "final_answer": "\\frac{290719}{80}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5de0a17955f1c0f5", "problem_id": "d0caa8abfaf83e96", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A store sells a backpack for $475. It is discounted by 35%. You buy 11 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 475 × (1 − 35/100) = 475 × 65/100 = 1235/4.\nSubtotal = 11 × 1235/4 = 13585/4.\nTotal with tax = 13585/4 × 107/100 = 290719/80.\nCheck: recomputing the formula yields 290719/80.\nFinal answer: $\\frac{290719}{80}$", "final_answer": "\\frac{290719}{80}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8b3e830448c31eee", "problem_id": "205bd23339f01d8c", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Two friends split $430 in the ratio 8:2. How much does each receive?", "output": "Total ratio parts = 8+2 = 10.\nEach part = 430 ÷ 10 = 43.\nFirst share = 8 × 43 = 344.\nSecond share = 2 × 43 = 86.\nCheck: 344+86=430.\nFinal answer: $86$", "final_answer": "86", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9228b470ecf38c1e", "problem_id": "205bd23339f01d8c", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Two friends split $430 in the ratio 8:2. How much does each receive?", "output": "Total ratio parts = 8+2 = 10.\nEach part = 430 ÷ 10 = 43.\nFirst share = 8 × 43 = 344.\nFinal answer: $86$", "final_answer": "86", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "34da71a531f1ddbe", "problem_id": "861556b92d03f7ab", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{3} x}{3} + \\frac{4 x}{3} - \\frac{2 \\sqrt{3}}{9} + \\frac{4}{9}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{4}{3} - \\frac{\\sqrt{3}}{3}$, c=$\\frac{4}{9} - \\frac{2 \\sqrt{3}}{9}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{1}{3}$, so √Δ = $\\frac{\\sqrt{3}}{3}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{2}{3} + \\frac{\\sqrt{3}}{3}$.\nx2 = $- \\frac{2}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{2}{3} + \\frac{\\sqrt{3}}{3}, - \\frac{2}{3}\\}$", "final_answer": "x \\in \\{- \\frac{2}{3} + \\frac{\\sqrt{3}}{3}, - \\frac{2}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2e2eb72d72f735b2", "problem_id": "861556b92d03f7ab", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{3} x}{3} + \\frac{4 x}{3} - \\frac{2 \\sqrt{3}}{9} + \\frac{4}{9}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{4}{3} - \\frac{\\sqrt{3}}{3}$, c=$\\frac{4}{9} - \\frac{2 \\sqrt{3}}{9}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{1}{3}$, so √Δ = $\\frac{\\sqrt{3}}{3}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{2}{3} + \\frac{\\sqrt{3}}{3}$.\n7) x2 = $- \\frac{2}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{2}{3} + \\frac{\\sqrt{3}}{3}, - \\frac{2}{3}\\}$", "final_answer": "x \\in \\{- \\frac{2}{3} + \\frac{\\sqrt{3}}{3}, - \\frac{2}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "af9196d86670ff79", "problem_id": "191a722471eebfba", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Two friends split $192 in the ratio 8:8. How much does each receive?", "output": "1) Total ratio parts = 8+8 = 16.\n2) Each part = 192 ÷ 16 = 12.\n3) First share = 8 × 12 = 96.\n4) Second share = 8 × 12 = 96.\nFinal answer: $96$", "final_answer": "96", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8a15e6afb6412dc2", "problem_id": "191a722471eebfba", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Two friends split $192 in the ratio 8:8. How much does each receive?", "output": "Total ratio parts = 8+8 = 16.\nEach part = 192 ÷ 16 = 12.\nFirst share = 8 × 12 = 96.\nSecond share = 8 × 12 = 96.\nCheck: 96+96=192.\nFinal answer: $96$", "final_answer": "96", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4d837314f21c89b7", "problem_id": "4b7fce675f4fcb00", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the multiplicative inverse of 29 modulo 31.", "output": "We want x such that 29x ≡ 1 (mod 31).\nUse extended Euclid to find s,t with 29s + 31t = 1.\nOne Bezout identity is: 29·(15) + 31·(-14) = 1.\nReduce s modulo 31: x ≡ 15 (mod 31).\nCheck: 29·15 mod 31 = 1.\nFinal answer: $x\\equiv 15\\pmod{31}$", "final_answer": "x\\equiv 15\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6dd30982729cc19a", "problem_id": "4b7fce675f4fcb00", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the multiplicative inverse of 29 modulo 31.", "output": "1) We want x such that 29x ≡ 1 (mod 31).\n2) Use extended Euclid to find s,t with 29s + 31t = 1.\n3) One Bezout identity is: 29·(15) + 31·(-14) = 1.\n4) Reduce s modulo 31: x ≡ 15 (mod 31).\nFinal answer: $x\\equiv 15\\pmod{31}$", "final_answer": "x\\equiv 15\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3144419d74c5c9cd", "problem_id": "ca2d5a6670e3f47f", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 10 red balls and 6 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(16,8).\nFavorable samples: choose k reds and n−k blues: C(10,3)·C(6,5).\nCompute totals: C(16,8) = 12870, and favorable = 720.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{8}{143}$.\nFinal answer: $\\frac{8}{143}$", "final_answer": "\\frac{8}{143}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f3450fc51c023434", "problem_id": "ca2d5a6670e3f47f", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "An urn has 10 red balls and 6 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(16,8).\n3) Favorable samples: choose k reds and n−k blues: C(10,3)·C(6,5).\n4) Compute totals: C(16,8) = 12870, and favorable = 720.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{8}{143}$", "final_answer": "\\frac{8}{143}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8958619dfb64266c", "problem_id": "20a52a658b02ba3a", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} x^{3} - 2 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $x^{3} - 2 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\nSubtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{\\pi^{3}}{12} - \\frac{\\pi^{2}}{8} + \\frac{\\pi^{4}}{64} + 2$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{\\pi^{3}}{12} - \\frac{\\pi^{2}}{8} + \\frac{\\pi^{4}}{64} + 2$", "final_answer": "- \\frac{\\pi^{3}}{12} - \\frac{\\pi^{2}}{8} + \\frac{\\pi^{4}}{64} + 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b14abb4fee2844a2", "problem_id": "20a52a658b02ba3a", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} x^{3} - 2 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $x^{3} - 2 x^{2} - x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)}$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\n3) Subtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{\\pi^{3}}{12} - \\frac{\\pi^{2}}{8} + \\frac{\\pi^{4}}{64} + 2$.\n4) Simplify the result.\nFinal answer: $- \\frac{\\pi^{3}}{12} - \\frac{\\pi^{2}}{8} + \\frac{\\pi^{4}}{64} + 2$", "final_answer": "- \\frac{\\pi^{3}}{12} - \\frac{\\pi^{2}}{8} + \\frac{\\pi^{4}}{64} + 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "92caaad7507b84aa", "problem_id": "b4fe96cd19f58237", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 11 times, what is the probability of exactly 7 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=7, p=$\\frac{1}{6}$ and simplify.\n4) Here C(11,7) = 330.\nFinal answer: $\\frac{34375}{60466176}$", "final_answer": "\\frac{34375}{60466176}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d58b95459743a2cc", "problem_id": "b4fe96cd19f58237", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 11 times, what is the probability of exactly 7 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=11, k=7, p=$\\frac{1}{6}$ and simplify.\nFinal answer: $\\frac{34375}{60466176}$", "final_answer": "\\frac{34375}{60466176}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cb0497cf53d75382", "problem_id": "2d767d5104b58c0c", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$8 x - 7 y$ = $-114$\n$4 x + 8 y$ = $12$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $32 x - 28 y$ = $-456$.\nMultiply the second equation by $8$: $32 x + 64 y$ = $96$.\nSubtract to eliminate x: $- 92 y$ = $-552$.\nSolve for y: y = $6$.\nSubstitute back to find x: x = $-9$.\nCheck: plug (x,y)=(-9,6) into both equations.\nFinal answer: $(x,y)=\\left(-9, 6\\right)$", "final_answer": "(x,y)=\\left(-9, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ba66aeff02318003", "problem_id": "2d767d5104b58c0c", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$8 x - 7 y$ = $-114$\n$4 x + 8 y$ = $12$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $32 x - 28 y$ = $-456$.\n3) Multiply the second equation by $8$: $32 x + 64 y$ = $96$.\n4) Subtract to eliminate x: $- 92 y$ = $-552$.\n5) Solve for y: y = $6$.\n6) Substitute back to find x: x = $-9$.\nFinal answer: $(x,y)=\\left(-9, 6\\right)$", "final_answer": "(x,y)=\\left(-9, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "55742f2f50fc41fd", "problem_id": "99b06a6bb87dc2b6", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 0.\n3) Then the limit is 0.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 2.\nFinal answer: $2$", "final_answer": "2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "bb31beb1931f26f4", "problem_id": "7228c077100530a1", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 12 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=6, p=$\\frac{3}{8}$ and simplify.\n4) Here C(12,6) = 924.\nFinal answer: $\\frac{2631234375}{17179869184}$", "final_answer": "\\frac{2631234375}{17179869184}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d695a035a951e315", "problem_id": "ed6e7940c5cae4b0", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $65 in the ratio 4:1. How much does each receive?", "output": "1) Total ratio parts = 4+1 = 5.\n2) Each part = 65 ÷ 5 = 13.\n3) First share = 4 × 13 = 52.\n4) Second share = 1 × 13 = 13.\nFinal answer: $13$", "final_answer": "13", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cb2ff1b33b65b3e0", "problem_id": "ed6e7940c5cae4b0", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Two friends split $65 in the ratio 4:1. How much does each receive?", "output": "Total ratio parts = 4+1 = 5.\nEach part = 65 ÷ 5 = 13.\nFirst share = 4 × 13 = 52.\nSecond share = 1 × 13 = 13.\nCheck: 52+13=65.\nFinal answer: $13$", "final_answer": "13", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "357dae5f4dea1a38", "problem_id": "d5e905e5efd5690e", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$- 2 x + 7 y$ = $37$\n$4 x - 2 y$ = $10$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $- 8 x + 28 y$ = $148$.\n3) Multiply the second equation by $-2$: $- 8 x + 4 y$ = $-20$.\n4) Subtract to eliminate x: $24 y$ = $168$.\n5) Solve for y: y = $7$.\n6) Substitute back to find x: x = $6$.\nFinal answer: $(x,y)=\\left(6, 7\\right)$", "final_answer": "(x,y)=\\left(6, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dae073ceb571c173", "problem_id": "d5e905e5efd5690e", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- 2 x + 7 y$ = $37$\n$4 x - 2 y$ = $10$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $- 8 x + 28 y$ = $148$.\nMultiply the second equation by $-2$: $- 8 x + 4 y$ = $-20$.\nSubtract to eliminate x: $24 y$ = $168$.\nSolve for y: y = $7$.\nSubstitute back to find x: x = $6$.\nCheck: plug (x,y)=(6,7) into both equations.\nFinal answer: $(x,y)=\\left(6, 7\\right)$", "final_answer": "(x,y)=\\left(6, 7\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3654b5062ed72b50", "problem_id": "843e6276611ec637", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_7=21$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_7=7.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=7 and k=7.\n4) Compute C(13,6) = 1716.\nFinal answer: $1716$", "final_answer": "1716", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9981532183c669c4", "problem_id": "843e6276611ec637", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_7=21$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_7=7.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=7 and k=7.\nCompute C(13,6) = 1716.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $1716$", "final_answer": "1716", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "722d5551918b0ca1", "problem_id": "782dc68c089f05ef", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 18 x + 24$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} - 18 x + 24$ = $3 \\left(x - 4\\right) \\left(x - 2\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $4$.\n5) From the second factor: x = $2$.\nFinal answer: $x \\in \\{4, 2\\}$", "final_answer": "x \\in \\{4, 2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1457782ac9379117", "problem_id": "782dc68c089f05ef", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 18 x + 24$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} - 18 x + 24$ = $3 \\left(x - 4\\right) \\left(x - 2\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFinal answer: $x \\in \\{4, 2\\}$", "final_answer": "x \\in \\{4, 2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f47a45b90af0dd57", "problem_id": "7bfc2cae6f68214e", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$5 x$ = $-25$\n$8 x + 3 y$ = $-46$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $8$: $40 x$ = $-200$.\n3) Multiply the second equation by $5$: $40 x + 15 y$ = $-230$.\n4) Subtract to eliminate x: $- 15 y$ = $30$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, -2\\right)$", "final_answer": "(x,y)=\\left(-5, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8bda35b67451f8a6", "problem_id": "7bfc2cae6f68214e", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$5 x$ = $-25$\n$8 x + 3 y$ = $-46$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $8$: $40 x$ = $-200$.\nMultiply the second equation by $5$: $40 x + 15 y$ = $-230$.\nSubtract to eliminate x: $- 15 y$ = $30$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $-5$.\nCheck: plug (x,y)=(-5,-2) into both equations.\nFinal answer: $(x,y)=\\left(-5, -2\\right)$", "final_answer": "(x,y)=\\left(-5, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fc8cc7c74d26d592", "problem_id": "95060afadbab5ff5", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- 4 x - 2 y$ = $-6$\n$4 x - 2 y$ = $-14$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $- 16 x - 8 y$ = $-24$.\nMultiply the second equation by $-4$: $- 16 x + 8 y$ = $56$.\nSubtract to eliminate x: $- 16 y$ = $-80$.\nSolve for y: y = $5$.\nSubstitute back to find x: x = $-1$.\nCheck: plug (x,y)=(-1,5) into both equations.\nFinal answer: $(x,y)=\\left(-1, 5\\right)$", "final_answer": "(x,y)=\\left(-1, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4238cd226c0da7ec", "problem_id": "95060afadbab5ff5", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the linear system:\n$- 4 x - 2 y$ = $-6$\n$4 x - 2 y$ = $-14$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $- 16 x - 8 y$ = $-24$.\nMultiply the second equation by $-4$: $- 16 x + 8 y$ = $56$.\nFinal answer: $(x,y)=\\left(-1, 5\\right)$", "final_answer": "(x,y)=\\left(-1, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cf74d810752d4f7a", "problem_id": "c1c3605553bf0fbe", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "You want 19 liters of a 30% salt solution. You have a 20% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 19−x liters is 50% solution.\nSalt equation: 20x + 50(19−x) = 30·19.\n(20-50)x = (30-50)·19 ⇒ x = 38/3.\nThen 19−x = 19/3.\nCheck: 20·38/3 + 50·19/3 = 30·19.\nFinal answer: $38/3 L of 20% and 19/3 L of 50%$", "final_answer": "38/3 L of 20% and 19/3 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "19ac709090bc2feb", "problem_id": "c1c3605553bf0fbe", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "You want 19 liters of a 30% salt solution. You have a 20% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 19−x liters is 50% solution.\n2) Salt equation: 20x + 50(19−x) = 30·19.\n3) (20-50)x = (30-50)·19 ⇒ x = 38/3.\n4) Then 19−x = 19/3.\nFinal answer: $38/3 L of 20% and 19/3 L of 50%$", "final_answer": "38/3 L of 20% and 19/3 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ad3a9540564b449e", "problem_id": "755a7e23efb2939a", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} - x^{2} + 4 x - 16$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{3} - 1$ and a = -4.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-48$", "final_answer": "-48", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0ca9eaee63afa008", "problem_id": "755a7e23efb2939a", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} - x^{2} + 4 x - 16$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{3} - 1$ and a = -4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 3 x^{2}$.\n5) Evaluate at a: p'(-4) = $-48$.\nFinal answer: $-48$", "final_answer": "-48", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "38e0d2e3844d9830", "problem_id": "51a614cd6dd8062c", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A store sells a backpack for $382. It is discounted by 20%. You buy 2 backpacks. What is the total cost?", "output": "1) Discounted price = 382 × (1 − 20/100) = 382 × 80/100 = 1528/5.\n2) Subtotal = 2 × 1528/5 = 3056/5.\nFinal answer: $\\frac{3056}{5}$", "final_answer": "\\frac{3056}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ab6c5fdf28620ca7", "problem_id": "51a614cd6dd8062c", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A store sells a backpack for $382. It is discounted by 20%. You buy 2 backpacks. What is the total cost?", "output": "Discounted price = 382 × (1 − 20/100) = 382 × 80/100 = 1528/5.\nSubtotal = 2 × 1528/5 = 3056/5.\nCheck: recomputing the formula yields 3056/5.\nFinal answer: $\\frac{3056}{5}$", "final_answer": "\\frac{3056}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f1a58178e8c995cb", "problem_id": "e8d46fa965b6dbf9", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{2} - 2 x + 1$.", "output": "1) Differentiate term-by-term:\n2) d/dx(1) = 0.\n3) d/dx(- 3 x^{2}) = - 6 x.\n4) d/dx(- 2 x) = -2.\n5) Combine: f'(x) = 2 \\left(- 3 x - 1\\right).\nFinal answer: $- 6\\cdot x - 2$", "final_answer": "- 6\\cdot x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "52ebe9cf870cdd3f", "problem_id": "e8d46fa965b6dbf9", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{2} - 2 x + 1$.", "output": "Differentiate term-by-term:\nd/dx(1) = 0.\nd/dx(- 3 x^{2}) = - 6 x.\nd/dx(- 2 x) = -2.\nCombine: f'(x) = 2 \\left(- 3 x - 1\\right).\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 6\\cdot x - 2$", "final_answer": "- 6\\cdot x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ce81c1f5b6b61cca", "problem_id": "667a13669b7f7989", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A cyclist rides at 88 miles per hour for 2 hours, then at 85 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 88 × 2 = 176 miles.\n2) Second leg distance = 85 × 2 = 170 miles.\n3) Total distance = 176 + 170 = 346 miles.\nFinal answer: $346$", "final_answer": "346", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "76f695552f63c15d", "problem_id": "6cbc4c7a2f1b0b3b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} 2 x^{3} - x^{2} - 3 x - 1$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} - x^{2} - 4 x - 1.\n3) Then the limit is -4.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: -3.\nFinal answer: $-3$", "final_answer": "-3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "0185a2bc16e07ddc", "problem_id": "98cd826fd8630985", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A store sells a backpack for $78. It is discounted by 40%. You buy 10 backpacks. What is the total cost?", "output": "1) Discounted price = 78 × (1 − 40/100) = 78 × 60/100 = 234/5.\n2) Subtotal = 10 × 234/5 = 468.\nFinal answer: $468$", "final_answer": "468", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4485cf40114da575", "problem_id": "98cd826fd8630985", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $78. It is discounted by 40%. You buy 10 backpacks. What is the total cost?", "output": "Discounted price = 78 × (1 − 40/100) = 78 × 60/100 = 234/5.\nSubtotal = 10 × 234/5 = 468.\nCheck: recomputing the formula yields 468.\nFinal answer: $468$", "final_answer": "468", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "65c58864b4e9fcda", "problem_id": "d0971d09102f48a7", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Find the equation of the tangent line to $y=3 x^{3} - 3 x^{2} - x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=0$.", "output": "1) Differentiate: f'(x) = $9 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 1$.\n2) Slope at x=0: m = f'(0) = $- \\frac{1}{2}$.\n3) Point on curve: (0, f(0)) = (0, $0$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 0 = - \\frac{1}{2}(x − 0).\n6) Optionally expand: y = $- \\frac{x}{2}$.\nFinal answer: $y = - \\frac{x}{2}$", "final_answer": "y = - \\frac{x}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3aa9d3345eca1ee5", "problem_id": "d0971d09102f48a7", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} - 3 x^{2} - x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 2$ at $x=0$.", "output": "Differentiate: f'(x) = $9 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 1$.\nSlope at x=0: m = f'(0) = $- \\frac{1}{2}$.\nPoint on curve: (0, f(0)) = (0, $0$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 0 = - \\frac{1}{2}(x − 0).\nOptionally expand: y = $- \\frac{x}{2}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - \\frac{x}{2}$", "final_answer": "y = - \\frac{x}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9c7285422d60172a", "problem_id": "3df8b1a26ca92155", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 17 times, what is the probability of exactly 7 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=17, k=7, p=$\\frac{1}{5}$ and simplify.\n4) Here C(17,7) = 19448.\nFinal answer: $\\frac{20392706048}{762939453125}$", "final_answer": "\\frac{20392706048}{762939453125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0a0c1e2f15664c6a", "problem_id": "3df8b1a26ca92155", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 17 times, what is the probability of exactly 7 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=17, k=7, p=$\\frac{1}{5}$ and simplify.\nFinal answer: $\\frac{20392706048}{762939453125}$", "final_answer": "\\frac{20392706048}{762939453125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4f5d7a710735335f", "problem_id": "f634625e969e140e", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 11 red balls and 15 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(26,7).\n3) Favorable samples: choose k reds and n−k blues: C(11,3)·C(15,4).\n4) Compute totals: C(26,7) = 657800, and favorable = 225225.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{63}{184}$", "final_answer": "\\frac{63}{184}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b429523be2ee058e", "problem_id": "f634625e969e140e", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 11 red balls and 15 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(26,7).\nFavorable samples: choose k reds and n−k blues: C(11,3)·C(15,4).\nCompute totals: C(26,7) = 657800, and favorable = 225225.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{63}{184}$.\nFinal answer: $\\frac{63}{184}$", "final_answer": "\\frac{63}{184}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f2ac705777807af4", "problem_id": "699230ee195f6ac9", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Two friends split $1848 in the ratio 2:9. How much does each receive?", "output": "1) Total ratio parts = 2+9 = 11.\n2) Each part = 1848 ÷ 11 = 168.\n3) First share = 2 × 168 = 336.\n4) Second share = 9 × 168 = 1512.\nFinal answer: $1512$", "final_answer": "1512", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f726d8b5ee23652c", "problem_id": "699230ee195f6ac9", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $1848 in the ratio 2:9. How much does each receive?", "output": "Total ratio parts = 2+9 = 11.\nEach part = 1848 ÷ 11 = 168.\nFirst share = 2 × 168 = 336.\nSecond share = 9 × 168 = 1512.\nCheck: 336+1512=1848.\nFinal answer: $1512$", "final_answer": "1512", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f407b35fc698dde7", "problem_id": "b4e4dcd971f2b828", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the multiplicative inverse of 12 modulo 41.", "output": "1) We want x such that 12x ≡ 1 (mod 41).\n2) Use extended Euclid to find s,t with 12s + 41t = 1.\n3) One Bezout identity is: 12·(-17) + 41·(5) = 1.\n4) Reduce s modulo 41: x ≡ 24 (mod 41).\nFinal answer: $x\\equiv 24\\pmod{41}$", "final_answer": "x\\equiv 24\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1ea4a5ddac570c90", "problem_id": "b4e4dcd971f2b828", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the multiplicative inverse of 12 modulo 41.", "output": "We want x such that 12x ≡ 1 (mod 41).\nUse extended Euclid to find s,t with 12s + 41t = 1.\nOne Bezout identity is: 12·(-17) + 41·(5) = 1.\nReduce s modulo 41: x ≡ 24 (mod 41).\nCheck: 12·24 mod 41 = 1.\nFinal answer: $x\\equiv 24\\pmod{41}$", "final_answer": "x\\equiv 24\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0112ccde5717d74c", "problem_id": "e50a764b775360d1", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the congruences: x≡5 (mod 11), x≡0 (mod 8). Give the solution modulo 88.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 88.\nCompute the CRT solution to get x ≡ 16 (mod 88).\nFinal answer: $x\\equiv 16\\pmod{88}$", "final_answer": "x\\equiv 16\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d9d66f7a2802ba85", "problem_id": "e50a764b775360d1", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡5 (mod 11), x≡0 (mod 8). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 16 (mod 88).\nFinal answer: $x\\equiv 16\\pmod{88}$", "final_answer": "x\\equiv 16\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "67c464c18386063d", "problem_id": "5095868e7eb18e6b", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡1 (mod 7), x≡4 (mod 5). Give the solution modulo 35.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 35.\n3) Compute the CRT solution to get x ≡ 29 (mod 35).\nFinal answer: $x\\equiv 29\\pmod{35}$", "final_answer": "x\\equiv 29\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d36ce06244c37d0e", "problem_id": "5095868e7eb18e6b", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡1 (mod 7), x≡4 (mod 5). Give the solution modulo 35.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 35.\nCompute the CRT solution to get x ≡ 29 (mod 35).\nCheck: 29 mod 7=1 and mod 5=4.\nFinal answer: $x\\equiv 29\\pmod{35}$", "final_answer": "x\\equiv 29\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "376a308261e3e717", "problem_id": "552ce9f80be21247", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{2} + 3 x + 1$.", "output": "1) Differentiate term-by-term:\n2) d/dx(1) = 0.\n3) d/dx(- 3 x^{2}) = - 6 x.\n4) d/dx(3 x) = 3.\n5) Combine: f'(x) = 3 \\left(1 - 2 x\\right).\nFinal answer: $3 - 6\\cdot x$", "final_answer": "3 - 6\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e73817e0352ebf3a", "problem_id": "552ce9f80be21247", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{2} + 3 x + 1$.", "output": "Differentiate term-by-term:\nd/dx(1) = 0.\nd/dx(- 3 x^{2}) = - 6 x.\nFinal answer: $3 - 6\\cdot x$", "final_answer": "3 - 6\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "279774fc4ade5a3c", "problem_id": "10fdd11d5f25c0b9", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$6 x - y$ = $39$\n$x - y$ = $9$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $6 x - y$ = $39$.\nMultiply the second equation by $6$: $6 x - 6 y$ = $54$.\nSubtract to eliminate x: $5 y$ = $-15$.\nSolve for y: y = $-3$.\nSubstitute back to find x: x = $6$.\nCheck: plug (x,y)=(6,-3) into both equations.\nFinal answer: $(x,y)=\\left(6, -3\\right)$", "final_answer": "(x,y)=\\left(6, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "05e5763c3307d03b", "problem_id": "10fdd11d5f25c0b9", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the linear system:\n$6 x - y$ = $39$\n$x - y$ = $9$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $6 x - y$ = $39$.\n3) Multiply the second equation by $6$: $6 x - 6 y$ = $54$.\n4) Subtract to eliminate x: $5 y$ = $-15$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $6$.\nFinal answer: $(x,y)=\\left(6, -3\\right)$", "final_answer": "(x,y)=\\left(6, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2c7deb8419b187a6", "problem_id": "f2140ebd800804ff", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 21 red balls and 15 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(36,8).\n3) Favorable samples: choose k reds and n−k blues: C(21,3)·C(15,5).\n4) Compute totals: C(36,8) = 30260340, and favorable = 3993990.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{12103}{91698}$", "final_answer": "\\frac{12103}{91698}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "07e2435d14eb38df", "problem_id": "49f21da7ba95c98b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(x^{2} - 2 x + 3\\right) \\cos{\\left(x^{2} + x - 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x^{2} \\cos{\\left(\\left(x^{2} + x\\right) - 3 \\right)}) = - x^{2} \\sin{\\left(x^{2} + x - 3 \\right)}$.\n3) So $f'(x) = 3 x^{2} \\sin{\\left(x^{2} + x - 3 \\right)} + x e^{x} - 4 x \\sin{\\left(x^{2} + x - 3 \\right)} + 3 e^{x} - 3 \\sin{\\left(x^{2} + x - 3 \\right)} - 2 \\cos{\\left(\\left(x^{2} + x\\right) - 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x^{2} \\cos{\\left(\\left(x^{2} + x\\right) - 3 \\right)}) = - x^{2} \\left(2 x + 1\\right) \\sin{\\left(x^{2} + x - 3 \\right)} + 2 x \\cos{\\left(\\left(x^{2} + x\\right) - 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x - 1\\right) \\cos{\\left(\\left(x^{2} + x\\right) - 3 \\right)} + \\left(x + 2\\right) e^{x} - \\left(2 x + 1\\right) \\left(x \\left(x - 2\\right) + 3\\right) \\sin{\\left(x^{2} + x - 3 \\right)} + e^{x}$.\nFinal answer: $2 \\left(x - 1\\right) \\cos{\\left(\\left(x^{2} + x\\right) - 3 \\right)} + \\left(x + 2\\right) e^{x} - \\left(2 x + 1\\right) \\left(x \\left(x - 2\\right) + 3\\right) \\sin{\\left(x^{2} + x - 3 \\right)} + e^{x}$", "final_answer": "2 \\left(x - 1\\right) \\cos{\\left(\\left(x^{2} + x\\right) - 3 \\right)} + \\left(x + 2\\right) e^{x} - \\left(2 x + 1\\right) \\left(x \\left(x - 2\\right) + 3\\right) \\sin{\\left(x^{2} + x - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d8714f8cadb08292", "problem_id": "2a60ca0d7709be18", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Two friends split $656 in the ratio 8:8. How much does each receive?", "output": "1) Total ratio parts = 8+8 = 16.\n2) Each part = 656 ÷ 16 = 41.\n3) First share = 8 × 41 = 328.\n4) Second share = 8 × 41 = 328.\nFinal answer: $328$", "final_answer": "328", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "083204ccd496e510", "problem_id": "2a60ca0d7709be18", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $656 in the ratio 8:8. How much does each receive?", "output": "Total ratio parts = 8+8 = 16.\nEach part = 656 ÷ 16 = 41.\nFirst share = 8 × 41 = 328.\nSecond share = 8 × 41 = 328.\nCheck: 328+328=656.\nFinal answer: $328$", "final_answer": "328", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c76e45e6c83a0fe4", "problem_id": "bea478ecbfb3a297", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A cyclist rides at 40 miles per hour for 1/2 hours, then at 79 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 40 × 1/2 = 20 miles.\n2) Second leg distance = 79 × 2 = 158 miles.\n3) Total distance = 20 + 158 = 178 miles.\nFinal answer: $178$", "final_answer": "178", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c08dddf1198a9ca4", "problem_id": "bea478ecbfb3a297", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A cyclist rides at 40 miles per hour for 1/2 hours, then at 79 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 40 × 1/2 = 20 miles.\nSecond leg distance = 79 × 2 = 158 miles.\nTotal distance = 20 + 158 = 178 miles.\nFinal answer: $178$", "final_answer": "178", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3779c826ff6629e5", "problem_id": "ca632275531815dc", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} 2 x^{3} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $2 x^{3} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- 5 \\pi - 1 + \\frac{45 \\pi^{2}}{8} + \\frac{255 \\pi^{4}}{32}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- 5\\cdot \\pi - 1 + \\frac{45\\cdot \\pi^{2}}{8} + \\frac{255\\cdot \\pi^{4}}{32}$", "final_answer": "- 5\\cdot \\pi - 1 + \\frac{45\\cdot \\pi^{2}}{8} + \\frac{255\\cdot \\pi^{4}}{32}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "11959e04ca6fa2a0", "problem_id": "ca632275531815dc", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} 2 x^{3} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{3} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- 5 \\pi - 1 + \\frac{45 \\pi^{2}}{8} + \\frac{255 \\pi^{4}}{32}$.\n4) Simplify the result.\nFinal answer: $- 5\\cdot \\pi - 1 + \\frac{45\\cdot \\pi^{2}}{8} + \\frac{255\\cdot \\pi^{4}}{32}$", "final_answer": "- 5\\cdot \\pi - 1 + \\frac{45\\cdot \\pi^{2}}{8} + \\frac{255\\cdot \\pi^{4}}{32}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "90ec000f25f7d8d9", "problem_id": "c1354d9fd36a3cbd", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡4 (mod 11), x≡0 (mod 5). Give the solution modulo 55.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 55.\n3) Compute the CRT solution to get x ≡ 15 (mod 55).\nFinal answer: $x\\equiv 15\\pmod{55}$", "final_answer": "x\\equiv 15\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "873a8187f8a5768f", "problem_id": "5d6146d3e3d49489", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x \\sin{\\left(x \\right)}) = 2 x \\cos{\\left(x \\right)}$.\n3) So $f'(x) = 2 x^{2} \\cos{\\left(x \\right)} + 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x \\sin{\\left(x \\right)}) = 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 x^{2} \\cos{\\left(x \\right)} + 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$.\nFinal answer: $2 x^{2} \\cos{\\left(x \\right)} + 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}$", "final_answer": "2 x^{2} \\cos{\\left(x \\right)} + 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "859b334aad41afd2", "problem_id": "1e743ef7d8f3be4b", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 12 x - \\frac{27}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$12$, c=$- \\frac{27}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $225$, so √Δ = $15$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{1}{2}$.\n7) x2 = $- \\frac{9}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{1}{2}, - \\frac{9}{2}\\}$", "final_answer": "x \\in \\{\\frac{1}{2}, - \\frac{9}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "09b0e5523075c34f", "problem_id": "1e743ef7d8f3be4b", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 12 x - \\frac{27}{4}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$12$, c=$- \\frac{27}{4}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $225$, so √Δ = $15$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{1}{2}$.\nx2 = $- \\frac{9}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{1}{2}, - \\frac{9}{2}\\}$", "final_answer": "x \\in \\{\\frac{1}{2}, - \\frac{9}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "08365c671a3d2b49", "problem_id": "d37a8dd6d1d2c67d", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{3} - 3 x^{2} - 2 x - 3$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-3) = 0.\n3) d/dx(- x^{3}) = - 3 x^{2}.\n4) d/dx(- 3 x^{2}) = - 6 x.\n5) d/dx(- 2 x) = -2.\n6) Combine: f'(x) = - 3 x^{2} - 6 x - 2.\nFinal answer: $- 3\\cdot x^{2} - 6\\cdot x - 2$", "final_answer": "- 3\\cdot x^{2} - 6\\cdot x - 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "155617e06777d376", "problem_id": "61597971a8e26afc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - x^{2} - 2 x - 3$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} - 3 x - 3.\n3) Then the limit is -21.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: -18.\nFinal answer: $-18$", "final_answer": "-18", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7ec82ae9f69c430e", "problem_id": "757f713564c0a349", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=42$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=42 and k=10.\n4) Compute C(51,9) = 3042312350.\nFinal answer: $3042312350$", "final_answer": "3042312350", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e949b7e547d01657", "problem_id": "e1636336dac3aceb", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{4} - 2 x^{3} + 2 x^{2} + 4 x + 2$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(2 x^{4}) = 8 x^{4}$.\n3) So $f'(x) = 8 x^{4} - 6 x^{2} + 4 x + 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(2 x^{4}) = 8 x^{3}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(4 x^{3} - 3 x^{2} + 2 x + 2\\right)$.\nFinal answer: $2 \\left(4 x^{3} - 3 x^{2} + 2 x + 2\\right)$", "final_answer": "2 \\left(4 x^{3} - 3 x^{2} + 2 x + 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "20c7dd4f25b6c4a2", "problem_id": "9eed65d742995755", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "You want 27 liters of a 30% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 27−x liters is 50% solution.\n2) Salt equation: 25x + 50(27−x) = 30·27.\n3) (25-50)x = (30-50)·27 ⇒ x = 108/5.\n4) Then 27−x = 27/5.\nFinal answer: $108/5 L of 25% and 27/5 L of 50%$", "final_answer": "108/5 L of 25% and 27/5 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "edd8eef9af66ec8b", "problem_id": "9eed65d742995755", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "You want 27 liters of a 30% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 27−x liters is 50% solution.\nSalt equation: 25x + 50(27−x) = 30·27.\n(25-50)x = (30-50)·27 ⇒ x = 108/5.\nThen 27−x = 27/5.\nCheck: 25·108/5 + 50·27/5 = 30·27.\nFinal answer: $108/5 L of 25% and 27/5 L of 50%$", "final_answer": "108/5 L of 25% and 27/5 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5e3a2560d92ffab4", "problem_id": "40c1b32428a6ddd3", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡0 (mod 3), x≡8 (mod 11). Give the solution modulo 33.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 33.\n3) Compute the CRT solution to get x ≡ 30 (mod 33).\nFinal answer: $x\\equiv 30\\pmod{33}$", "final_answer": "x\\equiv 30\\pmod{33}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "65a227fe7ba44ee0", "problem_id": "4154d331602d7bf5", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} 3 x^{2} + 4 x + 3$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{2} + 3 x + 3.\n3) Then the limit is 9.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: 10.\nFinal answer: $10$", "final_answer": "10", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "da8cb67179eb31e0", "problem_id": "db5ebad0461bb95a", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$- 3 x - 3 y$ = $-6$\n$- 2 y$ = $6$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $-3$: $6 y$ = $-18$.\n4) Subtract to eliminate x: $- 6 y$ = $18$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, -3\\right)$", "final_answer": "(x,y)=\\left(5, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dcae73962315a26e", "problem_id": "db5ebad0461bb95a", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the linear system:\n$- 3 x - 3 y$ = $-6$\n$- 2 y$ = $6$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $-3$: $6 y$ = $-18$.\nFinal answer: $(x,y)=\\left(5, -3\\right)$", "final_answer": "(x,y)=\\left(5, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "474bc481a04beb7a", "problem_id": "c02f2f7beacb69fb", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int x^{4} + 2 x^{3} + 4 x^{2} - x\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $x^{4} + 2 x^{3} + 4 x^{2} - x$.\nCompute an antiderivative: F(x) = $\\frac{x^{5}}{5} + \\frac{x^{4}}{2} + \\frac{4 x^{3}}{3} - \\frac{x^{2}}{2}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{x^{5}}{5} + \\frac{x^{4}}{2} + \\frac{4\\cdot x^{3}}{3} - \\frac{x^{2}}{2}$", "final_answer": "C + \\frac{x^{5}}{5} + \\frac{x^{4}}{2} + \\frac{4\\cdot x^{3}}{3} - \\frac{x^{2}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cc788144bb4df61c", "problem_id": "c02f2f7beacb69fb", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int x^{4} + 2 x^{3} + 4 x^{2} - x\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $x^{4} + 2 x^{3} + 4 x^{2} - x$.\n3) Compute an antiderivative: F(x) = $\\frac{x^{5}}{5} + \\frac{x^{4}}{2} + \\frac{4 x^{3}}{3} - \\frac{x^{2}}{2}$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{x^{5}}{5} + \\frac{x^{4}}{2} + \\frac{4\\cdot x^{3}}{3} - \\frac{x^{2}}{2}$", "final_answer": "C + \\frac{x^{5}}{5} + \\frac{x^{4}}{2} + \\frac{4\\cdot x^{3}}{3} - \\frac{x^{2}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e866348bc90c9feb", "problem_id": "1fb52ca219b48ac7", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=25$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=25 and k=7.\nFinal answer: $736281$", "final_answer": "736281", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "50a6ae7398d04cb0", "problem_id": "1fb52ca219b48ac7", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=25$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=25 and k=7.\n4) Compute C(31,6) = 736281.\nFinal answer: $736281$", "final_answer": "736281", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "02a3bc10bb7a7f35", "problem_id": "423eb28ae810e642", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Worker A finishes a job in 5 hours and Worker B finishes it in 9 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/5 job/hour. B's rate = 1/9 job/hour.\n2) Combined rate = 1/5 + 1/9 = 14/45 job/hour.\n3) Time = 1 ÷ (combined rate) = 45/14 hours.\nFinal answer: $\\frac{45}{14}$", "final_answer": "\\frac{45}{14}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8426834c3e873315", "problem_id": "423eb28ae810e642", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Worker A finishes a job in 5 hours and Worker B finishes it in 9 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/5 job/hour. B's rate = 1/9 job/hour.\nCombined rate = 1/5 + 1/9 = 14/45 job/hour.\nTime = 1 ÷ (combined rate) = 45/14 hours.\nFinal answer: $\\frac{45}{14}$", "final_answer": "\\frac{45}{14}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1ebfb1e5493764ba", "problem_id": "4d66671352de87e2", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "You want 27 liters of a 35% salt solution. You have a 30% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 27−x liters is 40% solution.\nSalt equation: 30x + 40(27−x) = 35·27.\n(30-40)x = (35-40)·27 ⇒ x = 27/2.\nThen 27−x = 27/2.\nCheck: 30·27/2 + 40·27/2 = 35·27.\nFinal answer: $27/2 L of 30% and 27/2 L of 40%$", "final_answer": "27/2 L of 30% and 27/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "484d6390f32a5747", "problem_id": "4d66671352de87e2", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "You want 27 liters of a 35% salt solution. You have a 30% solution and a 40% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 30% solution, so 27−x liters is 40% solution.\n2) Salt equation: 30x + 40(27−x) = 35·27.\n3) (30-40)x = (35-40)·27 ⇒ x = 27/2.\n4) Then 27−x = 27/2.\nFinal answer: $27/2 L of 30% and 27/2 L of 40%$", "final_answer": "27/2 L of 30% and 27/2 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "705ebcb9dbbc0b67", "problem_id": "09726800bf81c01d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} + 4 x^{2} + 3 x$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(3 x) = 3 x$.\n3) So $f'(x) = 3 x^{2} + 11 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(3 x) = 3$.\nTherefore the correct derivative is $f'(x) = 3 x^{2} + 8 x + 3$.\nFinal answer: $3 x^{2} + 8 x + 3$", "final_answer": "3 x^{2} + 8 x + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "60199d9086e0897b", "problem_id": "ab6d828c420d30fe", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A cyclist rides at 71 miles per hour for 2 hours, then at 40 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 71 × 2 = 142 miles.\n2) Second leg distance = 40 × 3 = 120 miles.\n3) Total distance = 142 + 120 = 262 miles.\nFinal answer: $262$", "final_answer": "262", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7b37bd73b44a4538", "problem_id": "616de88dd6a8580f", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 4 x^{2} - 2 x + 2$ at $x=0$.", "output": "Differentiate: f'(x) = $- 8 x - 2$.\nSlope at x=0: m = f'(0) = $-2$.\nPoint on curve: (0, f(0)) = (0, $2$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 2 = -2(x − 0).\nOptionally expand: y = $2 - 2 x$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 2 - 2\\cdot x$", "final_answer": "y = 2 - 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6e9dcdcf5691d72b", "problem_id": "616de88dd6a8580f", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the equation of the tangent line to $y=- 4 x^{2} - 2 x + 2$ at $x=0$.", "output": "1) Differentiate: f'(x) = $- 8 x - 2$.\n2) Slope at x=0: m = f'(0) = $-2$.\n3) Point on curve: (0, f(0)) = (0, $2$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 2 = -2(x − 0).\n6) Optionally expand: y = $2 - 2 x$.\nFinal answer: $y = 2 - 2\\cdot x$", "final_answer": "y = 2 - 2\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e821bd1242a14507", "problem_id": "c576674aaded5ee4", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(x^{2} \\sin{\\left(x \\right)}) = x^{2} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = x^{2} \\cos{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(x^{2} \\sin{\\left(x \\right)}) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$.\nFinal answer: $x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$", "final_answer": "x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ca0f2bcc513e1f67", "problem_id": "159e9967148b2c0b", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 23 red balls and 26 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(49,11).\nFavorable samples: choose k reds and n−k blues: C(23,4)·C(26,7).\nCompute totals: C(49,11) = 29135916264, and favorable = 5824819000.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{347875}{1740081}$.\nFinal answer: $\\frac{347875}{1740081}$", "final_answer": "\\frac{347875}{1740081}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d6e44410a03d70ab", "problem_id": "5b83661aed86468b", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A cyclist rides at 45 miles per hour for 3/2 hours, then at 80 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 45 × 3/2 = 135/2 miles.\n2) Second leg distance = 80 × 1 = 80 miles.\n3) Total distance = 135/2 + 80 = 295/2 miles.\nFinal answer: $\\frac{295}{2}$", "final_answer": "\\frac{295}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b4ee4e4c979cc793", "problem_id": "5b83661aed86468b", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A cyclist rides at 45 miles per hour for 3/2 hours, then at 80 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 45 × 3/2 = 135/2 miles.\nSecond leg distance = 80 × 1 = 80 miles.\nTotal distance = 135/2 + 80 = 295/2 miles.\nCheck: adding both legs gives 295/2.\nFinal answer: $\\frac{295}{2}$", "final_answer": "\\frac{295}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4a3dfb6eadd1e494", "problem_id": "6e1b0ce4d1396cad", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "You want 46 liters of a 25% salt solution. You have a 10% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 46−x liters is 60% solution.\n2) Salt equation: 10x + 60(46−x) = 25·46.\n3) (10-60)x = (25-60)·46 ⇒ x = 161/5.\n4) Then 46−x = 69/5.\nFinal answer: $161/5 L of 10% and 69/5 L of 60%$", "final_answer": "161/5 L of 10% and 69/5 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3769ff89ac81b8fa", "problem_id": "6e1b0ce4d1396cad", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "You want 46 liters of a 25% salt solution. You have a 10% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 46−x liters is 60% solution.\nSalt equation: 10x + 60(46−x) = 25·46.\n(10-60)x = (25-60)·46 ⇒ x = 161/5.\nThen 46−x = 69/5.\nCheck: 10·161/5 + 60·69/5 = 25·46.\nFinal answer: $161/5 L of 10% and 69/5 L of 60%$", "final_answer": "161/5 L of 10% and 69/5 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f03d14333113124d", "problem_id": "d5e0ed802fdb9c6c", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the congruences: x≡0 (mod 7), x≡9 (mod 13). Give the solution modulo 91.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 91.\nCompute the CRT solution to get x ≡ 35 (mod 91).\nFinal answer: $x\\equiv 35\\pmod{91}$", "final_answer": "x\\equiv 35\\pmod{91}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c2084d73eca0fd78", "problem_id": "d5e0ed802fdb9c6c", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡0 (mod 7), x≡9 (mod 13). Give the solution modulo 91.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 91.\nCompute the CRT solution to get x ≡ 35 (mod 91).\nCheck: 35 mod 7=0 and mod 13=9.\nFinal answer: $x\\equiv 35\\pmod{91}$", "final_answer": "x\\equiv 35\\pmod{91}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7efe23f41139fada", "problem_id": "2ba99f8dd6b9bc78", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A store sells a backpack for $106. It is discounted by 10%. You buy 5 backpacks. What is the total cost?", "output": "Discounted price = 106 × (1 − 10/100) = 106 × 90/100 = 477/5.\nSubtotal = 5 × 477/5 = 477.\nCheck: recomputing the formula yields 477.\nFinal answer: $477$", "final_answer": "477", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fa7ee46f56c52206", "problem_id": "2ba99f8dd6b9bc78", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A store sells a backpack for $106. It is discounted by 10%. You buy 5 backpacks. What is the total cost?", "output": "1) Discounted price = 106 × (1 − 10/100) = 106 × 90/100 = 477/5.\n2) Subtotal = 5 × 477/5 = 477.\nFinal answer: $477$", "final_answer": "477", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e89939ebf635332c", "problem_id": "303abfbf507cb79d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{2} \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 3 x^{2} \\sin{\\left(x \\right)}) = - 3 x^{2} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 3 x^{2} \\cos{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 3 x^{2} \\sin{\\left(x \\right)}) = - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $- 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$", "final_answer": "- 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "27ec8bdcaf74d8e1", "problem_id": "205142eee7ab9351", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡2 (mod 3), x≡12 (mod 13). Give the solution modulo 39.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 39.\n3) Compute the CRT solution to get x ≡ 38 (mod 39).\nFinal answer: $x\\equiv 38\\pmod{39}$", "final_answer": "x\\equiv 38\\pmod{39}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72646cad042c5b07", "problem_id": "0517c6fca487fa0f", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Worker A finishes a job in 21 hours and Worker B finishes it in 16 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/21 job/hour. B's rate = 1/16 job/hour.\n2) Combined rate = 1/21 + 1/16 = 37/336 job/hour.\n3) Time = 1 ÷ (combined rate) = 336/37 hours.\nFinal answer: $\\frac{336}{37}$", "final_answer": "\\frac{336}{37}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0eaba323095f3eae", "problem_id": "0517c6fca487fa0f", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Worker A finishes a job in 21 hours and Worker B finishes it in 16 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/21 job/hour. B's rate = 1/16 job/hour.\nCombined rate = 1/21 + 1/16 = 37/336 job/hour.\nTime = 1 ÷ (combined rate) = 336/37 hours.\nFinal answer: $\\frac{336}{37}$", "final_answer": "\\frac{336}{37}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0bcbc649d42f43b4", "problem_id": "8605a01c83c9c1d7", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{10}$. If it is flipped 10 times, what is the probability of exactly 7 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=7, p=$\\frac{3}{10}$ and simplify.\n4) Here C(10,7) = 120.\nFinal answer: $\\frac{2250423}{250000000}$", "final_answer": "\\frac{2250423}{250000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ae06c8edc650afb1", "problem_id": "622f9230972b3858", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 3 x^{2} + 5 x - 4$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} + 4 x - 4.\n3) Then the limit is -11.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: -12.\nFinal answer: $-12$", "final_answer": "-12", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c2125df54581ee61", "problem_id": "c2beab8a129c5d2e", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} 3 x + 9$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{2}$ and a = 3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $6 x$.\n5) Evaluate at a: p'(3) = $18$.\nFinal answer: $18$", "final_answer": "18", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "705ebcb9dbbc0b67", "problem_id": "09726800bf81c01d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} + 4 x^{2} + 3 x$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(3 x) = 3 x$.\n3) So $f'(x) = 3 x^{2} + 11 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(3 x) = 3$.\nTherefore the correct derivative is $f'(x) = 3 x^{2} + 8 x + 3$.\nFinal answer: $3 x^{2} + 8 x + 3$", "final_answer": "3 x^{2} + 8 x + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "fd32c29e51c20f48", "problem_id": "53b6a033fee0142e", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 7 red balls and 6 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(13,8).\n3) Favorable samples: choose k reds and n−k blues: C(7,2)·C(6,6).\n4) Compute totals: C(13,8) = 1287, and favorable = 21.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{7}{429}$", "final_answer": "\\frac{7}{429}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7f96c071aa0bea70", "problem_id": "53b6a033fee0142e", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 7 red balls and 6 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(13,8).\nFavorable samples: choose k reds and n−k blues: C(7,2)·C(6,6).\nCompute totals: C(13,8) = 1287, and favorable = 21.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{7}{429}$.\nFinal answer: $\\frac{7}{429}$", "final_answer": "\\frac{7}{429}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9cb3f155f72c3ebd", "problem_id": "3d2aa1ab039b2edc", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡2 (mod 8), x≡1 (mod 7). Give the solution modulo 56.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 56.\n3) Compute the CRT solution to get x ≡ 50 (mod 56).\nFinal answer: $x\\equiv 50\\pmod{56}$", "final_answer": "x\\equiv 50\\pmod{56}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ae06c8edc650afb1", "problem_id": "622f9230972b3858", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 3 x^{2} + 5 x - 4$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} + 4 x - 4.\n3) Then the limit is -11.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: -12.\nFinal answer: $-12$", "final_answer": "-12", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "db655db7eabf6968", "problem_id": "f7d4f6af5c069b4a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{2} + 3 x - 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- x^{2}) = - 2 x^{2}$.\n3) So $f'(x) = 3 - 2 x^{2}$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- x^{2}) = - 2 x$.\nTherefore the correct derivative is $f'(x) = 3 - 2 x$.\nFinal answer: $3 - 2 x$", "final_answer": "3 - 2 x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ec7777974d1829e8", "problem_id": "a4ce9d27fb6da342", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{7}$. If it is flipped 12 times, what is the probability of exactly 7 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=7, p=$\\frac{4}{7}$ and simplify.\n4) Here C(12,7) = 792.\nFinal answer: $\\frac{3153199104}{13841287201}$", "final_answer": "\\frac{3153199104}{13841287201}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6ce01a7c19765a2c", "problem_id": "52c5dddb7ecc8806", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$8 x + 10 y$ = $-36$\n$- 6 x - 6 y$ = $30$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-6$: $- 48 x - 60 y$ = $216$.\n3) Multiply the second equation by $8$: $- 48 x - 48 y$ = $240$.\n4) Subtract to eliminate x: $- 12 y$ = $-24$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $-7$.\nFinal answer: $(x,y)=\\left(-7, 2\\right)$", "final_answer": "(x,y)=\\left(-7, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "30cdefa86b4fe89a", "problem_id": "52c5dddb7ecc8806", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the linear system:\n$8 x + 10 y$ = $-36$\n$- 6 x - 6 y$ = $30$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-6$: $- 48 x - 60 y$ = $216$.\nMultiply the second equation by $8$: $- 48 x - 48 y$ = $240$.\nFinal answer: $(x,y)=\\left(-7, 2\\right)$", "final_answer": "(x,y)=\\left(-7, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a2a36c5bf81f183", "problem_id": "5614bda5b7c9618e", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 12 times, what is the probability of exactly 7 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=7, p=$\\frac{2}{7}$ and simplify.\n4) Here C(12,7) = 792.\nFinal answer: $\\frac{316800000}{13841287201}$", "final_answer": "\\frac{316800000}{13841287201}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1bcf79010a9646ee", "problem_id": "93b58f8747b5969b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x^{3} + 5 x^{2} + 11 x + 25$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to x^{3} + 5 x^{2} + 10 x + 25.\n3) Then the limit is 73.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 75.\nFinal answer: $75$", "final_answer": "75", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ba57c60782d5d2cf", "problem_id": "6651b79ace2bf048", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 2 \\sqrt{7} x - x + \\sqrt{7}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$- 2 \\sqrt{7} - 1$, c=$\\sqrt{7}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $29 - 4 \\sqrt{7}$, so √Δ = $\\sqrt{29 - 4 \\sqrt{7}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{1}{4} + \\frac{\\sqrt{29 - 4 \\sqrt{7}}}{4} + \\frac{\\sqrt{7}}{2}$.\n7) x2 = $- \\frac{\\sqrt{29 - 4 \\sqrt{7}}}{4} + \\frac{1}{4} + \\frac{\\sqrt{7}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{1}{4} + \\frac{\\sqrt{29 - 4\\cdot \\sqrt{7}}}{4} + \\frac{\\sqrt{7}}{2}, - \\frac{\\sqrt{29 - 4\\cdot \\sqrt{7}}}{4} + \\frac{1}{4} + \\frac{\\sqrt{7}}{2}\\}$", "final_answer": "x \\in \\{\\frac{1}{4} + \\frac{\\sqrt{29 - 4\\cdot \\sqrt{7}}}{4} + \\frac{\\sqrt{7}}{2}, - \\frac{\\sqrt{29 - 4\\cdot \\sqrt{7}}}{4} + \\frac{1}{4} + \\frac{\\sqrt{7}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8b086bbdd5f14014", "problem_id": "6651b79ace2bf048", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 2 \\sqrt{7} x - x + \\sqrt{7}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$- 2 \\sqrt{7} - 1$, c=$\\sqrt{7}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $29 - 4 \\sqrt{7}$, so √Δ = $\\sqrt{29 - 4 \\sqrt{7}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{1}{4} + \\frac{\\sqrt{29 - 4 \\sqrt{7}}}{4} + \\frac{\\sqrt{7}}{2}$.\nx2 = $- \\frac{\\sqrt{29 - 4 \\sqrt{7}}}{4} + \\frac{1}{4} + \\frac{\\sqrt{7}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{1}{4} + \\frac{\\sqrt{29 - 4\\cdot \\sqrt{7}}}{4} + \\frac{\\sqrt{7}}{2}, - \\frac{\\sqrt{29 - 4\\cdot \\sqrt{7}}}{4} + \\frac{1}{4} + \\frac{\\sqrt{7}}{2}\\}$", "final_answer": "x \\in \\{\\frac{1}{4} + \\frac{\\sqrt{29 - 4\\cdot \\sqrt{7}}}{4} + \\frac{\\sqrt{7}}{2}, - \\frac{\\sqrt{29 - 4\\cdot \\sqrt{7}}}{4} + \\frac{1}{4} + \\frac{\\sqrt{7}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d17b2b2c2f9c19a1", "problem_id": "abc9c07fe4eaa7b5", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 29 red balls and 19 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(48,8).\nFavorable samples: choose k reds and n−k blues: C(29,2)·C(19,6).\nCompute totals: C(48,8) = 377348994, and favorable = 11015592.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1835932}{62891499}$.\nFinal answer: $\\frac{1835932}{62891499}$", "final_answer": "\\frac{1835932}{62891499}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "60dfff12ad62256e", "problem_id": "abc9c07fe4eaa7b5", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "An urn has 29 red balls and 19 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(48,8).\n3) Favorable samples: choose k reds and n−k blues: C(29,2)·C(19,6).\n4) Compute totals: C(48,8) = 377348994, and favorable = 11015592.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1835932}{62891499}$", "final_answer": "\\frac{1835932}{62891499}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7d7d9a2f66010278", "problem_id": "b2cb06c80c852e6d", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the congruences: x≡5 (mod 11), x≡0 (mod 5). Give the solution modulo 55.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 55.\n3) Compute the CRT solution to get x ≡ 5 (mod 55).\nFinal answer: $x\\equiv 5\\pmod{55}$", "final_answer": "x\\equiv 5\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c18124108f6be872", "problem_id": "b2cb06c80c852e6d", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the congruences: x≡5 (mod 11), x≡0 (mod 5). Give the solution modulo 55.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 55.\nCompute the CRT solution to get x ≡ 5 (mod 55).\nFinal answer: $x\\equiv 5\\pmod{55}$", "final_answer": "x\\equiv 5\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d94b51b64d3da294", "problem_id": "1b5b76fd2e4408b0", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the linear system:\n$4 x - 3 y$ = $-17$\n$3 y$ = $9$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $4$: $12 y$ = $36$.\n4) Subtract to eliminate x: $- 12 y$ = $-36$.\n5) Solve for y: y = $3$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, 3\\right)$", "final_answer": "(x,y)=\\left(-2, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c00bf86cb182e17b", "problem_id": "1b5b76fd2e4408b0", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the linear system:\n$4 x - 3 y$ = $-17$\n$3 y$ = $9$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $4$: $12 y$ = $36$.\nFinal answer: $(x,y)=\\left(-2, 3\\right)$", "final_answer": "(x,y)=\\left(-2, 3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b208d4506c346201", "problem_id": "17abbe5abd1ea33d", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "An urn has 8 red balls and 15 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(23,4).\n3) Favorable samples: choose k reds and n−k blues: C(8,2)·C(15,2).\n4) Compute totals: C(23,4) = 8855, and favorable = 2940.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{84}{253}$", "final_answer": "\\frac{84}{253}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a82edea5ed017aa9", "problem_id": "17abbe5abd1ea33d", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "An urn has 8 red balls and 15 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(23,4).\nFavorable samples: choose k reds and n−k blues: C(8,2)·C(15,2).\nFinal answer: $\\frac{84}{253}$", "final_answer": "\\frac{84}{253}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "70a1eec06ea0d274", "problem_id": "4969e57a48d938b1", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - 2 x^{2} - 8 x - 22$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 2 x^{3} - 2 x^{2} + 2 x + 2$ and a = 3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 6 x^{2} - 4 x + 2$.\n5) Evaluate at a: p'(3) = $-64$.\nFinal answer: $-64$", "final_answer": "-64", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7f1317ee4a708beb", "problem_id": "4969e57a48d938b1", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - 2 x^{2} - 8 x - 22$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 2 x^{3} - 2 x^{2} + 2 x + 2$ and a = 3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-64$", "final_answer": "-64", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0450cd9bb06a8e3f", "problem_id": "d9cbb50ad371dc1d", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$8 x + 7 y$ = $20$\n$8 x - 6 y$ = $-32$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $8$: $64 x + 56 y$ = $160$.\nMultiply the second equation by $8$: $64 x - 48 y$ = $-256$.\nSubtract to eliminate x: $104 y$ = $416$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $-1$.\nCheck: plug (x,y)=(-1,4) into both equations.\nFinal answer: $(x,y)=\\left(-1, 4\\right)$", "final_answer": "(x,y)=\\left(-1, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "39a867757c668f05", "problem_id": "d9cbb50ad371dc1d", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$8 x + 7 y$ = $20$\n$8 x - 6 y$ = $-32$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $8$: $64 x + 56 y$ = $160$.\n3) Multiply the second equation by $8$: $64 x - 48 y$ = $-256$.\n4) Subtract to eliminate x: $104 y$ = $416$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, 4\\right)$", "final_answer": "(x,y)=\\left(-1, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "685e8310f1078110", "problem_id": "fd7f52f1a95e6f97", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "A store sells a backpack for $382. It is discounted by 5%. You buy 11 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 382 × (1 − 5/100) = 382 × 95/100 = 3629/10.\nSubtotal = 11 × 3629/10 = 39919/10.\nTotal with tax = 39919/10 × 108/100 = 1077813/250.\nFinal answer: $\\frac{1077813}{250}$", "final_answer": "\\frac{1077813}{250}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b54417d2aca7cf3", "problem_id": "fd7f52f1a95e6f97", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $382. It is discounted by 5%. You buy 11 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 382 × (1 − 5/100) = 382 × 95/100 = 3629/10.\nSubtotal = 11 × 3629/10 = 39919/10.\nTotal with tax = 39919/10 × 108/100 = 1077813/250.\nCheck: recomputing the formula yields 1077813/250.\nFinal answer: $\\frac{1077813}{250}$", "final_answer": "\\frac{1077813}{250}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "698e39887200fb13", "problem_id": "08e7491ce8169c21", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 6 x - 2 \\sqrt{2} x + 4 \\sqrt{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$-6 - 2 \\sqrt{2}$, c=$4 \\sqrt{2}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $44 - 24 \\sqrt{2}$, so √Δ = $\\sqrt{44 - 24 \\sqrt{2}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{2}}{3} + \\frac{\\sqrt{11 - 6 \\sqrt{2}}}{3} + 1$.\nx2 = $- \\frac{\\sqrt{11 - 6 \\sqrt{2}}}{3} + \\frac{\\sqrt{2}}{3} + 1$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{2}}{3} + \\frac{\\sqrt{11 - 6\\cdot \\sqrt{2}}}{3} + 1, - \\frac{\\sqrt{11 - 6\\cdot \\sqrt{2}}}{3} + \\frac{\\sqrt{2}}{3} + 1\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{2}}{3} + \\frac{\\sqrt{11 - 6\\cdot \\sqrt{2}}}{3} + 1, - \\frac{\\sqrt{11 - 6\\cdot \\sqrt{2}}}{3} + \\frac{\\sqrt{2}}{3} + 1\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "34f8e106988e79ab", "problem_id": "344f33ecb0617f1c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} 3 x + 14$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to 2 x + 14.\n3) Then the limit is 22.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: 26.\nFinal answer: $26$", "final_answer": "26", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2b67ba34e13076dc", "problem_id": "ae98557da9436ccd", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Worker A finishes a job in 6 hours and Worker B finishes it in 22 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/6 job/hour. B's rate = 1/22 job/hour.\n2) Combined rate = 1/6 + 1/22 = 7/33 job/hour.\n3) Time = 1 ÷ (combined rate) = 33/7 hours.\nFinal answer: $\\frac{33}{7}$", "final_answer": "\\frac{33}{7}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "36f8e64711756e2f", "problem_id": "ae98557da9436ccd", "style": "answer_only", "instruction": "Final answer only.", "input": "Worker A finishes a job in 6 hours and Worker B finishes it in 22 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "$\\frac{33}{7}$", "final_answer": "\\frac{33}{7}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "edb21e4d085634c3", "problem_id": "8aa500c233f21ba4", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the multiplicative inverse of 8 modulo 47.", "output": "1) We want x such that 8x ≡ 1 (mod 47).\n2) Use extended Euclid to find s,t with 8s + 47t = 1.\n3) One Bezout identity is: 8·(6) + 47·(-1) = 1.\n4) Reduce s modulo 47: x ≡ 6 (mod 47).\nFinal answer: $x\\equiv 6\\pmod{47}$", "final_answer": "x\\equiv 6\\pmod{47}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cdc343dfb110f9a9", "problem_id": "8aa500c233f21ba4", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 8 modulo 47.", "output": "We want x such that 8x ≡ 1 (mod 47).\nUse extended Euclid to find s,t with 8s + 47t = 1.\nOne Bezout identity is: 8·(6) + 47·(-1) = 1.\nReduce s modulo 47: x ≡ 6 (mod 47).\nCheck: 8·6 mod 47 = 1.\nFinal answer: $x\\equiv 6\\pmod{47}$", "final_answer": "x\\equiv 6\\pmod{47}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fb5ad593410b419f", "problem_id": "49578745ecf6750f", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - 2 x - 4$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 2 x^{2} + 2 x + 3$ and a = 3.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-10$", "final_answer": "-10", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "114e066c87200cdd", "problem_id": "49578745ecf6750f", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} - 2 x - 4$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 2 x^{2} + 2 x + 3$ and a = 3.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $2 - 4 x$.\n5) Evaluate at a: p'(3) = $-10$.\nFinal answer: $-10$", "final_answer": "-10", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3ba8bb8981c75a26", "problem_id": "930d3d88126efc21", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 4 x^{4} + 3 x^{3} - 4 x^{2} + 3 x - 3\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $- 4 x^{4} + 3 x^{3} - 4 x^{2} + 3 x - 3$.\n3) Compute an antiderivative: F(x) = $- \\frac{4 x^{5}}{5} + \\frac{3 x^{4}}{4} - \\frac{4 x^{3}}{3} + \\frac{3 x^{2}}{2} - 3 x$.\n4) Include the constant of integration C.\nFinal answer: $C - \\frac{4\\cdot x^{5}}{5} + \\frac{3\\cdot x^{4}}{4} - \\frac{4\\cdot x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} - 3\\cdot x$", "final_answer": "C - \\frac{4\\cdot x^{5}}{5} + \\frac{3\\cdot x^{4}}{4} - \\frac{4\\cdot x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} - 3\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d3b6762d77a42aea", "problem_id": "930d3d88126efc21", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int - 4 x^{4} + 3 x^{3} - 4 x^{2} + 3 x - 3\\,dx$.", "output": "$C - \\frac{4\\cdot x^{5}}{5} + \\frac{3\\cdot x^{4}}{4} - \\frac{4\\cdot x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} - 3\\cdot x$", "final_answer": "C - \\frac{4\\cdot x^{5}}{5} + \\frac{3\\cdot x^{4}}{4} - \\frac{4\\cdot x^{3}}{3} + \\frac{3\\cdot x^{2}}{2} - 3\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d26ff362f1e207e2", "problem_id": "e51c209f43f83a65", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} - 2 x^{2} + 2 \\sin{\\left(x \\right)} + 3$ at $x=1$.", "output": "1) Differentiate: f'(x) = $- 3 x^{2} - 4 x + 2 \\cos{\\left(x \\right)}$.\n2) Slope at x=1: m = f'(1) = $-7 + 2 \\cos{\\left(1 \\right)}$.\n3) Point on curve: (1, f(1)) = (1, $2 \\sin{\\left(1 \\right)}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 2 \\sin{\\left(1 \\right)} = -7 + 2 \\cos{\\left(1 \\right)}(x − 1).\n6) Optionally expand: y = $- 7 x + 2 x \\cos{\\left(1 \\right)} - 2 \\cos{\\left(1 \\right)} + 2 \\sin{\\left(1 \\right)} + 7$.\nFinal answer: $y = - 7\\cdot x + 2\\cdot x\\cdot \\cos{\\left(1 \\right)} - 2\\cdot \\cos{\\left(1 \\right)} + 2\\cdot \\sin{\\left(1 \\right)} + 7$", "final_answer": "y = - 7\\cdot x + 2\\cdot x\\cdot \\cos{\\left(1 \\right)} - 2\\cdot \\cos{\\left(1 \\right)} + 2\\cdot \\sin{\\left(1 \\right)} + 7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "13a270704fe30880", "problem_id": "e51c209f43f83a65", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Find the equation of the tangent line to $y=- x^{3} - 2 x^{2} + 2 \\sin{\\left(x \\right)} + 3$ at $x=1$.", "output": "Differentiate: f'(x) = $- 3 x^{2} - 4 x + 2 \\cos{\\left(x \\right)}$.\nSlope at x=1: m = f'(1) = $-7 + 2 \\cos{\\left(1 \\right)}$.\nPoint on curve: (1, f(1)) = (1, $2 \\sin{\\left(1 \\right)}$).\nFinal answer: $y = - 7\\cdot x + 2\\cdot x\\cdot \\cos{\\left(1 \\right)} - 2\\cdot \\cos{\\left(1 \\right)} + 2\\cdot \\sin{\\left(1 \\right)} + 7$", "final_answer": "y = - 7\\cdot x + 2\\cdot x\\cdot \\cos{\\left(1 \\right)} - 2\\cdot \\cos{\\left(1 \\right)} + 2\\cdot \\sin{\\left(1 \\right)} + 7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1d627796c7967d08", "problem_id": "bbb92d67c0e97e7e", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(3 x^{2} + 3\\right) \\cos{\\left(x^{2} - 3 x \\right)}$.", "output": "Let f(x) = $\\left(x + 1\\right) e^{x} + \\left(3 x^{2} + 3\\right) \\cos{\\left(x^{2} - 3 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $6 x \\cos{\\left(x^{2} - 3 x \\right)} + \\left(x + 1\\right) e^{x} - 3 \\left(2 x - 3\\right) \\left(x^{2} + 1\\right) \\sin{\\left(x^{2} - 3 x \\right)} + e^{x}$.\nFinal answer: $6\\cdot x\\cdot \\cos{\\left(x^{2} - 3\\cdot x \\right)} + \\left(x + 1\\right)\\cdot e^{x} - \\left(6\\cdot x - 9\\right)\\cdot \\left(x^{2} + 1\\right)\\cdot \\sin{\\left(x^{2} - 3\\cdot x \\right)} + e^{x}$", "final_answer": "6\\cdot x\\cdot \\cos{\\left(x^{2} - 3\\cdot x \\right)} + \\left(x + 1\\right)\\cdot e^{x} - \\left(6\\cdot x - 9\\right)\\cdot \\left(x^{2} + 1\\right)\\cdot \\sin{\\left(x^{2} - 3\\cdot x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1a792ce1c2fabe24", "problem_id": "bbb92d67c0e97e7e", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(3 x^{2} + 3\\right) \\cos{\\left(x^{2} - 3 x \\right)}$.", "output": "1) Let f(x) = $\\left(x + 1\\right) e^{x} + \\left(3 x^{2} + 3\\right) \\cos{\\left(x^{2} - 3 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $6 x \\cos{\\left(x^{2} - 3 x \\right)} + \\left(x + 1\\right) e^{x} - 3 \\left(2 x - 3\\right) \\left(x^{2} + 1\\right) \\sin{\\left(x^{2} - 3 x \\right)} + e^{x}$.\nFinal answer: $6\\cdot x\\cdot \\cos{\\left(x^{2} - 3\\cdot x \\right)} + \\left(x + 1\\right)\\cdot e^{x} - \\left(6\\cdot x - 9\\right)\\cdot \\left(x^{2} + 1\\right)\\cdot \\sin{\\left(x^{2} - 3\\cdot x \\right)} + e^{x}$", "final_answer": "6\\cdot x\\cdot \\cos{\\left(x^{2} - 3\\cdot x \\right)} + \\left(x + 1\\right)\\cdot e^{x} - \\left(6\\cdot x - 9\\right)\\cdot \\left(x^{2} + 1\\right)\\cdot \\sin{\\left(x^{2} - 3\\cdot x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d8b9decf7f5c9a2c", "problem_id": "3a90f0fab25e131b", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the multiplicative inverse of 23 modulo 37.", "output": "1) We want x such that 23x ≡ 1 (mod 37).\n2) Use extended Euclid to find s,t with 23s + 37t = 1.\n3) One Bezout identity is: 23·(-8) + 37·(5) = 1.\n4) Reduce s modulo 37: x ≡ 29 (mod 37).\nFinal answer: $x\\equiv 29\\pmod{37}$", "final_answer": "x\\equiv 29\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ef234d4ea2c56a4a", "problem_id": "3a90f0fab25e131b", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Find the multiplicative inverse of 23 modulo 37.", "output": "We want x such that 23x ≡ 1 (mod 37).\nUse extended Euclid to find s,t with 23s + 37t = 1.\nOne Bezout identity is: 23·(-8) + 37·(5) = 1.\nFinal answer: $x\\equiv 29\\pmod{37}$", "final_answer": "x\\equiv 29\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e63076f28974aad", "problem_id": "2fdb7dfe6e808e40", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "You want 60 liters of a 35% salt solution. You have a 20% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 60−x liters is 60% solution.\nSalt equation: 20x + 60(60−x) = 35·60.\n(20-60)x = (35-60)·60 ⇒ x = 75/2.\nThen 60−x = 45/2.\nCheck: 20·75/2 + 60·45/2 = 35·60.\nFinal answer: $75/2 L of 20% and 45/2 L of 60%$", "final_answer": "75/2 L of 20% and 45/2 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0fdeea18e666892b", "problem_id": "2fdb7dfe6e808e40", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "You want 60 liters of a 35% salt solution. You have a 20% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 60−x liters is 60% solution.\n2) Salt equation: 20x + 60(60−x) = 35·60.\n3) (20-60)x = (35-60)·60 ⇒ x = 75/2.\n4) Then 60−x = 45/2.\nFinal answer: $75/2 L of 20% and 45/2 L of 60%$", "final_answer": "75/2 L of 20% and 45/2 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "af08e22514863887", "problem_id": "2f98b98631d9deb2", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} 2 x^{3} - 5 x^{2} + 2 x - 5$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} - 5 x^{2} + x - 5.\n3) Then the limit is -13.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: -14.\nFinal answer: $-14$", "final_answer": "-14", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "98cfb6b5f7791338", "problem_id": "78023e414ed8793b", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- 4 x - 8 y$ = $36$\n$- 7 x - 9 y$ = $53$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-7$: $28 x + 56 y$ = $-252$.\nMultiply the second equation by $-4$: $28 x + 36 y$ = $-212$.\nSubtract to eliminate x: $20 y$ = $-40$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $-5$.\nCheck: plug (x,y)=(-5,-2) into both equations.\nFinal answer: $(x,y)=\\left(-5, -2\\right)$", "final_answer": "(x,y)=\\left(-5, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c1325780bd9a2d3e", "problem_id": "b7af3b9fab48cd8f", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 22 times, what is the probability of exactly 9 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=22, k=9, p=$\\frac{1}{5}$ and simplify.\n4) Here C(22,9) = 497420.\nFinal answer: $\\frac{6676258226176}{476837158203125}$", "final_answer": "\\frac{6676258226176}{476837158203125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e6990d457aeb1934", "problem_id": "b7af3b9fab48cd8f", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 22 times, what is the probability of exactly 9 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=22, k=9, p=$\\frac{1}{5}$ and simplify.\nHere C(22,9) = 497420.\nCheck: expression simplifies to $\\frac{6676258226176}{476837158203125}$.\nFinal answer: $\\frac{6676258226176}{476837158203125}$", "final_answer": "\\frac{6676258226176}{476837158203125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "987bea5b901547fe", "problem_id": "8d9d2ea553630b32", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡4 (mod 5), x≡9 (mod 11). Give the solution modulo 55.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 55.\n3) Compute the CRT solution to get x ≡ 9 (mod 55).\nFinal answer: $x\\equiv 9\\pmod{55}$", "final_answer": "x\\equiv 9\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "22d748c621945c62", "problem_id": "8d9d2ea553630b32", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the congruences: x≡4 (mod 5), x≡9 (mod 11). Give the solution modulo 55.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 55.\nCompute the CRT solution to get x ≡ 9 (mod 55).\nFinal answer: $x\\equiv 9\\pmod{55}$", "final_answer": "x\\equiv 9\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "126ff34fcf5f568d", "problem_id": "cdd52f80c6c99668", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡0 (mod 4), x≡6 (mod 11). Give the solution modulo 44.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 44.\nCompute the CRT solution to get x ≡ 28 (mod 44).\nCheck: 28 mod 4=0 and mod 11=6.\nFinal answer: $x\\equiv 28\\pmod{44}$", "final_answer": "x\\equiv 28\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c3e8b82b273013df", "problem_id": "cdd52f80c6c99668", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡0 (mod 4), x≡6 (mod 11). Give the solution modulo 44.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 44.\n3) Compute the CRT solution to get x ≡ 28 (mod 44).\nFinal answer: $x\\equiv 28\\pmod{44}$", "final_answer": "x\\equiv 28\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bb32b5b2d32666bd", "problem_id": "dfce624489a18310", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$- 7 x + 8 y$ = $71$\n$x$ = $-9$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $- 7 x + 8 y$ = $71$.\nMultiply the second equation by $-7$: $- 7 x$ = $63$.\nSubtract to eliminate x: $8 y$ = $8$.\nSolve for y: y = $1$.\nSubstitute back to find x: x = $-9$.\nCheck: plug (x,y)=(-9,1) into both equations.\nFinal answer: $(x,y)=\\left(-9, 1\\right)$", "final_answer": "(x,y)=\\left(-9, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "96546f9bb95a88bc", "problem_id": "dfce624489a18310", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$- 7 x + 8 y$ = $71$\n$x$ = $-9$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $- 7 x + 8 y$ = $71$.\n3) Multiply the second equation by $-7$: $- 7 x$ = $63$.\n4) Subtract to eliminate x: $8 y$ = $8$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $-9$.\nFinal answer: $(x,y)=\\left(-9, 1\\right)$", "final_answer": "(x,y)=\\left(-9, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3ad1d8412344428a", "problem_id": "489b2f990336aa6b", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡0 (mod 5). Give the solution modulo 15.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 15.\nCompute the CRT solution to get x ≡ 10 (mod 15).\nCheck: 10 mod 3=1 and mod 5=0.\nFinal answer: $x\\equiv 10\\pmod{15}$", "final_answer": "x\\equiv 10\\pmod{15}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a3ee07344ac59049", "problem_id": "489b2f990336aa6b", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡0 (mod 5). Give the solution modulo 15.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 15.\n3) Compute the CRT solution to get x ≡ 10 (mod 15).\nFinal answer: $x\\equiv 10\\pmod{15}$", "final_answer": "x\\equiv 10\\pmod{15}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7332a080a22ef248", "problem_id": "49a83a7ce7700bd5", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{3}$. If it is flipped 19 times, what is the probability of exactly 11 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=19, k=11, p=$\\frac{1}{3}$ and simplify.\n4) Here C(19,11) = 75582.\nFinal answer: $\\frac{2149888}{129140163}$", "final_answer": "\\frac{2149888}{129140163}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b74d6108857ccff6", "problem_id": "49a83a7ce7700bd5", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{3}$. If it is flipped 19 times, what is the probability of exactly 11 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=19, k=11, p=$\\frac{1}{3}$ and simplify.\nHere C(19,11) = 75582.\nCheck: expression simplifies to $\\frac{2149888}{129140163}$.\nFinal answer: $\\frac{2149888}{129140163}$", "final_answer": "\\frac{2149888}{129140163}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8e8b7fec6b55b2de", "problem_id": "4154d331602d7bf5", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} 3 x^{2} + 4 x + 3$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $3 x^{3} + x^{2} - x - 3$ and a = 1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $10$", "final_answer": "10", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8b1a0daaeb04a30f", "problem_id": "4154d331602d7bf5", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} 3 x^{2} + 4 x + 3$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{3} + x^{2} - x - 3$ and a = 1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $9 x^{2} + 2 x - 1$.\n5) Evaluate at a: p'(1) = $10$.\nFinal answer: $10$", "final_answer": "10", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0557ef609a72ee21", "problem_id": "86ad5e3e1f9c98d9", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - x^{2} + 5 x - 10$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{3} + 3 x^{2} - 1$ and a = -2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 3 x^{2} + 6 x$.\n5) Evaluate at a: p'(-2) = $-24$.\nFinal answer: $-24$", "final_answer": "-24", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9f3171b4aab76772", "problem_id": "86ad5e3e1f9c98d9", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - x^{2} + 5 x - 10$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{3} + 3 x^{2} - 1$ and a = -2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 3 x^{2} + 6 x$.\nEvaluate at a: p'(-2) = $-24$.\nCheck: the identity gives $-24$.\nFinal answer: $-24$", "final_answer": "-24", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2e4fcbe69debab13", "problem_id": "1fc957bc038ad882", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} + 3 x^{2} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 3$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $6 x^{2} + 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\n2) Slope at x=-2: m = f'(-2) = $2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e} + 10$.\n3) Point on curve: (-2, f(-2)) = (-2, $-3 + \\cos{\\left(4 \\right)} + e^{-1}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -3 + \\cos{\\left(4 \\right)} + e^{-1} = 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e} + 10(x − -2).\n6) Optionally expand: y = $2 x \\sin{\\left(4 \\right)} + \\frac{x}{2 e} + 10 x + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 17$.\nFinal answer: $y = 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 10\\cdot x + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 17$", "final_answer": "y = 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 10\\cdot x + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 17", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "45bed78b2ebe76bf", "problem_id": "1fc957bc038ad882", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} + 3 x^{2} - 2 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} - 3$ at $x=-2$.", "output": "Differentiate: f'(x) = $6 x^{2} + 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 2$.\nSlope at x=-2: m = f'(-2) = $2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e} + 10$.\nPoint on curve: (-2, f(-2)) = (-2, $-3 + \\cos{\\left(4 \\right)} + e^{-1}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -3 + \\cos{\\left(4 \\right)} + e^{-1} = 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e} + 10(x − -2).\nOptionally expand: y = $2 x \\sin{\\left(4 \\right)} + \\frac{x}{2 e} + 10 x + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 17$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 10\\cdot x + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 17$", "final_answer": "y = 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 10\\cdot x + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 17", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "39b7c23d8c04ce34", "problem_id": "1ccf259f164ae01f", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{3} x}{2} - \\frac{7 x}{4} - \\frac{15}{4} - \\frac{15 \\sqrt{3}}{8}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{3 \\sqrt{3}}{2} - \\frac{7}{4}$, c=$- \\frac{15}{4} - \\frac{15 \\sqrt{3}}{8}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{51 \\sqrt{3}}{4} + \\frac{397}{16}$, so √Δ = $\\sqrt{\\frac{51 \\sqrt{3}}{4} + \\frac{397}{16}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{7}{8} + \\frac{3 \\sqrt{3}}{4} + \\frac{\\sqrt{204 \\sqrt{3} + 397}}{8}$.\n7) x2 = $- \\frac{\\sqrt{204 \\sqrt{3} + 397}}{8} + \\frac{7}{8} + \\frac{3 \\sqrt{3}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{7}{8} + \\frac{3\\cdot \\sqrt{3}}{4} + \\frac{\\sqrt{204\\cdot \\sqrt{3} + 397}}{8}, - \\frac{\\sqrt{204\\cdot \\sqrt{3} + 397}}{8} + \\frac{7}{8} + \\frac{3\\cdot \\sqrt{3}}{4}\\}$", "final_answer": "x \\in \\{\\frac{7}{8} + \\frac{3\\cdot \\sqrt{3}}{4} + \\frac{\\sqrt{204\\cdot \\sqrt{3} + 397}}{8}, - \\frac{\\sqrt{204\\cdot \\sqrt{3} + 397}}{8} + \\frac{7}{8} + \\frac{3\\cdot \\sqrt{3}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2bbbe0ec33663db9", "problem_id": "1ccf259f164ae01f", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{3} x}{2} - \\frac{7 x}{4} - \\frac{15}{4} - \\frac{15 \\sqrt{3}}{8}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{3 \\sqrt{3}}{2} - \\frac{7}{4}$, c=$- \\frac{15}{4} - \\frac{15 \\sqrt{3}}{8}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{51 \\sqrt{3}}{4} + \\frac{397}{16}$, so √Δ = $\\sqrt{\\frac{51 \\sqrt{3}}{4} + \\frac{397}{16}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{7}{8} + \\frac{3 \\sqrt{3}}{4} + \\frac{\\sqrt{204 \\sqrt{3} + 397}}{8}$.\nx2 = $- \\frac{\\sqrt{204 \\sqrt{3} + 397}}{8} + \\frac{7}{8} + \\frac{3 \\sqrt{3}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{7}{8} + \\frac{3\\cdot \\sqrt{3}}{4} + \\frac{\\sqrt{204\\cdot \\sqrt{3} + 397}}{8}, - \\frac{\\sqrt{204\\cdot \\sqrt{3} + 397}}{8} + \\frac{7}{8} + \\frac{3\\cdot \\sqrt{3}}{4}\\}$", "final_answer": "x \\in \\{\\frac{7}{8} + \\frac{3\\cdot \\sqrt{3}}{4} + \\frac{\\sqrt{204\\cdot \\sqrt{3} + 397}}{8}, - \\frac{\\sqrt{204\\cdot \\sqrt{3} + 397}}{8} + \\frac{7}{8} + \\frac{3\\cdot \\sqrt{3}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "302210f6f467ba8b", "problem_id": "734e74ab4fb4d3e7", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_5=39$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_5=29.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=29 and k=5.\n4) Compute C(33,4) = 40920.\nFinal answer: $40920$", "final_answer": "40920", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ed0a32be068e978", "problem_id": "734e74ab4fb4d3e7", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_5=39$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_5=29.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=29 and k=5.\nCompute C(33,4) = 40920.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $40920$", "final_answer": "40920", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a527ef1b91cc2f8a", "problem_id": "e59bff104421303f", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 20 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=20, k=5, p=$\\frac{4}{5}$ and simplify.\n4) Here C(20,5) = 15504.\nFinal answer: $\\frac{15876096}{95367431640625}$", "final_answer": "\\frac{15876096}{95367431640625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "73d8604ba790306e", "problem_id": "e59bff104421303f", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 20 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=20, k=5, p=$\\frac{4}{5}$ and simplify.\nFinal answer: $\\frac{15876096}{95367431640625}$", "final_answer": "\\frac{15876096}{95367431640625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "39c1d5a7cfe8cdc7", "problem_id": "7bb37e807261cd5b", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=26$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=26 and k=10.\nCompute C(35,9) = 70607460.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $70607460$", "final_answer": "70607460", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aa2c574178c928d9", "problem_id": "7bb37e807261cd5b", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=26$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=26 and k=10.\n4) Compute C(35,9) = 70607460.\nFinal answer: $70607460$", "final_answer": "70607460", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bca48f44622317fb", "problem_id": "eb92fba8fbd55830", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_9=27$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_9=18.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=18 and k=9.\n4) Compute C(26,8) = 1562275.\nFinal answer: $1562275$", "final_answer": "1562275", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5c16cdf19f8a0333", "problem_id": "eb92fba8fbd55830", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_9=27$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_9=18.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=18 and k=9.\nCompute C(26,8) = 1562275.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $1562275$", "final_answer": "1562275", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "06f3c1f8fff36809", "problem_id": "c07a883a13e0a6c0", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A cyclist rides at 76 miles per hour for 1 hours, then at 46 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 76 × 1 = 76 miles.\n2) Second leg distance = 46 × 3/2 = 69 miles.\n3) Total distance = 76 + 69 = 145 miles.\nFinal answer: $145$", "final_answer": "145", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c6f092e2ca0a0f00", "problem_id": "c07a883a13e0a6c0", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 76 miles per hour for 1 hours, then at 46 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 76 × 1 = 76 miles.\nSecond leg distance = 46 × 3/2 = 69 miles.\nTotal distance = 76 + 69 = 145 miles.\nCheck: adding both legs gives 145.\nFinal answer: $145$", "final_answer": "145", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b1d056e73cb7ae73", "problem_id": "aff99cb21133b028", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 27 x + 60$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} - 27 x + 60$ = $3 \\left(x - 5\\right) \\left(x - 4\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $5$.\n5) From the second factor: x = $4$.\nFinal answer: $x \\in \\{5, 4\\}$", "final_answer": "x \\in \\{5, 4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1089e3f849043611", "problem_id": "aff99cb21133b028", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 27 x + 60$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} - 27 x + 60$ = $3 \\left(x - 5\\right) \\left(x - 4\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $5$.\nFrom the second factor: x = $4$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{5, 4\\}$", "final_answer": "x \\in \\{5, 4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "74b04f10f2e8b56d", "problem_id": "ac7de62326996d20", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Worker A finishes a job in 5 hours and Worker B finishes it in 12 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/5 job/hour. B's rate = 1/12 job/hour.\nCombined rate = 1/5 + 1/12 = 17/60 job/hour.\nTime = 1 ÷ (combined rate) = 60/17 hours.\nCheck: (combined rate)×(time) = 17/60×60/17 = 1.\nFinal answer: $\\frac{60}{17}$", "final_answer": "\\frac{60}{17}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ac8d67b31d803bbb", "problem_id": "ac7de62326996d20", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Worker A finishes a job in 5 hours and Worker B finishes it in 12 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/5 job/hour. B's rate = 1/12 job/hour.\n2) Combined rate = 1/5 + 1/12 = 17/60 job/hour.\n3) Time = 1 ÷ (combined rate) = 60/17 hours.\nFinal answer: $\\frac{60}{17}$", "final_answer": "\\frac{60}{17}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1cc83edcfb1de8e6", "problem_id": "3dff45a6371918d4", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A cyclist rides at 24 miles per hour for 1 hours, then at 66 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 24 × 1 = 24 miles.\n2) Second leg distance = 66 × 1 = 66 miles.\n3) Total distance = 24 + 66 = 90 miles.\nFinal answer: $90$", "final_answer": "90", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8e9673db4c07f3b8", "problem_id": "3dff45a6371918d4", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A cyclist rides at 24 miles per hour for 1 hours, then at 66 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 24 × 1 = 24 miles.\nSecond leg distance = 66 × 1 = 66 miles.\nTotal distance = 24 + 66 = 90 miles.\nCheck: adding both legs gives 90.\nFinal answer: $90$", "final_answer": "90", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "510cf38cc7aa9930", "problem_id": "2c1ec7a34537f448", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_9=29$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=29 and k=9.\nCompute C(37,8) = 38608020.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $38608020$", "final_answer": "38608020", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d119f89cb6ffe6cd", "problem_id": "2c1ec7a34537f448", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_9=29$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=29 and k=9.\n4) Compute C(37,8) = 38608020.\nFinal answer: $38608020$", "final_answer": "38608020", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a2cc3488ca67850", "problem_id": "a2aa57395edce067", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "An urn has 18 red balls and 21 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(39,11).\n3) Favorable samples: choose k reds and n−k blues: C(18,5)·C(21,6).\n4) Compute totals: C(39,11) = 1676056044, and favorable = 464933952.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{119952}{432419}$", "final_answer": "\\frac{119952}{432419}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "849b01ba8d20d90e", "problem_id": "b25da1383a89c6c7", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{2} - 3 x + \\left(- x^{2} + 2 x + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{- x^{2} + 2 x + 6}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- x^{2} + 2 x\\right) + 6}) = \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{- x^{2} + 2 x + 6}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{4 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{8 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{8 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{16 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{18 x^{2}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} + \\frac{24 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} + \\frac{48 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{68 x}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{12 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 4 x^{3} - 8 x^{2} + 24 x + 36} + \\frac{36 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} + \\frac{72 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} - \\frac{40}{2 x^{4} - 8 x^{3} - 16 x^{2} + 48 x + 72} + \\frac{12 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 4 x^{3} - 8 x^{2} + 24 x + 36} + \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{- x^{2} + 2 x + 6} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- x^{2} + 2 x\\right) + 6} - \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(- x^{2} + 2 x\\right) + 6}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- x^{2} + 2 x\\right) + 6}) = \\frac{6 \\left(2 x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(- x^{2} + 2 x\\right) + 6\\right)^{2}} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- x^{2} + 2 x\\right) + 6} + \\frac{6 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(- x^{2} + 2 x\\right) + 6}$.\nTherefore the correct derivative is $f'(x) = \\frac{4 \\left(x - 1\\right) \\left(- 3 x^{2} - 3 x + \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(x \\left(2 - x\\right) + 6\\right) \\left(- 12 x + 4 \\left(1 - x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x \\left(2 - x\\right) + 6\\right)^{2}}$.\nFinal answer: $\\frac{4 \\left(x - 1\\right) \\left(- 3 x^{2} - 3 x + \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(x \\left(2 - x\\right) + 6\\right) \\left(- 12 x + 4 \\left(1 - x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x \\left(2 - x\\right) + 6\\right)^{2}}$", "final_answer": "\\frac{4 \\left(x - 1\\right) \\left(- 3 x^{2} - 3 x + \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1\\right) + \\left(x \\left(2 - x\\right) + 6\\right) \\left(- 12 x + 4 \\left(1 - x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 - x\\right) + 6\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 6\\right)}{2 \\left(x \\left(2 - x\\right) + 6\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ce2165e7548ecb68", "problem_id": "6fcfd1fdb6af651b", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A store sells a backpack for $895. It is discounted by 30%. You buy 8 backpacks. What is the total cost?", "output": "1) Discounted price = 895 × (1 − 30/100) = 895 × 70/100 = 1253/2.\n2) Subtotal = 8 × 1253/2 = 5012.\nFinal answer: $5012$", "final_answer": "5012", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bac66529ac2b970d", "problem_id": "6fcfd1fdb6af651b", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $895. It is discounted by 30%. You buy 8 backpacks. What is the total cost?", "output": "Discounted price = 895 × (1 − 30/100) = 895 × 70/100 = 1253/2.\nSubtotal = 8 × 1253/2 = 5012.\nCheck: recomputing the formula yields 5012.\nFinal answer: $5012$", "final_answer": "5012", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6614f26d3185a2d8", "problem_id": "e1b7ecde5d47ec57", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "An urn has 22 red balls and 22 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(44,12).\n3) Favorable samples: choose k reds and n−k blues: C(22,4)·C(22,8).\n4) Compute totals: C(44,12) = 21090682613, and favorable = 2339117550.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{94050}{848003}$", "final_answer": "\\frac{94050}{848003}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ac1b1570867e9af3", "problem_id": "e1b7ecde5d47ec57", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 22 red balls and 22 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(44,12).\nFavorable samples: choose k reds and n−k blues: C(22,4)·C(22,8).\nCompute totals: C(44,12) = 21090682613, and favorable = 2339117550.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{94050}{848003}$.\nFinal answer: $\\frac{94050}{848003}$", "final_answer": "\\frac{94050}{848003}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5e5bbe0e7777a663", "problem_id": "0c2640647e98cc32", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Two friends split $192 in the ratio 4:2. How much does each receive?", "output": "1) Total ratio parts = 4+2 = 6.\n2) Each part = 192 ÷ 6 = 32.\n3) First share = 4 × 32 = 128.\n4) Second share = 2 × 32 = 64.\nFinal answer: $64$", "final_answer": "64", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "57a96d225efa1f95", "problem_id": "0c2640647e98cc32", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $192 in the ratio 4:2. How much does each receive?", "output": "Total ratio parts = 4+2 = 6.\nEach part = 192 ÷ 6 = 32.\nFirst share = 4 × 32 = 128.\nSecond share = 2 × 32 = 64.\nCheck: 128+64=192.\nFinal answer: $64$", "final_answer": "64", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "509ccf09d62fb1d4", "problem_id": "7d6202384233eda8", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$- 6 x + 7 y$ = $-37$\n$5 x + 2 y$ = $23$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $5$: $- 30 x + 35 y$ = $-185$.\nMultiply the second equation by $-6$: $- 30 x - 12 y$ = $-138$.\nSubtract to eliminate x: $47 y$ = $-47$.\nSolve for y: y = $-1$.\nSubstitute back to find x: x = $5$.\nCheck: plug (x,y)=(5,-1) into both equations.\nFinal answer: $(x,y)=\\left(5, -1\\right)$", "final_answer": "(x,y)=\\left(5, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f30c5ea59364b93b", "problem_id": "7d6202384233eda8", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the linear system:\n$- 6 x + 7 y$ = $-37$\n$5 x + 2 y$ = $23$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $5$: $- 30 x + 35 y$ = $-185$.\n3) Multiply the second equation by $-6$: $- 30 x - 12 y$ = $-138$.\n4) Subtract to eliminate x: $47 y$ = $-47$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, -1\\right)$", "final_answer": "(x,y)=\\left(5, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "edf55aa866818c08", "problem_id": "7412015af408cad7", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A store sells a backpack for $320. It is discounted by 35%. You buy 3 backpacks. What is the total cost?", "output": "1) Discounted price = 320 × (1 − 35/100) = 320 × 65/100 = 208.\n2) Subtotal = 3 × 208 = 624.\nFinal answer: $624$", "final_answer": "624", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "96630f4d2dbfc61c", "problem_id": "7412015af408cad7", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A store sells a backpack for $320. It is discounted by 35%. You buy 3 backpacks. What is the total cost?", "output": "Discounted price = 320 × (1 − 35/100) = 320 × 65/100 = 208.\nSubtotal = 3 × 208 = 624.\nCheck: recomputing the formula yields 624.\nFinal answer: $624$", "final_answer": "624", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "71c478ed5e12050a", "problem_id": "1175d6abfc7b4e1a", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} - 3 x^{3} + x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{3} + x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{45 \\pi^{4}}{64} - \\frac{3 \\pi^{2}}{4} - 1 + 3 \\pi + \\frac{3 \\pi^{3}}{8}$.\n4) Simplify the result.\nFinal answer: $- \\frac{45\\cdot \\pi^{4}}{64} - \\frac{3\\cdot \\pi^{2}}{4} - 1 + 3\\cdot \\pi + \\frac{3\\cdot \\pi^{3}}{8}$", "final_answer": "- \\frac{45\\cdot \\pi^{4}}{64} - \\frac{3\\cdot \\pi^{2}}{4} - 1 + 3\\cdot \\pi + \\frac{3\\cdot \\pi^{3}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e57c4647e0d5203c", "problem_id": "1175d6abfc7b4e1a", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{\\pi} - 3 x^{3} + x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{3} + x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2$.\nEvaluate at the bounds: F(\\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(\\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{45 \\pi^{4}}{64} - \\frac{3 \\pi^{2}}{4} - 1 + 3 \\pi + \\frac{3 \\pi^{3}}{8}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{45\\cdot \\pi^{4}}{64} - \\frac{3\\cdot \\pi^{2}}{4} - 1 + 3\\cdot \\pi + \\frac{3\\cdot \\pi^{3}}{8}$", "final_answer": "- \\frac{45\\cdot \\pi^{4}}{64} - \\frac{3\\cdot \\pi^{2}}{4} - 1 + 3\\cdot \\pi + \\frac{3\\cdot \\pi^{3}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "321ca22daf170f4e", "problem_id": "7793d5c9944f202e", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the multiplicative inverse of 30 modulo 43.", "output": "1) We want x such that 30x ≡ 1 (mod 43).\n2) Use extended Euclid to find s,t with 30s + 43t = 1.\n3) One Bezout identity is: 30·(-10) + 43·(7) = 1.\n4) Reduce s modulo 43: x ≡ 33 (mod 43).\nFinal answer: $x\\equiv 33\\pmod{43}$", "final_answer": "x\\equiv 33\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1256f576ea15d31e", "problem_id": "a8b7f4ccc19b327f", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} - 2 x^{2} + 3 x - 12$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{2} + 2 x - 12.\n3) Then the limit is -36.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -39.\nFinal answer: $-39$", "final_answer": "-39", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d64d3b556c4a4de4", "problem_id": "f68bb3f571b72069", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "You want 26 liters of a 25% salt solution. You have a 20% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 26−x liters is 60% solution.\n2) Salt equation: 20x + 60(26−x) = 25·26.\n3) (20-60)x = (25-60)·26 ⇒ x = 91/4.\n4) Then 26−x = 13/4.\nFinal answer: $91/4 L of 20% and 13/4 L of 60%$", "final_answer": "91/4 L of 20% and 13/4 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6abd84e8287ca30d", "problem_id": "46a40bc3e7cbfc1d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(2 x^{2} - 2 x - 3\\right) \\cos{\\left(3 x^{2} + 3 x - 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 2 x \\cos{\\left(\\left(3 x^{2} + 3 x\\right) - 1 \\right)}) = 2 x \\sin{\\left(3 x^{2} + 3 x - 1 \\right)}$.\n3) So $f'(x) = - 12 x^{3} \\sin{\\left(3 x^{2} + 3 x - 1 \\right)} - 6 x^{2} \\sin{\\left(3 x^{2} + 3 x - 1 \\right)} + 2 x e^{x} + 20 x \\sin{\\left(3 x^{2} + 3 x - 1 \\right)} + 4 x \\cos{\\left(\\left(3 x^{2} + 3 x\\right) - 1 \\right)} + 6 e^{x} + 9 \\sin{\\left(3 x^{2} + 3 x - 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 2 x \\cos{\\left(\\left(3 x^{2} + 3 x\\right) - 1 \\right)}) = 2 x \\left(6 x + 3\\right) \\sin{\\left(3 x^{2} + 3 x - 1 \\right)} - 2 \\cos{\\left(\\left(3 x^{2} + 3 x\\right) - 1 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x + 2\\right) e^{x} + 2 \\left(2 x - 1\\right) \\cos{\\left(\\left(3 x^{2} + 3 x\\right) - 1 \\right)} - 3 \\left(2 x + 1\\right) \\left(2 x \\left(x - 1\\right) - 3\\right) \\sin{\\left(3 x^{2} + 3 x - 1 \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(x + 2\\right) e^{x} + 2 \\left(2 x - 1\\right) \\cos{\\left(\\left(3 x^{2} + 3 x\\right) - 1 \\right)} - 3 \\left(2 x + 1\\right) \\left(2 x \\left(x - 1\\right) - 3\\right) \\sin{\\left(3 x^{2} + 3 x - 1 \\right)} + 2 e^{x}$", "final_answer": "2 \\left(x + 2\\right) e^{x} + 2 \\left(2 x - 1\\right) \\cos{\\left(\\left(3 x^{2} + 3 x\\right) - 1 \\right)} - 3 \\left(2 x + 1\\right) \\left(2 x \\left(x - 1\\right) - 3\\right) \\sin{\\left(3 x^{2} + 3 x - 1 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "49c8dd8fcf5d09a2", "problem_id": "888ac461b1afacb4", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} 2 x^{2} + 4 x + 1$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{3} + 2 x^{2} - 3 x - 3$ and a = 1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $6 x^{2} + 4 x - 3$.\nEvaluate at a: p'(1) = $7$.\nCheck: the identity gives $7$.\nFinal answer: $7$", "final_answer": "7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e1d8267360542d6", "problem_id": "127845561a322a32", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{3} + 4 x^{2} + 7 x + 11$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} + 4 x^{2} + 6 x + 11.\n3) Then the limit is 55.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 57.\nFinal answer: $57$", "final_answer": "57", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c6f77eecc9c97def", "problem_id": "98d3bf0cd2d9522c", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡7 (mod 9), x≡2 (mod 7). Give the solution modulo 63.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 63.\n3) Compute the CRT solution to get x ≡ 16 (mod 63).\nFinal answer: $x\\equiv 16\\pmod{63}$", "final_answer": "x\\equiv 16\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6c9e7be678edef92", "problem_id": "98d3bf0cd2d9522c", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡7 (mod 9), x≡2 (mod 7). Give the solution modulo 63.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 63.\nCompute the CRT solution to get x ≡ 16 (mod 63).\nCheck: 16 mod 9=7 and mod 7=2.\nFinal answer: $x\\equiv 16\\pmod{63}$", "final_answer": "x\\equiv 16\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "daf1b2d7bae07f46", "problem_id": "13c9931fc03174e0", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 14 red balls and 6 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(20,7).\n3) Favorable samples: choose k reds and n−k blues: C(14,3)·C(6,4).\n4) Compute totals: C(20,7) = 77520, and favorable = 5460.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{91}{1292}$", "final_answer": "\\frac{91}{1292}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5883c2ba2379c56c", "problem_id": "13c9931fc03174e0", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 14 red balls and 6 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(20,7).\nFavorable samples: choose k reds and n−k blues: C(14,3)·C(6,4).\nCompute totals: C(20,7) = 77520, and favorable = 5460.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{91}{1292}$.\nFinal answer: $\\frac{91}{1292}$", "final_answer": "\\frac{91}{1292}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1fb98cb3840291f5", "problem_id": "db6d6fc837e96b9e", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "You want 14 liters of a 25% salt solution. You have a 15% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 14−x liters is 70% solution.\nSalt equation: 15x + 70(14−x) = 25·14.\n(15-70)x = (25-70)·14 ⇒ x = 126/11.\nThen 14−x = 28/11.\nCheck: 15·126/11 + 70·28/11 = 25·14.\nFinal answer: $126/11 L of 15% and 28/11 L of 70%$", "final_answer": "126/11 L of 15% and 28/11 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "879737e14049a65d", "problem_id": "db6d6fc837e96b9e", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "You want 14 liters of a 25% salt solution. You have a 15% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 14−x liters is 70% solution.\n2) Salt equation: 15x + 70(14−x) = 25·14.\n3) (15-70)x = (25-70)·14 ⇒ x = 126/11.\n4) Then 14−x = 28/11.\nFinal answer: $126/11 L of 15% and 28/11 L of 70%$", "final_answer": "126/11 L of 15% and 28/11 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c33ae7ecf1012519", "problem_id": "5fb511748522caac", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡1 (mod 13). Give the solution modulo 39.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 39.\n3) Compute the CRT solution to get x ≡ 1 (mod 39).\nFinal answer: $x\\equiv 1\\pmod{39}$", "final_answer": "x\\equiv 1\\pmod{39}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1b3df82e646cff6f", "problem_id": "5fb511748522caac", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡1 (mod 13). Give the solution modulo 39.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 39.\nCompute the CRT solution to get x ≡ 1 (mod 39).\nFinal answer: $x\\equiv 1\\pmod{39}$", "final_answer": "x\\equiv 1\\pmod{39}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b73a52fffd8e227f", "problem_id": "dbd610d86c917096", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{3}$. If it is flipped 9 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=9, k=4, p=$\\frac{1}{3}$ and simplify.\n4) Here C(9,4) = 126.\nFinal answer: $\\frac{448}{2187}$", "final_answer": "\\frac{448}{2187}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e607887abea0b29f", "problem_id": "dbd610d86c917096", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{3}$. If it is flipped 9 times, what is the probability of exactly 4 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=9, k=4, p=$\\frac{1}{3}$ and simplify.\nHere C(9,4) = 126.\nCheck: expression simplifies to $\\frac{448}{2187}$.\nFinal answer: $\\frac{448}{2187}$", "final_answer": "\\frac{448}{2187}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f31029e4a8a9a383", "problem_id": "f19adbad6523e379", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} - 2 x^{3} + 4 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{3} + 4 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $\\frac{3 \\pi}{2} + 5 + \\frac{15 \\pi^{4}}{32} + \\frac{3 \\pi^{3}}{2}$.\n4) Simplify the result.\nFinal answer: $\\frac{3\\cdot \\pi}{2} + 5 + \\frac{15\\cdot \\pi^{4}}{32} + \\frac{3\\cdot \\pi^{3}}{2}$", "final_answer": "\\frac{3\\cdot \\pi}{2} + 5 + \\frac{15\\cdot \\pi^{4}}{32} + \\frac{3\\cdot \\pi^{3}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5066c877c47aa8e5", "problem_id": "f19adbad6523e379", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} - 2 x^{3} + 4 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{3} + 4 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $\\frac{3 \\pi}{2} + 5 + \\frac{15 \\pi^{4}}{32} + \\frac{3 \\pi^{3}}{2}$.\nFinal answer: $\\frac{3\\cdot \\pi}{2} + 5 + \\frac{15\\cdot \\pi^{4}}{32} + \\frac{3\\cdot \\pi^{3}}{2}$", "final_answer": "\\frac{3\\cdot \\pi}{2} + 5 + \\frac{15\\cdot \\pi^{4}}{32} + \\frac{3\\cdot \\pi^{3}}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6301ac213f083cbc", "problem_id": "aaf533374e288e52", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A store sells a backpack for $338. It is discounted by 35%. You buy 3 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 338 × (1 − 35/100) = 338 × 65/100 = 2197/10.\nSubtotal = 3 × 2197/10 = 6591/10.\nTotal with tax = 6591/10 × 109/100 = 718419/1000.\nCheck: recomputing the formula yields 718419/1000.\nFinal answer: $\\frac{718419}{1000}$", "final_answer": "\\frac{718419}{1000}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4a235fdf8b21749b", "problem_id": "aaf533374e288e52", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A store sells a backpack for $338. It is discounted by 35%. You buy 3 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 338 × (1 − 35/100) = 338 × 65/100 = 2197/10.\n2) Subtotal = 3 × 2197/10 = 6591/10.\n3) Total with tax = 6591/10 × 109/100 = 718419/1000.\nFinal answer: $\\frac{718419}{1000}$", "final_answer": "\\frac{718419}{1000}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "006147ef57465013", "problem_id": "15d7d6b74a82a81e", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 21 modulo 59.", "output": "We want x such that 21x ≡ 1 (mod 59).\nUse extended Euclid to find s,t with 21s + 59t = 1.\nOne Bezout identity is: 21·(-14) + 59·(5) = 1.\nReduce s modulo 59: x ≡ 45 (mod 59).\nCheck: 21·45 mod 59 = 1.\nFinal answer: $x\\equiv 45\\pmod{59}$", "final_answer": "x\\equiv 45\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4114ed52a5099d80", "problem_id": "15d7d6b74a82a81e", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the multiplicative inverse of 21 modulo 59.", "output": "1) We want x such that 21x ≡ 1 (mod 59).\n2) Use extended Euclid to find s,t with 21s + 59t = 1.\n3) One Bezout identity is: 21·(-14) + 59·(5) = 1.\n4) Reduce s modulo 59: x ≡ 45 (mod 59).\nFinal answer: $x\\equiv 45\\pmod{59}$", "final_answer": "x\\equiv 45\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9bf9a2469ee12328", "problem_id": "cd6f3f178182d0bd", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 21 times, what is the probability of exactly 8 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=21, k=8, p=$\\frac{1}{4}$ and simplify.\n4) Here C(21,8) = 203490.\nFinal answer: $\\frac{162214393635}{2199023255552}$", "final_answer": "\\frac{162214393635}{2199023255552}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "faccf687ad4737ba", "problem_id": "4515d91ae660cecc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} - 2 x^{2} + 3 x - 8$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{2} + 2 x - 8.\n3) Then the limit is -20.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -22.\nFinal answer: $-22$", "final_answer": "-22", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "63103d42dc35631e", "problem_id": "2a51514b9206b915", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A cyclist rides at 72 miles per hour for 1 hours, then at 69 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 72 × 1 = 72 miles.\n2) Second leg distance = 69 × 3 = 207 miles.\n3) Total distance = 72 + 207 = 279 miles.\nFinal answer: $279$", "final_answer": "279", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c498098df38773ee", "problem_id": "2a51514b9206b915", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 72 miles per hour for 1 hours, then at 69 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 72 × 1 = 72 miles.\nSecond leg distance = 69 × 3 = 207 miles.\nTotal distance = 72 + 207 = 279 miles.\nCheck: adding both legs gives 279.\nFinal answer: $279$", "final_answer": "279", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a282e278e99d00c1", "problem_id": "2693a4c0b96890d8", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_10=24$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_10=4.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=4 and k=10.\n4) Compute C(13,9) = 715.\nFinal answer: $715$", "final_answer": "715", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "387136e446e027ad", "problem_id": "dfe45db6d235951c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\cos{\\left(2 x \\right)}) = - \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\cos{\\left(2 x \\right)}) = - 2 \\sin{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$", "final_answer": "- 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7e540a82ea96eca9", "problem_id": "60c29980ba7784f4", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 21 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=21, k=5, p=$\\frac{1}{10}$ and simplify.\n4) Here C(21,5) = 20349.\nFinal answer: $\\frac{37707107822946112509}{1000000000000000000000}$", "final_answer": "\\frac{37707107822946112509}{1000000000000000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f6569430a6dbc0fb", "problem_id": "60c29980ba7784f4", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 21 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=21, k=5, p=$\\frac{1}{10}$ and simplify.\nHere C(21,5) = 20349.\nCheck: expression simplifies to $\\frac{37707107822946112509}{1000000000000000000000}$.\nFinal answer: $\\frac{37707107822946112509}{1000000000000000000000}$", "final_answer": "\\frac{37707107822946112509}{1000000000000000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f8a7ce1c7ae4e161", "problem_id": "4dc7351d3207582c", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} 2 x^{2} + 9 x + 34$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $2 x^{3} + x^{2} - 2 x - 1$ and a = 4.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $6 x^{2} + 2 x - 2$.\nEvaluate at a: p'(4) = $102$.\nCheck: the identity gives $102$.\nFinal answer: $102$", "final_answer": "102", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "20f32edef04d4911", "problem_id": "edc00fcb23b7330a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} x^{3} + 7 x^{2} + 27 x + 109$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to x^{3} + 7 x^{2} + 26 x + 109.\n3) Then the limit is 389.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: 393.\nFinal answer: $393$", "final_answer": "393", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "001bea9984908d6f", "problem_id": "d254f1114f02e847", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $714 in the ratio 9:8. How much does each receive?", "output": "1) Total ratio parts = 9+8 = 17.\n2) Each part = 714 ÷ 17 = 42.\n3) First share = 9 × 42 = 378.\n4) Second share = 8 × 42 = 336.\nFinal answer: $336$", "final_answer": "336", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2e69697e510fecca", "problem_id": "d254f1114f02e847", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Two friends split $714 in the ratio 9:8. How much does each receive?", "output": "Total ratio parts = 9+8 = 17.\nEach part = 714 ÷ 17 = 42.\nFirst share = 9 × 42 = 378.\nSecond share = 8 × 42 = 336.\nCheck: 378+336=714.\nFinal answer: $336$", "final_answer": "336", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a2af2ce696cb2625", "problem_id": "4eec0d2cb3b92e90", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $1452 in the ratio 3:8. How much does each receive?", "output": "1) Total ratio parts = 3+8 = 11.\n2) Each part = 1452 ÷ 11 = 132.\n3) First share = 3 × 132 = 396.\n4) Second share = 8 × 132 = 1056.\nFinal answer: $1056$", "final_answer": "1056", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b054135c7f0eca44", "problem_id": "4eec0d2cb3b92e90", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Two friends split $1452 in the ratio 3:8. How much does each receive?", "output": "Total ratio parts = 3+8 = 11.\nEach part = 1452 ÷ 11 = 132.\nFirst share = 3 × 132 = 396.\nFinal answer: $1056$", "final_answer": "1056", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2c826f541c85af14", "problem_id": "400213e0843d149a", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the congruences: x≡0 (mod 8), x≡1 (mod 9). Give the solution modulo 72.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 72.\n3) Compute the CRT solution to get x ≡ 64 (mod 72).\nFinal answer: $x\\equiv 64\\pmod{72}$", "final_answer": "x\\equiv 64\\pmod{72}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "efa43f2b21abd110", "problem_id": "400213e0843d149a", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡0 (mod 8), x≡1 (mod 9). Give the solution modulo 72.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 72.\nCompute the CRT solution to get x ≡ 64 (mod 72).\nCheck: 64 mod 8=0 and mod 9=1.\nFinal answer: $x\\equiv 64\\pmod{72}$", "final_answer": "x\\equiv 64\\pmod{72}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1bb2ac5160052c48", "problem_id": "6f7150d9f63180fe", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the multiplicative inverse of 19 modulo 37.", "output": "We want x such that 19x ≡ 1 (mod 37).\nUse extended Euclid to find s,t with 19s + 37t = 1.\nOne Bezout identity is: 19·(2) + 37·(-1) = 1.\nReduce s modulo 37: x ≡ 2 (mod 37).\nCheck: 19·2 mod 37 = 1.\nFinal answer: $x\\equiv 2\\pmod{37}$", "final_answer": "x\\equiv 2\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "269e497fc9bd66ad", "problem_id": "6f7150d9f63180fe", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Find the multiplicative inverse of 19 modulo 37.", "output": "1) We want x such that 19x ≡ 1 (mod 37).\n2) Use extended Euclid to find s,t with 19s + 37t = 1.\n3) One Bezout identity is: 19·(2) + 37·(-1) = 1.\n4) Reduce s modulo 37: x ≡ 2 (mod 37).\nFinal answer: $x\\equiv 2\\pmod{37}$", "final_answer": "x\\equiv 2\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7d564c21b1c5f636", "problem_id": "bbf61dbfd9e0b8ff", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "An urn has 13 red balls and 11 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(24,8).\n3) Favorable samples: choose k reds and n−k blues: C(13,2)·C(11,6).\n4) Compute totals: C(24,8) = 735471, and favorable = 36036.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{364}{7429}$", "final_answer": "\\frac{364}{7429}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cecc7329a927b0bc", "problem_id": "bbf61dbfd9e0b8ff", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "An urn has 13 red balls and 11 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(24,8).\nFavorable samples: choose k reds and n−k blues: C(13,2)·C(11,6).\nFinal answer: $\\frac{364}{7429}$", "final_answer": "\\frac{364}{7429}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aa14e7b2e0e4d9c9", "problem_id": "74d954a61e6625c1", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 30 red balls and 19 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(49,11).\n3) Favorable samples: choose k reds and n−k blues: C(30,3)·C(19,8).\n4) Compute totals: C(49,11) = 29135916264, and favorable = 306862920.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{140505}{13340621}$", "final_answer": "\\frac{140505}{13340621}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "39f676c45823689e", "problem_id": "74d954a61e6625c1", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 30 red balls and 19 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(49,11).\nFavorable samples: choose k reds and n−k blues: C(30,3)·C(19,8).\nCompute totals: C(49,11) = 29135916264, and favorable = 306862920.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{140505}{13340621}$.\nFinal answer: $\\frac{140505}{13340621}$", "final_answer": "\\frac{140505}{13340621}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6fcdbcb5dd5c59fa", "problem_id": "a00373c18df48976", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $3 x^{2} \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $3 \\left(x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)}\\right)$.\nFinal answer: $3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)}$", "final_answer": "3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x\\cdot \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3e66ee65ef9e2521", "problem_id": "39c50ddcf2498768", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x - 1$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to -1.\n3) Then the limit is -1.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 1.\nFinal answer: $1$", "final_answer": "1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "1a11127ef1cd6179", "problem_id": "c6036cc9ed008741", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} 4 x^{4} - 4 x^{3} + 3 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $4 x^{4} - 4 x^{3} + 3 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nEvaluate at the bounds: F(2 \\pi) and F(0).\nSubtract: F(2 \\pi) − F(0) = $\\pi^{2} \\left(- 16 \\pi^{2} - 6 + 8 \\pi + \\frac{128 \\pi^{3}}{5}\\right)$.\nFinal answer: $\\pi^{2}\\cdot \\left(- 16\\cdot \\pi^{2} - 6 + 8\\cdot \\pi + \\frac{128\\cdot \\pi^{3}}{5}\\right)$", "final_answer": "\\pi^{2}\\cdot \\left(- 16\\cdot \\pi^{2} - 6 + 8\\cdot \\pi + \\frac{128\\cdot \\pi^{3}}{5}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "febb818e67e0c0b5", "problem_id": "c6036cc9ed008741", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} 4 x^{4} - 4 x^{3} + 3 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $4 x^{4} - 4 x^{3} + 3 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\n2) Evaluate at the bounds: F(2 \\pi) and F(0).\n3) Subtract: F(2 \\pi) − F(0) = $\\pi^{2} \\left(- 16 \\pi^{2} - 6 + 8 \\pi + \\frac{128 \\pi^{3}}{5}\\right)$.\n4) Simplify the result.\nFinal answer: $\\pi^{2}\\cdot \\left(- 16\\cdot \\pi^{2} - 6 + 8\\cdot \\pi + \\frac{128\\cdot \\pi^{3}}{5}\\right)$", "final_answer": "\\pi^{2}\\cdot \\left(- 16\\cdot \\pi^{2} - 6 + 8\\cdot \\pi + \\frac{128\\cdot \\pi^{3}}{5}\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "967a9c14d5a9a3b5", "problem_id": "b6919d05747282d0", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 3 x^{2} + 5 x + 11$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{3} - x^{2} + x + 3$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $9 x^{2} - 2 x + 1$.\n5) Evaluate at a: p'(2) = $33$.\nFinal answer: $33$", "final_answer": "33", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "613c7fe33774a146", "problem_id": "608fff4ecd21142c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} - 4 x$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(2 x^{2}) = 4 x^{2}$.\n3) So $f'(x) = 4 x^{2} - 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(2 x^{2}) = 4 x$.\nTherefore the correct derivative is $f'(x) = 4 \\left(x - 1\\right)$.\nFinal answer: $4 \\left(x - 1\\right)$", "final_answer": "4 \\left(x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "8260530f60a0c29e", "problem_id": "b7fc1838779a86cc", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡1 (mod 9), x≡1 (mod 10). Give the solution modulo 90.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 90.\n3) Compute the CRT solution to get x ≡ 1 (mod 90).\nFinal answer: $x\\equiv 1\\pmod{90}$", "final_answer": "x\\equiv 1\\pmod{90}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1df86b77212da723", "problem_id": "b7fc1838779a86cc", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Solve the congruences: x≡1 (mod 9), x≡1 (mod 10). Give the solution modulo 90.", "output": "$x\\equiv 1\\pmod{90}$", "final_answer": "x\\equiv 1\\pmod{90}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "160f4548f41d9e0b", "problem_id": "a8ac96f3e086a7e2", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 8 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=8, k=2, p=$\\frac{2}{7}$ and simplify.\n4) Here C(8,2) = 28.\nFinal answer: $\\frac{250000}{823543}$", "final_answer": "\\frac{250000}{823543}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "90f908971a6d8095", "problem_id": "a8ac96f3e086a7e2", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 8 times, what is the probability of exactly 2 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=8, k=2, p=$\\frac{2}{7}$ and simplify.\nHere C(8,2) = 28.\nCheck: expression simplifies to $\\frac{250000}{823543}$.\nFinal answer: $\\frac{250000}{823543}$", "final_answer": "\\frac{250000}{823543}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6017fa289523fb8c", "problem_id": "bbcc5d519e0c0ca6", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_8=30$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=30 and k=8.\n4) Compute C(37,7) = 10295472.\nFinal answer: $10295472$", "final_answer": "10295472", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "74a32b1a4d411919", "problem_id": "bbcc5d519e0c0ca6", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_8=30$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=30 and k=8.\nFinal answer: $10295472$", "final_answer": "10295472", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f0765f2514eb34e", "problem_id": "12c67600cf41c0e4", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "An urn has 29 red balls and 10 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(39,11).\n3) Favorable samples: choose k reds and n−k blues: C(29,3)·C(10,8).\n4) Compute totals: C(39,11) = 1676056044, and favorable = 164430.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{945}{9632506}$", "final_answer": "\\frac{945}{9632506}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "20f32edef04d4911", "problem_id": "edc00fcb23b7330a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} x^{3} + 7 x^{2} + 27 x + 109$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to x^{3} + 7 x^{2} + 26 x + 109.\n3) Then the limit is 389.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: 393.\nFinal answer: $393$", "final_answer": "393", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "9a54fb081e4c4369", "problem_id": "4dd4cc37b96485d5", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} + x - 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(2 x^{2}) = 4 x^{2}$.\n3) So $f'(x) = 4 x^{2} + 1$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(2 x^{2}) = 4 x$.\nTherefore the correct derivative is $f'(x) = 4 x + 1$.\nFinal answer: $4 x + 1$", "final_answer": "4 x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "bda064fd719dc932", "problem_id": "d66abe22734a4229", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_6=42$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_6=24.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=24 and k=6.\n4) Compute C(29,5) = 118755.\nFinal answer: $118755$", "final_answer": "118755", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "19bf7f84082d3071", "problem_id": "06a581739d7c01da", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{3}$. If it is flipped 19 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=19, k=3, p=$\\frac{1}{3}$ and simplify.\n4) Here C(19,3) = 969.\nFinal answer: $\\frac{21168128}{387420489}$", "final_answer": "\\frac{21168128}{387420489}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0f3fc5cf97dbae24", "problem_id": "06a581739d7c01da", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{3}$. If it is flipped 19 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=19, k=3, p=$\\frac{1}{3}$ and simplify.\nFinal answer: $\\frac{21168128}{387420489}$", "final_answer": "\\frac{21168128}{387420489}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bb0b5b871af29866", "problem_id": "8d928bf2c814be3d", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} 4 x^{3} + 2 x^{2} + 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $4 x^{3} + 2 x^{2} + 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nEvaluate at the bounds: F(\\pi) and F(- \\pi).\nSubtract: F(\\pi) − F(- \\pi) = $\\frac{4 \\pi^{3}}{3}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{4\\cdot \\pi^{3}}{3}$", "final_answer": "\\frac{4\\cdot \\pi^{3}}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d04c07f4901e6447", "problem_id": "8d928bf2c814be3d", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} 4 x^{3} + 2 x^{2} + 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $4 x^{3} + 2 x^{2} + 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\n2) Evaluate at the bounds: F(\\pi) and F(- \\pi).\n3) Subtract: F(\\pi) − F(- \\pi) = $\\frac{4 \\pi^{3}}{3}$.\n4) Simplify the result.\nFinal answer: $\\frac{4\\cdot \\pi^{3}}{3}$", "final_answer": "\\frac{4\\cdot \\pi^{3}}{3}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "19626dd40d5bf4e6", "problem_id": "4303602b72d4f841", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} 3 x^{3} - 11 x^{2} + 47 x - 191$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{4} + x^{3} + 3 x^{2} - 3 x + 2$ and a = -4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $12 x^{3} + 3 x^{2} + 6 x - 3$.\n5) Evaluate at a: p'(-4) = $-747$.\nFinal answer: $-747$", "final_answer": "-747", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "56956f19682eeb35", "problem_id": "4303602b72d4f841", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} 3 x^{3} - 11 x^{2} + 47 x - 191$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $3 x^{4} + x^{3} + 3 x^{2} - 3 x + 2$ and a = -4.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-747$", "final_answer": "-747", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a0594cfd5d54c190", "problem_id": "c413ddc41aa5ecb3", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 18 x + \\frac{81}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$-18$, c=$\\frac{81}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $81$, so √Δ = $9$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{9}{2}$.\n7) x2 = $\\frac{3}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{9}{2}, \\frac{3}{2}\\}$", "final_answer": "x \\in \\{\\frac{9}{2}, \\frac{3}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cd20197cf36f2a64", "problem_id": "c413ddc41aa5ecb3", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 18 x + \\frac{81}{4}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$3$, b=$-18$, c=$\\frac{81}{4}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{9}{2}, \\frac{3}{2}\\}$", "final_answer": "x \\in \\{\\frac{9}{2}, \\frac{3}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e923729d308a36f6", "problem_id": "e3d3da7c198668cb", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{2} + 4 x - 3$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $4 - 6 x$.\n2) Slope at x=-2: m = f'(-2) = $16$.\n3) Point on curve: (-2, f(-2)) = (-2, $-23$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -23 = 16(x − -2).\n6) Optionally expand: y = $16 x + 9$.\nFinal answer: $y = 16\\cdot x + 9$", "final_answer": "y = 16\\cdot x + 9", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ba2593879e760384", "problem_id": "e3d3da7c198668cb", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{2} + 4 x - 3$ at $x=-2$.", "output": "Differentiate: f'(x) = $4 - 6 x$.\nSlope at x=-2: m = f'(-2) = $16$.\nPoint on curve: (-2, f(-2)) = (-2, $-23$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -23 = 16(x − -2).\nOptionally expand: y = $16 x + 9$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 16\\cdot x + 9$", "final_answer": "y = 16\\cdot x + 9", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b40d89e095ae65b0", "problem_id": "a2bf3fe63bd02258", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A store sells a backpack for $206. It is discounted by 25%. You buy 6 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 206 × (1 − 25/100) = 206 × 75/100 = 309/2.\n2) Subtotal = 6 × 309/2 = 927.\n3) Total with tax = 927 × 109/100 = 101043/100.\nFinal answer: $\\frac{101043}{100}$", "final_answer": "\\frac{101043}{100}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dfb938ab26250c0a", "problem_id": "a2bf3fe63bd02258", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A store sells a backpack for $206. It is discounted by 25%. You buy 6 backpacks. After the discount, a sales tax of 9% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 206 × (1 − 25/100) = 206 × 75/100 = 309/2.\nSubtotal = 6 × 309/2 = 927.\nTotal with tax = 927 × 109/100 = 101043/100.\nCheck: recomputing the formula yields 101043/100.\nFinal answer: $\\frac{101043}{100}$", "final_answer": "\\frac{101043}{100}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "18cc4504d1d838e9", "problem_id": "b4fc9c66fdbfd048", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 22 times, what is the probability of exactly 10 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=22, k=10, p=$\\frac{4}{5}$ and simplify.\nHere C(22,10) = 646646.\nCheck: expression simplifies to $\\frac{678057476096}{2384185791015625}$.\nFinal answer: $\\frac{678057476096}{2384185791015625}$", "final_answer": "\\frac{678057476096}{2384185791015625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c3896cc5dd0464dc", "problem_id": "b4fc9c66fdbfd048", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 22 times, what is the probability of exactly 10 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=22, k=10, p=$\\frac{4}{5}$ and simplify.\n4) Here C(22,10) = 646646.\nFinal answer: $\\frac{678057476096}{2384185791015625}$", "final_answer": "\\frac{678057476096}{2384185791015625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "80df23930c688eda", "problem_id": "2c75fee84492ce89", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$- x + 7 y$ = $1$\n$- 4 x + y$ = $4$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $4 x - 28 y$ = $-4$.\n3) Multiply the second equation by $-1$: $4 x - y$ = $-4$.\n4) Subtract to eliminate x: $- 27 y$ = $0$.\n5) Solve for y: y = $0$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, 0\\right)$", "final_answer": "(x,y)=\\left(-1, 0\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3642976a3e5799b8", "problem_id": "2eae6a3d1f410b8c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} \\sin{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 \\sin{\\left(x \\right)}) = 3 \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} + 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 \\sin{\\left(x \\right)}) = 3 \\cos{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} + 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} + 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$", "final_answer": "- 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} + 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2ff54267d4da0543", "problem_id": "3a8424efc87b676f", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{5}$. If it is flipped 10 times, what is the probability of exactly 4 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=4, p=$\\frac{3}{5}$ and simplify.\nHere C(10,4) = 210.\nCheck: expression simplifies to $\\frac{217728}{1953125}$.\nFinal answer: $\\frac{217728}{1953125}$", "final_answer": "\\frac{217728}{1953125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0bea2306d1f776d4", "problem_id": "3a8424efc87b676f", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{5}$. If it is flipped 10 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=4, p=$\\frac{3}{5}$ and simplify.\n4) Here C(10,4) = 210.\nFinal answer: $\\frac{217728}{1953125}$", "final_answer": "\\frac{217728}{1953125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "798ecef37864dc64", "problem_id": "682a57b379476b23", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the multiplicative inverse of 7 modulo 17.", "output": "1) We want x such that 7x ≡ 1 (mod 17).\n2) Use extended Euclid to find s,t with 7s + 17t = 1.\n3) One Bezout identity is: 7·(5) + 17·(-2) = 1.\n4) Reduce s modulo 17: x ≡ 5 (mod 17).\nFinal answer: $x\\equiv 5\\pmod{17}$", "final_answer": "x\\equiv 5\\pmod{17}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4089051880b195b4", "problem_id": "682a57b379476b23", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 7 modulo 17.", "output": "We want x such that 7x ≡ 1 (mod 17).\nUse extended Euclid to find s,t with 7s + 17t = 1.\nOne Bezout identity is: 7·(5) + 17·(-2) = 1.\nReduce s modulo 17: x ≡ 5 (mod 17).\nCheck: 7·5 mod 17 = 1.\nFinal answer: $x\\equiv 5\\pmod{17}$", "final_answer": "x\\equiv 5\\pmod{17}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cf03e64196967240", "problem_id": "ea97e8c8211295b0", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Two friends split $220 in the ratio 3:1. How much does each receive?", "output": "1) Total ratio parts = 3+1 = 4.\n2) Each part = 220 ÷ 4 = 55.\n3) First share = 3 × 55 = 165.\n4) Second share = 1 × 55 = 55.\nFinal answer: $55$", "final_answer": "55", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1ee123d51aae8401", "problem_id": "3bd33c59fe1be88f", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 2 x^{2} - 2 x + \\left(- 3 x^{2} + x + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{- 3 x^{2} + x + 2}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{3 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- 3 x^{2} + x\\right) + 2}) = \\frac{3 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x^{2} - x - 2}$.\n3) So $f'(x) = \\frac{9 x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 22 x^{2} + 8 x + 8} + \\frac{18 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 22 x^{2} + 8 x + 8} - \\frac{6 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 22 x^{2} + 8 x + 8} - \\frac{12 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 22 x^{2} + 8 x + 8} + \\frac{18 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{4} - 6 x^{3} - 11 x^{2} + 4 x + 4} - \\frac{11 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 22 x^{2} + 8 x + 8} - \\frac{22 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 22 x^{2} + 8 x + 8} - \\frac{16 x^{2}}{18 x^{4} - 12 x^{3} - 22 x^{2} + 8 x + 8} - \\frac{3 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{4} - 6 x^{3} - 11 x^{2} + 4 x + 4} + \\frac{3 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x^{2} - x - 2} + \\frac{3 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{- 6 x^{2} + 2 x + 4} + \\frac{3 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(- 3 x^{2} + x\\right) + 2} + \\frac{4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 22 x^{2} + 8 x + 8} + \\frac{8 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 22 x^{2} + 8 x + 8} - \\frac{52 x}{18 x^{4} - 12 x^{3} - 22 x^{2} + 8 x + 8} + \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- 3 x^{2} + x\\right) + 2} + \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 22 x^{2} + 8 x + 8} + \\frac{8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} - 12 x^{3} - 22 x^{2} + 8 x + 8} - \\frac{2}{18 x^{4} - 12 x^{3} - 22 x^{2} + 8 x + 8}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{3 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- 3 x^{2} + x\\right) + 2}) = - \\frac{3 x^{2} \\left(6 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(- 3 x^{2} + x\\right) + 2\\right)^{2}} - \\frac{3 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(- 3 x^{2} + x\\right) + 2\\right)} - \\frac{3 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(- 3 x^{2} + x\\right) + 2} - \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(- 3 x^{2} + x\\right) + 2}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(6 x - 1\\right) \\left(- 2 x^{2} - 2 x + \\left(x \\left(1 - 3 x\\right) + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x \\left(1 - 3 x\\right) + 2\\right) \\left(- 8 x + 2 \\left(1 - 6 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(1 - 3 x\\right) + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(1 - 3 x\\right) + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 4\\right)}{2 \\left(x \\left(1 - 3 x\\right) + 2\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(6 x - 1\\right) \\left(- 2 x^{2} - 2 x + \\left(x \\left(1 - 3 x\\right) + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x \\left(1 - 3 x\\right) + 2\\right) \\left(- 8 x + 2 \\left(1 - 6 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(1 - 3 x\\right) + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(1 - 3 x\\right) + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 4\\right)}{2 \\left(x \\left(1 - 3 x\\right) + 2\\right)^{2}}$", "final_answer": "\\frac{2 \\left(6 x - 1\\right) \\left(- 2 x^{2} - 2 x + \\left(x \\left(1 - 3 x\\right) + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3\\right) + \\left(x \\left(1 - 3 x\\right) + 2\\right) \\left(- 8 x + 2 \\left(1 - 6 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(1 - 3 x\\right) + 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(1 - 3 x\\right) + 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 4\\right)}{2 \\left(x \\left(1 - 3 x\\right) + 2\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "9c88927b660f0521", "problem_id": "ac28d829b2d1eb76", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the equation of the tangent line to $y=3 x^{2} - x - 4$ at $x=1$.", "output": "1) Differentiate: f'(x) = $6 x - 1$.\n2) Slope at x=1: m = f'(1) = $5$.\n3) Point on curve: (1, f(1)) = (1, $-2$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -2 = 5(x − 1).\n6) Optionally expand: y = $5 x - 7$.\nFinal answer: $y = 5\\cdot x - 7$", "final_answer": "y = 5\\cdot x - 7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8cb194a119e3e649", "problem_id": "ac28d829b2d1eb76", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{2} - x - 4$ at $x=1$.", "output": "Differentiate: f'(x) = $6 x - 1$.\nSlope at x=1: m = f'(1) = $5$.\nPoint on curve: (1, f(1)) = (1, $-2$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -2 = 5(x − 1).\nOptionally expand: y = $5 x - 7$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 5\\cdot x - 7$", "final_answer": "y = 5\\cdot x - 7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8a0979dd47268be7", "problem_id": "344f33ecb0617f1c", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} 3 x + 14$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{2} + 2 x + 1$ and a = 4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $6 x + 2$.\n5) Evaluate at a: p'(4) = $26$.\nFinal answer: $26$", "final_answer": "26", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "615bd32218f213c1", "problem_id": "344f33ecb0617f1c", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} 3 x + 14$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $3 x^{2} + 2 x + 1$ and a = 4.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $6 x + 2$.\nEvaluate at a: p'(4) = $26$.\nCheck: the identity gives $26$.\nFinal answer: $26$", "final_answer": "26", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c2c3e4783146e2cf", "problem_id": "37ecf5f605acfe9d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{3 x^{3} + 2 x^{2} - x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{3 x + 4}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 4}) = \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 4}$.\n3) So $f'(x) = \\frac{36 x^{3}}{18 x^{2} + 48 x + 32} + \\frac{9 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{18 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{84 x^{2}}{18 x^{2} + 48 x + 32} + \\frac{24 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{48 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{32 x}{18 x^{2} + 48 x + 32} + \\frac{9 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{2} + 24 x + 16} - \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x + 8} + \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 4} - \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 4} + \\frac{16 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{32 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{4}{18 x^{2} + 48 x + 32} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 4}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 4}) = \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(3 x + 4\\right)} + \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 4} - \\frac{9 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(3 x + 4\\right)^{2}} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 4}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 18 x^{3} - 12 x^{2} + 6 x + \\left(3 x + 4\\right) \\left(18 x^{2} + 8 x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 6 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 12}{2 \\left(3 x + 4\\right)^{2}}$.\nFinal answer: $\\frac{- 18 x^{3} - 12 x^{2} + 6 x + \\left(3 x + 4\\right) \\left(18 x^{2} + 8 x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 6 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 12}{2 \\left(3 x + 4\\right)^{2}}$", "final_answer": "\\frac{- 18 x^{3} - 12 x^{2} + 6 x + \\left(3 x + 4\\right) \\left(18 x^{2} + 8 x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) - 6 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 12}{2 \\left(3 x + 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e367a0bf1dcf428b", "problem_id": "97479518539e8c26", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A cyclist rides at 51 miles per hour for 1 hours, then at 51 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 51 × 1 = 51 miles.\nSecond leg distance = 51 × 1 = 51 miles.\nTotal distance = 51 + 51 = 102 miles.\nCheck: adding both legs gives 102.\nFinal answer: $102$", "final_answer": "102", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ab55cefc4a405113", "problem_id": "007cdbf03ae2103a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A cyclist rides at 47 miles per hour for 1/2 hours, then at 80 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 47 × 1/2 = 47/2 miles.\nSecond leg distance = 80 × 2 = 160 miles.\nTotal distance = 47/2 + 160 = 367/2 miles.\nCheck: adding both legs gives 367/2.\nFinal answer: $\\frac{367}{2}$", "final_answer": "\\frac{367}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ded026314b505c06", "problem_id": "007cdbf03ae2103a", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A cyclist rides at 47 miles per hour for 1/2 hours, then at 80 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 47 × 1/2 = 47/2 miles.\n2) Second leg distance = 80 × 2 = 160 miles.\n3) Total distance = 47/2 + 160 = 367/2 miles.\nFinal answer: $\\frac{367}{2}$", "final_answer": "\\frac{367}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cc0d98229d871254", "problem_id": "fcba67bdeb309fc2", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the linear system:\n$x - 2 y$ = $-13$\n$- 3 x - 6 y$ = $-9$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $- 3 x + 6 y$ = $39$.\n3) Multiply the second equation by $1$: $- 3 x - 6 y$ = $-9$.\n4) Subtract to eliminate x: $12 y$ = $48$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, 4\\right)$", "final_answer": "(x,y)=\\left(-5, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bc57201c9d27a2fb", "problem_id": "fcba67bdeb309fc2", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$x - 2 y$ = $-13$\n$- 3 x - 6 y$ = $-9$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-3$: $- 3 x + 6 y$ = $39$.\nMultiply the second equation by $1$: $- 3 x - 6 y$ = $-9$.\nSubtract to eliminate x: $12 y$ = $48$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $-5$.\nCheck: plug (x,y)=(-5,4) into both equations.\nFinal answer: $(x,y)=\\left(-5, 4\\right)$", "final_answer": "(x,y)=\\left(-5, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3afdf2f863e672b0", "problem_id": "46088c634a742072", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡6 (mod 8), x≡9 (mod 11). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 86 (mod 88).\nFinal answer: $x\\equiv 86\\pmod{88}$", "final_answer": "x\\equiv 86\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "055cd6b7070e58ba", "problem_id": "46088c634a742072", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡6 (mod 8), x≡9 (mod 11). Give the solution modulo 88.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 88.\nCompute the CRT solution to get x ≡ 86 (mod 88).\nCheck: 86 mod 8=6 and mod 11=9.\nFinal answer: $x\\equiv 86\\pmod{88}$", "final_answer": "x\\equiv 86\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "11dcd9d4542a09e8", "problem_id": "2eee1eda5cd95041", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "An urn has 29 red balls and 18 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(47,11).\n3) Favorable samples: choose k reds and n−k blues: C(29,3)·C(18,8).\n4) Compute totals: C(47,11) = 17417133617, and favorable = 159891732.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{12299364}{1339779509}$", "final_answer": "\\frac{12299364}{1339779509}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f2b908773cbba811", "problem_id": "2eee1eda5cd95041", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "An urn has 29 red balls and 18 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(47,11).\nFavorable samples: choose k reds and n−k blues: C(29,3)·C(18,8).\nFinal answer: $\\frac{12299364}{1339779509}$", "final_answer": "\\frac{12299364}{1339779509}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a8fc4357dd715ab4", "problem_id": "a45030850dbc75c8", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{10}$. If it is flipped 17 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=17, k=3, p=$\\frac{3}{10}$ and simplify.\n4) Here C(17,3) = 680.\nFinal answer: $\\frac{311304390437691}{2500000000000000}$", "final_answer": "\\frac{311304390437691}{2500000000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "faffab1e9f028e86", "problem_id": "a45030850dbc75c8", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{10}$. If it is flipped 17 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=17, k=3, p=$\\frac{3}{10}$ and simplify.\nFinal answer: $\\frac{311304390437691}{2500000000000000}$", "final_answer": "\\frac{311304390437691}{2500000000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8579a7cb6d9853a3", "problem_id": "df7cb126eaa89338", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡9 (mod 11), x≡5 (mod 9). Give the solution modulo 99.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 99.\n3) Compute the CRT solution to get x ≡ 86 (mod 99).\nFinal answer: $x\\equiv 86\\pmod{99}$", "final_answer": "x\\equiv 86\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "85bfb351f3056f80", "problem_id": "df7cb126eaa89338", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡9 (mod 11), x≡5 (mod 9). Give the solution modulo 99.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 99.\nCompute the CRT solution to get x ≡ 86 (mod 99).\nCheck: 86 mod 11=9 and mod 9=5.\nFinal answer: $x\\equiv 86\\pmod{99}$", "final_answer": "x\\equiv 86\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "29c360a9a62e355b", "problem_id": "86b258ce2d1631ea", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(- x^{3} - 2 x^{2} + 3\\right) \\cos{\\left(x + 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x^{3} \\cos{\\left(x + 1 \\right)}) = x^{3} \\sin{\\left(x + 1 \\right)}$.\n3) So $f'(x) = x^{3} \\sin{\\left(x + 1 \\right)} + 2 x^{2} \\sin{\\left(x + 1 \\right)} + x e^{x} - 4 x \\cos{\\left(x + 1 \\right)} - 2 e^{x} - 3 \\sin{\\left(x + 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x^{3} \\cos{\\left(x + 1 \\right)}) = x^{3} \\sin{\\left(x + 1 \\right)} - 3 x^{2} \\cos{\\left(x + 1 \\right)}$.\nTherefore the correct derivative is $f'(x) = x \\left(- 3 x - 4\\right) \\cos{\\left(x + 1 \\right)} + \\left(x - 3\\right) e^{x} - \\left(x^{2} \\left(- x - 2\\right) + 3\\right) \\sin{\\left(x + 1 \\right)} + e^{x}$.\nFinal answer: $x \\left(- 3 x - 4\\right) \\cos{\\left(x + 1 \\right)} + \\left(x - 3\\right) e^{x} - \\left(x^{2} \\left(- x - 2\\right) + 3\\right) \\sin{\\left(x + 1 \\right)} + e^{x}$", "final_answer": "x \\left(- 3 x - 4\\right) \\cos{\\left(x + 1 \\right)} + \\left(x - 3\\right) e^{x} - \\left(x^{2} \\left(- x - 2\\right) + 3\\right) \\sin{\\left(x + 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f18ea012b6de420c", "problem_id": "bf0d4ab3dbf006f9", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=28$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=28 and k=7.\n4) Compute C(34,6) = 1344904.\nFinal answer: $1344904$", "final_answer": "1344904", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5c816f4a5e345c92", "problem_id": "eba248c430fcccb9", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡0 (mod 5), x≡0 (mod 13). Give the solution modulo 65.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 65.\n3) Compute the CRT solution to get x ≡ 0 (mod 65).\nFinal answer: $x\\equiv 0\\pmod{65}$", "final_answer": "x\\equiv 0\\pmod{65}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8e7551906444502b", "problem_id": "eba248c430fcccb9", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the congruences: x≡0 (mod 5), x≡0 (mod 13). Give the solution modulo 65.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 65.\nCompute the CRT solution to get x ≡ 0 (mod 65).\nFinal answer: $x\\equiv 0\\pmod{65}$", "final_answer": "x\\equiv 0\\pmod{65}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "57cd2e8224e0e74b", "problem_id": "ee391dd8555cbd05", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 x^{3} - 7 x^{2} + 11 x - 21$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} - 7 x^{2} + 10 x - 21.\n3) Then the limit is -93.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -95.\nFinal answer: $-95$", "final_answer": "-95", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5a7d751371500d2b", "problem_id": "c7dec00959aa3551", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{4}{7}$. If it is flipped 11 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=5, p=$\\frac{4}{7}$ and simplify.\n4) Here C(11,5) = 462.\nFinal answer: $\\frac{49268736}{282475249}$", "final_answer": "\\frac{49268736}{282475249}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d27b6098692e2207", "problem_id": "d9c7560e78bac255", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many binary strings of length 21 contain exactly 7 ones with no two ones adjacent?", "output": "1) Reserve 6 mandatory zeros between the 7 ones.\n2) This leaves n−k+1 = 15 valid slots for the ones.\n3) Choose the slots: C(15,7) = 6435.\nFinal answer: $6435$", "final_answer": "6435", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ced3080db3ebd465", "problem_id": "d9c7560e78bac255", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "How many binary strings of length 21 contain exactly 7 ones with no two ones adjacent?", "output": "Reserve 6 mandatory zeros between the 7 ones.\nThis leaves n−k+1 = 15 valid slots for the ones.\nChoose the slots: C(15,7) = 6435.\nFinal answer: $6435$", "final_answer": "6435", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9380c8bd009fa6bb", "problem_id": "6203a50b5e875cc8", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 9 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=9, k=5, p=$\\frac{1}{2}$ and simplify.\nHere C(9,5) = 126.\nCheck: expression simplifies to $\\frac{63}{256}$.\nFinal answer: $\\frac{63}{256}$", "final_answer": "\\frac{63}{256}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b986aaacecb811a", "problem_id": "6203a50b5e875cc8", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 9 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=9, k=5, p=$\\frac{1}{2}$ and simplify.\n4) Here C(9,5) = 126.\nFinal answer: $\\frac{63}{256}$", "final_answer": "\\frac{63}{256}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "30ef9c7ee45d95e1", "problem_id": "939ede2e1a317e20", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $- 3 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $- 3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot x\\cdot \\cos{\\left(x \\right)} - 2\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot x\\cdot \\cos{\\left(x \\right)} - 2\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "892d8b004b483892", "problem_id": "939ede2e1a317e20", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $- 3 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot x\\cdot \\cos{\\left(x \\right)} - 2\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "- 3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot x\\cdot \\cos{\\left(x \\right)} - 2\\cdot \\sin{\\left(x \\right)} - 2\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ac5466a58fb23db", "problem_id": "1dc698c868f502c0", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$x$ = $-1$\n$- x - 6 y$ = $37$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-1$: $- x$ = $1$.\nMultiply the second equation by $1$: $- x - 6 y$ = $37$.\nSubtract to eliminate x: $6 y$ = $-36$.\nSolve for y: y = $-6$.\nSubstitute back to find x: x = $-1$.\nCheck: plug (x,y)=(-1,-6) into both equations.\nFinal answer: $(x,y)=\\left(-1, -6\\right)$", "final_answer": "(x,y)=\\left(-1, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ba6119402b5a1480", "problem_id": "1dc698c868f502c0", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the linear system:\n$x$ = $-1$\n$- x - 6 y$ = $37$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-1$: $- x$ = $1$.\n3) Multiply the second equation by $1$: $- x - 6 y$ = $37$.\n4) Subtract to eliminate x: $6 y$ = $-36$.\n5) Solve for y: y = $-6$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, -6\\right)$", "final_answer": "(x,y)=\\left(-1, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d7ce36c68340f032", "problem_id": "bb13328851182325", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 18 x - 48$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} - 18 x - 48$ = $3 \\left(x - 8\\right) \\left(x + 2\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $8$.\n5) From the second factor: x = $-2$.\nFinal answer: $x \\in \\{8, -2\\}$", "final_answer": "x \\in \\{8, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "16254a8431d48c1f", "problem_id": "bb13328851182325", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 18 x - 48$ = 0.", "output": "$x \\in \\{8, -2\\}$", "final_answer": "x \\in \\{8, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "871249bb4c23a43c", "problem_id": "1c1ca30ef3ab81be", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "You want 52 liters of a 30% salt solution. You have a 15% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 52−x liters is 60% solution.\n2) Salt equation: 15x + 60(52−x) = 30·52.\n3) (15-60)x = (30-60)·52 ⇒ x = 104/3.\n4) Then 52−x = 52/3.\nFinal answer: $104/3 L of 15% and 52/3 L of 60%$", "final_answer": "104/3 L of 15% and 52/3 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ce6ca64ab7f2903d", "problem_id": "1c1ca30ef3ab81be", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "You want 52 liters of a 30% salt solution. You have a 15% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 52−x liters is 60% solution.\nSalt equation: 15x + 60(52−x) = 30·52.\n(15-60)x = (30-60)·52 ⇒ x = 104/3.\nThen 52−x = 52/3.\nCheck: 15·104/3 + 60·52/3 = 30·52.\nFinal answer: $104/3 L of 15% and 52/3 L of 60%$", "final_answer": "104/3 L of 15% and 52/3 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aec4a93211017228", "problem_id": "87dabad4d9800aa8", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$x - 3 y$ = $-11$\n$2 x + 2 y$ = $10$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $2 x - 6 y$ = $-22$.\n3) Multiply the second equation by $1$: $2 x + 2 y$ = $10$.\n4) Subtract to eliminate x: $- 8 y$ = $-32$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $1$.\nFinal answer: $(x,y)=\\left(1, 4\\right)$", "final_answer": "(x,y)=\\left(1, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cb8684aab7596435", "problem_id": "87dabad4d9800aa8", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$x - 3 y$ = $-11$\n$2 x + 2 y$ = $10$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $2$: $2 x - 6 y$ = $-22$.\nMultiply the second equation by $1$: $2 x + 2 y$ = $10$.\nSubtract to eliminate x: $- 8 y$ = $-32$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $1$.\nCheck: plug (x,y)=(1,4) into both equations.\nFinal answer: $(x,y)=\\left(1, 4\\right)$", "final_answer": "(x,y)=\\left(1, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a4b93bacc34734fc", "problem_id": "e2df7b887c237dcf", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{3} x - \\frac{9}{4} + \\frac{3 \\sqrt{3}}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\sqrt{3}$, c=$- \\frac{9}{4} + \\frac{3 \\sqrt{3}}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $12 - 6 \\sqrt{3}$, so √Δ = $\\sqrt{12 - 6 \\sqrt{3}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{\\sqrt{12 - 6 \\sqrt{3}}}{2} + \\frac{\\sqrt{3}}{2}$.\nx2 = $- \\frac{\\sqrt{12 - 6 \\sqrt{3}}}{2} + \\frac{\\sqrt{3}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{\\sqrt{12 - 6\\cdot \\sqrt{3}}}{2} + \\frac{\\sqrt{3}}{2}, - \\frac{\\sqrt{12 - 6\\cdot \\sqrt{3}}}{2} + \\frac{\\sqrt{3}}{2}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{12 - 6\\cdot \\sqrt{3}}}{2} + \\frac{\\sqrt{3}}{2}, - \\frac{\\sqrt{12 - 6\\cdot \\sqrt{3}}}{2} + \\frac{\\sqrt{3}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d02adc2dc217ed7c", "problem_id": "e2df7b887c237dcf", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{3} x - \\frac{9}{4} + \\frac{3 \\sqrt{3}}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\sqrt{3}$, c=$- \\frac{9}{4} + \\frac{3 \\sqrt{3}}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $12 - 6 \\sqrt{3}$, so √Δ = $\\sqrt{12 - 6 \\sqrt{3}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{12 - 6 \\sqrt{3}}}{2} + \\frac{\\sqrt{3}}{2}$.\n7) x2 = $- \\frac{\\sqrt{12 - 6 \\sqrt{3}}}{2} + \\frac{\\sqrt{3}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{12 - 6\\cdot \\sqrt{3}}}{2} + \\frac{\\sqrt{3}}{2}, - \\frac{\\sqrt{12 - 6\\cdot \\sqrt{3}}}{2} + \\frac{\\sqrt{3}}{2}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{12 - 6\\cdot \\sqrt{3}}}{2} + \\frac{\\sqrt{3}}{2}, - \\frac{\\sqrt{12 - 6\\cdot \\sqrt{3}}}{2} + \\frac{\\sqrt{3}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fe24415f5eb08e92", "problem_id": "ec90fa0f665eba6e", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{4}{7}$. If it is flipped 8 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=8, k=6, p=$\\frac{4}{7}$ and simplify.\n4) Here C(8,6) = 28.\nFinal answer: $\\frac{147456}{823543}$", "final_answer": "\\frac{147456}{823543}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "afaa19fd1b14cdfd", "problem_id": "ec90fa0f665eba6e", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{7}$. If it is flipped 8 times, what is the probability of exactly 6 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=8, k=6, p=$\\frac{4}{7}$ and simplify.\nHere C(8,6) = 28.\nCheck: expression simplifies to $\\frac{147456}{823543}$.\nFinal answer: $\\frac{147456}{823543}$", "final_answer": "\\frac{147456}{823543}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "84254364999b7513", "problem_id": "ed600c22c4d61fd6", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_3=27$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=27 and k=3.\n4) Compute C(29,2) = 406.\nFinal answer: $406$", "final_answer": "406", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a3261c1f11161156", "problem_id": "ed600c22c4d61fd6", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_3=27$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=27 and k=3.\nFinal answer: $406$", "final_answer": "406", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "00906fd3b713ab87", "problem_id": "a76a49351a68bb40", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 7 red balls and 16 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(23,7).\nFavorable samples: choose k reds and n−k blues: C(7,2)·C(16,5).\nCompute totals: C(23,7) = 245157, and favorable = 91728.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{30576}{81719}$.\nFinal answer: $\\frac{30576}{81719}$", "final_answer": "\\frac{30576}{81719}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f2e13c85c6650fff", "problem_id": "a76a49351a68bb40", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "An urn has 7 red balls and 16 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(23,7).\n3) Favorable samples: choose k reds and n−k blues: C(7,2)·C(16,5).\n4) Compute totals: C(23,7) = 245157, and favorable = 91728.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{30576}{81719}$", "final_answer": "\\frac{30576}{81719}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0b0ef6b2382a01ad", "problem_id": "9d6cc78c3c7ee641", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - x^{3} - x^{2} - 7 x - 31$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to - x^{3} - x^{2} - 8 x - 31.\n3) Then the limit is -143.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: -139.\nFinal answer: $-139$", "final_answer": "-139", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "443ee5f2f5707b59", "problem_id": "2b31d10fc5b15763", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A cyclist rides at 71 miles per hour for 2 hours, then at 23 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 71 × 2 = 142 miles.\n2) Second leg distance = 23 × 2 = 46 miles.\n3) Total distance = 142 + 46 = 188 miles.\nFinal answer: $188$", "final_answer": "188", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fe5ac37755b1fbf6", "problem_id": "5dcb6ca09f6caf0d", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡2 (mod 5), x≡11 (mod 13). Give the solution modulo 65.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 65.\n3) Compute the CRT solution to get x ≡ 37 (mod 65).\nFinal answer: $x\\equiv 37\\pmod{65}$", "final_answer": "x\\equiv 37\\pmod{65}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cf00bab76273ce7c", "problem_id": "5dcb6ca09f6caf0d", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 5), x≡11 (mod 13). Give the solution modulo 65.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 65.\nCompute the CRT solution to get x ≡ 37 (mod 65).\nCheck: 37 mod 5=2 and mod 13=11.\nFinal answer: $x\\equiv 37\\pmod{65}$", "final_answer": "x\\equiv 37\\pmod{65}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c6100d6d0dd3cd1a", "problem_id": "a1f650c53ce275bf", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 x^{2} - 9 x + 21$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{3} - 3 x^{2} + 3 x + 3$ and a = -2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $9 x^{2} - 6 x + 3$.\n5) Evaluate at a: p'(-2) = $51$.\nFinal answer: $51$", "final_answer": "51", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2da1b58f2d5c5dd7", "problem_id": "a1f650c53ce275bf", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 x^{2} - 9 x + 21$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $3 x^{3} - 3 x^{2} + 3 x + 3$ and a = -2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $9 x^{2} - 6 x + 3$.\nEvaluate at a: p'(-2) = $51$.\nCheck: the identity gives $51$.\nFinal answer: $51$", "final_answer": "51", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ed057716bc94b303", "problem_id": "112052a7cf7789cd", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $329 in the ratio 4:3. How much does each receive?", "output": "Total ratio parts = 4+3 = 7.\nEach part = 329 ÷ 7 = 47.\nFirst share = 4 × 47 = 188.\nSecond share = 3 × 47 = 141.\nCheck: 188+141=329.\nFinal answer: $141$", "final_answer": "141", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bd8aacdd2483be38", "problem_id": "112052a7cf7789cd", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $329 in the ratio 4:3. How much does each receive?", "output": "1) Total ratio parts = 4+3 = 7.\n2) Each part = 329 ÷ 7 = 47.\n3) First share = 4 × 47 = 188.\n4) Second share = 3 × 47 = 141.\nFinal answer: $141$", "final_answer": "141", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8c03874002c0c967", "problem_id": "edca00c03f1d3a93", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the multiplicative inverse of 41 modulo 59.", "output": "1) We want x such that 41x ≡ 1 (mod 59).\n2) Use extended Euclid to find s,t with 41s + 59t = 1.\n3) One Bezout identity is: 41·(-23) + 59·(16) = 1.\n4) Reduce s modulo 59: x ≡ 36 (mod 59).\nFinal answer: $x\\equiv 36\\pmod{59}$", "final_answer": "x\\equiv 36\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e4c489426f5a80f9", "problem_id": "edca00c03f1d3a93", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the multiplicative inverse of 41 modulo 59.", "output": "We want x such that 41x ≡ 1 (mod 59).\nUse extended Euclid to find s,t with 41s + 59t = 1.\nOne Bezout identity is: 41·(-23) + 59·(16) = 1.\nReduce s modulo 59: x ≡ 36 (mod 59).\nCheck: 41·36 mod 59 = 1.\nFinal answer: $x\\equiv 36\\pmod{59}$", "final_answer": "x\\equiv 36\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "16c4fe7a0e6b1d98", "problem_id": "002c0776a2411346", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 16 red balls and 11 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(27,6).\nFavorable samples: choose k reds and n−k blues: C(16,2)·C(11,4).\nCompute totals: C(27,6) = 296010, and favorable = 39600.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{40}{299}$.\nFinal answer: $\\frac{40}{299}$", "final_answer": "\\frac{40}{299}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0cf98d37860a1b5a", "problem_id": "002c0776a2411346", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "An urn has 16 red balls and 11 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(27,6).\nFavorable samples: choose k reds and n−k blues: C(16,2)·C(11,4).\nFinal answer: $\\frac{40}{299}$", "final_answer": "\\frac{40}{299}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b89e3951df23256b", "problem_id": "d32bda4a9c8def86", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the linear system:\n$x - 2 y$ = $1$\n$2 x + 2 y$ = $14$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $2 x - 4 y$ = $2$.\n3) Multiply the second equation by $1$: $2 x + 2 y$ = $14$.\n4) Subtract to eliminate x: $- 6 y$ = $-12$.\n5) Solve for y: y = $2$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, 2\\right)$", "final_answer": "(x,y)=\\left(5, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6cd394577f80733b", "problem_id": "d32bda4a9c8def86", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$x - 2 y$ = $1$\n$2 x + 2 y$ = $14$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $2$: $2 x - 4 y$ = $2$.\nMultiply the second equation by $1$: $2 x + 2 y$ = $14$.\nSubtract to eliminate x: $- 6 y$ = $-12$.\nSolve for y: y = $2$.\nSubstitute back to find x: x = $5$.\nCheck: plug (x,y)=(5,2) into both equations.\nFinal answer: $(x,y)=\\left(5, 2\\right)$", "final_answer": "(x,y)=\\left(5, 2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "30ceaabcc26b8839", "problem_id": "8919ccea006b2968", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 16 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=16, k=3, p=$\\frac{1}{2}$ and simplify.\nHere C(16,3) = 560.\nCheck: expression simplifies to $\\frac{35}{4096}$.\nFinal answer: $\\frac{35}{4096}$", "final_answer": "\\frac{35}{4096}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1503819f6041f5f0", "problem_id": "8919ccea006b2968", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 16 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=16, k=3, p=$\\frac{1}{2}$ and simplify.\n4) Here C(16,3) = 560.\nFinal answer: $\\frac{35}{4096}$", "final_answer": "\\frac{35}{4096}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ebd16c902c2d46fb", "problem_id": "ffa999b9c7598305", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 11 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=11, k=5, p=$\\frac{1}{6}$ and simplify.\nHere C(11,5) = 462.\nCheck: expression simplifies to $\\frac{1203125}{60466176}$.\nFinal answer: $\\frac{1203125}{60466176}$", "final_answer": "\\frac{1203125}{60466176}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "51e3f04fdf25f3dc", "problem_id": "ffa999b9c7598305", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 11 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=5, p=$\\frac{1}{6}$ and simplify.\n4) Here C(11,5) = 462.\nFinal answer: $\\frac{1203125}{60466176}$", "final_answer": "\\frac{1203125}{60466176}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "18996bd92a81e20d", "problem_id": "162efc4fa9e7c3d9", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} 3 x^{3} + 3 x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{3} + 3 x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\nSubtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{\\pi^{2}}{8} + \\frac{\\pi}{2} + \\frac{\\pi^{3}}{8} + \\frac{3 \\pi^{4}}{64} + 5$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{\\pi^{2}}{8} + \\frac{\\pi}{2} + \\frac{\\pi^{3}}{8} + \\frac{3\\cdot \\pi^{4}}{64} + 5$", "final_answer": "- \\frac{\\pi^{2}}{8} + \\frac{\\pi}{2} + \\frac{\\pi^{3}}{8} + \\frac{3\\cdot \\pi^{4}}{64} + 5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "36443c53c73bf604", "problem_id": "162efc4fa9e7c3d9", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} 3 x^{3} + 3 x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{3} + 3 x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 1$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\n3) Subtract: F(\\frac{\\pi}{2}) − F(0) = $- \\frac{\\pi^{2}}{8} + \\frac{\\pi}{2} + \\frac{\\pi^{3}}{8} + \\frac{3 \\pi^{4}}{64} + 5$.\n4) Simplify the result.\nFinal answer: $- \\frac{\\pi^{2}}{8} + \\frac{\\pi}{2} + \\frac{\\pi^{3}}{8} + \\frac{3\\cdot \\pi^{4}}{64} + 5$", "final_answer": "- \\frac{\\pi^{2}}{8} + \\frac{\\pi}{2} + \\frac{\\pi^{3}}{8} + \\frac{3\\cdot \\pi^{4}}{64} + 5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e12a84451ff7b7d0", "problem_id": "3f379de88b94927f", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$4 x + 8 y$ = $40$\n$- 2 x + 8 y$ = $88$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $- 8 x - 16 y$ = $-80$.\n3) Multiply the second equation by $4$: $- 8 x + 32 y$ = $352$.\n4) Subtract to eliminate x: $- 48 y$ = $-432$.\n5) Solve for y: y = $9$.\n6) Substitute back to find x: x = $-8$.\nFinal answer: $(x,y)=\\left(-8, 9\\right)$", "final_answer": "(x,y)=\\left(-8, 9\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9adfba7fbf0d47ee", "problem_id": "0d1ed7df721c833d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(3 x^{2} + x + 1\\right) \\cos{\\left(x + 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\cos{\\left(x + 3 \\right)}) = - \\sin{\\left(x + 3 \\right)}$.\n3) So $f'(x) = - 3 x^{2} \\sin{\\left(x + 3 \\right)} + 2 x e^{x} - x \\sin{\\left(x + 3 \\right)} + 6 x \\cos{\\left(x + 3 \\right)} - 4 e^{x} - \\sin{\\left(x + 3 \\right)} + \\cos{\\left(x + 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\cos{\\left(x + 3 \\right)}) = - \\sin{\\left(x + 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x - 3\\right) e^{x} + \\left(6 x + 1\\right) \\cos{\\left(x + 3 \\right)} - \\left(x \\left(3 x + 1\\right) + 1\\right) \\sin{\\left(x + 3 \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(x - 3\\right) e^{x} + \\left(6 x + 1\\right) \\cos{\\left(x + 3 \\right)} - \\left(x \\left(3 x + 1\\right) + 1\\right) \\sin{\\left(x + 3 \\right)} + 2 e^{x}$", "final_answer": "2 \\left(x - 3\\right) e^{x} + \\left(6 x + 1\\right) \\cos{\\left(x + 3 \\right)} - \\left(x \\left(3 x + 1\\right) + 1\\right) \\sin{\\left(x + 3 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "418351403b3246ba", "problem_id": "71005ff4c4d4f98d", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 22 times, what is the probability of exactly 11 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=22, k=11, p=$\\frac{2}{3}$ and simplify.\n4) Here C(22,11) = 705432.\nFinal answer: $\\frac{481574912}{10460353203}$", "final_answer": "\\frac{481574912}{10460353203}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5e3ebf2fdd69f8e5", "problem_id": "71005ff4c4d4f98d", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 22 times, what is the probability of exactly 11 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=22, k=11, p=$\\frac{2}{3}$ and simplify.\nHere C(22,11) = 705432.\nCheck: expression simplifies to $\\frac{481574912}{10460353203}$.\nFinal answer: $\\frac{481574912}{10460353203}$", "final_answer": "\\frac{481574912}{10460353203}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8a34fa51c7606bc2", "problem_id": "a895f9b1fb6c169b", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{19 x}{4} + \\frac{21}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{19}{4}$, c=$\\frac{21}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{25}{16}$, so √Δ = $\\frac{5}{4}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $3$.\n7) x2 = $\\frac{7}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{3, \\frac{7}{4}\\}$", "final_answer": "x \\in \\{3, \\frac{7}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "98d1c0096bc08cae", "problem_id": "a895f9b1fb6c169b", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{19 x}{4} + \\frac{21}{4}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{19}{4}$, c=$\\frac{21}{4}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{25}{16}$, so √Δ = $\\frac{5}{4}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $3$.\nx2 = $\\frac{7}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{3, \\frac{7}{4}\\}$", "final_answer": "x \\in \\{3, \\frac{7}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ccead066f701d37d", "problem_id": "d53a40ae055a7c16", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 23 red balls and 18 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(41,11).\n3) Favorable samples: choose k reds and n−k blues: C(23,2)·C(18,9).\n4) Compute totals: C(41,11) = 3159461968, and favorable = 12300860.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{13915}{3574052}$", "final_answer": "\\frac{13915}{3574052}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ea7404eba9239377", "problem_id": "d53a40ae055a7c16", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 23 red balls and 18 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(41,11).\nFavorable samples: choose k reds and n−k blues: C(23,2)·C(18,9).\nCompute totals: C(41,11) = 3159461968, and favorable = 12300860.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{13915}{3574052}$.\nFinal answer: $\\frac{13915}{3574052}$", "final_answer": "\\frac{13915}{3574052}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ee2684b869d91667", "problem_id": "90254d9381cdeeed", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 x^{3} - 3 x^{2} + 9 x - 21$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $3 x^{4} + 3 x^{3} + 3 x^{2} - 3 x$ and a = -2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $12 x^{3} + 9 x^{2} + 6 x - 3$.\n5) Evaluate at a: p'(-2) = $-75$.\nFinal answer: $-75$", "final_answer": "-75", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c2583667ca8cc90f", "problem_id": "90254d9381cdeeed", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 x^{3} - 3 x^{2} + 9 x - 21$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $3 x^{4} + 3 x^{3} + 3 x^{2} - 3 x$ and a = -2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-75$", "final_answer": "-75", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a00f7faf6d9f45c6", "problem_id": "adfe202ac964bfdc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{4} + 3 x^{3} + 3 x^{2} - 3 x + 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(3 x^{2}) = 6 x^{2}$.\n3) So $f'(x) = 16 x^{3} + 15 x^{2} - 3$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(3 x^{2}) = 6 x$.\nTherefore the correct derivative is $f'(x) = 16 x^{3} + 9 x^{2} + 6 x - 3$.\nFinal answer: $16 x^{3} + 9 x^{2} + 6 x - 3$", "final_answer": "16 x^{3} + 9 x^{2} + 6 x - 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3506e047ea2a71e0", "problem_id": "309e433ee8036fee", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡2 (mod 3), x≡3 (mod 8). Give the solution modulo 24.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 24.\n3) Compute the CRT solution to get x ≡ 11 (mod 24).\nFinal answer: $x\\equiv 11\\pmod{24}$", "final_answer": "x\\equiv 11\\pmod{24}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7bb9cd5fbfce9070", "problem_id": "acbdd243270b4553", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} + x^{2} - 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 2$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 6 x^{2} + 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 3$.\n2) Slope at x=2: m = f'(2) = $-23 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\n3) Point on curve: (2, f(2)) = (2, $-16 + \\cos{\\left(4 \\right)} + e$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -16 + \\cos{\\left(4 \\right)} + e = -23 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\n6) Optionally expand: y = $- 23 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 30$.\nFinal answer: $y = - 23\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 30$", "final_answer": "y = - 23\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 30", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6bc9959798981a88", "problem_id": "acbdd243270b4553", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} + x^{2} - 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 2$ at $x=2$.", "output": "Differentiate: f'(x) = $- 6 x^{2} + 2 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 3$.\nSlope at x=2: m = f'(2) = $-23 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}$.\nPoint on curve: (2, f(2)) = (2, $-16 + \\cos{\\left(4 \\right)} + e$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -16 + \\cos{\\left(4 \\right)} + e = -23 + \\frac{e}{2} - 2 \\sin{\\left(4 \\right)}(x − 2).\nOptionally expand: y = $- 23 x + \\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 30$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = - 23\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 30$", "final_answer": "y = - 23\\cdot x + \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + 30", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c85760f8ddffb3fb", "problem_id": "bcf2ae03f37f627e", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$- x + 3 y$ = $11$\n$5 x$ = $35$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $5$: $- 5 x + 15 y$ = $55$.\n3) Multiply the second equation by $-1$: $- 5 x$ = $-35$.\n4) Subtract to eliminate x: $15 y$ = $90$.\n5) Solve for y: y = $6$.\n6) Substitute back to find x: x = $7$.\nFinal answer: $(x,y)=\\left(7, 6\\right)$", "final_answer": "(x,y)=\\left(7, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dd5132c27d15c81e", "problem_id": "81bdf5f170e55a9a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{3} + x^{2} - x + \\left(2 x^{2} + x - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{2 x^{2} + x - 2}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{2 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x^{2} + x\\right) - 2}) = \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + x - 2}$.\n3) So $f'(x) = \\frac{4 x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 14 x^{2} - 8 x + 8} + \\frac{8 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 14 x^{2} - 8 x + 8} - \\frac{4 x^{4}}{8 x^{4} + 8 x^{3} - 14 x^{2} - 8 x + 8} + \\frac{4 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 14 x^{2} - 8 x + 8} + \\frac{8 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 14 x^{2} - 8 x + 8} - \\frac{4 x^{3}}{8 x^{4} + 8 x^{3} - 14 x^{2} - 8 x + 8} + \\frac{8 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{4 x^{4} + 4 x^{3} - 7 x^{2} - 4 x + 4} - \\frac{7 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 14 x^{2} - 8 x + 8} - \\frac{14 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 14 x^{2} - 8 x + 8} + \\frac{18 x^{2}}{8 x^{4} + 8 x^{3} - 14 x^{2} - 8 x + 8} + \\frac{2 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{4 x^{4} + 4 x^{3} - 7 x^{2} - 4 x + 4} + \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} + x - 2} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x^{2} + x\\right) - 2} - \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(2 x^{2} + x\\right) - 2} - \\frac{4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 14 x^{2} - 8 x + 8} - \\frac{8 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 14 x^{2} - 8 x + 8} - \\frac{32 x}{8 x^{4} + 8 x^{3} - 14 x^{2} - 8 x + 8} - \\frac{4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x^{2} + x\\right) - 2} + \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 14 x^{2} - 8 x + 8} + \\frac{8 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} + 8 x^{3} - 14 x^{2} - 8 x + 8} - \\frac{2}{8 x^{4} + 8 x^{3} - 14 x^{2} - 8 x + 8}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{2 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x^{2} + x\\right) - 2}) = \\frac{2 x^{2} \\left(- 4 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(2 x^{2} + x\\right) - 2\\right)^{2}} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x^{2} + x\\right) - 2} + \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(2 x^{2} + x\\right) - 2} + \\frac{4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x^{2} + x\\right) - 2}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(- 4 x - 1\\right) \\left(- x^{3} + x^{2} - x + \\left(x \\left(2 x + 1\\right) - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(2 x + 1\\right) - 2\\right) \\left(- 6 x^{2} + 4 x + 2 \\left(4 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 x + 1\\right) - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 x + 1\\right) - 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x \\left(2 x + 1\\right) - 2\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(- 4 x - 1\\right) \\left(- x^{3} + x^{2} - x + \\left(x \\left(2 x + 1\\right) - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(2 x + 1\\right) - 2\\right) \\left(- 6 x^{2} + 4 x + 2 \\left(4 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 x + 1\\right) - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 x + 1\\right) - 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x \\left(2 x + 1\\right) - 2\\right)^{2}}$", "final_answer": "\\frac{2 \\left(- 4 x - 1\\right) \\left(- x^{3} + x^{2} - x + \\left(x \\left(2 x + 1\\right) - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(2 x + 1\\right) - 2\\right) \\left(- 6 x^{2} + 4 x + 2 \\left(4 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 x + 1\\right) - 2\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 x + 1\\right) - 2\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(x \\left(2 x + 1\\right) - 2\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "05e567c3f890c51a", "problem_id": "238746f570f06d4a", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡10 (mod 11), x≡12 (mod 13). Give the solution modulo 143.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 143.\n3) Compute the CRT solution to get x ≡ 142 (mod 143).\nFinal answer: $x\\equiv 142\\pmod{143}$", "final_answer": "x\\equiv 142\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "86dc6ce75e0d32c0", "problem_id": "238746f570f06d4a", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the congruences: x≡10 (mod 11), x≡12 (mod 13). Give the solution modulo 143.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 143.\nCompute the CRT solution to get x ≡ 142 (mod 143).\nFinal answer: $x\\equiv 142\\pmod{143}$", "final_answer": "x\\equiv 142\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2dd6f6a56274a1b6", "problem_id": "351cd1acb41e8e0f", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 15 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=15, k=2, p=$\\frac{1}{6}$ and simplify.\n4) Here C(15,2) = 105.\nFinal answer: $\\frac{42724609375}{156728328192}$", "final_answer": "\\frac{42724609375}{156728328192}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b1ca7e6022bfe7c9", "problem_id": "351cd1acb41e8e0f", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 15 times, what is the probability of exactly 2 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=15, k=2, p=$\\frac{1}{6}$ and simplify.\nFinal answer: $\\frac{42724609375}{156728328192}$", "final_answer": "\\frac{42724609375}{156728328192}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5bb91d53b03bd25f", "problem_id": "3ff52399bc922d50", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{4} - x^{3} + 3 x - 1$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-1) = 0.\n3) d/dx(- x^{3}) = - 3 x^{2}.\n4) d/dx(3 x) = 3.\n5) d/dx(4 x^{4}) = 16 x^{3}.\n6) Combine: f'(x) = 16 x^{3} - 3 x^{2} + 3.\nFinal answer: $16\\cdot x^{3} - 3\\cdot x^{2} + 3$", "final_answer": "16\\cdot x^{3} - 3\\cdot x^{2} + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b9a50028ebc1b766", "problem_id": "3ff52399bc922d50", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{4} - x^{3} + 3 x - 1$.", "output": "Differentiate term-by-term:\nd/dx(-1) = 0.\nd/dx(- x^{3}) = - 3 x^{2}.\nd/dx(3 x) = 3.\nd/dx(4 x^{4}) = 16 x^{3}.\nCombine: f'(x) = 16 x^{3} - 3 x^{2} + 3.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $16\\cdot x^{3} - 3\\cdot x^{2} + 3$", "final_answer": "16\\cdot x^{3} - 3\\cdot x^{2} + 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3c5ed041c9748c74", "problem_id": "f6645e937cf52825", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_10=43$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_10=33.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=33 and k=10.\nCompute C(42,9) = 445891810.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $445891810$", "final_answer": "445891810", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dcc941e0e30714be", "problem_id": "f6645e937cf52825", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_10=43$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_10=33.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=33 and k=10.\n4) Compute C(42,9) = 445891810.\nFinal answer: $445891810$", "final_answer": "445891810", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "512bf0c534f496ff", "problem_id": "ec095f44ad8370a0", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 30 red balls and 27 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(57,8).\n3) Favorable samples: choose k reds and n−k blues: C(30,2)·C(27,6).\n4) Compute totals: C(57,8) = 1652411475, and favorable = 128764350.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1334}{17119}$", "final_answer": "\\frac{1334}{17119}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2003483f3aa261f9", "problem_id": "ec095f44ad8370a0", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 30 red balls and 27 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(57,8).\nFavorable samples: choose k reds and n−k blues: C(30,2)·C(27,6).\nCompute totals: C(57,8) = 1652411475, and favorable = 128764350.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1334}{17119}$.\nFinal answer: $\\frac{1334}{17119}$", "final_answer": "\\frac{1334}{17119}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "23f413a543124ffe", "problem_id": "fce51105bf7cf726", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{3} \\sin{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 x^{2} \\sin{\\left(x \\right)}) = 3 x^{2} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = 3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + 3 x^{2} \\cos{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 x^{2} \\sin{\\left(x \\right)}) = 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$", "final_answer": "3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} + 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "dc8ccf9b80c21e8a", "problem_id": "d18c2d6b6b64466d", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{7}$. If it is flipped 13 times, what is the probability of exactly 9 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=13, k=9, p=$\\frac{1}{7}$ and simplify.\n4) Here C(13,9) = 715.\nFinal answer: $\\frac{926640}{96889010407}$", "final_answer": "\\frac{926640}{96889010407}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec7a0a04bece7e0d", "problem_id": "727bec7ffd87c9ba", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} + 2 x + \\left(2 x^{2} - x + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{2 x^{2} - x + 5}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x^{2} - x\\right) + 5}) = \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - x + 5}$.\n3) So $f'(x) = \\frac{4 x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} - 8 x^{3} + 42 x^{2} - 20 x + 50} + \\frac{8 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} - 8 x^{3} + 42 x^{2} - 20 x + 50} - \\frac{4 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} - 8 x^{3} + 42 x^{2} - 20 x + 50} - \\frac{8 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} - 8 x^{3} + 42 x^{2} - 20 x + 50} + \\frac{21 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} - 8 x^{3} + 42 x^{2} - 20 x + 50} + \\frac{42 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} - 8 x^{3} + 42 x^{2} - 20 x + 50} - \\frac{6 x^{2}}{8 x^{4} - 8 x^{3} + 42 x^{2} - 20 x + 50} - \\frac{10 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} - 8 x^{3} + 42 x^{2} - 20 x + 50} - \\frac{20 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} - 8 x^{3} + 42 x^{2} - 20 x + 50} - \\frac{4 x}{8 x^{4} - 8 x^{3} + 42 x^{2} - 20 x + 50} + \\frac{20 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{4 x^{4} - 4 x^{3} + 21 x^{2} - 10 x + 25} + \\frac{25 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{8 x^{4} - 8 x^{3} + 42 x^{2} - 20 x + 50} + \\frac{50 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{8 x^{4} - 8 x^{3} + 42 x^{2} - 20 x + 50} + \\frac{16}{8 x^{4} - 8 x^{3} + 42 x^{2} - 20 x + 50} - \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{4 x^{4} - 4 x^{3} + 21 x^{2} - 10 x + 25} - \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{4 x^{2} - 2 x + 10} + \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - x + 5} - \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(2 x^{2} - x\\right) + 5}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(2 x^{2} - x\\right) + 5}) = \\frac{5 \\left(1 - 4 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(2 x^{2} - x\\right) + 5\\right)^{2}} + \\frac{5 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(2 x^{2} - x\\right) + 5\\right)} + \\frac{5 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(2 x^{2} - x\\right) + 5}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(1 - 4 x\\right) \\left(- x^{2} + 2 x + \\left(x \\left(2 x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(x \\left(2 x - 1\\right) + 5\\right) \\left(- 4 x + 2 \\left(4 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(2 x - 1\\right) + 5\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(1 - 4 x\\right) \\left(- x^{2} + 2 x + \\left(x \\left(2 x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(x \\left(2 x - 1\\right) + 5\\right) \\left(- 4 x + 2 \\left(4 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(2 x - 1\\right) + 5\\right)^{2}}$", "final_answer": "\\frac{2 \\left(1 - 4 x\\right) \\left(- x^{2} + 2 x + \\left(x \\left(2 x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(x \\left(2 x - 1\\right) + 5\\right) \\left(- 4 x + 2 \\left(4 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(2 x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(2 x - 1\\right) + 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4\\right)}{2 \\left(x \\left(2 x - 1\\right) + 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "31a1533c1c2ecdee", "problem_id": "c8ae16ebebfa4cef", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 24 x + 21$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} + 24 x + 21$ = $3 \\left(x + 1\\right) \\left(x + 7\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $-1$.\n5) From the second factor: x = $-7$.\nFinal answer: $x \\in \\{-1, -7\\}$", "final_answer": "x \\in \\{-1, -7\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2c0ecc0e448bac4a", "problem_id": "1e03313c159a96aa", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 19 red balls and 28 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(47,8).\nFavorable samples: choose k reds and n−k blues: C(19,4)·C(28,4).\nCompute totals: C(47,8) = 314457495, and favorable = 79361100.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{5290740}{20963833}$.\nFinal answer: $\\frac{5290740}{20963833}$", "final_answer": "\\frac{5290740}{20963833}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bc180edffc704ea9", "problem_id": "1e03313c159a96aa", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "An urn has 19 red balls and 28 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(47,8).\n3) Favorable samples: choose k reds and n−k blues: C(19,4)·C(28,4).\n4) Compute totals: C(47,8) = 314457495, and favorable = 79361100.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{5290740}{20963833}$", "final_answer": "\\frac{5290740}{20963833}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b79de4b000698146", "problem_id": "96bbc54f67487b22", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 15 times, what is the probability of exactly 6 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=15, k=6, p=$\\frac{1}{5}$ and simplify.\nHere C(15,6) = 5005.\nCheck: expression simplifies to $\\frac{262406144}{6103515625}$.\nFinal answer: $\\frac{262406144}{6103515625}$", "final_answer": "\\frac{262406144}{6103515625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "34832ac935ea52c8", "problem_id": "96bbc54f67487b22", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 15 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=15, k=6, p=$\\frac{1}{5}$ and simplify.\n4) Here C(15,6) = 5005.\nFinal answer: $\\frac{262406144}{6103515625}$", "final_answer": "\\frac{262406144}{6103515625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2e81807e273c9544", "problem_id": "9a5311ce98124512", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡0 (mod 11). Give the solution modulo 33.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 33.\n3) Compute the CRT solution to get x ≡ 22 (mod 33).\nFinal answer: $x\\equiv 22\\pmod{33}$", "final_answer": "x\\equiv 22\\pmod{33}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6f7e969aa03af628", "problem_id": "9a5311ce98124512", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡0 (mod 11). Give the solution modulo 33.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 33.\nCompute the CRT solution to get x ≡ 22 (mod 33).\nFinal answer: $x\\equiv 22\\pmod{33}$", "final_answer": "x\\equiv 22\\pmod{33}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "26d4679d1f29c477", "problem_id": "4701d8855a458549", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{2 \\sqrt{2} x}{3} + 4 x - \\frac{14 \\sqrt{2}}{9} + \\frac{35}{9}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$4 - \\frac{2 \\sqrt{2}}{3}$, c=$\\frac{35}{9} - \\frac{14 \\sqrt{2}}{9}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{8 \\sqrt{2}}{9} + \\frac{4}{3}$, so √Δ = $\\sqrt{\\frac{8 \\sqrt{2}}{9} + \\frac{4}{3}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $-2 + \\frac{\\sqrt{2}}{3} + \\frac{\\sqrt{2 \\sqrt{2} + 3}}{3}$.\n7) x2 = $-2 - \\frac{\\sqrt{2 \\sqrt{2} + 3}}{3} + \\frac{\\sqrt{2}}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{-2 + \\frac{\\sqrt{2}}{3} + \\frac{\\sqrt{2\\cdot \\sqrt{2} + 3}}{3}, -2 - \\frac{\\sqrt{2\\cdot \\sqrt{2} + 3}}{3} + \\frac{\\sqrt{2}}{3}\\}$", "final_answer": "x \\in \\{-2 + \\frac{\\sqrt{2}}{3} + \\frac{\\sqrt{2\\cdot \\sqrt{2} + 3}}{3}, -2 - \\frac{\\sqrt{2\\cdot \\sqrt{2} + 3}}{3} + \\frac{\\sqrt{2}}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bd210d4e1c54893d", "problem_id": "4701d8855a458549", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{2 \\sqrt{2} x}{3} + 4 x - \\frac{14 \\sqrt{2}}{9} + \\frac{35}{9}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$4 - \\frac{2 \\sqrt{2}}{3}$, c=$\\frac{35}{9} - \\frac{14 \\sqrt{2}}{9}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{8 \\sqrt{2}}{9} + \\frac{4}{3}$, so √Δ = $\\sqrt{\\frac{8 \\sqrt{2}}{9} + \\frac{4}{3}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $-2 + \\frac{\\sqrt{2}}{3} + \\frac{\\sqrt{2 \\sqrt{2} + 3}}{3}$.\nx2 = $-2 - \\frac{\\sqrt{2 \\sqrt{2} + 3}}{3} + \\frac{\\sqrt{2}}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{-2 + \\frac{\\sqrt{2}}{3} + \\frac{\\sqrt{2\\cdot \\sqrt{2} + 3}}{3}, -2 - \\frac{\\sqrt{2\\cdot \\sqrt{2} + 3}}{3} + \\frac{\\sqrt{2}}{3}\\}$", "final_answer": "x \\in \\{-2 + \\frac{\\sqrt{2}}{3} + \\frac{\\sqrt{2\\cdot \\sqrt{2} + 3}}{3}, -2 - \\frac{\\sqrt{2\\cdot \\sqrt{2} + 3}}{3} + \\frac{\\sqrt{2}}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cee9a3f072f18163", "problem_id": "bdfc7ed1e6d5f604", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(x^{3} + x^{2} + 3\\right) \\cos{\\left(x \\right)}$.", "output": "1) Let f(x) = $\\left(x - 3\\right) e^{x} + \\left(x^{3} + x^{2} + 3\\right) \\cos{\\left(x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $x \\left(3 x + 2\\right) \\cos{\\left(x \\right)} + \\left(x - 3\\right) e^{x} - \\left(x^{3} + x^{2} + 3\\right) \\sin{\\left(x \\right)} + e^{x}$.\nFinal answer: $x\\cdot \\left(3\\cdot x + 2\\right)\\cdot \\cos{\\left(x \\right)} + \\left(x - 3\\right)\\cdot e^{x} - \\left(x^{3} + x^{2} + 3\\right)\\cdot \\sin{\\left(x \\right)} + e^{x}$", "final_answer": "x\\cdot \\left(3\\cdot x + 2\\right)\\cdot \\cos{\\left(x \\right)} + \\left(x - 3\\right)\\cdot e^{x} - \\left(x^{3} + x^{2} + 3\\right)\\cdot \\sin{\\left(x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "118ff821d20a52ea", "problem_id": "bdfc7ed1e6d5f604", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(x^{3} + x^{2} + 3\\right) \\cos{\\left(x \\right)}$.", "output": "Let f(x) = $\\left(x - 3\\right) e^{x} + \\left(x^{3} + x^{2} + 3\\right) \\cos{\\left(x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $x \\left(3 x + 2\\right) \\cos{\\left(x \\right)} + \\left(x - 3\\right) e^{x} - \\left(x^{3} + x^{2} + 3\\right) \\sin{\\left(x \\right)} + e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $x\\cdot \\left(3\\cdot x + 2\\right)\\cdot \\cos{\\left(x \\right)} + \\left(x - 3\\right)\\cdot e^{x} - \\left(x^{3} + x^{2} + 3\\right)\\cdot \\sin{\\left(x \\right)} + e^{x}$", "final_answer": "x\\cdot \\left(3\\cdot x + 2\\right)\\cdot \\cos{\\left(x \\right)} + \\left(x - 3\\right)\\cdot e^{x} - \\left(x^{3} + x^{2} + 3\\right)\\cdot \\sin{\\left(x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d970849db00c4187", "problem_id": "590b2f85c6e0faad", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡1 (mod 4), x≡5 (mod 7). Give the solution modulo 28.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 28.\n3) Compute the CRT solution to get x ≡ 5 (mod 28).\nFinal answer: $x\\equiv 5\\pmod{28}$", "final_answer": "x\\equiv 5\\pmod{28}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ae30c7e3746c0e0", "problem_id": "590b2f85c6e0faad", "style": "answer_only", "instruction": "Final answer only.", "input": "Solve the congruences: x≡1 (mod 4), x≡5 (mod 7). Give the solution modulo 28.", "output": "$x\\equiv 5\\pmod{28}$", "final_answer": "x\\equiv 5\\pmod{28}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "71f13ad080549e50", "problem_id": "b7b505935386b9f6", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 10 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=2, p=$\\frac{4}{5}$ and simplify.\n4) Here C(10,2) = 45.\nFinal answer: $\\frac{144}{1953125}$", "final_answer": "\\frac{144}{1953125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "289fe5336f11e411", "problem_id": "b7b505935386b9f6", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 10 times, what is the probability of exactly 2 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=2, p=$\\frac{4}{5}$ and simplify.\nHere C(10,2) = 45.\nCheck: expression simplifies to $\\frac{144}{1953125}$.\nFinal answer: $\\frac{144}{1953125}$", "final_answer": "\\frac{144}{1953125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "656ca20b3f2b9514", "problem_id": "3494192fd84ded94", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$- 2 x$ = $4$\n$- 2 x + 2 y$ = $-2$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-2$: $4 x$ = $-8$.\n3) Multiply the second equation by $-2$: $4 x - 4 y$ = $4$.\n4) Subtract to eliminate x: $4 y$ = $-12$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, -3\\right)$", "final_answer": "(x,y)=\\left(-2, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e1f712783f135777", "problem_id": "3494192fd84ded94", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$- 2 x$ = $4$\n$- 2 x + 2 y$ = $-2$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-2$: $4 x$ = $-8$.\nMultiply the second equation by $-2$: $4 x - 4 y$ = $4$.\nSubtract to eliminate x: $4 y$ = $-12$.\nSolve for y: y = $-3$.\nSubstitute back to find x: x = $-2$.\nCheck: plug (x,y)=(-2,-3) into both equations.\nFinal answer: $(x,y)=\\left(-2, -3\\right)$", "final_answer": "(x,y)=\\left(-2, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b59b202890b7e89f", "problem_id": "73ada385b7dd5d86", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "An urn has 17 red balls and 14 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(31,8).\n3) Favorable samples: choose k reds and n−k blues: C(17,4)·C(14,4).\n4) Compute totals: C(31,8) = 7888725, and favorable = 2382380.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{36652}{121365}$", "final_answer": "\\frac{36652}{121365}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5645321a0e1f76f3", "problem_id": "73ada385b7dd5d86", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 17 red balls and 14 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(31,8).\nFavorable samples: choose k reds and n−k blues: C(17,4)·C(14,4).\nCompute totals: C(31,8) = 7888725, and favorable = 2382380.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{36652}{121365}$.\nFinal answer: $\\frac{36652}{121365}$", "final_answer": "\\frac{36652}{121365}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e55a3def34049184", "problem_id": "cfb17c3485b8e4c5", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Two friends split $341 in the ratio 9:2. How much does each receive?", "output": "1) Total ratio parts = 9+2 = 11.\n2) Each part = 341 ÷ 11 = 31.\n3) First share = 9 × 31 = 279.\n4) Second share = 2 × 31 = 62.\nFinal answer: $62$", "final_answer": "62", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "70ce058e435e7eac", "problem_id": "cfb17c3485b8e4c5", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $341 in the ratio 9:2. How much does each receive?", "output": "Total ratio parts = 9+2 = 11.\nEach part = 341 ÷ 11 = 31.\nFirst share = 9 × 31 = 279.\nSecond share = 2 × 31 = 62.\nCheck: 279+62=341.\nFinal answer: $62$", "final_answer": "62", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4735b9c31b1981da", "problem_id": "a050fb3c0749220f", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the multiplicative inverse of 12 modulo 19.", "output": "1) We want x such that 12x ≡ 1 (mod 19).\n2) Use extended Euclid to find s,t with 12s + 19t = 1.\n3) One Bezout identity is: 12·(8) + 19·(-5) = 1.\n4) Reduce s modulo 19: x ≡ 8 (mod 19).\nFinal answer: $x\\equiv 8\\pmod{19}$", "final_answer": "x\\equiv 8\\pmod{19}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "52270e9a23e4c544", "problem_id": "a050fb3c0749220f", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the multiplicative inverse of 12 modulo 19.", "output": "We want x such that 12x ≡ 1 (mod 19).\nUse extended Euclid to find s,t with 12s + 19t = 1.\nOne Bezout identity is: 12·(8) + 19·(-5) = 1.\nReduce s modulo 19: x ≡ 8 (mod 19).\nCheck: 12·8 mod 19 = 1.\nFinal answer: $x\\equiv 8\\pmod{19}$", "final_answer": "x\\equiv 8\\pmod{19}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dfdfbd1a20ce422a", "problem_id": "bcdb2f98282042ff", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $1352 in the ratio 3:5. How much does each receive?", "output": "Total ratio parts = 3+5 = 8.\nEach part = 1352 ÷ 8 = 169.\nFirst share = 3 × 169 = 507.\nSecond share = 5 × 169 = 845.\nCheck: 507+845=1352.\nFinal answer: $845$", "final_answer": "845", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6b1cdafeb19194ad", "problem_id": "bcdb2f98282042ff", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Two friends split $1352 in the ratio 3:5. How much does each receive?", "output": "1) Total ratio parts = 3+5 = 8.\n2) Each part = 1352 ÷ 8 = 169.\n3) First share = 3 × 169 = 507.\n4) Second share = 5 × 169 = 845.\nFinal answer: $845$", "final_answer": "845", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a5ae138238eca426", "problem_id": "b85a48890c788430", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_9=34$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_9=16.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=16 and k=9.\n4) Compute C(24,8) = 735471.\nFinal answer: $735471$", "final_answer": "735471", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e63bdcdf1f2088c6", "problem_id": "b85a48890c788430", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_9=34$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_9=16.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=16 and k=9.\nFinal answer: $735471$", "final_answer": "735471", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8895321bf97a7a41", "problem_id": "f4ddf3f75cd93879", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 25 red balls and 17 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(42,12).\n3) Favorable samples: choose k reds and n−k blues: C(25,5)·C(17,7).\n4) Compute totals: C(42,12) = 11058116888, and favorable = 1033272240.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{83490}{893513}$", "final_answer": "\\frac{83490}{893513}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "44834e35887d2389", "problem_id": "f4ddf3f75cd93879", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 25 red balls and 17 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(42,12).\nFavorable samples: choose k reds and n−k blues: C(25,5)·C(17,7).\nCompute totals: C(42,12) = 11058116888, and favorable = 1033272240.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{83490}{893513}$.\nFinal answer: $\\frac{83490}{893513}$", "final_answer": "\\frac{83490}{893513}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3b30a15837f92870", "problem_id": "cbdfca8c3b009096", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 24 red balls and 26 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(50,7).\n3) Favorable samples: choose k reds and n−k blues: C(24,1)·C(26,6).\n4) Compute totals: C(50,7) = 99884400, and favorable = 5525520.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{13}{235}$", "final_answer": "\\frac{13}{235}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b5dab1bf9c86ca3d", "problem_id": "cbdfca8c3b009096", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 24 red balls and 26 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(50,7).\nFavorable samples: choose k reds and n−k blues: C(24,1)·C(26,6).\nCompute totals: C(50,7) = 99884400, and favorable = 5525520.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{13}{235}$.\nFinal answer: $\\frac{13}{235}$", "final_answer": "\\frac{13}{235}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "95a19b0c23fa9ee2", "problem_id": "39ca62fd5e9fef91", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_9=30$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_9=3.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=3 and k=9.\n4) Compute C(11,8) = 165.\nFinal answer: $165$", "final_answer": "165", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b3d354133c07afde", "problem_id": "abbc71750aa82b49", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - x^{2} - 5 x - 11$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} - 6 x - 11.\n3) Then the limit is -27.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -25.\nFinal answer: $-25$", "final_answer": "-25", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a2d54a8401880553", "problem_id": "4e9d8488c2b09192", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Worker A finishes a job in 21 hours and Worker B finishes it in 20 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/21 job/hour. B's rate = 1/20 job/hour.\n2) Combined rate = 1/21 + 1/20 = 41/420 job/hour.\n3) Time = 1 ÷ (combined rate) = 420/41 hours.\nFinal answer: $\\frac{420}{41}$", "final_answer": "\\frac{420}{41}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "943966d16fcf0501", "problem_id": "4e9d8488c2b09192", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Worker A finishes a job in 21 hours and Worker B finishes it in 20 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/21 job/hour. B's rate = 1/20 job/hour.\nCombined rate = 1/21 + 1/20 = 41/420 job/hour.\nTime = 1 ÷ (combined rate) = 420/41 hours.\nCheck: (combined rate)×(time) = 41/420×420/41 = 1.\nFinal answer: $\\frac{420}{41}$", "final_answer": "\\frac{420}{41}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b2dd663021e7ee9e", "problem_id": "4421fe05fab477f0", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{3} + 4 x^{2} + 4 x - 3$.", "output": "Differentiate term-by-term:\nd/dx(-3) = 0.\nd/dx(3 x^{3}) = 9 x^{2}.\nFinal answer: $9\\cdot x^{2} + 8\\cdot x + 4$", "final_answer": "9\\cdot x^{2} + 8\\cdot x + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dc202d20917f0241", "problem_id": "4421fe05fab477f0", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{3} + 4 x^{2} + 4 x - 3$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-3) = 0.\n3) d/dx(3 x^{3}) = 9 x^{2}.\n4) d/dx(4 x) = 4.\n5) d/dx(4 x^{2}) = 8 x.\n6) Combine: f'(x) = 9 x^{2} + 8 x + 4.\nFinal answer: $9\\cdot x^{2} + 8\\cdot x + 4$", "final_answer": "9\\cdot x^{2} + 8\\cdot x + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4266c4ff02d0dc3c", "problem_id": "282ef99c18e4d5e4", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A store sells a backpack for $865. It is discounted by 20%. You buy 8 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 865 × (1 − 20/100) = 865 × 80/100 = 692.\n2) Subtotal = 8 × 692 = 5536.\n3) Total with tax = 5536 × 107/100 = 148088/25.\nFinal answer: $\\frac{148088}{25}$", "final_answer": "\\frac{148088}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d4add95da870c26b", "problem_id": "e3962e367ed3f02a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{2} + x - 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(4 x^{2}) = 8 x^{2}$.\n3) So $f'(x) = 8 x^{2} + 1$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(4 x^{2}) = 8 x$.\nTherefore the correct derivative is $f'(x) = 8 x + 1$.\nFinal answer: $8 x + 1$", "final_answer": "8 x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e9090cddb986feba", "problem_id": "32d8d20afd531b76", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{3} + x^{2} - \\frac{x}{2} + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - \\frac{1}{2}}{x - 4}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}) = \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4}$.\n3) So $f'(x) = \\frac{4 x^{3}}{2 x^{2} - 16 x + 32} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{22 x^{2}}{2 x^{2} - 16 x + 32} - \\frac{8 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{16 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{16 x}{2 x^{2} - 16 x + 32} + \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 8 x + 16} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x - 8} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} - \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} + \\frac{16 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{32 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{5}{2 x^{2} - 16 x + 32} - \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}) = \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x - 4\\right)} + \\frac{x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} - \\frac{x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x - 4\\right)^{2}} + \\frac{e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 2 x^{3} - 2 x^{2} + x + \\left(x - 4\\right) \\left(6 x^{2} + 4 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{2 \\left(x - 4\\right)^{2}}$.\nFinal answer: $\\frac{- 2 x^{3} - 2 x^{2} + x + \\left(x - 4\\right) \\left(6 x^{2} + 4 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{2 \\left(x - 4\\right)^{2}}$", "final_answer": "\\frac{- 2 x^{3} - 2 x^{2} + x + \\left(x - 4\\right) \\left(6 x^{2} + 4 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 1\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{2 \\left(x - 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5b46e3545f1574a8", "problem_id": "65580e9af284e66f", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "An urn has 15 red balls and 20 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(35,11).\n3) Favorable samples: choose k reds and n−k blues: C(15,4)·C(20,7).\n4) Compute totals: C(35,11) = 417225900, and favorable = 105814800.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{228}{899}$", "final_answer": "\\frac{228}{899}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b2b56f5efa60c8ef", "problem_id": "28c806ec14894a1c", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=37$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=37 and k=6.\n4) Compute C(42,5) = 850668.\nFinal answer: $850668$", "final_answer": "850668", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b4a5d8c8735dd4be", "problem_id": "28c806ec14894a1c", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=37$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=37 and k=6.\nCompute C(42,5) = 850668.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $850668$", "final_answer": "850668", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d16144b0b627f2d", "problem_id": "44b5df590c930f00", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 15 times, what is the probability of exactly 12 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=15, k=12, p=$\\frac{1}{4}$ and simplify.\n4) Here C(15,12) = 455.\nFinal answer: $\\frac{12285}{1073741824}$", "final_answer": "\\frac{12285}{1073741824}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fa896d5ab72a316c", "problem_id": "24d123f4406e305d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{3} - x^{2} - \\frac{3 x}{2} + \\left(x^{2} - x + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\frac{3}{2}}{x^{2} - x + 3}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 3}) = \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 3}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18} + \\frac{2 x^{4}}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18} - \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18} - \\frac{4 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18} - \\frac{4 x^{3}}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18} + \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 7 x^{2} - 6 x + 9} + \\frac{7 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18} + \\frac{14 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18} + \\frac{23 x^{2}}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 2 x^{3} + 7 x^{2} - 6 x + 9} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 2 x + 6} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - x + 3} - \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 3} - \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18} - \\frac{12 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18} - \\frac{18 x}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18} - \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 3} + \\frac{9 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18} + \\frac{18 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18} - \\frac{6}{2 x^{4} - 4 x^{3} + 14 x^{2} - 12 x + 18}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 3}) = \\frac{x^{2} \\left(1 - 2 x\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(x^{2} - x\\right) + 3\\right)^{2}} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(x^{2} - x\\right) + 3\\right)} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 3} + \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - x\\right) + 3}$.\nTherefore the correct derivative is $f'(x) = \\frac{\\left(1 - 2 x\\right) \\left(2 x^{3} - 2 x^{2} - 3 x + 2 \\left(x \\left(x - 1\\right) + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(x - 1\\right) + 3\\right) \\left(6 x^{2} - 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 3\\right)}{2 \\left(x \\left(x - 1\\right) + 3\\right)^{2}}$.\nFinal answer: $\\frac{\\left(1 - 2 x\\right) \\left(2 x^{3} - 2 x^{2} - 3 x + 2 \\left(x \\left(x - 1\\right) + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(x - 1\\right) + 3\\right) \\left(6 x^{2} - 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 3\\right)}{2 \\left(x \\left(x - 1\\right) + 3\\right)^{2}}$", "final_answer": "\\frac{\\left(1 - 2 x\\right) \\left(2 x^{3} - 2 x^{2} - 3 x + 2 \\left(x \\left(x - 1\\right) + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(x \\left(x - 1\\right) + 3\\right) \\left(6 x^{2} - 4 x + 2 \\left(2 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(x - 1\\right) + 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(x - 1\\right) + 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 3\\right)}{2 \\left(x \\left(x - 1\\right) + 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c99cdb6eeedfadd5", "problem_id": "e40340743539c035", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$- 6 x + 7 y$ = $14$\n$- 4 x - 6 y$ = $52$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $24 x - 28 y$ = $-56$.\nMultiply the second equation by $-6$: $24 x + 36 y$ = $-312$.\nSubtract to eliminate x: $- 64 y$ = $256$.\nSolve for y: y = $-4$.\nSubstitute back to find x: x = $-7$.\nCheck: plug (x,y)=(-7,-4) into both equations.\nFinal answer: $(x,y)=\\left(-7, -4\\right)$", "final_answer": "(x,y)=\\left(-7, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fedf6ae4c3a0aea8", "problem_id": "e40340743539c035", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$- 6 x + 7 y$ = $14$\n$- 4 x - 6 y$ = $52$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $24 x - 28 y$ = $-56$.\n3) Multiply the second equation by $-6$: $24 x + 36 y$ = $-312$.\n4) Subtract to eliminate x: $- 64 y$ = $256$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $-7$.\nFinal answer: $(x,y)=\\left(-7, -4\\right)$", "final_answer": "(x,y)=\\left(-7, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f696a6670ddb9fd7", "problem_id": "c35d849537b25acb", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡3 (mod 9), x≡4 (mod 11). Give the solution modulo 99.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 99.\nCompute the CRT solution to get x ≡ 48 (mod 99).\nCheck: 48 mod 9=3 and mod 11=4.\nFinal answer: $x\\equiv 48\\pmod{99}$", "final_answer": "x\\equiv 48\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "37c538f9f04dbbb8", "problem_id": "c35d849537b25acb", "style": "answer_only", "instruction": "Final answer only.", "input": "Solve the congruences: x≡3 (mod 9), x≡4 (mod 11). Give the solution modulo 99.", "output": "$x\\equiv 48\\pmod{99}$", "final_answer": "x\\equiv 48\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "170be3346428ad89", "problem_id": "4229de537d9f241f", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} - 2 x^{2} + x + 2 \\sin{\\left(x \\right)} - 3$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $6 x^{2} - 4 x + 2 \\cos{\\left(x \\right)} + 1$.\n2) Slope at x=-2: m = f'(-2) = $2 \\cos{\\left(2 \\right)} + 33$.\n3) Point on curve: (-2, f(-2)) = (-2, $-29 - 2 \\sin{\\left(2 \\right)}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -29 - 2 \\sin{\\left(2 \\right)} = 2 \\cos{\\left(2 \\right)} + 33(x − -2).\n6) Optionally expand: y = $2 x \\cos{\\left(2 \\right)} + 33 x - 2 \\sin{\\left(2 \\right)} + 4 \\cos{\\left(2 \\right)} + 37$.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 33\\cdot x - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)} + 37$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 33\\cdot x - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)} + 37", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "45223ae695ff3780", "problem_id": "4229de537d9f241f", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} - 2 x^{2} + x + 2 \\sin{\\left(x \\right)} - 3$ at $x=-2$.", "output": "Differentiate: f'(x) = $6 x^{2} - 4 x + 2 \\cos{\\left(x \\right)} + 1$.\nSlope at x=-2: m = f'(-2) = $2 \\cos{\\left(2 \\right)} + 33$.\nPoint on curve: (-2, f(-2)) = (-2, $-29 - 2 \\sin{\\left(2 \\right)}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -29 - 2 \\sin{\\left(2 \\right)} = 2 \\cos{\\left(2 \\right)} + 33(x − -2).\nOptionally expand: y = $2 x \\cos{\\left(2 \\right)} + 33 x - 2 \\sin{\\left(2 \\right)} + 4 \\cos{\\left(2 \\right)} + 37$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 33\\cdot x - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)} + 37$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 33\\cdot x - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)} + 37", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db99c42dca548ea3", "problem_id": "83045f3da2a9778f", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 7 red balls and 14 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(21,6).\nFavorable samples: choose k reds and n−k blues: C(7,3)·C(14,3).\nCompute totals: C(21,6) = 54264, and favorable = 12740.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{455}{1938}$.\nFinal answer: $\\frac{455}{1938}$", "final_answer": "\\frac{455}{1938}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eeb93b8a24695166", "problem_id": "83045f3da2a9778f", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 7 red balls and 14 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(21,6).\n3) Favorable samples: choose k reds and n−k blues: C(7,3)·C(14,3).\n4) Compute totals: C(21,6) = 54264, and favorable = 12740.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{455}{1938}$", "final_answer": "\\frac{455}{1938}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "52e32e75fcc80404", "problem_id": "85406dd5cc145844", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=21$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=21 and k=10.\nFinal answer: $14307150$", "final_answer": "14307150", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "47cb13e6a03aa9f8", "problem_id": "85406dd5cc145844", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_10=21$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=21 and k=10.\n4) Compute C(30,9) = 14307150.\nFinal answer: $14307150$", "final_answer": "14307150", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8cabf0f80b98a41f", "problem_id": "40b41ca7c7cf2269", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 17 times, what is the probability of exactly 13 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=17, k=13, p=$\\frac{1}{4}$ and simplify.\n4) Here C(17,13) = 2380.\nFinal answer: $\\frac{48195}{4294967296}$", "final_answer": "\\frac{48195}{4294967296}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4fa2af0835379f39", "problem_id": "40b41ca7c7cf2269", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 17 times, what is the probability of exactly 13 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=17, k=13, p=$\\frac{1}{4}$ and simplify.\nHere C(17,13) = 2380.\nCheck: expression simplifies to $\\frac{48195}{4294967296}$.\nFinal answer: $\\frac{48195}{4294967296}$", "final_answer": "\\frac{48195}{4294967296}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b48336048ec20f6a", "problem_id": "1a770c2e00b4b5a6", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Two friends split $384 in the ratio 3:5. How much does each receive?", "output": "1) Total ratio parts = 3+5 = 8.\n2) Each part = 384 ÷ 8 = 48.\n3) First share = 3 × 48 = 144.\n4) Second share = 5 × 48 = 240.\nFinal answer: $240$", "final_answer": "240", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8c74ba5d67af8c2c", "problem_id": "1a770c2e00b4b5a6", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $384 in the ratio 3:5. How much does each receive?", "output": "Total ratio parts = 3+5 = 8.\nEach part = 384 ÷ 8 = 48.\nFirst share = 3 × 48 = 144.\nSecond share = 5 × 48 = 240.\nCheck: 144+240=384.\nFinal answer: $240$", "final_answer": "240", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "382d4ac13a9cadda", "problem_id": "cbf32737f907bae0", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡0 (mod 3), x≡4 (mod 7). Give the solution modulo 21.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 21.\nCompute the CRT solution to get x ≡ 18 (mod 21).\nCheck: 18 mod 3=0 and mod 7=4.\nFinal answer: $x\\equiv 18\\pmod{21}$", "final_answer": "x\\equiv 18\\pmod{21}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ea1a0bd85e3c1fc2", "problem_id": "cbf32737f907bae0", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the congruences: x≡0 (mod 3), x≡4 (mod 7). Give the solution modulo 21.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 21.\n3) Compute the CRT solution to get x ≡ 18 (mod 21).\nFinal answer: $x\\equiv 18\\pmod{21}$", "final_answer": "x\\equiv 18\\pmod{21}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4065000a8242c917", "problem_id": "7ad602b7d6879177", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- 3 x^{2} - 3 x - 3\\right) \\cos{\\left(2 x + 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x e^{x}) = 2 x e^{x}$.\n3) So $f'(x) = 6 x^{2} \\sin{\\left(2 x + 1 \\right)} + 2 x e^{x} + 6 x \\sin{\\left(2 x + 1 \\right)} - 6 x \\cos{\\left(2 x + 1 \\right)} + 2 e^{x} + 6 \\sin{\\left(2 x + 1 \\right)} - 3 \\cos{\\left(2 x + 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x e^{x}) = 2 x e^{x} + 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = 3 \\left(- 2 x - 1\\right) \\cos{\\left(2 x + 1 \\right)} + 2 \\left(x + 1\\right) e^{x} - 6 \\left(x \\left(- x - 1\\right) - 1\\right) \\sin{\\left(2 x + 1 \\right)} + 2 e^{x}$.\nFinal answer: $3 \\left(- 2 x - 1\\right) \\cos{\\left(2 x + 1 \\right)} + 2 \\left(x + 1\\right) e^{x} - 6 \\left(x \\left(- x - 1\\right) - 1\\right) \\sin{\\left(2 x + 1 \\right)} + 2 e^{x}$", "final_answer": "3 \\left(- 2 x - 1\\right) \\cos{\\left(2 x + 1 \\right)} + 2 \\left(x + 1\\right) e^{x} - 6 \\left(x \\left(- x - 1\\right) - 1\\right) \\sin{\\left(2 x + 1 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d9a2bf34ca75bb74", "problem_id": "6874f732d7fe3c5f", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{7}$. If it is flipped 12 times, what is the probability of exactly 8 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=8, p=$\\frac{1}{7}$ and simplify.\n4) Here C(12,8) = 495.\nFinal answer: $\\frac{641520}{13841287201}$", "final_answer": "\\frac{641520}{13841287201}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7fa013c73c2a84fa", "problem_id": "2454d65ac00f7605", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "An urn has 30 red balls and 16 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(46,8).\n3) Favorable samples: choose k reds and n−k blues: C(30,3)·C(16,5).\n4) Compute totals: C(46,8) = 260932815, and favorable = 17734080.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{90944}{1338117}$", "final_answer": "\\frac{90944}{1338117}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fc4d3ef9d33f9d3b", "problem_id": "2454d65ac00f7605", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "An urn has 30 red balls and 16 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(46,8).\nFavorable samples: choose k reds and n−k blues: C(30,3)·C(16,5).\nFinal answer: $\\frac{90944}{1338117}$", "final_answer": "\\frac{90944}{1338117}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "195ae4c0946f8dbf", "problem_id": "b24656916fd7923f", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡3 (mod 9), x≡4 (mod 7). Give the solution modulo 63.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 63.\n3) Compute the CRT solution to get x ≡ 39 (mod 63).\nFinal answer: $x\\equiv 39\\pmod{63}$", "final_answer": "x\\equiv 39\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "25cdff0c93e2ddd4", "problem_id": "b24656916fd7923f", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the congruences: x≡3 (mod 9), x≡4 (mod 7). Give the solution modulo 63.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 63.\nCompute the CRT solution to get x ≡ 39 (mod 63).\nFinal answer: $x\\equiv 39\\pmod{63}$", "final_answer": "x\\equiv 39\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "adc6d6d276ffea45", "problem_id": "3075d3cc726489b8", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Find the multiplicative inverse of 25 modulo 47.", "output": "1) We want x such that 25x ≡ 1 (mod 47).\n2) Use extended Euclid to find s,t with 25s + 47t = 1.\n3) One Bezout identity is: 25·(-15) + 47·(8) = 1.\n4) Reduce s modulo 47: x ≡ 32 (mod 47).\nFinal answer: $x\\equiv 32\\pmod{47}$", "final_answer": "x\\equiv 32\\pmod{47}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "28d5035587157ef4", "problem_id": "3075d3cc726489b8", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 25 modulo 47.", "output": "We want x such that 25x ≡ 1 (mod 47).\nUse extended Euclid to find s,t with 25s + 47t = 1.\nOne Bezout identity is: 25·(-15) + 47·(8) = 1.\nReduce s modulo 47: x ≡ 32 (mod 47).\nCheck: 25·32 mod 47 = 1.\nFinal answer: $x\\equiv 32\\pmod{47}$", "final_answer": "x\\equiv 32\\pmod{47}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "70ab040fb58a28d5", "problem_id": "79c0d192e1d2c9d7", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "An urn has 13 red balls and 16 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(29,8).\n3) Favorable samples: choose k reds and n−k blues: C(13,4)·C(16,4).\n4) Compute totals: C(29,8) = 4292145, and favorable = 1301300.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1820}{6003}$", "final_answer": "\\frac{1820}{6003}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "042028deb4995422", "problem_id": "79c0d192e1d2c9d7", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 13 red balls and 16 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(29,8).\nFavorable samples: choose k reds and n−k blues: C(13,4)·C(16,4).\nCompute totals: C(29,8) = 4292145, and favorable = 1301300.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1820}{6003}$.\nFinal answer: $\\frac{1820}{6003}$", "final_answer": "\\frac{1820}{6003}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "29b35f2e9c93054e", "problem_id": "31e7fedd861c159c", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $618 in the ratio 1:5. How much does each receive?", "output": "Total ratio parts = 1+5 = 6.\nEach part = 618 ÷ 6 = 103.\nFirst share = 1 × 103 = 103.\nSecond share = 5 × 103 = 515.\nCheck: 103+515=618.\nFinal answer: $515$", "final_answer": "515", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1c817e180496e7ac", "problem_id": "31e7fedd861c159c", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Two friends split $618 in the ratio 1:5. How much does each receive?", "output": "1) Total ratio parts = 1+5 = 6.\n2) Each part = 618 ÷ 6 = 103.\n3) First share = 1 × 103 = 103.\n4) Second share = 5 × 103 = 515.\nFinal answer: $515$", "final_answer": "515", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "adffc087823c38d3", "problem_id": "dfec43a743c8addc", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the congruences: x≡0 (mod 11), x≡2 (mod 5). Give the solution modulo 55.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 55.\n3) Compute the CRT solution to get x ≡ 22 (mod 55).\nFinal answer: $x\\equiv 22\\pmod{55}$", "final_answer": "x\\equiv 22\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "08ea569620bb43cb", "problem_id": "dfec43a743c8addc", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡0 (mod 11), x≡2 (mod 5). Give the solution modulo 55.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 55.\nCompute the CRT solution to get x ≡ 22 (mod 55).\nCheck: 22 mod 11=0 and mod 5=2.\nFinal answer: $x\\equiv 22\\pmod{55}$", "final_answer": "x\\equiv 22\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a9c9ea0359447695", "problem_id": "7e76661bd4579029", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the multiplicative inverse of 42 modulo 47.", "output": "1) We want x such that 42x ≡ 1 (mod 47).\n2) Use extended Euclid to find s,t with 42s + 47t = 1.\n3) One Bezout identity is: 42·(-19) + 47·(17) = 1.\n4) Reduce s modulo 47: x ≡ 28 (mod 47).\nFinal answer: $x\\equiv 28\\pmod{47}$", "final_answer": "x\\equiv 28\\pmod{47}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d303e65d31d8452d", "problem_id": "3e52b4d472b1c18d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 \\cos{\\left(2 x \\right)}) = - 3 \\sin{\\left(2 x \\right)}$.\n3) So $f'(x) = - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 3 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 \\cos{\\left(2 x \\right)}) = - 6 \\sin{\\left(2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $- x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$", "final_answer": "- x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2b9164be908b44c0", "problem_id": "35b4f19f1e022695", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{11} x - \\frac{3 x}{2} - \\frac{9}{2} + 3 \\sqrt{11}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\sqrt{11} - \\frac{3}{2}$, c=$- \\frac{9}{2} + 3 \\sqrt{11}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{\\sqrt{125 - 36\\cdot \\sqrt{11}}}{4} + \\frac{3}{4} + \\frac{\\sqrt{11}}{2}, - \\frac{\\sqrt{125 - 36\\cdot \\sqrt{11}}}{4} + \\frac{3}{4} + \\frac{\\sqrt{11}}{2}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{125 - 36\\cdot \\sqrt{11}}}{4} + \\frac{3}{4} + \\frac{\\sqrt{11}}{2}, - \\frac{\\sqrt{125 - 36\\cdot \\sqrt{11}}}{4} + \\frac{3}{4} + \\frac{\\sqrt{11}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0708b05d42846fd0", "problem_id": "35b4f19f1e022695", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{11} x - \\frac{3 x}{2} - \\frac{9}{2} + 3 \\sqrt{11}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\sqrt{11} - \\frac{3}{2}$, c=$- \\frac{9}{2} + 3 \\sqrt{11}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{125}{4} - 9 \\sqrt{11}$, so √Δ = $\\sqrt{\\frac{125}{4} - 9 \\sqrt{11}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{125 - 36 \\sqrt{11}}}{4} + \\frac{3}{4} + \\frac{\\sqrt{11}}{2}$.\n7) x2 = $- \\frac{\\sqrt{125 - 36 \\sqrt{11}}}{4} + \\frac{3}{4} + \\frac{\\sqrt{11}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{125 - 36\\cdot \\sqrt{11}}}{4} + \\frac{3}{4} + \\frac{\\sqrt{11}}{2}, - \\frac{\\sqrt{125 - 36\\cdot \\sqrt{11}}}{4} + \\frac{3}{4} + \\frac{\\sqrt{11}}{2}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{125 - 36\\cdot \\sqrt{11}}}{4} + \\frac{3}{4} + \\frac{\\sqrt{11}}{2}, - \\frac{\\sqrt{125 - 36\\cdot \\sqrt{11}}}{4} + \\frac{3}{4} + \\frac{\\sqrt{11}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "24f5f2fcaf69d1a1", "problem_id": "35f2ee3e09e86391", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 18 red balls and 21 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(39,12).\n3) Favorable samples: choose k reds and n−k blues: C(18,5)·C(21,7).\n4) Compute totals: C(39,12) = 3910797436, and favorable = 996287040.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{110160}{432419}$", "final_answer": "\\frac{110160}{432419}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "969880b90c9c6266", "problem_id": "35f2ee3e09e86391", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 18 red balls and 21 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(39,12).\nFavorable samples: choose k reds and n−k blues: C(18,5)·C(21,7).\nCompute totals: C(39,12) = 3910797436, and favorable = 996287040.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{110160}{432419}$.\nFinal answer: $\\frac{110160}{432419}$", "final_answer": "\\frac{110160}{432419}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fd83fd750f6a4f80", "problem_id": "a78eb1998df123ac", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A cyclist rides at 60 miles per hour for 1/2 hours, then at 39 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 60 × 1/2 = 30 miles.\n2) Second leg distance = 39 × 5/2 = 195/2 miles.\n3) Total distance = 30 + 195/2 = 255/2 miles.\nFinal answer: $\\frac{255}{2}$", "final_answer": "\\frac{255}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "21d612a5d0099f1d", "problem_id": "a78eb1998df123ac", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A cyclist rides at 60 miles per hour for 1/2 hours, then at 39 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 60 × 1/2 = 30 miles.\nSecond leg distance = 39 × 5/2 = 195/2 miles.\nTotal distance = 30 + 195/2 = 255/2 miles.\nCheck: adding both legs gives 255/2.\nFinal answer: $\\frac{255}{2}$", "final_answer": "\\frac{255}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "96eb579a3f059f47", "problem_id": "2d9453a41cd9dde4", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=40$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=40 and k=5.\n4) Compute C(44,4) = 135751.\nFinal answer: $135751$", "final_answer": "135751", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e0a4407b8d0950c2", "problem_id": "47c0687fe4ea8f77", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} \\sin{\\left(x \\right)} + 3 x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 x \\sin{\\left(x \\right)}) = 3 x \\cos{\\left(x \\right)}$.\n3) So $f'(x) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 x \\sin{\\left(x \\right)}) = 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.\nFinal answer: $x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$", "final_answer": "x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7ebfffbff7e73a36", "problem_id": "929050f10f545aaa", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(3 x^{3} + 2 x^{2} + 3 x\\right) \\cos{\\left(2 x + 2 \\right)}$.", "output": "1) Let f(x) = $2 \\left(x + 2\\right) e^{x} + \\left(3 x^{3} + 2 x^{2} + 3 x\\right) \\cos{\\left(2 x + 2 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- 2 x \\left(3 x^{2} + 2 x + 3\\right) \\sin{\\left(2 x + 2 \\right)} + 2 \\left(x + 2\\right) e^{x} + \\left(9 x^{2} + 4 x + 3\\right) \\cos{\\left(2 x + 2 \\right)} + 2 e^{x}$.\nFinal answer: $- 2\\cdot x\\cdot \\left(3\\cdot x^{2} + 2\\cdot x + 3\\right)\\cdot \\sin{\\left(2\\cdot x + 2 \\right)} + \\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(9\\cdot x^{2} + 4\\cdot x + 3\\right)\\cdot \\cos{\\left(2\\cdot x + 2 \\right)} + 2\\cdot e^{x}$", "final_answer": "- 2\\cdot x\\cdot \\left(3\\cdot x^{2} + 2\\cdot x + 3\\right)\\cdot \\sin{\\left(2\\cdot x + 2 \\right)} + \\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(9\\cdot x^{2} + 4\\cdot x + 3\\right)\\cdot \\cos{\\left(2\\cdot x + 2 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e0f0a20a3f8e7a30", "problem_id": "929050f10f545aaa", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(3 x^{3} + 2 x^{2} + 3 x\\right) \\cos{\\left(2 x + 2 \\right)}$.", "output": "Let f(x) = $2 \\left(x + 2\\right) e^{x} + \\left(3 x^{3} + 2 x^{2} + 3 x\\right) \\cos{\\left(2 x + 2 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- 2 x \\left(3 x^{2} + 2 x + 3\\right) \\sin{\\left(2 x + 2 \\right)} + 2 \\left(x + 2\\right) e^{x} + \\left(9 x^{2} + 4 x + 3\\right) \\cos{\\left(2 x + 2 \\right)} + 2 e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 2\\cdot x\\cdot \\left(3\\cdot x^{2} + 2\\cdot x + 3\\right)\\cdot \\sin{\\left(2\\cdot x + 2 \\right)} + \\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(9\\cdot x^{2} + 4\\cdot x + 3\\right)\\cdot \\cos{\\left(2\\cdot x + 2 \\right)} + 2\\cdot e^{x}$", "final_answer": "- 2\\cdot x\\cdot \\left(3\\cdot x^{2} + 2\\cdot x + 3\\right)\\cdot \\sin{\\left(2\\cdot x + 2 \\right)} + \\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(9\\cdot x^{2} + 4\\cdot x + 3\\right)\\cdot \\cos{\\left(2\\cdot x + 2 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a1ff8503bafb9287", "problem_id": "bec8276e4a2401a6", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_9=25$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_9=7.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=7 and k=9.\n4) Compute C(15,8) = 6435.\nFinal answer: $6435$", "final_answer": "6435", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1acd4cfca27bea08", "problem_id": "bec8276e4a2401a6", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_9=25$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_9=7.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=7 and k=9.\nCompute C(15,8) = 6435.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $6435$", "final_answer": "6435", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "389debcbd37b686a", "problem_id": "e622a085a477ad6f", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 22 red balls and 27 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(49,10).\nFavorable samples: choose k reds and n−k blues: C(22,5)·C(27,5).\nCompute totals: C(49,10) = 8217822536, and favorable = 2125943820.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{300105}{1160054}$.\nFinal answer: $\\frac{300105}{1160054}$", "final_answer": "\\frac{300105}{1160054}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bd704ec9c5079b39", "problem_id": "e622a085a477ad6f", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 22 red balls and 27 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(49,10).\n3) Favorable samples: choose k reds and n−k blues: C(22,5)·C(27,5).\n4) Compute totals: C(49,10) = 8217822536, and favorable = 2125943820.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{300105}{1160054}$", "final_answer": "\\frac{300105}{1160054}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1c92a5358d57146a", "problem_id": "2ae2078df32bec1c", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 61 miles per hour for 3/2 hours, then at 79 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 61 × 3/2 = 183/2 miles.\nSecond leg distance = 79 × 3 = 237 miles.\nTotal distance = 183/2 + 237 = 657/2 miles.\nCheck: adding both legs gives 657/2.\nFinal answer: $\\frac{657}{2}$", "final_answer": "\\frac{657}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "34de96420a4c4b25", "problem_id": "2ae2078df32bec1c", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A cyclist rides at 61 miles per hour for 3/2 hours, then at 79 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 61 × 3/2 = 183/2 miles.\n2) Second leg distance = 79 × 3 = 237 miles.\n3) Total distance = 183/2 + 237 = 657/2 miles.\nFinal answer: $\\frac{657}{2}$", "final_answer": "\\frac{657}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "70784258861ad235", "problem_id": "f7c17f59dd11440c", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡4 (mod 7), x≡3 (mod 13). Give the solution modulo 91.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 91.\n3) Compute the CRT solution to get x ≡ 81 (mod 91).\nFinal answer: $x\\equiv 81\\pmod{91}$", "final_answer": "x\\equiv 81\\pmod{91}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4d76dd143ef547f2", "problem_id": "f7c17f59dd11440c", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the congruences: x≡4 (mod 7), x≡3 (mod 13). Give the solution modulo 91.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 91.\nCompute the CRT solution to get x ≡ 81 (mod 91).\nFinal answer: $x\\equiv 81\\pmod{91}$", "final_answer": "x\\equiv 81\\pmod{91}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "15ee826a7002c699", "problem_id": "19d224a19f416658", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 x^{3} - 7 x^{2} + 11 x - 23$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} - 7 x^{2} + 10 x - 23.\n3) Then the limit is -95.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -97.\nFinal answer: $-97$", "final_answer": "-97", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d2d3b51e52d36451", "problem_id": "70f798d96723e3f3", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the multiplicative inverse of 57 modulo 59.", "output": "1) We want x such that 57x ≡ 1 (mod 59).\n2) Use extended Euclid to find s,t with 57s + 59t = 1.\n3) One Bezout identity is: 57·(29) + 59·(-28) = 1.\n4) Reduce s modulo 59: x ≡ 29 (mod 59).\nFinal answer: $x\\equiv 29\\pmod{59}$", "final_answer": "x\\equiv 29\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f3f602c451dc1988", "problem_id": "f720e692f4a78b81", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A cyclist rides at 84 miles per hour for 2 hours, then at 38 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 84 × 2 = 168 miles.\n2) Second leg distance = 38 × 1 = 38 miles.\n3) Total distance = 168 + 38 = 206 miles.\nFinal answer: $206$", "final_answer": "206", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4c8d880341fc8df4", "problem_id": "f720e692f4a78b81", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A cyclist rides at 84 miles per hour for 2 hours, then at 38 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 84 × 2 = 168 miles.\nSecond leg distance = 38 × 1 = 38 miles.\nTotal distance = 168 + 38 = 206 miles.\nCheck: adding both legs gives 206.\nFinal answer: $206$", "final_answer": "206", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e3d631f694fc1bcc", "problem_id": "81064fae64d5a3f1", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "How many binary strings of length 26 contain exactly 7 ones with no two ones adjacent?", "output": "Reserve 6 mandatory zeros between the 7 ones.\nThis leaves n−k+1 = 20 valid slots for the ones.\nChoose the slots: C(20,7) = 77520.\nFinal answer: $77520$", "final_answer": "77520", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0224e8e058a77b02", "problem_id": "81064fae64d5a3f1", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many binary strings of length 26 contain exactly 7 ones with no two ones adjacent?", "output": "1) Reserve 6 mandatory zeros between the 7 ones.\n2) This leaves n−k+1 = 20 valid slots for the ones.\n3) Choose the slots: C(20,7) = 77520.\nFinal answer: $77520$", "final_answer": "77520", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ff096d07f42000a", "problem_id": "0e7582750d0cf0d3", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 12 x - 15$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} + 12 x - 15$ = $3 \\left(x - 1\\right) \\left(x + 5\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFinal answer: $x \\in \\{1, -5\\}$", "final_answer": "x \\in \\{1, -5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "267baf02e4053e26", "problem_id": "0e7582750d0cf0d3", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} + 12 x - 15$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} + 12 x - 15$ = $3 \\left(x - 1\\right) \\left(x + 5\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $1$.\n5) From the second factor: x = $-5$.\nFinal answer: $x \\in \\{1, -5\\}$", "final_answer": "x \\in \\{1, -5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9dadadc366254fb2", "problem_id": "8a664c9fd8fde4b4", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Two friends split $676 in the ratio 6:7. How much does each receive?", "output": "1) Total ratio parts = 6+7 = 13.\n2) Each part = 676 ÷ 13 = 52.\n3) First share = 6 × 52 = 312.\n4) Second share = 7 × 52 = 364.\nFinal answer: $364$", "final_answer": "364", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e1a3148ff7a2c7a4", "problem_id": "130dadf694997c19", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- 2 x^{2} + 2 x + 1\\right) \\cos{\\left(x - 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\cos{\\left(x - 1 \\right)}) = - \\sin{\\left(x - 1 \\right)}$.\n3) So $f'(x) = 2 x^{2} \\sin{\\left(x - 1 \\right)} + 2 x e^{x} - 2 x \\sin{\\left(x - 1 \\right)} - 4 x \\cos{\\left(x - 1 \\right)} + 4 e^{x} - \\sin{\\left(x - 1 \\right)} + 2 \\cos{\\left(x - 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\cos{\\left(x - 1 \\right)}) = - \\sin{\\left(x - 1 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(1 - 2 x\\right) \\cos{\\left(x - 1 \\right)} + 2 \\left(x + 1\\right) e^{x} - \\left(2 x \\left(1 - x\\right) + 1\\right) \\sin{\\left(x - 1 \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(1 - 2 x\\right) \\cos{\\left(x - 1 \\right)} + 2 \\left(x + 1\\right) e^{x} - \\left(2 x \\left(1 - x\\right) + 1\\right) \\sin{\\left(x - 1 \\right)} + 2 e^{x}$", "final_answer": "2 \\left(1 - 2 x\\right) \\cos{\\left(x - 1 \\right)} + 2 \\left(x + 1\\right) e^{x} - \\left(2 x \\left(1 - x\\right) + 1\\right) \\sin{\\left(x - 1 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c63ab25b9f1f94c9", "problem_id": "95deba79439d3564", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Find the multiplicative inverse of 15 modulo 43.", "output": "1) We want x such that 15x ≡ 1 (mod 43).\n2) Use extended Euclid to find s,t with 15s + 43t = 1.\n3) One Bezout identity is: 15·(-20) + 43·(7) = 1.\n4) Reduce s modulo 43: x ≡ 23 (mod 43).\nFinal answer: $x\\equiv 23\\pmod{43}$", "final_answer": "x\\equiv 23\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "60271a49e6655128", "problem_id": "95deba79439d3564", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Find the multiplicative inverse of 15 modulo 43.", "output": "We want x such that 15x ≡ 1 (mod 43).\nUse extended Euclid to find s,t with 15s + 43t = 1.\nOne Bezout identity is: 15·(-20) + 43·(7) = 1.\nFinal answer: $x\\equiv 23\\pmod{43}$", "final_answer": "x\\equiv 23\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2b590e4075b327a9", "problem_id": "338a75b59a3d4938", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 22 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=22, k=2, p=$\\frac{1}{5}$ and simplify.\n4) Here C(22,2) = 231.\nFinal answer: $\\frac{253987186016256}{2384185791015625}$", "final_answer": "\\frac{253987186016256}{2384185791015625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dc52cd6d6d03d6d2", "problem_id": "338a75b59a3d4938", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 22 times, what is the probability of exactly 2 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=22, k=2, p=$\\frac{1}{5}$ and simplify.\nHere C(22,2) = 231.\nCheck: expression simplifies to $\\frac{253987186016256}{2384185791015625}$.\nFinal answer: $\\frac{253987186016256}{2384185791015625}$", "final_answer": "\\frac{253987186016256}{2384185791015625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c4b25c0e1fa55ad4", "problem_id": "f7bd28d78700979b", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$- 12 x + 7 y$ = $71$\n$- 10 x + y$ = $35$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-10$: $120 x - 70 y$ = $-710$.\n3) Multiply the second equation by $-12$: $120 x - 12 y$ = $-420$.\n4) Subtract to eliminate x: $- 58 y$ = $-290$.\n5) Solve for y: y = $5$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, 5\\right)$", "final_answer": "(x,y)=\\left(-3, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e2cd72e86c5be777", "problem_id": "f7bd28d78700979b", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- 12 x + 7 y$ = $71$\n$- 10 x + y$ = $35$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-10$: $120 x - 70 y$ = $-710$.\nMultiply the second equation by $-12$: $120 x - 12 y$ = $-420$.\nSubtract to eliminate x: $- 58 y$ = $-290$.\nSolve for y: y = $5$.\nSubstitute back to find x: x = $-3$.\nCheck: plug (x,y)=(-3,5) into both equations.\nFinal answer: $(x,y)=\\left(-3, 5\\right)$", "final_answer": "(x,y)=\\left(-3, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "58fbd101412d4de8", "problem_id": "4cd938055689bf32", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Worker A finishes a job in 12 hours and Worker B finishes it in 16 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/12 job/hour. B's rate = 1/16 job/hour.\n2) Combined rate = 1/12 + 1/16 = 7/48 job/hour.\n3) Time = 1 ÷ (combined rate) = 48/7 hours.\nFinal answer: $\\frac{48}{7}$", "final_answer": "\\frac{48}{7}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2e91b66dd7d62d2e", "problem_id": "4cd938055689bf32", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Worker A finishes a job in 12 hours and Worker B finishes it in 16 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/12 job/hour. B's rate = 1/16 job/hour.\nCombined rate = 1/12 + 1/16 = 7/48 job/hour.\nTime = 1 ÷ (combined rate) = 48/7 hours.\nCheck: (combined rate)×(time) = 7/48×48/7 = 1.\nFinal answer: $\\frac{48}{7}$", "final_answer": "\\frac{48}{7}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f3a568a3eaded6d3", "problem_id": "e2240b0f38ef7e49", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "An urn has 7 red balls and 12 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(19,8).\n3) Favorable samples: choose k reds and n−k blues: C(7,1)·C(12,7).\n4) Compute totals: C(19,8) = 75582, and favorable = 5544.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{308}{4199}$", "final_answer": "\\frac{308}{4199}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bdfd9e597785e185", "problem_id": "e2240b0f38ef7e49", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 7 red balls and 12 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(19,8).\nFavorable samples: choose k reds and n−k blues: C(7,1)·C(12,7).\nCompute totals: C(19,8) = 75582, and favorable = 5544.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{308}{4199}$.\nFinal answer: $\\frac{308}{4199}$", "final_answer": "\\frac{308}{4199}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b8bc9613cebe1abb", "problem_id": "d7033354dd81255e", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 11), x≡5 (mod 7). Give the solution modulo 77.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 77.\nCompute the CRT solution to get x ≡ 68 (mod 77).\nCheck: 68 mod 11=2 and mod 7=5.\nFinal answer: $x\\equiv 68\\pmod{77}$", "final_answer": "x\\equiv 68\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3cce7acd7c53292a", "problem_id": "d7033354dd81255e", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡2 (mod 11), x≡5 (mod 7). Give the solution modulo 77.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 77.\n3) Compute the CRT solution to get x ≡ 68 (mod 77).\nFinal answer: $x\\equiv 68\\pmod{77}$", "final_answer": "x\\equiv 68\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e2dc06ac6564338b", "problem_id": "b2113700c9731f86", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$- x + 3 y$ = $0$\n$- 4 y$ = $-4$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $0$: $0$ = $0$.\n3) Multiply the second equation by $-1$: $4 y$ = $4$.\n4) Subtract to eliminate x: $- 4 y$ = $-4$.\n5) Solve for y: y = $1$.\n6) Substitute back to find x: x = $3$.\nFinal answer: $(x,y)=\\left(3, 1\\right)$", "final_answer": "(x,y)=\\left(3, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fee529a88d9a6f9a", "problem_id": "b2113700c9731f86", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the linear system:\n$- x + 3 y$ = $0$\n$- 4 y$ = $-4$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $0$: $0$ = $0$.\nMultiply the second equation by $-1$: $4 y$ = $4$.\nSubtract to eliminate x: $- 4 y$ = $-4$.\nSolve for y: y = $1$.\nSubstitute back to find x: x = $3$.\nCheck: plug (x,y)=(3,1) into both equations.\nFinal answer: $(x,y)=\\left(3, 1\\right)$", "final_answer": "(x,y)=\\left(3, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e755dafd7bd1c489", "problem_id": "15da61464e0604a4", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "You want 46 liters of a 25% salt solution. You have a 10% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 46−x liters is 70% solution.\n2) Salt equation: 10x + 70(46−x) = 25·46.\n3) (10-70)x = (25-70)·46 ⇒ x = 69/2.\n4) Then 46−x = 23/2.\nFinal answer: $69/2 L of 10% and 23/2 L of 70%$", "final_answer": "69/2 L of 10% and 23/2 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "77114c46a5a1fff0", "problem_id": "9047de539922cc99", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(- 3 x^{2} - x - 1\\right) \\cos{\\left(x - 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x \\cos{\\left(x - 1 \\right)}) = x \\sin{\\left(x - 1 \\right)}$.\n3) So $f'(x) = 3 x^{2} \\sin{\\left(x - 1 \\right)} + x e^{x} + x \\sin{\\left(x - 1 \\right)} - 6 x \\cos{\\left(x - 1 \\right)} - 2 e^{x} + \\sin{\\left(x - 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x \\cos{\\left(x - 1 \\right)}) = x \\sin{\\left(x - 1 \\right)} - \\cos{\\left(x - 1 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(- 6 x - 1\\right) \\cos{\\left(x - 1 \\right)} + \\left(x - 3\\right) e^{x} - \\left(x \\left(- 3 x - 1\\right) - 1\\right) \\sin{\\left(x - 1 \\right)} + e^{x}$.\nFinal answer: $\\left(- 6 x - 1\\right) \\cos{\\left(x - 1 \\right)} + \\left(x - 3\\right) e^{x} - \\left(x \\left(- 3 x - 1\\right) - 1\\right) \\sin{\\left(x - 1 \\right)} + e^{x}$", "final_answer": "\\left(- 6 x - 1\\right) \\cos{\\left(x - 1 \\right)} + \\left(x - 3\\right) e^{x} - \\left(x \\left(- 3 x - 1\\right) - 1\\right) \\sin{\\left(x - 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "fe71b456fa5667b0", "problem_id": "96d2eb4db63bcbf5", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡3 (mod 4), x≡9 (mod 11). Give the solution modulo 44.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 44.\n3) Compute the CRT solution to get x ≡ 31 (mod 44).\nFinal answer: $x\\equiv 31\\pmod{44}$", "final_answer": "x\\equiv 31\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d5572b09c2be5a34", "problem_id": "96d2eb4db63bcbf5", "style": "answer_only", "instruction": "Final answer only.", "input": "Solve the congruences: x≡3 (mod 4), x≡9 (mod 11). Give the solution modulo 44.", "output": "$x\\equiv 31\\pmod{44}$", "final_answer": "x\\equiv 31\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9d1f0b2f083b157a", "problem_id": "622f9230972b3858", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 3 x^{2} + 5 x - 4$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 3 x^{3} + 2 x^{2} + x + 2$ and a = -1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 9 x^{2} + 4 x + 1$.\n5) Evaluate at a: p'(-1) = $-12$.\nFinal answer: $-12$", "final_answer": "-12", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "64f4aadfb20954a2", "problem_id": "622f9230972b3858", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 3 x^{2} + 5 x - 4$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 3 x^{3} + 2 x^{2} + x + 2$ and a = -1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 9 x^{2} + 4 x + 1$.\nEvaluate at a: p'(-1) = $-12$.\nCheck: the identity gives $-12$.\nFinal answer: $-12$", "final_answer": "-12", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "02c718bdd9b264b6", "problem_id": "f00d1fd7f7474d42", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 10 times, what is the probability of exactly 7 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=7, p=$\\frac{3}{7}$ and simplify.\n4) Here C(10,7) = 120.\nFinal answer: $\\frac{16796160}{282475249}$", "final_answer": "\\frac{16796160}{282475249}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "649995a54f8984f8", "problem_id": "f00d1fd7f7474d42", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 10 times, what is the probability of exactly 7 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=7, p=$\\frac{3}{7}$ and simplify.\nFinal answer: $\\frac{16796160}{282475249}$", "final_answer": "\\frac{16796160}{282475249}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "451d98d3a13a88ed", "problem_id": "cd914184f029ce8f", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the equation of the tangent line to $y=- x^{2} - x + 3$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 2 x - 1$.\n2) Slope at x=2: m = f'(2) = $-5$.\n3) Point on curve: (2, f(2)) = (2, $-3$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -3 = -5(x − 2).\n6) Optionally expand: y = $7 - 5 x$.\nFinal answer: $y = 7 - 5\\cdot x$", "final_answer": "y = 7 - 5\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6d8bc0794be8406a", "problem_id": "af630a815e14be2a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 \\sin{\\left(x \\right)}) = 2 \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 \\sin{\\left(x \\right)}) = 2 \\cos{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $- 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$", "final_answer": "- 2 x^{3} \\cos{\\left(x \\right)} - 6 x^{2} \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "75c16b86e6ac4da7", "problem_id": "e350de75a78e04b7", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 27 red balls and 23 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(50,11).\nFavorable samples: choose k reds and n−k blues: C(27,4)·C(23,7).\nCompute totals: C(50,11) = 37353738800, and favorable = 4302505350.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{3741309}{32481512}$.\nFinal answer: $\\frac{3741309}{32481512}$", "final_answer": "\\frac{3741309}{32481512}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "67d9d754276adb35", "problem_id": "e350de75a78e04b7", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 27 red balls and 23 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(50,11).\n3) Favorable samples: choose k reds and n−k blues: C(27,4)·C(23,7).\n4) Compute totals: C(50,11) = 37353738800, and favorable = 4302505350.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{3741309}{32481512}$", "final_answer": "\\frac{3741309}{32481512}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "08469ccd4154e345", "problem_id": "b1bea00589ab8cb2", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{10}$. If it is flipped 8 times, what is the probability of exactly 4 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=8, k=4, p=$\\frac{3}{10}$ and simplify.\nHere C(8,4) = 70.\nCheck: expression simplifies to $\\frac{1361367}{10000000}$.\nFinal answer: $\\frac{1361367}{10000000}$", "final_answer": "\\frac{1361367}{10000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0d0331b8637d3771", "problem_id": "b1bea00589ab8cb2", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{3}{10}$. If it is flipped 8 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=8, k=4, p=$\\frac{3}{10}$ and simplify.\n4) Here C(8,4) = 70.\nFinal answer: $\\frac{1361367}{10000000}$", "final_answer": "\\frac{1361367}{10000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8d4beb4ba73bb82c", "problem_id": "ab6bcbf69af090d7", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_8=27$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_8=19.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=19 and k=8.\n4) Compute C(26,7) = 657800.\nFinal answer: $657800$", "final_answer": "657800", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d66091b77801a17f", "problem_id": "ab6bcbf69af090d7", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_8=27$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_8=19.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=19 and k=8.\nFinal answer: $657800$", "final_answer": "657800", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "18b204a27799b1f8", "problem_id": "fccb0731acc05359", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡4 (mod 8), x≡4 (mod 13). Give the solution modulo 104.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 104.\n3) Compute the CRT solution to get x ≡ 4 (mod 104).\nFinal answer: $x\\equiv 4\\pmod{104}$", "final_answer": "x\\equiv 4\\pmod{104}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bb4303c3a5651097", "problem_id": "fccb0731acc05359", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡4 (mod 8), x≡4 (mod 13). Give the solution modulo 104.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 104.\nCompute the CRT solution to get x ≡ 4 (mod 104).\nCheck: 4 mod 8=4 and mod 13=4.\nFinal answer: $x\\equiv 4\\pmod{104}$", "final_answer": "x\\equiv 4\\pmod{104}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "afdbf9dc885fe606", "problem_id": "903922fb2368996e", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 5 x + 2$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$3$, b=$-5$, c=$2$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $1$, so √Δ = $1$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $1$.\n7) x2 = $\\frac{2}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{1, \\frac{2}{3}\\}$", "final_answer": "x \\in \\{1, \\frac{2}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1d58bf54c4848217", "problem_id": "903922fb2368996e", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 5 x + 2$ = 0.", "output": "$x \\in \\{1, \\frac{2}{3}\\}$", "final_answer": "x \\in \\{1, \\frac{2}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c058ce26fd8164af", "problem_id": "8245a3b341c408a5", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡8 (mod 11), x≡9 (mod 13). Give the solution modulo 143.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 143.\n3) Compute the CRT solution to get x ≡ 74 (mod 143).\nFinal answer: $x\\equiv 74\\pmod{143}$", "final_answer": "x\\equiv 74\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "37be484e7ee77f99", "problem_id": "c70b1fa0bce0719e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} 2 - x$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to 2 - 2 x.\n3) Then the limit is 10.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: 6.\nFinal answer: $6$", "final_answer": "6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "86992c0347ef80a4", "problem_id": "9f9ec76328efdbc0", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Solve the congruences: x≡3 (mod 4), x≡7 (mod 11). Give the solution modulo 44.", "output": "$x\\equiv 7\\pmod{44}$", "final_answer": "x\\equiv 7\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "60056092cf054713", "problem_id": "9f9ec76328efdbc0", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the congruences: x≡3 (mod 4), x≡7 (mod 11). Give the solution modulo 44.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 44.\nCompute the CRT solution to get x ≡ 7 (mod 44).\nFinal answer: $x\\equiv 7\\pmod{44}$", "final_answer": "x\\equiv 7\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7b097513eef00b80", "problem_id": "0f7fe4911e9f9749", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 7 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=7, k=2, p=$\\frac{1}{10}$ and simplify.\n4) Here C(7,2) = 21.\nFinal answer: $\\frac{1240029}{10000000}$", "final_answer": "\\frac{1240029}{10000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d5aa6299eb951609", "problem_id": "f7c0b0635d9ddf78", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} - 3 x - 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x^{2}) = - 4 x^{2}$.\n3) So $f'(x) = - 4 x^{2} - 3$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x^{2}) = - 4 x$.\nTherefore the correct derivative is $f'(x) = - 4 x - 3$.\nFinal answer: $- 4 x - 3$", "final_answer": "- 4 x - 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ac2c35f8745d534c", "problem_id": "c2ebc71f7b95e0fa", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A store sells a backpack for $395. It is discounted by 40%. You buy 10 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 395 × (1 − 40/100) = 395 × 60/100 = 237.\n2) Subtotal = 10 × 237 = 2370.\n3) Total with tax = 2370 × 107/100 = 25359/10.\nFinal answer: $\\frac{25359}{10}$", "final_answer": "\\frac{25359}{10}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "98cec08ffb2f4d4a", "problem_id": "c2ebc71f7b95e0fa", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "A store sells a backpack for $395. It is discounted by 40%. You buy 10 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 395 × (1 − 40/100) = 395 × 60/100 = 237.\nSubtotal = 10 × 237 = 2370.\nTotal with tax = 2370 × 107/100 = 25359/10.\nFinal answer: $\\frac{25359}{10}$", "final_answer": "\\frac{25359}{10}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6ba9d0605a380220", "problem_id": "75b784e72331b122", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$- 4 x + 4 y$ = $-24$\n$4 x - y$ = $21$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $4$: $- 16 x + 16 y$ = $-96$.\n3) Multiply the second equation by $-4$: $- 16 x + 4 y$ = $-84$.\n4) Subtract to eliminate x: $12 y$ = $-12$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, -1\\right)$", "final_answer": "(x,y)=\\left(5, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "276c56f7dd143936", "problem_id": "75b784e72331b122", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$- 4 x + 4 y$ = $-24$\n$4 x - y$ = $21$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $4$: $- 16 x + 16 y$ = $-96$.\nMultiply the second equation by $-4$: $- 16 x + 4 y$ = $-84$.\nSubtract to eliminate x: $12 y$ = $-12$.\nSolve for y: y = $-1$.\nSubstitute back to find x: x = $5$.\nCheck: plug (x,y)=(5,-1) into both equations.\nFinal answer: $(x,y)=\\left(5, -1\\right)$", "final_answer": "(x,y)=\\left(5, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a9e39776ca0901bf", "problem_id": "0026b7d057122a39", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡5 (mod 11), x≡3 (mod 5). Give the solution modulo 55.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 55.\n3) Compute the CRT solution to get x ≡ 38 (mod 55).\nFinal answer: $x\\equiv 38\\pmod{55}$", "final_answer": "x\\equiv 38\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d70f65445fa126d6", "problem_id": "0026b7d057122a39", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡5 (mod 11), x≡3 (mod 5). Give the solution modulo 55.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 55.\nCompute the CRT solution to get x ≡ 38 (mod 55).\nCheck: 38 mod 11=5 and mod 5=3.\nFinal answer: $x\\equiv 38\\pmod{55}$", "final_answer": "x\\equiv 38\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fc42cf331a2ae5f4", "problem_id": "9cbc09078430482e", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 11 red balls and 12 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(23,12).\n3) Favorable samples: choose k reds and n−k blues: C(11,1)·C(12,11).\n4) Compute totals: C(23,12) = 1352078, and favorable = 132.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{66}{676039}$", "final_answer": "\\frac{66}{676039}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5c27862767b2a7ea", "problem_id": "9cbc09078430482e", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 11 red balls and 12 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(23,12).\nFavorable samples: choose k reds and n−k blues: C(11,1)·C(12,11).\nCompute totals: C(23,12) = 1352078, and favorable = 132.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{66}{676039}$.\nFinal answer: $\\frac{66}{676039}$", "final_answer": "\\frac{66}{676039}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0c80c05be4e9507b", "problem_id": "ffb704bf640040c4", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the linear system:\n$- 8 x - 4 y$ = $80$\n$3 x - 10 y$ = $62$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $3$: $- 24 x - 12 y$ = $240$.\n3) Multiply the second equation by $-8$: $- 24 x + 80 y$ = $-496$.\n4) Subtract to eliminate x: $- 92 y$ = $736$.\n5) Solve for y: y = $-8$.\n6) Substitute back to find x: x = $-6$.\nFinal answer: $(x,y)=\\left(-6, -8\\right)$", "final_answer": "(x,y)=\\left(-6, -8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "019a830f3d6d5a7c", "problem_id": "ffb704bf640040c4", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- 8 x - 4 y$ = $80$\n$3 x - 10 y$ = $62$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $3$: $- 24 x - 12 y$ = $240$.\nMultiply the second equation by $-8$: $- 24 x + 80 y$ = $-496$.\nSubtract to eliminate x: $- 92 y$ = $736$.\nSolve for y: y = $-8$.\nSubstitute back to find x: x = $-6$.\nCheck: plug (x,y)=(-6,-8) into both equations.\nFinal answer: $(x,y)=\\left(-6, -8\\right)$", "final_answer": "(x,y)=\\left(-6, -8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "052caa3c5013dff3", "problem_id": "950c9a6c5c33852f", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 13 times, what is the probability of exactly 11 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=13, k=11, p=$\\frac{1}{2}$ and simplify.\nHere C(13,11) = 78.\nCheck: expression simplifies to $\\frac{39}{4096}$.\nFinal answer: $\\frac{39}{4096}$", "final_answer": "\\frac{39}{4096}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c87ab4487f3e293c", "problem_id": "950c9a6c5c33852f", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 13 times, what is the probability of exactly 11 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=13, k=11, p=$\\frac{1}{2}$ and simplify.\n4) Here C(13,11) = 78.\nFinal answer: $\\frac{39}{4096}$", "final_answer": "\\frac{39}{4096}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cf03fd7774c647b0", "problem_id": "f8243139dedde385", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the linear system:\n$- 7 x - 6 y$ = $97$\n$- 12 x + 2 y$ = $68$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-12$: $84 x + 72 y$ = $-1164$.\n3) Multiply the second equation by $-7$: $84 x - 14 y$ = $-476$.\n4) Subtract to eliminate x: $86 y$ = $-688$.\n5) Solve for y: y = $-8$.\n6) Substitute back to find x: x = $-7$.\nFinal answer: $(x,y)=\\left(-7, -8\\right)$", "final_answer": "(x,y)=\\left(-7, -8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1e5905b2b05d7684", "problem_id": "f8243139dedde385", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$- 7 x - 6 y$ = $97$\n$- 12 x + 2 y$ = $68$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-12$: $84 x + 72 y$ = $-1164$.\nMultiply the second equation by $-7$: $84 x - 14 y$ = $-476$.\nSubtract to eliminate x: $86 y$ = $-688$.\nSolve for y: y = $-8$.\nSubstitute back to find x: x = $-7$.\nCheck: plug (x,y)=(-7,-8) into both equations.\nFinal answer: $(x,y)=\\left(-7, -8\\right)$", "final_answer": "(x,y)=\\left(-7, -8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "56860e663b1e0276", "problem_id": "e3e63fa51b52e5ba", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A store sells a backpack for $615. It is discounted by 20%. You buy 12 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 615 × (1 − 20/100) = 615 × 80/100 = 492.\nSubtotal = 12 × 492 = 5904.\nTotal with tax = 5904 × 108/100 = 159408/25.\nCheck: recomputing the formula yields 159408/25.\nFinal answer: $\\frac{159408}{25}$", "final_answer": "\\frac{159408}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "996b77e3fd576838", "problem_id": "e3e63fa51b52e5ba", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A store sells a backpack for $615. It is discounted by 20%. You buy 12 backpacks. After the discount, a sales tax of 8% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 615 × (1 − 20/100) = 615 × 80/100 = 492.\n2) Subtotal = 12 × 492 = 5904.\n3) Total with tax = 5904 × 108/100 = 159408/25.\nFinal answer: $\\frac{159408}{25}$", "final_answer": "\\frac{159408}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "efb150dc54071deb", "problem_id": "d1731117ba71e546", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Worker A finishes a job in 11 hours and Worker B finishes it in 10 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/11 job/hour. B's rate = 1/10 job/hour.\n2) Combined rate = 1/11 + 1/10 = 21/110 job/hour.\n3) Time = 1 ÷ (combined rate) = 110/21 hours.\nFinal answer: $\\frac{110}{21}$", "final_answer": "\\frac{110}{21}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2b5b2fe0510c269b", "problem_id": "d1731117ba71e546", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Worker A finishes a job in 11 hours and Worker B finishes it in 10 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/11 job/hour. B's rate = 1/10 job/hour.\nCombined rate = 1/11 + 1/10 = 21/110 job/hour.\nTime = 1 ÷ (combined rate) = 110/21 hours.\nCheck: (combined rate)×(time) = 21/110×110/21 = 1.\nFinal answer: $\\frac{110}{21}$", "final_answer": "\\frac{110}{21}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ddf859f2cde522e7", "problem_id": "2e0d80074ebb10d8", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 28 red balls and 20 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(48,12).\nFavorable samples: choose k reds and n−k blues: C(28,2)·C(20,10).\nCompute totals: C(48,12) = 69668534468, and favorable = 69837768.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{70686}{70514711}$.\nFinal answer: $\\frac{70686}{70514711}$", "final_answer": "\\frac{70686}{70514711}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c91b9f8ca4e4345e", "problem_id": "2e0d80074ebb10d8", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "An urn has 28 red balls and 20 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(48,12).\n3) Favorable samples: choose k reds and n−k blues: C(28,2)·C(20,10).\n4) Compute totals: C(48,12) = 69668534468, and favorable = 69837768.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{70686}{70514711}$", "final_answer": "\\frac{70686}{70514711}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "882565ac3b53b2d2", "problem_id": "bd9c82ffe02b07b4", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 27 red balls and 21 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(48,12).\n3) Favorable samples: choose k reds and n−k blues: C(27,1)·C(21,11).\n4) Compute totals: C(48,12) = 69668534468, and favorable = 9523332.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{9639}{70514711}$", "final_answer": "\\frac{9639}{70514711}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3be5c290019858ac", "problem_id": "bd9c82ffe02b07b4", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 27 red balls and 21 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(48,12).\nFavorable samples: choose k reds and n−k blues: C(27,1)·C(21,11).\nCompute totals: C(48,12) = 69668534468, and favorable = 9523332.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{9639}{70514711}$.\nFinal answer: $\\frac{9639}{70514711}$", "final_answer": "\\frac{9639}{70514711}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "01f1831880e4d728", "problem_id": "4f34ebf0b592a4e5", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A store sells a backpack for $111. It is discounted by 20%. You buy 12 backpacks. What is the total cost?", "output": "1) Discounted price = 111 × (1 − 20/100) = 111 × 80/100 = 444/5.\n2) Subtotal = 12 × 444/5 = 5328/5.\nFinal answer: $\\frac{5328}{5}$", "final_answer": "\\frac{5328}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a5f35e884bc9409d", "problem_id": "4f34ebf0b592a4e5", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $111. It is discounted by 20%. You buy 12 backpacks. What is the total cost?", "output": "Discounted price = 111 × (1 − 20/100) = 111 × 80/100 = 444/5.\nSubtotal = 12 × 444/5 = 5328/5.\nCheck: recomputing the formula yields 5328/5.\nFinal answer: $\\frac{5328}{5}$", "final_answer": "\\frac{5328}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a4f862c7780e9ee4", "problem_id": "f56c0f4f63420932", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} x - 1$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $x^{2} - 1$ and a = -1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nFinal answer: $-2$", "final_answer": "-2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "29caf940be3a93cc", "problem_id": "f56c0f4f63420932", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} x - 1$.", "output": "$-2$", "final_answer": "-2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4b6d6e139424f658", "problem_id": "f3c5ad0f91d4a7ee", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $- 3 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.\nFinal answer: $- 3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - x\\cdot \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - \\cos{\\left(x \\right)}$", "final_answer": "- 3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - x\\cdot \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "20d70a0caff9bd70", "problem_id": "f3c5ad0f91d4a7ee", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $- 3 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.\nFinal answer: $- 3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - x\\cdot \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - \\cos{\\left(x \\right)}$", "final_answer": "- 3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} - 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - x\\cdot \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0c0627189d880382", "problem_id": "5d13ff0b51171b12", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 30 red balls and 28 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(58,7).\n3) Favorable samples: choose k reds and n−k blues: C(30,3)·C(28,4).\n4) Compute totals: C(58,7) = 300674088, and favorable = 83128500.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{6125}{22154}$", "final_answer": "\\frac{6125}{22154}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8798901fa8a5fc92", "problem_id": "f0edbf272a0e2609", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{3} + x^{2} - 2 x + 4$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x^{3}) = - 6 x^{3}$.\n3) So $f'(x) = - 6 x^{3} + 2 x - 2$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x^{3}) = - 6 x^{2}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- 3 x^{2} + x - 1\\right)$.\nFinal answer: $2 \\left(- 3 x^{2} + x - 1\\right)$", "final_answer": "2 \\left(- 3 x^{2} + x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c051fb0e8216e822", "problem_id": "9e22fdbc2bee1b08", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡0 (mod 7), x≡2 (mod 9). Give the solution modulo 63.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 63.\n3) Compute the CRT solution to get x ≡ 56 (mod 63).\nFinal answer: $x\\equiv 56\\pmod{63}$", "final_answer": "x\\equiv 56\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7cbf57cc1a2e5e40", "problem_id": "9e22fdbc2bee1b08", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡0 (mod 7), x≡2 (mod 9). Give the solution modulo 63.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 63.\nCompute the CRT solution to get x ≡ 56 (mod 63).\nCheck: 56 mod 7=0 and mod 9=2.\nFinal answer: $x\\equiv 56\\pmod{63}$", "final_answer": "x\\equiv 56\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "00caebeee09a7bbd", "problem_id": "f792d2e623894ff4", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 13 red balls and 29 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(42,11).\n3) Favorable samples: choose k reds and n−k blues: C(13,4)·C(29,7).\n4) Compute totals: C(42,11) = 4280561376, and favorable = 1115957700.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{7153575}{27439496}$", "final_answer": "\\frac{7153575}{27439496}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "42b6217fca002305", "problem_id": "f792d2e623894ff4", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "An urn has 13 red balls and 29 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(42,11).\nFavorable samples: choose k reds and n−k blues: C(13,4)·C(29,7).\nFinal answer: $\\frac{7153575}{27439496}$", "final_answer": "\\frac{7153575}{27439496}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7197521ab009a3a0", "problem_id": "12eaa8eea36da858", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the linear system:\n$- 3 x + 4 y$ = $-15$\n$7 x - 8 y$ = $27$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $7$: $- 21 x + 28 y$ = $-105$.\nMultiply the second equation by $-3$: $- 21 x + 24 y$ = $-81$.\nFinal answer: $(x,y)=\\left(-3, -6\\right)$", "final_answer": "(x,y)=\\left(-3, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1bb1cd70674b6d87", "problem_id": "12eaa8eea36da858", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the linear system:\n$- 3 x + 4 y$ = $-15$\n$7 x - 8 y$ = $27$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $7$: $- 21 x + 28 y$ = $-105$.\n3) Multiply the second equation by $-3$: $- 21 x + 24 y$ = $-81$.\n4) Subtract to eliminate x: $4 y$ = $-24$.\n5) Solve for y: y = $-6$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, -6\\right)$", "final_answer": "(x,y)=\\left(-3, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aebb1789ff236924", "problem_id": "75ec828a8e4e0411", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡5 (mod 11), x≡6 (mod 9). Give the solution modulo 99.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 99.\n3) Compute the CRT solution to get x ≡ 60 (mod 99).\nFinal answer: $x\\equiv 60\\pmod{99}$", "final_answer": "x\\equiv 60\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "763d3d20e10dc5db", "problem_id": "75ec828a8e4e0411", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡5 (mod 11), x≡6 (mod 9). Give the solution modulo 99.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 99.\nCompute the CRT solution to get x ≡ 60 (mod 99).\nCheck: 60 mod 11=5 and mod 9=6.\nFinal answer: $x\\equiv 60\\pmod{99}$", "final_answer": "x\\equiv 60\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a79f245be0eb6a14", "problem_id": "85b2dc1c11ab7709", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 17 times, what is the probability of exactly 12 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=17, k=12, p=$\\frac{2}{5}$ and simplify.\n4) Here C(17,12) = 6188.\nFinal answer: $\\frac{6159089664}{762939453125}$", "final_answer": "\\frac{6159089664}{762939453125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "47b9cd88a10db5e0", "problem_id": "85b2dc1c11ab7709", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 17 times, what is the probability of exactly 12 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=17, k=12, p=$\\frac{2}{5}$ and simplify.\nHere C(17,12) = 6188.\nCheck: expression simplifies to $\\frac{6159089664}{762939453125}$.\nFinal answer: $\\frac{6159089664}{762939453125}$", "final_answer": "\\frac{6159089664}{762939453125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d9fdb9b820faa6f2", "problem_id": "30d830ad4bad8714", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_8=27$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=27 and k=8.\nCompute C(34,7) = 5379616.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $5379616$", "final_answer": "5379616", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "378d3b5e7565344c", "problem_id": "30d830ad4bad8714", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_8=27$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=27 and k=8.\n4) Compute C(34,7) = 5379616.\nFinal answer: $5379616$", "final_answer": "5379616", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "418484bffddf6622", "problem_id": "2a3eac3612928203", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_3=14$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=14 and k=3.\n4) Compute C(16,2) = 120.\nFinal answer: $120$", "final_answer": "120", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b17028bf1b4f5b47", "problem_id": "2a3eac3612928203", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_3=14$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=14 and k=3.\nFinal answer: $120$", "final_answer": "120", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8654fa890ced22d1", "problem_id": "4a7b357280ef675a", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{3}$. If it is flipped 21 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=21, k=3, p=$\\frac{1}{3}$ and simplify.\nFinal answer: $\\frac{348651520}{10460353203}$", "final_answer": "\\frac{348651520}{10460353203}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "42bc9c15e829618c", "problem_id": "4a7b357280ef675a", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{3}$. If it is flipped 21 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=21, k=3, p=$\\frac{1}{3}$ and simplify.\nHere C(21,3) = 1330.\nCheck: expression simplifies to $\\frac{348651520}{10460353203}$.\nFinal answer: $\\frac{348651520}{10460353203}$", "final_answer": "\\frac{348651520}{10460353203}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f69c14915ac1822d", "problem_id": "9f1d8e72f497f57c", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Worker A finishes a job in 12 hours and Worker B finishes it in 22 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/12 job/hour. B's rate = 1/22 job/hour.\n2) Combined rate = 1/12 + 1/22 = 17/132 job/hour.\n3) Time = 1 ÷ (combined rate) = 132/17 hours.\nFinal answer: $\\frac{132}{17}$", "final_answer": "\\frac{132}{17}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ac8402065516f3ad", "problem_id": "9f1d8e72f497f57c", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Worker A finishes a job in 12 hours and Worker B finishes it in 22 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/12 job/hour. B's rate = 1/22 job/hour.\nCombined rate = 1/12 + 1/22 = 17/132 job/hour.\nTime = 1 ÷ (combined rate) = 132/17 hours.\nFinal answer: $\\frac{132}{17}$", "final_answer": "\\frac{132}{17}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ee6407eb88e814bd", "problem_id": "0610f60037a10aa3", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Worker A finishes a job in 13 hours and Worker B finishes it in 16 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/13 job/hour. B's rate = 1/16 job/hour.\n2) Combined rate = 1/13 + 1/16 = 29/208 job/hour.\n3) Time = 1 ÷ (combined rate) = 208/29 hours.\nFinal answer: $\\frac{208}{29}$", "final_answer": "\\frac{208}{29}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "32e5d156206f4ba6", "problem_id": "0610f60037a10aa3", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Worker A finishes a job in 13 hours and Worker B finishes it in 16 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/13 job/hour. B's rate = 1/16 job/hour.\nCombined rate = 1/13 + 1/16 = 29/208 job/hour.\nTime = 1 ÷ (combined rate) = 208/29 hours.\nCheck: (combined rate)×(time) = 29/208×208/29 = 1.\nFinal answer: $\\frac{208}{29}$", "final_answer": "\\frac{208}{29}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "95d16788be656d39", "problem_id": "63d8b716118e5065", "style": "answer_only", "instruction": "Final answer only.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\pi} - 3 x^{4} - 3 x^{3} + 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2\\,dx$.", "output": "$- \\frac{6\\cdot \\pi^{5}}{5} - 4\\cdot \\pi$", "final_answer": "- \\frac{6\\cdot \\pi^{5}}{5} - 4\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a3a20aa8f740ae51", "problem_id": "d6bb2ee279f828cb", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} + x + 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(3 x^{2}) = 6 x^{2}$.\n3) So $f'(x) = 6 x^{2} + 1$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(3 x^{2}) = 6 x$.\nTherefore the correct derivative is $f'(x) = 6 x + 1$.\nFinal answer: $6 x + 1$", "final_answer": "6 x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "cdeeae53e97e38b4", "problem_id": "d1748de08158ee5d", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 14 times, what is the probability of exactly 7 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=14, k=7, p=$\\frac{4}{5}$ and simplify.\nFinal answer: $\\frac{56229888}{6103515625}$", "final_answer": "\\frac{56229888}{6103515625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3ff930d1d9ca981f", "problem_id": "d1748de08158ee5d", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 14 times, what is the probability of exactly 7 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=14, k=7, p=$\\frac{4}{5}$ and simplify.\n4) Here C(14,7) = 3432.\nFinal answer: $\\frac{56229888}{6103515625}$", "final_answer": "\\frac{56229888}{6103515625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6085b0fc02f2ecd4", "problem_id": "59fab7f7aa903462", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡7 (mod 8), x≡2 (mod 11). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 79 (mod 88).\nFinal answer: $x\\equiv 79\\pmod{88}$", "final_answer": "x\\equiv 79\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f530cb8eefccce64", "problem_id": "59fab7f7aa903462", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the congruences: x≡7 (mod 8), x≡2 (mod 11). Give the solution modulo 88.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 88.\nCompute the CRT solution to get x ≡ 79 (mod 88).\nFinal answer: $x\\equiv 79\\pmod{88}$", "final_answer": "x\\equiv 79\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8dbd92902080a28a", "problem_id": "2122a8006b5361f9", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡4 (mod 5), x≡7 (mod 8). Give the solution modulo 40.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 40.\n3) Compute the CRT solution to get x ≡ 39 (mod 40).\nFinal answer: $x\\equiv 39\\pmod{40}$", "final_answer": "x\\equiv 39\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c8dd67883592c751", "problem_id": "2122a8006b5361f9", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡4 (mod 5), x≡7 (mod 8). Give the solution modulo 40.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 40.\nCompute the CRT solution to get x ≡ 39 (mod 40).\nCheck: 39 mod 5=4 and mod 8=7.\nFinal answer: $x\\equiv 39\\pmod{40}$", "final_answer": "x\\equiv 39\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8d0ee846eae85e9e", "problem_id": "a3d8b30ad2fb2a08", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 12 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=4, p=$\\frac{1}{5}$ and simplify.\n4) Here C(12,4) = 495.\nFinal answer: $\\frac{6488064}{48828125}$", "final_answer": "\\frac{6488064}{48828125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "13ea1630cb404a94", "problem_id": "a3d8b30ad2fb2a08", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 12 times, what is the probability of exactly 4 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=12, k=4, p=$\\frac{1}{5}$ and simplify.\nHere C(12,4) = 495.\nCheck: expression simplifies to $\\frac{6488064}{48828125}$.\nFinal answer: $\\frac{6488064}{48828125}$", "final_answer": "\\frac{6488064}{48828125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db127abc17f609ca", "problem_id": "3e0204c59c66d327", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡2 (mod 9), x≡4 (mod 7). Give the solution modulo 63.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 63.\n3) Compute the CRT solution to get x ≡ 11 (mod 63).\nFinal answer: $x\\equiv 11\\pmod{63}$", "final_answer": "x\\equiv 11\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "53c7bf050725a198", "problem_id": "3e0204c59c66d327", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡2 (mod 9), x≡4 (mod 7). Give the solution modulo 63.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 63.\nCompute the CRT solution to get x ≡ 11 (mod 63).\nCheck: 11 mod 9=2 and mod 7=4.\nFinal answer: $x\\equiv 11\\pmod{63}$", "final_answer": "x\\equiv 11\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e8d91e911b85e69", "problem_id": "d8ba5ecd6ddbe0bf", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 3 x^{3} - 6 x^{2} - 9 x - 15$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{3} - 6 x^{2} - 10 x - 15.\n3) Then the limit is -83.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -81.\nFinal answer: $-81$", "final_answer": "-81", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "4bcf6e1a5cae6c4f", "problem_id": "2dca719869ee0153", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "You want 35 liters of a 30% salt solution. You have a 15% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 35−x liters is 60% solution.\n2) Salt equation: 15x + 60(35−x) = 30·35.\n3) (15-60)x = (30-60)·35 ⇒ x = 70/3.\n4) Then 35−x = 35/3.\nFinal answer: $70/3 L of 15% and 35/3 L of 60%$", "final_answer": "70/3 L of 15% and 35/3 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f0e4215857463888", "problem_id": "a6ec28fdb0b163bc", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "An urn has 20 red balls and 26 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(46,12).\nFavorable samples: choose k reds and n−k blues: C(20,2)·C(26,10).\nFinal answer: $\\frac{35530}{1369851}$", "final_answer": "\\frac{35530}{1369851}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2e49bdaa82732652", "problem_id": "a6ec28fdb0b163bc", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 20 red balls and 26 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(46,12).\n3) Favorable samples: choose k reds and n−k blues: C(20,2)·C(26,10).\n4) Compute totals: C(46,12) = 38910617655, and favorable = 1009229650.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{35530}{1369851}$", "final_answer": "\\frac{35530}{1369851}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aea10f4ecf2095fb", "problem_id": "bbb103ce83294033", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A store sells a backpack for $695. It is discounted by 25%. You buy 5 backpacks. What is the total cost?", "output": "1) Discounted price = 695 × (1 − 25/100) = 695 × 75/100 = 2085/4.\n2) Subtotal = 5 × 2085/4 = 10425/4.\nFinal answer: $\\frac{10425}{4}$", "final_answer": "\\frac{10425}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "55c44a10ba672e44", "problem_id": "bbb103ce83294033", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "A store sells a backpack for $695. It is discounted by 25%. You buy 5 backpacks. What is the total cost?", "output": "Discounted price = 695 × (1 − 25/100) = 695 × 75/100 = 2085/4.\nSubtotal = 5 × 2085/4 = 10425/4.\nCheck: recomputing the formula yields 10425/4.\nFinal answer: $\\frac{10425}{4}$", "final_answer": "\\frac{10425}{4}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e6d44a7432c59e4a", "problem_id": "37161c531fe3d70e", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "You want 51 liters of a 30% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "136/3 L of 25% and 17/3 L of 70%", "final_answer": "136/3 L of 25% and 17/3 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "75ee1472b273f768", "problem_id": "37161c531fe3d70e", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "You want 51 liters of a 30% salt solution. You have a 25% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 51−x liters is 70% solution.\nSalt equation: 25x + 70(51−x) = 30·51.\n(25-70)x = (30-70)·51 ⇒ x = 136/3.\nFinal answer: $136/3 L of 25% and 17/3 L of 70%$", "final_answer": "136/3 L of 25% and 17/3 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bad07f214ad01823", "problem_id": "4b292d4ce9a87790", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$- 4 x - y$ = $-25$\n$x$ = $5$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $1$: $- 4 x - y$ = $-25$.\nMultiply the second equation by $-4$: $- 4 x$ = $-20$.\nSubtract to eliminate x: $- y$ = $-5$.\nSolve for y: y = $5$.\nSubstitute back to find x: x = $5$.\nCheck: plug (x,y)=(5,5) into both equations.\nFinal answer: $(x,y)=\\left(5, 5\\right)$", "final_answer": "(x,y)=\\left(5, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2ce515638f44a552", "problem_id": "4b292d4ce9a87790", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$- 4 x - y$ = $-25$\n$x$ = $5$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $1$: $- 4 x - y$ = $-25$.\n3) Multiply the second equation by $-4$: $- 4 x$ = $-20$.\n4) Subtract to eliminate x: $- y$ = $-5$.\n5) Solve for y: y = $5$.\n6) Substitute back to find x: x = $5$.\nFinal answer: $(x,y)=\\left(5, 5\\right)$", "final_answer": "(x,y)=\\left(5, 5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bc1b970440e6cd2a", "problem_id": "33eb293dd4fbf1e2", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 x - \\frac{\\sqrt{5} x}{2} + \\frac{3 \\sqrt{5}}{4} + \\frac{9}{4}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$-3 - \\frac{\\sqrt{5}}{2}$, c=$\\frac{3 \\sqrt{5}}{4} + \\frac{9}{4}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{5}{4}$, so √Δ = $\\frac{\\sqrt{5}}{2}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{\\sqrt{5}}{2} + \\frac{3}{2}$.\n7) x2 = $\\frac{3}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{\\sqrt{5}}{2} + \\frac{3}{2}, \\frac{3}{2}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{5}}{2} + \\frac{3}{2}, \\frac{3}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "474b71318056a93b", "problem_id": "33eb293dd4fbf1e2", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - 3 x - \\frac{\\sqrt{5} x}{2} + \\frac{3 \\sqrt{5}}{4} + \\frac{9}{4}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$-3 - \\frac{\\sqrt{5}}{2}$, c=$\\frac{3 \\sqrt{5}}{4} + \\frac{9}{4}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{\\sqrt{5}}{2} + \\frac{3}{2}, \\frac{3}{2}\\}$", "final_answer": "x \\in \\{\\frac{\\sqrt{5}}{2} + \\frac{3}{2}, \\frac{3}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "add28f8ede9e0496", "problem_id": "76ce2554a5b2c569", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Two friends split $255 in the ratio 2:3. How much does each receive?", "output": "Total ratio parts = 2+3 = 5.\nEach part = 255 ÷ 5 = 51.\nFirst share = 2 × 51 = 102.\nSecond share = 3 × 51 = 153.\nCheck: 102+153=255.\nFinal answer: $153$", "final_answer": "153", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4cf80cf132774f93", "problem_id": "76ce2554a5b2c569", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Two friends split $255 in the ratio 2:3. How much does each receive?", "output": "1) Total ratio parts = 2+3 = 5.\n2) Each part = 255 ÷ 5 = 51.\n3) First share = 2 × 51 = 102.\n4) Second share = 3 × 51 = 153.\nFinal answer: $153$", "final_answer": "153", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2b981397454a5e54", "problem_id": "386832c8926b71b4", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$7 x - y$ = $-27$\n$- 5 x - 3 y$ = $-3$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-5$: $- 35 x + 5 y$ = $135$.\nMultiply the second equation by $7$: $- 35 x - 21 y$ = $-21$.\nSubtract to eliminate x: $26 y$ = $156$.\nSolve for y: y = $6$.\nSubstitute back to find x: x = $-3$.\nCheck: plug (x,y)=(-3,6) into both equations.\nFinal answer: $(x,y)=\\left(-3, 6\\right)$", "final_answer": "(x,y)=\\left(-3, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "57250fc32fe41fac", "problem_id": "386832c8926b71b4", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$7 x - y$ = $-27$\n$- 5 x - 3 y$ = $-3$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-5$: $- 35 x + 5 y$ = $135$.\n3) Multiply the second equation by $7$: $- 35 x - 21 y$ = $-21$.\n4) Subtract to eliminate x: $26 y$ = $156$.\n5) Solve for y: y = $6$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, 6\\right)$", "final_answer": "(x,y)=\\left(-3, 6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0f705dc72efbc700", "problem_id": "2cf1f9a8c2aefc0b", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the congruences: x≡1 (mod 4), x≡4 (mod 5). Give the solution modulo 20.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 20.\nCompute the CRT solution to get x ≡ 9 (mod 20).\nFinal answer: $x\\equiv 9\\pmod{20}$", "final_answer": "x\\equiv 9\\pmod{20}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "926def7db2a04253", "problem_id": "2cf1f9a8c2aefc0b", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡1 (mod 4), x≡4 (mod 5). Give the solution modulo 20.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 20.\n3) Compute the CRT solution to get x ≡ 9 (mod 20).\nFinal answer: $x\\equiv 9\\pmod{20}$", "final_answer": "x\\equiv 9\\pmod{20}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b7fe98cbe248fbff", "problem_id": "ec526b6b0dd9ac68", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "An urn has 16 red balls and 11 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(27,8).\n3) Favorable samples: choose k reds and n−k blues: C(16,1)·C(11,7).\n4) Compute totals: C(27,8) = 2220075, and favorable = 5280.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{32}{13455}$", "final_answer": "\\frac{32}{13455}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "42f900b8ebd4496c", "problem_id": "ec526b6b0dd9ac68", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "An urn has 16 red balls and 11 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(27,8).\nFavorable samples: choose k reds and n−k blues: C(16,1)·C(11,7).\nFinal answer: $\\frac{32}{13455}$", "final_answer": "\\frac{32}{13455}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d7b423aa84e46263", "problem_id": "63250a61649eb4f1", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $2100 in the ratio 8:6. How much does each receive?", "output": "1) Total ratio parts = 8+6 = 14.\n2) Each part = 2100 ÷ 14 = 150.\n3) First share = 8 × 150 = 1200.\n4) Second share = 6 × 150 = 900.\nFinal answer: $900$", "final_answer": "900", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6ff7987e49eb467c", "problem_id": "63250a61649eb4f1", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Two friends split $2100 in the ratio 8:6. How much does each receive?", "output": "Total ratio parts = 8+6 = 14.\nEach part = 2100 ÷ 14 = 150.\nFirst share = 8 × 150 = 1200.\nFinal answer: $900$", "final_answer": "900", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e949b7e547d01657", "problem_id": "e1636336dac3aceb", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{4} - 2 x^{3} + 2 x^{2} + 4 x + 2$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(2 x^{4}) = 8 x^{4}$.\n3) So $f'(x) = 8 x^{4} - 6 x^{2} + 4 x + 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(2 x^{4}) = 8 x^{3}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(4 x^{3} - 3 x^{2} + 2 x + 2\\right)$.\nFinal answer: $2 \\left(4 x^{3} - 3 x^{2} + 2 x + 2\\right)$", "final_answer": "2 \\left(4 x^{3} - 3 x^{2} + 2 x + 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f80d2f197e3ed26b", "problem_id": "b08353fa8dae09ce", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 21 times, what is the probability of exactly 16 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=21, k=16, p=$\\frac{2}{5}$ and simplify.\n4) Here C(21,16) = 20349.\nFinal answer: $\\frac{324062871552}{476837158203125}$", "final_answer": "\\frac{324062871552}{476837158203125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "40bcd769985117ed", "problem_id": "1567de312e331523", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A store sells a backpack for $142. It is discounted by 10%. You buy 11 backpacks. What is the total cost?", "output": "1) Discounted price = 142 × (1 − 10/100) = 142 × 90/100 = 639/5.\n2) Subtotal = 11 × 639/5 = 7029/5.\nFinal answer: $\\frac{7029}{5}$", "final_answer": "\\frac{7029}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "97d5347ad2edbd0f", "problem_id": "1567de312e331523", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A store sells a backpack for $142. It is discounted by 10%. You buy 11 backpacks. What is the total cost?", "output": "Discounted price = 142 × (1 − 10/100) = 142 × 90/100 = 639/5.\nSubtotal = 11 × 639/5 = 7029/5.\nCheck: recomputing the formula yields 7029/5.\nFinal answer: $\\frac{7029}{5}$", "final_answer": "\\frac{7029}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0c9e2f25bc2ec44d", "problem_id": "0a4f776d6c85505d", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 x}{2} - 1$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{3}{2}$, c=$-1$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{25}{4}$, so √Δ = $\\frac{5}{2}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $2$.\n7) x2 = $- \\frac{1}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{2, - \\frac{1}{2}\\}$", "final_answer": "x \\in \\{2, - \\frac{1}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a5b53c5fac0f22ad", "problem_id": "0a4f776d6c85505d", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 x}{2} - 1$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{3}{2}$, c=$-1$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{2, - \\frac{1}{2}\\}$", "final_answer": "x \\in \\{2, - \\frac{1}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d6c32913608a5ebb", "problem_id": "5d840ece5c3c05ae", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Two friends split $2324 in the ratio 9:5. How much does each receive?", "output": "Total ratio parts = 9+5 = 14.\nEach part = 2324 ÷ 14 = 166.\nFirst share = 9 × 166 = 1494.\nSecond share = 5 × 166 = 830.\nCheck: 1494+830=2324.\nFinal answer: $830$", "final_answer": "830", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "87b97187c3c561ff", "problem_id": "5d840ece5c3c05ae", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Two friends split $2324 in the ratio 9:5. How much does each receive?", "output": "1) Total ratio parts = 9+5 = 14.\n2) Each part = 2324 ÷ 14 = 166.\n3) First share = 9 × 166 = 1494.\n4) Second share = 5 × 166 = 830.\nFinal answer: $830$", "final_answer": "830", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b8d315924602c61f", "problem_id": "1ac6ccfea27a5216", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$- x + 4 y$ = $19$\n$2 x - 4 y$ = $-22$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $2$: $- 2 x + 8 y$ = $38$.\n3) Multiply the second equation by $-1$: $- 2 x + 4 y$ = $22$.\n4) Subtract to eliminate x: $4 y$ = $16$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $-3$.\nFinal answer: $(x,y)=\\left(-3, 4\\right)$", "final_answer": "(x,y)=\\left(-3, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7666a8454ee9d399", "problem_id": "1ac6ccfea27a5216", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the linear system:\n$- x + 4 y$ = $19$\n$2 x - 4 y$ = $-22$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $2$: $- 2 x + 8 y$ = $38$.\nMultiply the second equation by $-1$: $- 2 x + 4 y$ = $22$.\nFinal answer: $(x,y)=\\left(-3, 4\\right)$", "final_answer": "(x,y)=\\left(-3, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "92b0cd62231122e2", "problem_id": "730608faa9f27cd3", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{13} x}{2} + \\frac{9 x}{2} - \\frac{3 \\sqrt{13}}{2} + \\frac{9}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{9}{2} - \\frac{\\sqrt{13}}{2}$, c=$\\frac{9}{2} - \\frac{3 \\sqrt{13}}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{3 \\sqrt{13}}{2} + \\frac{11}{2}$, so √Δ = $\\sqrt{\\frac{3 \\sqrt{13}}{2} + \\frac{11}{2}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{9}{4} + \\frac{\\sqrt{13}}{4} + \\frac{\\sqrt{6 \\sqrt{13} + 22}}{4}$.\n7) x2 = $- \\frac{9}{4} - \\frac{\\sqrt{6 \\sqrt{13} + 22}}{4} + \\frac{\\sqrt{13}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{9}{4} + \\frac{\\sqrt{13}}{4} + \\frac{\\sqrt{6\\cdot \\sqrt{13} + 22}}{4}, - \\frac{9}{4} - \\frac{\\sqrt{6\\cdot \\sqrt{13} + 22}}{4} + \\frac{\\sqrt{13}}{4}\\}$", "final_answer": "x \\in \\{- \\frac{9}{4} + \\frac{\\sqrt{13}}{4} + \\frac{\\sqrt{6\\cdot \\sqrt{13} + 22}}{4}, - \\frac{9}{4} - \\frac{\\sqrt{6\\cdot \\sqrt{13} + 22}}{4} + \\frac{\\sqrt{13}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2338134f91f86302", "problem_id": "d3faf64d698f0068", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 3} x + 5$.\n\nStudent solution:\n1) Substitute x=3 into the expression.\n2) (Mistake) Simplify the expression to 5.\n3) Then the limit is 5.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=3 (polynomial is continuous).\nCorrect limit value: 8.\nFinal answer: $8$", "final_answer": "8", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "4381d0ff94539769", "problem_id": "bb328e43b13d48eb", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 30 red balls and 27 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(57,12).\n3) Favorable samples: choose k reds and n−k blues: C(30,2)·C(27,10).\n4) Compute totals: C(57,12) = 707285522580, and favorable = 3669783975.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{43065}{8300012}$", "final_answer": "\\frac{43065}{8300012}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e486333bfb303e1", "problem_id": "bb328e43b13d48eb", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 30 red balls and 27 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(57,12).\nFavorable samples: choose k reds and n−k blues: C(30,2)·C(27,10).\nCompute totals: C(57,12) = 707285522580, and favorable = 3669783975.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{43065}{8300012}$.\nFinal answer: $\\frac{43065}{8300012}$", "final_answer": "\\frac{43065}{8300012}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "400d8c204a618e15", "problem_id": "24c797ad33062b20", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 14 red balls and 8 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(22,4).\n3) Favorable samples: choose k reds and n−k blues: C(14,1)·C(8,3).\n4) Compute totals: C(22,4) = 7315, and favorable = 784.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{112}{1045}$", "final_answer": "\\frac{112}{1045}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d8b4bc347842115f", "problem_id": "24c797ad33062b20", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 14 red balls and 8 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(22,4).\nFavorable samples: choose k reds and n−k blues: C(14,1)·C(8,3).\nCompute totals: C(22,4) = 7315, and favorable = 784.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{112}{1045}$.\nFinal answer: $\\frac{112}{1045}$", "final_answer": "\\frac{112}{1045}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6943a27c0ab7ac6f", "problem_id": "3fc314e625b5e9f7", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_5=29$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_5=24.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=24 and k=5.\nCompute C(28,4) = 20475.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $20475$", "final_answer": "20475", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a2332492b49f19ea", "problem_id": "3fc314e625b5e9f7", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_5=29$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_5=24.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=24 and k=5.\n4) Compute C(28,4) = 20475.\nFinal answer: $20475$", "final_answer": "20475", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "27180f06620b3dfe", "problem_id": "a828736bc3af145e", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{2} - x + 2 \\sin{\\left(x \\right)} - 3$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $- 6 x + 2 \\cos{\\left(x \\right)} - 1$.\n2) Slope at x=-2: m = f'(-2) = $2 \\cos{\\left(2 \\right)} + 11$.\n3) Point on curve: (-2, f(-2)) = (-2, $-13 - 2 \\sin{\\left(2 \\right)}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -13 - 2 \\sin{\\left(2 \\right)} = 2 \\cos{\\left(2 \\right)} + 11(x − -2).\n6) Optionally expand: y = $2 x \\cos{\\left(2 \\right)} + 11 x - 2 \\sin{\\left(2 \\right)} + 4 \\cos{\\left(2 \\right)} + 9$.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 11\\cdot x - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)} + 9$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 11\\cdot x - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)} + 9", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c7eada5b712d9eec", "problem_id": "a828736bc3af145e", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the equation of the tangent line to $y=- 3 x^{2} - x + 2 \\sin{\\left(x \\right)} - 3$ at $x=-2$.", "output": "Differentiate: f'(x) = $- 6 x + 2 \\cos{\\left(x \\right)} - 1$.\nSlope at x=-2: m = f'(-2) = $2 \\cos{\\left(2 \\right)} + 11$.\nPoint on curve: (-2, f(-2)) = (-2, $-13 - 2 \\sin{\\left(2 \\right)}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -13 - 2 \\sin{\\left(2 \\right)} = 2 \\cos{\\left(2 \\right)} + 11(x − -2).\nOptionally expand: y = $2 x \\cos{\\left(2 \\right)} + 11 x - 2 \\sin{\\left(2 \\right)} + 4 \\cos{\\left(2 \\right)} + 9$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 11\\cdot x - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)} + 9$", "final_answer": "y = 2\\cdot x\\cdot \\cos{\\left(2 \\right)} + 11\\cdot x - 2\\cdot \\sin{\\left(2 \\right)} + 4\\cdot \\cos{\\left(2 \\right)} + 9", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a66712ce65f188cd", "problem_id": "427b0bfb4573698e", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 10 x - 72$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 10 x - 72$ = $2 \\left(x - 9\\right) \\left(x + 4\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $9$.\n5) From the second factor: x = $-4$.\nFinal answer: $x \\in \\{9, -4\\}$", "final_answer": "x \\in \\{9, -4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b5d4d852bcaa25c", "problem_id": "427b0bfb4573698e", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 10 x - 72$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} - 10 x - 72$ = $2 \\left(x - 9\\right) \\left(x + 4\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $9$.\nFrom the second factor: x = $-4$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{9, -4\\}$", "final_answer": "x \\in \\{9, -4\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cb6f4564120f4b62", "problem_id": "8435069dc1b0a7d8", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A cyclist rides at 21 miles per hour for 1 hours, then at 73 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 21 × 1 = 21 miles.\nSecond leg distance = 73 × 3 = 219 miles.\nTotal distance = 21 + 219 = 240 miles.\nFinal answer: $240$", "final_answer": "240", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b690ca58c182010a", "problem_id": "8435069dc1b0a7d8", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A cyclist rides at 21 miles per hour for 1 hours, then at 73 miles per hour for 3 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 21 × 1 = 21 miles.\n2) Second leg distance = 73 × 3 = 219 miles.\n3) Total distance = 21 + 219 = 240 miles.\nFinal answer: $240$", "final_answer": "240", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eb8d8924a551fab2", "problem_id": "021e1a5c808cbde7", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the linear system:\n$3 x + 8 y$ = $-69$\n$- 8 x - 7 y$ = $98$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-8$: $- 24 x - 64 y$ = $552$.\n3) Multiply the second equation by $3$: $- 24 x - 21 y$ = $294$.\n4) Subtract to eliminate x: $- 43 y$ = $258$.\n5) Solve for y: y = $-6$.\n6) Substitute back to find x: x = $-7$.\nFinal answer: $(x,y)=\\left(-7, -6\\right)$", "final_answer": "(x,y)=\\left(-7, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "60e42bcc7d2fac2d", "problem_id": "021e1a5c808cbde7", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "Solve the linear system:\n$3 x + 8 y$ = $-69$\n$- 8 x - 7 y$ = $98$", "output": "$(x,y)=\\left(-7, -6\\right)$", "final_answer": "(x,y)=\\left(-7, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "98fe69e158f7f822", "problem_id": "8e5141de3c51c0b9", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡3 (mod 8), x≡5 (mod 13). Give the solution modulo 104.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 104.\n3) Compute the CRT solution to get x ≡ 83 (mod 104).\nFinal answer: $x\\equiv 83\\pmod{104}$", "final_answer": "x\\equiv 83\\pmod{104}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bb01542707174661", "problem_id": "cb09f9741b7199b8", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(2 x^{2} - 3 x + 2\\right) \\cos{\\left(2 x + 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x^{2} \\cos{\\left(2 x + 1 \\right)}) = - 2 x^{2} \\sin{\\left(2 x + 1 \\right)}$.\n3) So $f'(x) = - 2 x^{2} \\sin{\\left(2 x + 1 \\right)} + x e^{x} + 6 x \\sin{\\left(2 x + 1 \\right)} + 3 e^{x} - 4 \\sin{\\left(2 x + 1 \\right)} - 3 \\cos{\\left(2 x + 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x^{2} \\cos{\\left(2 x + 1 \\right)}) = - 4 x^{2} \\sin{\\left(2 x + 1 \\right)} + 4 x \\cos{\\left(2 x + 1 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 2\\right) e^{x} + \\left(4 x - 3\\right) \\cos{\\left(2 x + 1 \\right)} - 2 \\left(x \\left(2 x - 3\\right) + 2\\right) \\sin{\\left(2 x + 1 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 2\\right) e^{x} + \\left(4 x - 3\\right) \\cos{\\left(2 x + 1 \\right)} - 2 \\left(x \\left(2 x - 3\\right) + 2\\right) \\sin{\\left(2 x + 1 \\right)} + e^{x}$", "final_answer": "\\left(x + 2\\right) e^{x} + \\left(4 x - 3\\right) \\cos{\\left(2 x + 1 \\right)} - 2 \\left(x \\left(2 x - 3\\right) + 2\\right) \\sin{\\left(2 x + 1 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "da636e9b1a67f8a2", "problem_id": "ab6a9986a1c86393", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 20 red balls and 17 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(37,9).\n3) Favorable samples: choose k reds and n−k blues: C(20,1)·C(17,8).\n4) Compute totals: C(37,9) = 124403620, and favorable = 486200.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{130}{33263}$", "final_answer": "\\frac{130}{33263}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1e918ffb9c50cbb1", "problem_id": "aed1a0eaae1d9cef", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{3} + x^{2} + 4 x + 6$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} + x^{2} + 3 x + 6.\n3) Then the limit is 32.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 34.\nFinal answer: $34$", "final_answer": "34", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2b53d9bde062fba0", "problem_id": "7a0c13decb66c1c9", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{7} x}{3} + \\frac{7 x}{15} - \\frac{2}{15} + \\frac{\\sqrt{7}}{15}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{7}{15} - \\frac{\\sqrt{7}}{3}$, c=$- \\frac{2}{15} + \\frac{\\sqrt{7}}{15}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{344}{225} - \\frac{26 \\sqrt{7}}{45}$, so √Δ = $\\sqrt{\\frac{344}{225} - \\frac{26 \\sqrt{7}}{45}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{7}{30} + \\frac{\\sqrt{344 - 130 \\sqrt{7}}}{30} + \\frac{\\sqrt{7}}{6}$.\n7) x2 = $- \\frac{7}{30} - \\frac{\\sqrt{344 - 130 \\sqrt{7}}}{30} + \\frac{\\sqrt{7}}{6}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{7}{30} + \\frac{\\sqrt{344 - 130\\cdot \\sqrt{7}}}{30} + \\frac{\\sqrt{7}}{6}, - \\frac{7}{30} - \\frac{\\sqrt{344 - 130\\cdot \\sqrt{7}}}{30} + \\frac{\\sqrt{7}}{6}\\}$", "final_answer": "x \\in \\{- \\frac{7}{30} + \\frac{\\sqrt{344 - 130\\cdot \\sqrt{7}}}{30} + \\frac{\\sqrt{7}}{6}, - \\frac{7}{30} - \\frac{\\sqrt{344 - 130\\cdot \\sqrt{7}}}{30} + \\frac{\\sqrt{7}}{6}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "85d3549e3f822552", "problem_id": "7a0c13decb66c1c9", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{\\sqrt{7} x}{3} + \\frac{7 x}{15} - \\frac{2}{15} + \\frac{\\sqrt{7}}{15}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{7}{15} - \\frac{\\sqrt{7}}{3}$, c=$- \\frac{2}{15} + \\frac{\\sqrt{7}}{15}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{344}{225} - \\frac{26 \\sqrt{7}}{45}$, so √Δ = $\\sqrt{\\frac{344}{225} - \\frac{26 \\sqrt{7}}{45}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{7}{30} + \\frac{\\sqrt{344 - 130 \\sqrt{7}}}{30} + \\frac{\\sqrt{7}}{6}$.\nx2 = $- \\frac{7}{30} - \\frac{\\sqrt{344 - 130 \\sqrt{7}}}{30} + \\frac{\\sqrt{7}}{6}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{7}{30} + \\frac{\\sqrt{344 - 130\\cdot \\sqrt{7}}}{30} + \\frac{\\sqrt{7}}{6}, - \\frac{7}{30} - \\frac{\\sqrt{344 - 130\\cdot \\sqrt{7}}}{30} + \\frac{\\sqrt{7}}{6}\\}$", "final_answer": "x \\in \\{- \\frac{7}{30} + \\frac{\\sqrt{344 - 130\\cdot \\sqrt{7}}}{30} + \\frac{\\sqrt{7}}{6}, - \\frac{7}{30} - \\frac{\\sqrt{344 - 130\\cdot \\sqrt{7}}}{30} + \\frac{\\sqrt{7}}{6}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f67edf8c5f000658", "problem_id": "cfec818f96101131", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A store sells a backpack for $46. It is discounted by 10%. You buy 14 backpacks. What is the total cost?", "output": "1) Discounted price = 46 × (1 − 10/100) = 46 × 90/100 = 207/5.\n2) Subtotal = 14 × 207/5 = 2898/5.\nFinal answer: $\\frac{2898}{5}$", "final_answer": "\\frac{2898}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f866d9ab3724969a", "problem_id": "cfec818f96101131", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A store sells a backpack for $46. It is discounted by 10%. You buy 14 backpacks. What is the total cost?", "output": "Discounted price = 46 × (1 − 10/100) = 46 × 90/100 = 207/5.\nSubtotal = 14 × 207/5 = 2898/5.\nFinal answer: $\\frac{2898}{5}$", "final_answer": "\\frac{2898}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1b115d4844a30f77", "problem_id": "760e17c8987210d7", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many binary strings of length 20 contain exactly 6 ones with no two ones adjacent?", "output": "Reserve 5 mandatory zeros between the 6 ones.\nThis leaves n−k+1 = 15 valid slots for the ones.\nChoose the slots: C(15,6) = 5005.\nCheck: standard gap method gives C(n−k+1,k).\nFinal answer: $5005$", "final_answer": "5005", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b8f6bca41a6998d4", "problem_id": "760e17c8987210d7", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many binary strings of length 20 contain exactly 6 ones with no two ones adjacent?", "output": "1) Reserve 5 mandatory zeros between the 6 ones.\n2) This leaves n−k+1 = 15 valid slots for the ones.\n3) Choose the slots: C(15,6) = 5005.\nFinal answer: $5005$", "final_answer": "5005", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1bd305b18eda99db", "problem_id": "fadf6ccd06988224", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Two friends split $704 in the ratio 8:8. How much does each receive?", "output": "Total ratio parts = 8+8 = 16.\nEach part = 704 ÷ 16 = 44.\nFirst share = 8 × 44 = 352.\nFinal answer: $352$", "final_answer": "352", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a74e5c7224a9cea", "problem_id": "fadf6ccd06988224", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Two friends split $704 in the ratio 8:8. How much does each receive?", "output": "Total ratio parts = 8+8 = 16.\nEach part = 704 ÷ 16 = 44.\nFirst share = 8 × 44 = 352.\nSecond share = 8 × 44 = 352.\nCheck: 352+352=704.\nFinal answer: $352$", "final_answer": "352", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3487de3053a1529c", "problem_id": "f02e5ac610543f7c", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A cyclist rides at 60 miles per hour for 2 hours, then at 42 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 60 × 2 = 120 miles.\nSecond leg distance = 42 × 1 = 42 miles.\nTotal distance = 120 + 42 = 162 miles.\nCheck: adding both legs gives 162.\nFinal answer: $162$", "final_answer": "162", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e1294cb1065bd308", "problem_id": "f02e5ac610543f7c", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A cyclist rides at 60 miles per hour for 2 hours, then at 42 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 60 × 2 = 120 miles.\n2) Second leg distance = 42 × 1 = 42 miles.\n3) Total distance = 120 + 42 = 162 miles.\nFinal answer: $162$", "final_answer": "162", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ab1cc02bfe3a77ff", "problem_id": "94af261b55c09eca", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡3 (mod 4), x≡1 (mod 11). Give the solution modulo 44.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 44.\n3) Compute the CRT solution to get x ≡ 23 (mod 44).\nFinal answer: $x\\equiv 23\\pmod{44}$", "final_answer": "x\\equiv 23\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "141e65061c6886f8", "problem_id": "94af261b55c09eca", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡3 (mod 4), x≡1 (mod 11). Give the solution modulo 44.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 44.\nCompute the CRT solution to get x ≡ 23 (mod 44).\nCheck: 23 mod 4=3 and mod 11=1.\nFinal answer: $x\\equiv 23\\pmod{44}$", "final_answer": "x\\equiv 23\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "15ee826a7002c699", "problem_id": "19d224a19f416658", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 x^{3} - 7 x^{2} + 11 x - 23$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{3} - 7 x^{2} + 10 x - 23.\n3) Then the limit is -95.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: -97.\nFinal answer: $-97$", "final_answer": "-97", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7143153564841aff", "problem_id": "14375a9c513dc2d4", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡5 (mod 9), x≡4 (mod 7). Give the solution modulo 63.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 63.\n3) Compute the CRT solution to get x ≡ 32 (mod 63).\nFinal answer: $x\\equiv 32\\pmod{63}$", "final_answer": "x\\equiv 32\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8b6019e5bf4031db", "problem_id": "f4bb53485da72686", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{4} - 3 x^{2} + x - 4$.", "output": "Differentiate term-by-term:\nd/dx(-4) = 0.\nd/dx(x) = 1.\nd/dx(- 3 x^{2}) = - 6 x.\nd/dx(4 x^{4}) = 16 x^{3}.\nCombine: f'(x) = 16 x^{3} - 6 x + 1.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $16\\cdot x^{3} - 6\\cdot x + 1$", "final_answer": "16\\cdot x^{3} - 6\\cdot x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6d4f25b0807adcf8", "problem_id": "f4bb53485da72686", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{4} - 3 x^{2} + x - 4$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-4) = 0.\n3) d/dx(x) = 1.\n4) d/dx(- 3 x^{2}) = - 6 x.\n5) d/dx(4 x^{4}) = 16 x^{3}.\n6) Combine: f'(x) = 16 x^{3} - 6 x + 1.\nFinal answer: $16\\cdot x^{3} - 6\\cdot x + 1$", "final_answer": "16\\cdot x^{3} - 6\\cdot x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8854e7d1aa5609a8", "problem_id": "95af1c4ebe466b2a", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 12 red balls and 15 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(27,8).\n3) Favorable samples: choose k reds and n−k blues: C(12,2)·C(15,6).\n4) Compute totals: C(27,8) = 2220075, and favorable = 330330.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{154}{1035}$", "final_answer": "\\frac{154}{1035}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0e9a6c921c6f02fa", "problem_id": "2aff119ccb94f44f", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 \\sin{\\left(x \\right)}) = 3 \\cos{\\left(x \\right)}$.\n3) So $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 \\sin{\\left(x \\right)}) = 3 \\cos{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$", "final_answer": "x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "1048a3bc7c33d89c", "problem_id": "37f0be849b85c177", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "An urn has 13 red balls and 14 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(27,4).\nFavorable samples: choose k reds and n−k blues: C(13,1)·C(14,3).\nFinal answer: $\\frac{182}{675}$", "final_answer": "\\frac{182}{675}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d3bcdce30b9b4c37", "problem_id": "37f0be849b85c177", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 13 red balls and 14 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(27,4).\n3) Favorable samples: choose k reds and n−k blues: C(13,1)·C(14,3).\n4) Compute totals: C(27,4) = 17550, and favorable = 4732.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{182}{675}$", "final_answer": "\\frac{182}{675}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "27ed0b805889b739", "problem_id": "d54f72c07c949c2c", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x^{2} - 3 x - 9$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 2 x^{3} + x^{2} - 3 x$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 6 x^{2} + 2 x - 3$.\nEvaluate at a: p'(2) = $-23$.\nCheck: the identity gives $-23$.\nFinal answer: $-23$", "final_answer": "-23", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "66bba20651822430", "problem_id": "d54f72c07c949c2c", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x^{2} - 3 x - 9$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 2 x^{3} + x^{2} - 3 x$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 6 x^{2} + 2 x - 3$.\n5) Evaluate at a: p'(2) = $-23$.\nFinal answer: $-23$", "final_answer": "-23", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1219044733ffd1ab", "problem_id": "1f0602dc9ff57fdb", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡4 (mod 7), x≡1 (mod 5). Give the solution modulo 35.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 35.\nCompute the CRT solution to get x ≡ 11 (mod 35).\nCheck: 11 mod 7=4 and mod 5=1.\nFinal answer: $x\\equiv 11\\pmod{35}$", "final_answer": "x\\equiv 11\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0afa5cd49c8da620", "problem_id": "f66393517368d768", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x^{2} - 2 x - 2$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{2} - 3 x - 2.\n3) Then the limit is -16.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -14.\nFinal answer: $-14$", "final_answer": "-14", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "548bc9b402645c98", "problem_id": "e6fd5f3032e39b5e", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 18 red balls and 14 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(32,12).\n3) Favorable samples: choose k reds and n−k blues: C(18,4)·C(14,8).\n4) Compute totals: C(32,12) = 225792840, and favorable = 9189180.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1683}{41354}$", "final_answer": "\\frac{1683}{41354}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bc76e21de37b9cff", "problem_id": "e6fd5f3032e39b5e", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 18 red balls and 14 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(32,12).\nFavorable samples: choose k reds and n−k blues: C(18,4)·C(14,8).\nCompute totals: C(32,12) = 225792840, and favorable = 9189180.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1683}{41354}$.\nFinal answer: $\\frac{1683}{41354}$", "final_answer": "\\frac{1683}{41354}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f8740f95ac1aa619", "problem_id": "97cbb6047e4471c5", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 3 x^{2} + 9 x + 18$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{2} + 8 x + 18.\n3) Then the limit is 46.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 48.\nFinal answer: $48$", "final_answer": "48", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ba29c4ac60a640bd", "problem_id": "81b02b5605fe9a8d", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡4 (mod 9), x≡0 (mod 5). Give the solution modulo 45.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 45.\n3) Compute the CRT solution to get x ≡ 40 (mod 45).\nFinal answer: $x\\equiv 40\\pmod{45}$", "final_answer": "x\\equiv 40\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "197d2b2476d9c744", "problem_id": "2125b01081905f28", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the congruences: x≡6 (mod 11), x≡8 (mod 13). Give the solution modulo 143.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 143.\n3) Compute the CRT solution to get x ≡ 138 (mod 143).\nFinal answer: $x\\equiv 138\\pmod{143}$", "final_answer": "x\\equiv 138\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d6dbe6a6d661228", "problem_id": "ee2a845187ed539e", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x + 2$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 2.\n3) Then the limit is 2.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 4.\nFinal answer: $4$", "final_answer": "4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "920e78dec4690e4d", "problem_id": "0eec8941e9c14f0d", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 14 times, what is the probability of exactly 12 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=14, k=12, p=$\\frac{3}{8}$ and simplify.\n4) Here C(14,12) = 91.\nFinal answer: $\\frac{1209028275}{4398046511104}$", "final_answer": "\\frac{1209028275}{4398046511104}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ac99fa84c47d7cab", "problem_id": "846deecb9de432dc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(x^{3} + 3 x^{2} - x - 1\\right) \\cos{\\left(x^{2} - 3 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\cos{\\left(x^{2} - 3 \\right)}) = \\sin{\\left(x^{2} - 3 \\right)}$.\n3) So $f'(x) = - 2 x^{4} \\sin{\\left(x^{2} - 3 \\right)} - 6 x^{3} \\sin{\\left(x^{2} - 3 \\right)} + 2 x^{2} \\sin{\\left(x^{2} - 3 \\right)} + 3 x^{2} \\cos{\\left(x^{2} - 3 \\right)} + x e^{x} + 6 x \\cos{\\left(x^{2} - 3 \\right)} + 2 e^{x} + \\sin{\\left(x^{2} - 3 \\right)} - \\cos{\\left(x^{2} - 3 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\cos{\\left(x^{2} - 3 \\right)}) = 2 x \\sin{\\left(x^{2} - 3 \\right)}$.\nTherefore the correct derivative is $f'(x) = - 2 x \\left(x \\left(x \\left(x + 3\\right) - 1\\right) - 1\\right) \\sin{\\left(x^{2} - 3 \\right)} + \\left(x + 1\\right) e^{x} + \\left(3 x^{2} + 6 x - 1\\right) \\cos{\\left(x^{2} - 3 \\right)} + e^{x}$.\nFinal answer: $- 2 x \\left(x \\left(x \\left(x + 3\\right) - 1\\right) - 1\\right) \\sin{\\left(x^{2} - 3 \\right)} + \\left(x + 1\\right) e^{x} + \\left(3 x^{2} + 6 x - 1\\right) \\cos{\\left(x^{2} - 3 \\right)} + e^{x}$", "final_answer": "- 2 x \\left(x \\left(x \\left(x + 3\\right) - 1\\right) - 1\\right) \\sin{\\left(x^{2} - 3 \\right)} + \\left(x + 1\\right) e^{x} + \\left(3 x^{2} + 6 x - 1\\right) \\cos{\\left(x^{2} - 3 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "85cd3e0749061d89", "problem_id": "941d908021c2439b", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡0 (mod 3), x≡2 (mod 11). Give the solution modulo 33.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 33.\n3) Compute the CRT solution to get x ≡ 24 (mod 33).\nFinal answer: $x\\equiv 24\\pmod{33}$", "final_answer": "x\\equiv 24\\pmod{33}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "89b08f0a305cdffa", "problem_id": "941d908021c2439b", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the congruences: x≡0 (mod 3), x≡2 (mod 11). Give the solution modulo 33.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 33.\nCompute the CRT solution to get x ≡ 24 (mod 33).\nFinal answer: $x\\equiv 24\\pmod{33}$", "final_answer": "x\\equiv 24\\pmod{33}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bde4e24b115cbccc", "problem_id": "125c8590bc481b64", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 18 times, what is the probability of exactly 13 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=18, k=13, p=$\\frac{2}{5}$ and simplify.\n4) Here C(18,13) = 8568.\nFinal answer: $\\frac{17055940608}{3814697265625}$", "final_answer": "\\frac{17055940608}{3814697265625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "41f5c5d072257cfd", "problem_id": "125c8590bc481b64", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 18 times, what is the probability of exactly 13 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=18, k=13, p=$\\frac{2}{5}$ and simplify.\nFinal answer: $\\frac{17055940608}{3814697265625}$", "final_answer": "\\frac{17055940608}{3814697265625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9cdbeec7d6d937b1", "problem_id": "7a159ee5bb93f37e", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "You want 61 liters of a 35% salt solution. You have a 15% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 61−x liters is 60% solution.\n2) Salt equation: 15x + 60(61−x) = 35·61.\n3) (15-60)x = (35-60)·61 ⇒ x = 305/9.\n4) Then 61−x = 244/9.\nFinal answer: $305/9 L of 15% and 244/9 L of 60%$", "final_answer": "305/9 L of 15% and 244/9 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8c6cf5c7ced626ac", "problem_id": "7a159ee5bb93f37e", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "You want 61 liters of a 35% salt solution. You have a 15% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 61−x liters is 60% solution.\nSalt equation: 15x + 60(61−x) = 35·61.\n(15-60)x = (35-60)·61 ⇒ x = 305/9.\nThen 61−x = 244/9.\nCheck: 15·305/9 + 60·244/9 = 35·61.\nFinal answer: $305/9 L of 15% and 244/9 L of 60%$", "final_answer": "305/9 L of 15% and 244/9 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "677aafb9eab21361", "problem_id": "9d6cc78c3c7ee641", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - x^{3} - x^{2} - 7 x - 31$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- x^{4} + 3 x^{3} - 3 x^{2} - 3 x + 2$ and a = 4.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 4 x^{3} + 9 x^{2} - 6 x - 3$.\n5) Evaluate at a: p'(4) = $-139$.\nFinal answer: $-139$", "final_answer": "-139", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4df2533d51d0d71e", "problem_id": "9d6cc78c3c7ee641", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} - x^{3} - x^{2} - 7 x - 31$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- x^{4} + 3 x^{3} - 3 x^{2} - 3 x + 2$ and a = 4.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 4 x^{3} + 9 x^{2} - 6 x - 3$.\nEvaluate at a: p'(4) = $-139$.\nCheck: the identity gives $-139$.\nFinal answer: $-139$", "final_answer": "-139", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "718b728ac13501d1", "problem_id": "872bd532dfe469eb", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the congruences: x≡2 (mod 7), x≡1 (mod 5). Give the solution modulo 35.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 35.\n3) Compute the CRT solution to get x ≡ 16 (mod 35).\nFinal answer: $x\\equiv 16\\pmod{35}$", "final_answer": "x\\equiv 16\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7251e055aa2bf505", "problem_id": "872bd532dfe469eb", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 7), x≡1 (mod 5). Give the solution modulo 35.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 35.\nCompute the CRT solution to get x ≡ 16 (mod 35).\nCheck: 16 mod 7=2 and mod 5=1.\nFinal answer: $x\\equiv 16\\pmod{35}$", "final_answer": "x\\equiv 16\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "add0672a91b91418", "problem_id": "4dc7351d3207582c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 4} 2 x^{2} + 9 x + 34$.\n\nStudent solution:\n1) Substitute x=4 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{2} + 8 x + 34.\n3) Then the limit is 98.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=4 (polynomial is continuous).\nCorrect limit value: 102.\nFinal answer: $102$", "final_answer": "102", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "69fab3392b712312", "problem_id": "6b33060ebb3c03e4", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡6 (mod 7). Give the solution modulo 21.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 21.\nCompute the CRT solution to get x ≡ 13 (mod 21).\nCheck: 13 mod 3=1 and mod 7=6.\nFinal answer: $x\\equiv 13\\pmod{21}$", "final_answer": "x\\equiv 13\\pmod{21}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d8d7e72327e2b556", "problem_id": "af7b0f60f2a2007c", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{3} - 3 x^{2} + x - 1$.", "output": "1) Differentiate term-by-term:\n2) d/dx(-1) = 0.\n3) d/dx(x) = 1.\n4) d/dx(- 3 x^{2}) = - 6 x.\n5) d/dx(4 x^{3}) = 12 x^{2}.\n6) Combine: f'(x) = 12 x^{2} - 6 x + 1.\nFinal answer: $12\\cdot x^{2} - 6\\cdot x + 1$", "final_answer": "12\\cdot x^{2} - 6\\cdot x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8f12aa1b9b18bf2f", "problem_id": "af7b0f60f2a2007c", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{3} - 3 x^{2} + x - 1$.", "output": "Differentiate term-by-term:\nd/dx(-1) = 0.\nd/dx(x) = 1.\nd/dx(- 3 x^{2}) = - 6 x.\nd/dx(4 x^{3}) = 12 x^{2}.\nCombine: f'(x) = 12 x^{2} - 6 x + 1.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $12\\cdot x^{2} - 6\\cdot x + 1$", "final_answer": "12\\cdot x^{2} - 6\\cdot x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ef5cd1200d1edef5", "problem_id": "5760c1791afd92bd", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 19 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=19, k=4, p=$\\frac{2}{3}$ and simplify.\n4) Here C(19,4) = 3876.\nFinal answer: $\\frac{20672}{387420489}$", "final_answer": "\\frac{20672}{387420489}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c50e95b828a4255d", "problem_id": "5760c1791afd92bd", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 19 times, what is the probability of exactly 4 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=19, k=4, p=$\\frac{2}{3}$ and simplify.\nHere C(19,4) = 3876.\nCheck: expression simplifies to $\\frac{20672}{387420489}$.\nFinal answer: $\\frac{20672}{387420489}$", "final_answer": "\\frac{20672}{387420489}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "234fb117ab0a3af3", "problem_id": "4292633ccafd8e3e", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "An urn has 10 red balls and 13 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(23,4).\n3) Favorable samples: choose k reds and n−k blues: C(10,1)·C(13,3).\n4) Compute totals: C(23,4) = 8855, and favorable = 2860.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{52}{161}$", "final_answer": "\\frac{52}{161}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a7b0a0a280cdccea", "problem_id": "4292633ccafd8e3e", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "An urn has 10 red balls and 13 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(23,4).\nFavorable samples: choose k reds and n−k blues: C(10,1)·C(13,3).\nFinal answer: $\\frac{52}{161}$", "final_answer": "\\frac{52}{161}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "440502fa96bef908", "problem_id": "871278ec09a88e30", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $352 in the ratio 6:5. How much does each receive?", "output": "1) Total ratio parts = 6+5 = 11.\n2) Each part = 352 ÷ 11 = 32.\n3) First share = 6 × 32 = 192.\n4) Second share = 5 × 32 = 160.\nFinal answer: $160$", "final_answer": "160", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ae4b0f06f428036f", "problem_id": "871278ec09a88e30", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Two friends split $352 in the ratio 6:5. How much does each receive?", "output": "Total ratio parts = 6+5 = 11.\nEach part = 352 ÷ 11 = 32.\nFirst share = 6 × 32 = 192.\nSecond share = 5 × 32 = 160.\nCheck: 192+160=352.\nFinal answer: $160$", "final_answer": "160", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8259792415fa0145", "problem_id": "ac9ab70ed03ef901", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} - x^{4} - 3 x^{3} + 3 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{4} - 3 x^{3} + 3 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $\\frac{3 \\pi \\left(- 44 \\pi^{4} - 75 \\pi^{3} - 30 \\pi - 40 + 60 \\pi^{2}\\right)}{20}$.\n4) Simplify the result.\nFinal answer: $\\frac{3\\cdot \\pi\\cdot \\left(- 44\\cdot \\pi^{4} - 75\\cdot \\pi^{3} - 30\\cdot \\pi - 40 + 60\\cdot \\pi^{2}\\right)}{20}$", "final_answer": "\\frac{3\\cdot \\pi\\cdot \\left(- 44\\cdot \\pi^{4} - 75\\cdot \\pi^{3} - 30\\cdot \\pi - 40 + 60\\cdot \\pi^{2}\\right)}{20}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a4b3549daec73e2", "problem_id": "ac9ab70ed03ef901", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} - x^{4} - 3 x^{3} + 3 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{4} - 3 x^{3} + 3 x^{2} - 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 2$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $\\frac{3 \\pi \\left(- 44 \\pi^{4} - 75 \\pi^{3} - 30 \\pi - 40 + 60 \\pi^{2}\\right)}{20}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{3\\cdot \\pi\\cdot \\left(- 44\\cdot \\pi^{4} - 75\\cdot \\pi^{3} - 30\\cdot \\pi - 40 + 60\\cdot \\pi^{2}\\right)}{20}$", "final_answer": "\\frac{3\\cdot \\pi\\cdot \\left(- 44\\cdot \\pi^{4} - 75\\cdot \\pi^{3} - 30\\cdot \\pi - 40 + 60\\cdot \\pi^{2}\\right)}{20}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "26e0944075062e71", "problem_id": "b87a4a80258c11c5", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{x}{3} - 5$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$\\frac{1}{3}$, c=$-5$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{361}{9}$, so √Δ = $\\frac{19}{3}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{3}{2}$.\nx2 = $- \\frac{5}{3}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{3}{2}, - \\frac{5}{3}\\}$", "final_answer": "x \\in \\{\\frac{3}{2}, - \\frac{5}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c00398fe788ea5f3", "problem_id": "b87a4a80258c11c5", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{x}{3} - 5$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$\\frac{1}{3}$, c=$-5$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{361}{9}$, so √Δ = $\\frac{19}{3}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{3}{2}$.\n7) x2 = $- \\frac{5}{3}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{3}{2}, - \\frac{5}{3}\\}$", "final_answer": "x \\in \\{\\frac{3}{2}, - \\frac{5}{3}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d704e1e9a94a2d46", "problem_id": "42034a134607d592", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "An urn has 14 red balls and 16 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(30,10).\nFavorable samples: choose k reds and n−k blues: C(14,2)·C(16,8).\nFinal answer: $\\frac{26}{667}$", "final_answer": "\\frac{26}{667}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6e9940386c031904", "problem_id": "42034a134607d592", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 14 red balls and 16 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(30,10).\n3) Favorable samples: choose k reds and n−k blues: C(14,2)·C(16,8).\n4) Compute totals: C(30,10) = 30045015, and favorable = 1171170.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{26}{667}$", "final_answer": "\\frac{26}{667}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6d3fe4f7bc22b456", "problem_id": "7d6442faf276e17f", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Worker A finishes a job in 17 hours and Worker B finishes it in 8 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/17 job/hour. B's rate = 1/8 job/hour.\n2) Combined rate = 1/17 + 1/8 = 25/136 job/hour.\n3) Time = 1 ÷ (combined rate) = 136/25 hours.\nFinal answer: $\\frac{136}{25}$", "final_answer": "\\frac{136}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7e4bc13fc2bb0d5e", "problem_id": "7d6442faf276e17f", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Worker A finishes a job in 17 hours and Worker B finishes it in 8 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/17 job/hour. B's rate = 1/8 job/hour.\nCombined rate = 1/17 + 1/8 = 25/136 job/hour.\nTime = 1 ÷ (combined rate) = 136/25 hours.\nFinal answer: $\\frac{136}{25}$", "final_answer": "\\frac{136}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "14c2b1c12be76ac2", "problem_id": "c272449f3b029e48", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 16 red balls and 15 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(31,6).\n3) Favorable samples: choose k reds and n−k blues: C(16,1)·C(15,5).\n4) Compute totals: C(31,6) = 736281, and favorable = 48048.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{176}{2697}$", "final_answer": "\\frac{176}{2697}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8d2df13cc514871c", "problem_id": "c272449f3b029e48", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 16 red balls and 15 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(31,6).\nFavorable samples: choose k reds and n−k blues: C(16,1)·C(15,5).\nCompute totals: C(31,6) = 736281, and favorable = 48048.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{176}{2697}$.\nFinal answer: $\\frac{176}{2697}$", "final_answer": "\\frac{176}{2697}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0b75e90119351bfc", "problem_id": "a2da4052c7b37d0d", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "An urn has 8 red balls and 7 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(15,8).\n3) Favorable samples: choose k reds and n−k blues: C(8,2)·C(7,6).\n4) Compute totals: C(15,8) = 6435, and favorable = 196.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{196}{6435}$", "final_answer": "\\frac{196}{6435}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c2b4ba874489e428", "problem_id": "a2da4052c7b37d0d", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "An urn has 8 red balls and 7 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(15,8).\nFavorable samples: choose k reds and n−k blues: C(8,2)·C(7,6).\nFinal answer: $\\frac{196}{6435}$", "final_answer": "\\frac{196}{6435}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0540c52236367a7b", "problem_id": "e9908592a9b6969b", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 16 red balls and 27 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(43,12).\n3) Favorable samples: choose k reds and n−k blues: C(16,5)·C(27,7).\n4) Compute totals: C(43,12) = 15338678264, and favorable = 3878915040.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{5328180}{21069613}$", "final_answer": "\\frac{5328180}{21069613}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e7a6135960852e1c", "problem_id": "adfe202ac964bfdc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{4} + 3 x^{3} + 3 x^{2} - 3 x + 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(3 x^{3}) = 9 x^{3}$.\n3) So $f'(x) = 25 x^{3} + 6 x - 3$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(3 x^{3}) = 9 x^{2}$.\nTherefore the correct derivative is $f'(x) = 16 x^{3} + 9 x^{2} + 6 x - 3$.\nFinal answer: $16 x^{3} + 9 x^{2} + 6 x - 3$", "final_answer": "16 x^{3} + 9 x^{2} + 6 x - 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "509323114ccb8970", "problem_id": "e1cd11471ec54fab", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$- 2 x - 8 y$ = $-10$\n$- 4 x + 3 y$ = $-39$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $8 x + 32 y$ = $40$.\nMultiply the second equation by $-2$: $8 x - 6 y$ = $78$.\nSubtract to eliminate x: $38 y$ = $-38$.\nSolve for y: y = $-1$.\nSubstitute back to find x: x = $9$.\nCheck: plug (x,y)=(9,-1) into both equations.\nFinal answer: $(x,y)=\\left(9, -1\\right)$", "final_answer": "(x,y)=\\left(9, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "615d7b30519b441a", "problem_id": "e1cd11471ec54fab", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the linear system:\n$- 2 x - 8 y$ = $-10$\n$- 4 x + 3 y$ = $-39$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $8 x + 32 y$ = $40$.\n3) Multiply the second equation by $-2$: $8 x - 6 y$ = $78$.\n4) Subtract to eliminate x: $38 y$ = $-38$.\n5) Solve for y: y = $-1$.\n6) Substitute back to find x: x = $9$.\nFinal answer: $(x,y)=\\left(9, -1\\right)$", "final_answer": "(x,y)=\\left(9, -1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d96d858c05460e4a", "problem_id": "af49589028df5abc", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$x - 8 y$ = $14$\n$- 6 x + 8 y$ = $-4$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-6$: $- 6 x + 48 y$ = $-84$.\n3) Multiply the second equation by $1$: $- 6 x + 8 y$ = $-4$.\n4) Subtract to eliminate x: $40 y$ = $-80$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $-2$.\nFinal answer: $(x,y)=\\left(-2, -2\\right)$", "final_answer": "(x,y)=\\left(-2, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c274513dab40274c", "problem_id": "48dbd6030f3b25ea", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 x^{3} \\sin{\\left(x \\right)}) = 3 x^{3} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = 3 x^{3} \\cos{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 x^{3} \\sin{\\left(x \\right)}) = 3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)}$.\nFinal answer: $3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)}$", "final_answer": "3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ca4f7a7beb40d49e", "problem_id": "9637c747028a3726", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 44 liters of a 35% salt solution. You have a 30% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 44−x liters is 40% solution.\nSalt equation: 30x + 40(44−x) = 35·44.\n(30-40)x = (35-40)·44 ⇒ x = 22.\nThen 44−x = 22.\nCheck: 30·22 + 40·22 = 35·44.\nFinal answer: $22 L of 30% and 22 L of 40%$", "final_answer": "22 L of 30% and 22 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "53c88c8cb14eaa7a", "problem_id": "9637c747028a3726", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "You want 44 liters of a 35% salt solution. You have a 30% solution and a 40% solution. How many liters of each should you mix?", "output": "Let x = liters of 30% solution, so 44−x liters is 40% solution.\nSalt equation: 30x + 40(44−x) = 35·44.\n(30-40)x = (35-40)·44 ⇒ x = 22.\nFinal answer: $22 L of 30% and 22 L of 40%$", "final_answer": "22 L of 30% and 22 L of 40%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "561ceb6235837905", "problem_id": "e3c1868197425aeb", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{2} - x + \\left(2 x^{2} + 2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{2 x^{2} + 2 x - 5}$.", "output": "Let f(x) = $\\frac{x^{2} - x + \\left(2 x^{2} + 2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{2 x^{2} + 2 x - 5}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{4 \\left(- 2 x - 1\\right) \\left(x^{2} - x + \\left(2 x^{2} + 2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(2 x^{2} + 2 x - 5\\right) \\left(4 x + 4 \\left(2 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(2 x^{2} + 2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x^{2} + 2 x - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(2 x^{2} + 2 x - 5\\right)^{2}}$.\nFinal answer: $\\frac{\\left(- 8\\cdot x - 4\\right)\\cdot \\left(x^{2} - x + \\left(2\\cdot x^{2} + 2\\cdot x - 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 3\\right) + \\left(2\\cdot x^{2} + 2\\cdot x - 5\\right)\\cdot \\left(4\\cdot x + \\left(8\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 2\\cdot x - 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x^{2} + 4\\cdot x - 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 2\\right)}{2\\cdot \\left(2\\cdot x^{2} + 2\\cdot x - 5\\right)^{2}}$", "final_answer": "\\frac{\\left(- 8\\cdot x - 4\\right)\\cdot \\left(x^{2} - x + \\left(2\\cdot x^{2} + 2\\cdot x - 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 3\\right) + \\left(2\\cdot x^{2} + 2\\cdot x - 5\\right)\\cdot \\left(4\\cdot x + \\left(8\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 2\\cdot x - 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x^{2} + 4\\cdot x - 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 2\\right)}{2\\cdot \\left(2\\cdot x^{2} + 2\\cdot x - 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "076239939cf40875", "problem_id": "e3c1868197425aeb", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{x^{2} - x + \\left(2 x^{2} + 2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{2 x^{2} + 2 x - 5}$.", "output": "1) Let f(x) = $\\frac{x^{2} - x + \\left(2 x^{2} + 2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3}{2 x^{2} + 2 x - 5}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{4 \\left(- 2 x - 1\\right) \\left(x^{2} - x + \\left(2 x^{2} + 2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 3\\right) + \\left(2 x^{2} + 2 x - 5\\right) \\left(4 x + 4 \\left(2 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(2 x^{2} + 2 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x^{2} + 2 x - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 2\\right)}{2 \\left(2 x^{2} + 2 x - 5\\right)^{2}}$.\nFinal answer: $\\frac{\\left(- 8\\cdot x - 4\\right)\\cdot \\left(x^{2} - x + \\left(2\\cdot x^{2} + 2\\cdot x - 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 3\\right) + \\left(2\\cdot x^{2} + 2\\cdot x - 5\\right)\\cdot \\left(4\\cdot x + \\left(8\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 2\\cdot x - 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x^{2} + 4\\cdot x - 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 2\\right)}{2\\cdot \\left(2\\cdot x^{2} + 2\\cdot x - 5\\right)^{2}}$", "final_answer": "\\frac{\\left(- 8\\cdot x - 4\\right)\\cdot \\left(x^{2} - x + \\left(2\\cdot x^{2} + 2\\cdot x - 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 3\\right) + \\left(2\\cdot x^{2} + 2\\cdot x - 5\\right)\\cdot \\left(4\\cdot x + \\left(8\\cdot x + 4\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(2\\cdot x^{2} + 2\\cdot x - 5\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x^{2} + 4\\cdot x - 10\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} - 2\\right)}{2\\cdot \\left(2\\cdot x^{2} + 2\\cdot x - 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a2ee7c9b74662846", "problem_id": "11e017c705d52658", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 2 x^{3} + 3 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{3} + 3 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $\\frac{3 \\pi \\left(\\pi + 6 + 6 \\pi^{2} + 5 \\pi^{3}\\right)}{2}$.\n4) Simplify the result.\nFinal answer: $\\frac{3\\cdot \\pi\\cdot \\left(\\pi + 6 + 6\\cdot \\pi^{2} + 5\\cdot \\pi^{3}\\right)}{2}$", "final_answer": "\\frac{3\\cdot \\pi\\cdot \\left(\\pi + 6 + 6\\cdot \\pi^{2} + 5\\cdot \\pi^{3}\\right)}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ea1fd56e64f97a89", "problem_id": "11e017c705d52658", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} 2 x^{3} + 3 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $2 x^{3} + 3 x^{2} + x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 3$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $\\frac{3 \\pi \\left(\\pi + 6 + 6 \\pi^{2} + 5 \\pi^{3}\\right)}{2}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $\\frac{3\\cdot \\pi\\cdot \\left(\\pi + 6 + 6\\cdot \\pi^{2} + 5\\cdot \\pi^{3}\\right)}{2}$", "final_answer": "\\frac{3\\cdot \\pi\\cdot \\left(\\pi + 6 + 6\\cdot \\pi^{2} + 5\\cdot \\pi^{3}\\right)}{2}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "32b8be57198bbbdc", "problem_id": "515d1eaae18d1dbd", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 9 x - 30$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} - 9 x - 30$ = $3 \\left(x - 5\\right) \\left(x + 2\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $5$.\n5) From the second factor: x = $-2$.\nFinal answer: $x \\in \\{5, -2\\}$", "final_answer": "x \\in \\{5, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6fe728647c01265f", "problem_id": "515d1eaae18d1dbd", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 9 x - 30$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $3 x^{2} - 9 x - 30$ = $3 \\left(x - 5\\right) \\left(x + 2\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $5$.\nFrom the second factor: x = $-2$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{5, -2\\}$", "final_answer": "x \\in \\{5, -2\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f7196f8b2b6118b0", "problem_id": "ba6ea727953770b5", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_7=26$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_7=5.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=5 and k=7.\n4) Compute C(11,6) = 462.\nFinal answer: $462$", "final_answer": "462", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c7bc862bbcb85909", "problem_id": "ba6ea727953770b5", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_7=26$?", "output": "Let y_i=x_i-3, then y_i≥0 and y_1+…+y_7=5.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=5 and k=7.\nFinal answer: $462$", "final_answer": "462", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c81d1bdbae26a6aa", "problem_id": "4de46a39da1882c5", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 2\\right) e^{x} + \\left(- 2 x^{3} - 3\\right) \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\n3) So $f'(x) = 4 x^{3} \\sin{\\left(2 x \\right)} - 6 x^{2} \\cos{\\left(2 x \\right)} + x e^{x} + 3 e^{x} + 6 \\sin{\\left(2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 e^{x}) = 2 e^{x}$.\nTherefore the correct derivative is $f'(x) = - 6 x^{2} \\cos{\\left(2 x \\right)} + \\left(x + 2\\right) e^{x} - 2 \\left(- 2 x^{3} - 3\\right) \\sin{\\left(2 x \\right)} + e^{x}$.\nFinal answer: $- 6 x^{2} \\cos{\\left(2 x \\right)} + \\left(x + 2\\right) e^{x} - 2 \\left(- 2 x^{3} - 3\\right) \\sin{\\left(2 x \\right)} + e^{x}$", "final_answer": "- 6 x^{2} \\cos{\\left(2 x \\right)} + \\left(x + 2\\right) e^{x} - 2 \\left(- 2 x^{3} - 3\\right) \\sin{\\left(2 x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "dad4de917db60835", "problem_id": "0d11294334ff8db5", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 7 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=7, k=5, p=$\\frac{3}{7}$ and simplify.\n4) Here C(7,5) = 21.\nFinal answer: $\\frac{11664}{117649}$", "final_answer": "\\frac{11664}{117649}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eaf54e353571ceee", "problem_id": "adb88e9f788ad181", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "You want 50 liters of a 30% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 25% solution, so 50−x liters is 50% solution.\nSalt equation: 25x + 50(50−x) = 30·50.\n(25-50)x = (30-50)·50 ⇒ x = 40.\nThen 50−x = 10.\nCheck: 25·40 + 50·10 = 30·50.\nFinal answer: $40 L of 25% and 10 L of 50%$", "final_answer": "40 L of 25% and 10 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f795516bfd51cf68", "problem_id": "adb88e9f788ad181", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "You want 50 liters of a 30% salt solution. You have a 25% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 50−x liters is 50% solution.\n2) Salt equation: 25x + 50(50−x) = 30·50.\n3) (25-50)x = (30-50)·50 ⇒ x = 40.\n4) Then 50−x = 10.\nFinal answer: $40 L of 25% and 10 L of 50%$", "final_answer": "40 L of 25% and 10 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8c1f65695f22a6c9", "problem_id": "13463460c307d536", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡2 (mod 5), x≡2 (mod 11). Give the solution modulo 55.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 55.\n3) Compute the CRT solution to get x ≡ 2 (mod 55).\nFinal answer: $x\\equiv 2\\pmod{55}$", "final_answer": "x\\equiv 2\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3f6ff909aad5846a", "problem_id": "13463460c307d536", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡2 (mod 5), x≡2 (mod 11). Give the solution modulo 55.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 55.\nCompute the CRT solution to get x ≡ 2 (mod 55).\nCheck: 2 mod 5=2 and mod 11=2.\nFinal answer: $x\\equiv 2\\pmod{55}$", "final_answer": "x\\equiv 2\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bfe55b0a4ebcafa4", "problem_id": "45bdb23f53f5fafe", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A store sells a backpack for $650. It is discounted by 10%. You buy 11 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 650 × (1 − 10/100) = 650 × 90/100 = 585.\n2) Subtotal = 11 × 585 = 6435.\n3) Total with tax = 6435 × 107/100 = 137709/20.\nFinal answer: $\\frac{137709}{20}$", "final_answer": "\\frac{137709}{20}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "23f41ea4a8ec22f3", "problem_id": "45bdb23f53f5fafe", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "A store sells a backpack for $650. It is discounted by 10%. You buy 11 backpacks. After the discount, a sales tax of 7% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 650 × (1 − 10/100) = 650 × 90/100 = 585.\nSubtotal = 11 × 585 = 6435.\nTotal with tax = 6435 × 107/100 = 137709/20.\nFinal answer: $\\frac{137709}{20}$", "final_answer": "\\frac{137709}{20}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "091791b24a3bb8d8", "problem_id": "810ae96f382e4297", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the linear system:\n$3 x$ = $12$\n$2 x - 4 y$ = $4$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $2$: $6 x$ = $24$.\nMultiply the second equation by $3$: $6 x - 12 y$ = $12$.\nFinal answer: $(x,y)=\\left(4, 1\\right)$", "final_answer": "(x,y)=\\left(4, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e032338d188cc2b5", "problem_id": "810ae96f382e4297", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$3 x$ = $12$\n$2 x - 4 y$ = $4$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $2$: $6 x$ = $24$.\nMultiply the second equation by $3$: $6 x - 12 y$ = $12$.\nSubtract to eliminate x: $12 y$ = $12$.\nSolve for y: y = $1$.\nSubstitute back to find x: x = $4$.\nCheck: plug (x,y)=(4,1) into both equations.\nFinal answer: $(x,y)=\\left(4, 1\\right)$", "final_answer": "(x,y)=\\left(4, 1\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "648c126571426962", "problem_id": "c731bebfbc384f2b", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 16 red balls and 6 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(22,7).\nFavorable samples: choose k reds and n−k blues: C(16,2)·C(6,5).\nCompute totals: C(22,7) = 170544, and favorable = 720.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{15}{3553}$.\nFinal answer: $\\frac{15}{3553}$", "final_answer": "\\frac{15}{3553}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a0aef433af8284f", "problem_id": "c731bebfbc384f2b", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 16 red balls and 6 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(22,7).\n3) Favorable samples: choose k reds and n−k blues: C(16,2)·C(6,5).\n4) Compute totals: C(22,7) = 170544, and favorable = 720.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{15}{3553}$", "final_answer": "\\frac{15}{3553}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c4489eac839f16cb", "problem_id": "b49cfc7fbc06a527", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "A cyclist rides at 74 miles per hour for 1 hours, then at 68 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 74 × 1 = 74 miles.\nSecond leg distance = 68 × 2 = 136 miles.\nTotal distance = 74 + 136 = 210 miles.\nFinal answer: $210$", "final_answer": "210", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "448d86a659b6e3f4", "problem_id": "b49cfc7fbc06a527", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A cyclist rides at 74 miles per hour for 1 hours, then at 68 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 74 × 1 = 74 miles.\n2) Second leg distance = 68 × 2 = 136 miles.\n3) Total distance = 74 + 136 = 210 miles.\nFinal answer: $210$", "final_answer": "210", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e58161110b68ff11", "problem_id": "0477376dc038561e", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Two friends split $923 in the ratio 6:7. How much does each receive?", "output": "1) Total ratio parts = 6+7 = 13.\n2) Each part = 923 ÷ 13 = 71.\n3) First share = 6 × 71 = 426.\n4) Second share = 7 × 71 = 497.\nFinal answer: $497$", "final_answer": "497", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cc4d9ff68fa8fcd6", "problem_id": "0477376dc038561e", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $923 in the ratio 6:7. How much does each receive?", "output": "Total ratio parts = 6+7 = 13.\nEach part = 923 ÷ 13 = 71.\nFirst share = 6 × 71 = 426.\nSecond share = 7 × 71 = 497.\nCheck: 426+497=923.\nFinal answer: $497$", "final_answer": "497", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ac8ec3ce9785458", "problem_id": "05cb621287c2e579", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 18 times, what is the probability of exactly 4 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=18, k=4, p=$\\frac{1}{6}$ and simplify.\n4) Here C(18,4) = 3060.\nFinal answer: $\\frac{518798828125}{2821109907456}$", "final_answer": "\\frac{518798828125}{2821109907456}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d8fa584acbaeb569", "problem_id": "05cb621287c2e579", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 18 times, what is the probability of exactly 4 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=18, k=4, p=$\\frac{1}{6}$ and simplify.\nHere C(18,4) = 3060.\nCheck: expression simplifies to $\\frac{518798828125}{2821109907456}$.\nFinal answer: $\\frac{518798828125}{2821109907456}$", "final_answer": "\\frac{518798828125}{2821109907456}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "25f6fbb29de30000", "problem_id": "3729e07e1e6309f9", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the congruences: x≡2 (mod 3), x≡6 (mod 11). Give the solution modulo 33.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 33.\n3) Compute the CRT solution to get x ≡ 17 (mod 33).\nFinal answer: $x\\equiv 17\\pmod{33}$", "final_answer": "x\\equiv 17\\pmod{33}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7bf5fd6a68b96999", "problem_id": "3729e07e1e6309f9", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 3), x≡6 (mod 11). Give the solution modulo 33.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 33.\nCompute the CRT solution to get x ≡ 17 (mod 33).\nCheck: 17 mod 3=2 and mod 11=6.\nFinal answer: $x\\equiv 17\\pmod{33}$", "final_answer": "x\\equiv 17\\pmod{33}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ebd38c74c9e817f3", "problem_id": "c20113ee2dc781b4", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 11 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=5, p=$\\frac{2}{7}$ and simplify.\n4) Here C(11,5) = 462.\nFinal answer: $\\frac{33000000}{282475249}$", "final_answer": "\\frac{33000000}{282475249}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "33d54dfb4d9fd7c8", "problem_id": "c20113ee2dc781b4", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 11 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=11, k=5, p=$\\frac{2}{7}$ and simplify.\nHere C(11,5) = 462.\nCheck: expression simplifies to $\\frac{33000000}{282475249}$.\nFinal answer: $\\frac{33000000}{282475249}$", "final_answer": "\\frac{33000000}{282475249}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6f0fa0fbe5b5c1c4", "problem_id": "dd7c942eb7d1845c", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{x}{5} - \\frac{21}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$\\frac{1}{5}$, c=$- \\frac{21}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{841}{25}$, so √Δ = $\\frac{29}{5}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{7}{5}$.\n7) x2 = $- \\frac{3}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{7}{5}, - \\frac{3}{2}\\}$", "final_answer": "x \\in \\{\\frac{7}{5}, - \\frac{3}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1a5e7106d38dc668", "problem_id": "dd7c942eb7d1845c", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + \\frac{x}{5} - \\frac{21}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$\\frac{1}{5}$, c=$- \\frac{21}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{7}{5}, - \\frac{3}{2}\\}$", "final_answer": "x \\in \\{\\frac{7}{5}, - \\frac{3}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "31fbec7af3fe7fe7", "problem_id": "ae835e87f127d218", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 2 x - 112$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} + 2 x - 112$ = $2 \\left(x - 7\\right) \\left(x + 8\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFinal answer: $x \\in \\{7, -8\\}$", "final_answer": "x \\in \\{7, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7c80359226461f1c", "problem_id": "ae835e87f127d218", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} + 2 x - 112$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} + 2 x - 112$ = $2 \\left(x - 7\\right) \\left(x + 8\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $7$.\nFrom the second factor: x = $-8$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{7, -8\\}$", "final_answer": "x \\in \\{7, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "250fea39151d79f5", "problem_id": "3378f9a313ef898f", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 6 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=6, k=2, p=$\\frac{2}{7}$ and simplify.\n4) Here C(6,2) = 15.\nFinal answer: $\\frac{37500}{117649}$", "final_answer": "\\frac{37500}{117649}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "841fbd3763458865", "problem_id": "f4bb53485da72686", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{4} - 3 x^{2} + x - 4$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(x) = x$.\n3) So $f'(x) = 16 x^{3} - 5 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(x) = 1$.\nTherefore the correct derivative is $f'(x) = 16 x^{3} - 6 x + 1$.\nFinal answer: $16 x^{3} - 6 x + 1$", "final_answer": "16 x^{3} - 6 x + 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "40c3251d268f3d74", "problem_id": "bbc41260b59c631c", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Two friends split $240 in the ratio 3:7. How much does each receive?", "output": "1) Total ratio parts = 3+7 = 10.\n2) Each part = 240 ÷ 10 = 24.\n3) First share = 3 × 24 = 72.\n4) Second share = 7 × 24 = 168.\nFinal answer: $168$", "final_answer": "168", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fb4f3d78644e4e97", "problem_id": "bbc41260b59c631c", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Two friends split $240 in the ratio 3:7. How much does each receive?", "output": "Total ratio parts = 3+7 = 10.\nEach part = 240 ÷ 10 = 24.\nFirst share = 3 × 24 = 72.\nSecond share = 7 × 24 = 168.\nCheck: 72+168=240.\nFinal answer: $168$", "final_answer": "168", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a68f876e43433a8", "problem_id": "798fa5b4e4c4f411", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} \\frac{- 2 x^{4} + 3 x^{2} + 20}{x + 2}$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 2 x^{4} + 3 x^{2} - 1$ and a = -2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 8 x^{3} + 6 x$.\n5) Evaluate at a: p'(-2) = $52$.\nFinal answer: $52$", "final_answer": "52", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "117d99320e7130d5", "problem_id": "798fa5b4e4c4f411", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} \\frac{- 2 x^{4} + 3 x^{2} + 20}{x + 2}$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 2 x^{4} + 3 x^{2} - 1$ and a = -2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 8 x^{3} + 6 x$.\nEvaluate at a: p'(-2) = $52$.\nCheck: the identity gives $52$.\nFinal answer: $52$", "final_answer": "52", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "73bfeb58abac0a19", "problem_id": "404bffc1498ac192", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 14 red balls and 14 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(28,9).\n3) Favorable samples: choose k reds and n−k blues: C(14,3)·C(14,6).\n4) Compute totals: C(28,9) = 6906900, and favorable = 1093092.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{91}{575}$", "final_answer": "\\frac{91}{575}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "51b4a09bcd4633d8", "problem_id": "404bffc1498ac192", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "An urn has 14 red balls and 14 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "$\\frac{91}{575}$", "final_answer": "\\frac{91}{575}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "602a2b462f22b089", "problem_id": "207ee8628793f26b", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A store sells a backpack for $272. It is discounted by 10%. You buy 8 backpacks. What is the total cost?", "output": "1) Discounted price = 272 × (1 − 10/100) = 272 × 90/100 = 1224/5.\n2) Subtotal = 8 × 1224/5 = 9792/5.\nFinal answer: $\\frac{9792}{5}$", "final_answer": "\\frac{9792}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "81c7136162980b42", "problem_id": "207ee8628793f26b", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A store sells a backpack for $272. It is discounted by 10%. You buy 8 backpacks. What is the total cost?", "output": "Discounted price = 272 × (1 − 10/100) = 272 × 90/100 = 1224/5.\nSubtotal = 8 × 1224/5 = 9792/5.\nFinal answer: $\\frac{9792}{5}$", "final_answer": "\\frac{9792}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "201b338f186167d8", "problem_id": "28f0f25570adc202", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡4 (mod 11), x≡3 (mod 10). Give the solution modulo 110.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 110.\n3) Compute the CRT solution to get x ≡ 103 (mod 110).\nFinal answer: $x\\equiv 103\\pmod{110}$", "final_answer": "x\\equiv 103\\pmod{110}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fa025efd66aecdcc", "problem_id": "28f0f25570adc202", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the congruences: x≡4 (mod 11), x≡3 (mod 10). Give the solution modulo 110.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 110.\nCompute the CRT solution to get x ≡ 103 (mod 110).\nFinal answer: $x\\equiv 103\\pmod{110}$", "final_answer": "x\\equiv 103\\pmod{110}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cbba19f45b425e83", "problem_id": "930c991debe3b5fe", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "An urn has 11 red balls and 15 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(26,7).\n3) Favorable samples: choose k reds and n−k blues: C(11,2)·C(15,5).\n4) Compute totals: C(26,7) = 657800, and favorable = 165165.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{231}{920}$", "final_answer": "\\frac{231}{920}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a226cd4a3c1280d3", "problem_id": "930c991debe3b5fe", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 11 red balls and 15 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(26,7).\nFavorable samples: choose k reds and n−k blues: C(11,2)·C(15,5).\nCompute totals: C(26,7) = 657800, and favorable = 165165.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{231}{920}$.\nFinal answer: $\\frac{231}{920}$", "final_answer": "\\frac{231}{920}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72e7f00f6c4f09dc", "problem_id": "a73fafd1d20fabcd", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A cyclist rides at 25 miles per hour for 3/2 hours, then at 32 miles per hour for 1 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 25 × 3/2 = 75/2 miles.\n2) Second leg distance = 32 × 1 = 32 miles.\n3) Total distance = 75/2 + 32 = 139/2 miles.\nFinal answer: $\\frac{139}{2}$", "final_answer": "\\frac{139}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "475cc78c267d0d72", "problem_id": "9d4cd9c05c99030c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x^{3} + 4 x^{2} + 7 x + 13$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to x^{3} + 4 x^{2} + 6 x + 13.\n3) Then the limit is 49.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 51.\nFinal answer: $51$", "final_answer": "51", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "349b2d8460e5d02c", "problem_id": "ef10f1326eb7abed", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} - 2 x^{2} - 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 3$ at $x=2$.", "output": "1) Differentiate: f'(x) = $6 x^{2} - 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 3$.\n2) Slope at x=2: m = f'(2) = $\\frac{e}{2} - 2 \\sin{\\left(4 \\right)} + 13$.\n3) Point on curve: (2, f(2)) = (2, $\\cos{\\left(4 \\right)} + e + 5$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(4 \\right)} + e + 5 = \\frac{e}{2} - 2 \\sin{\\left(4 \\right)} + 13(x − 2).\n6) Optionally expand: y = $\\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 13 x - 21 + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$.\nFinal answer: $y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 13\\cdot x - 21 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$", "final_answer": "y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 13\\cdot x - 21 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "48a106dabc1d7e42", "problem_id": "ef10f1326eb7abed", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the equation of the tangent line to $y=2 x^{3} - 2 x^{2} - 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 3$ at $x=2$.", "output": "Differentiate: f'(x) = $6 x^{2} - 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 3$.\nSlope at x=2: m = f'(2) = $\\frac{e}{2} - 2 \\sin{\\left(4 \\right)} + 13$.\nPoint on curve: (2, f(2)) = (2, $\\cos{\\left(4 \\right)} + e + 5$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − \\cos{\\left(4 \\right)} + e + 5 = \\frac{e}{2} - 2 \\sin{\\left(4 \\right)} + 13(x − 2).\nOptionally expand: y = $\\frac{e x}{2} - 2 x \\sin{\\left(4 \\right)} + 13 x - 21 + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 13\\cdot x - 21 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}$", "final_answer": "y = \\frac{e\\cdot x}{2} - 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + 13\\cdot x - 21 + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "787ed01cc2efd3d5", "problem_id": "bf2932e59e34e5b5", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 22 times, what is the probability of exactly 17 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=22, k=17, p=$\\frac{1}{2}$ and simplify.\n4) Here C(22,17) = 26334.\nFinal answer: $\\frac{13167}{2097152}$", "final_answer": "\\frac{13167}{2097152}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9b76599ee44576e1", "problem_id": "bf2932e59e34e5b5", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 22 times, what is the probability of exactly 17 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=22, k=17, p=$\\frac{1}{2}$ and simplify.\nFinal answer: $\\frac{13167}{2097152}$", "final_answer": "\\frac{13167}{2097152}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7459c29f4c444e1b", "problem_id": "3f94836bddce978f", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{10} x}{2} - \\frac{3 x}{2} - \\frac{9}{2} + \\frac{9 \\sqrt{10}}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{3 \\sqrt{10}}{2} - \\frac{3}{2}$, c=$- \\frac{9}{2} + \\frac{9 \\sqrt{10}}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{171}{4} - \\frac{27 \\sqrt{10}}{2}$, so √Δ = $\\sqrt{\\frac{171}{4} - \\frac{27 \\sqrt{10}}{2}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{3 \\sqrt{19 - 6 \\sqrt{10}}}{4} + \\frac{3}{4} + \\frac{3 \\sqrt{10}}{4}$.\n7) x2 = $- \\frac{3 \\sqrt{19 - 6 \\sqrt{10}}}{4} + \\frac{3}{4} + \\frac{3 \\sqrt{10}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{3\\cdot \\sqrt{19 - 6\\cdot \\sqrt{10}}}{4} + \\frac{3}{4} + \\frac{3\\cdot \\sqrt{10}}{4}, - \\frac{3\\cdot \\sqrt{19 - 6\\cdot \\sqrt{10}}}{4} + \\frac{3}{4} + \\frac{3\\cdot \\sqrt{10}}{4}\\}$", "final_answer": "x \\in \\{\\frac{3\\cdot \\sqrt{19 - 6\\cdot \\sqrt{10}}}{4} + \\frac{3}{4} + \\frac{3\\cdot \\sqrt{10}}{4}, - \\frac{3\\cdot \\sqrt{19 - 6\\cdot \\sqrt{10}}}{4} + \\frac{3}{4} + \\frac{3\\cdot \\sqrt{10}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5ef76dbd5543bbe3", "problem_id": "3f94836bddce978f", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{10} x}{2} - \\frac{3 x}{2} - \\frac{9}{2} + \\frac{9 \\sqrt{10}}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{3 \\sqrt{10}}{2} - \\frac{3}{2}$, c=$- \\frac{9}{2} + \\frac{9 \\sqrt{10}}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{171}{4} - \\frac{27 \\sqrt{10}}{2}$, so √Δ = $\\sqrt{\\frac{171}{4} - \\frac{27 \\sqrt{10}}{2}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $\\frac{3 \\sqrt{19 - 6 \\sqrt{10}}}{4} + \\frac{3}{4} + \\frac{3 \\sqrt{10}}{4}$.\nx2 = $- \\frac{3 \\sqrt{19 - 6 \\sqrt{10}}}{4} + \\frac{3}{4} + \\frac{3 \\sqrt{10}}{4}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{\\frac{3\\cdot \\sqrt{19 - 6\\cdot \\sqrt{10}}}{4} + \\frac{3}{4} + \\frac{3\\cdot \\sqrt{10}}{4}, - \\frac{3\\cdot \\sqrt{19 - 6\\cdot \\sqrt{10}}}{4} + \\frac{3}{4} + \\frac{3\\cdot \\sqrt{10}}{4}\\}$", "final_answer": "x \\in \\{\\frac{3\\cdot \\sqrt{19 - 6\\cdot \\sqrt{10}}}{4} + \\frac{3}{4} + \\frac{3\\cdot \\sqrt{10}}{4}, - \\frac{3\\cdot \\sqrt{19 - 6\\cdot \\sqrt{10}}}{4} + \\frac{3}{4} + \\frac{3\\cdot \\sqrt{10}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c178a2eb7d1825df", "problem_id": "4d2d03fe850a20ac", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 8 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=8, k=2, p=$\\frac{4}{5}$ and simplify.\n4) Here C(8,2) = 28.\nFinal answer: $\\frac{448}{390625}$", "final_answer": "\\frac{448}{390625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4e5a6e04ebcd635a", "problem_id": "4d2d03fe850a20ac", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{5}$. If it is flipped 8 times, what is the probability of exactly 2 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=8, k=2, p=$\\frac{4}{5}$ and simplify.\nHere C(8,2) = 28.\nCheck: expression simplifies to $\\frac{448}{390625}$.\nFinal answer: $\\frac{448}{390625}$", "final_answer": "\\frac{448}{390625}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "78f4347a322a1311", "problem_id": "6016bc3936473c82", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 13 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=13, k=3, p=$\\frac{3}{7}$ and simplify.\n4) Here C(13,3) = 286.\nFinal answer: $\\frac{8097103872}{96889010407}$", "final_answer": "\\frac{8097103872}{96889010407}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1e0478f1d7cf86b2", "problem_id": "6016bc3936473c82", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 13 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=13, k=3, p=$\\frac{3}{7}$ and simplify.\nHere C(13,3) = 286.\nCheck: expression simplifies to $\\frac{8097103872}{96889010407}$.\nFinal answer: $\\frac{8097103872}{96889010407}$", "final_answer": "\\frac{8097103872}{96889010407}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c09d1bee93c02db0", "problem_id": "4e0be0c7eda18651", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_3=25$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=25 and k=3.\n4) Compute C(27,2) = 351.\nFinal answer: $351$", "final_answer": "351", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "240e1b84b34680c6", "problem_id": "4e0be0c7eda18651", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_3=25$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=25 and k=3.\nCompute C(27,2) = 351.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $351$", "final_answer": "351", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "56ef401368304594", "problem_id": "43257001b1c13a5c", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Worker A finishes a job in 13 hours and Worker B finishes it in 17 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/13 job/hour. B's rate = 1/17 job/hour.\n2) Combined rate = 1/13 + 1/17 = 30/221 job/hour.\n3) Time = 1 ÷ (combined rate) = 221/30 hours.\nFinal answer: $\\frac{221}{30}$", "final_answer": "\\frac{221}{30}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "37ecc3844731ce33", "problem_id": "43257001b1c13a5c", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Worker A finishes a job in 13 hours and Worker B finishes it in 17 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/13 job/hour. B's rate = 1/17 job/hour.\nCombined rate = 1/13 + 1/17 = 30/221 job/hour.\nTime = 1 ÷ (combined rate) = 221/30 hours.\nCheck: (combined rate)×(time) = 30/221×221/30 = 1.\nFinal answer: $\\frac{221}{30}$", "final_answer": "\\frac{221}{30}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cf58d89bc043535b", "problem_id": "68889e65f803daf9", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "You want 60 liters of a 45% salt solution. You have a 10% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 60−x liters is 60% solution.\n2) Salt equation: 10x + 60(60−x) = 45·60.\n3) (10-60)x = (45-60)·60 ⇒ x = 18.\n4) Then 60−x = 42.\nFinal answer: $18 L of 10% and 42 L of 60%$", "final_answer": "18 L of 10% and 42 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a4b681aaffde1fdf", "problem_id": "68889e65f803daf9", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "You want 60 liters of a 45% salt solution. You have a 10% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 10% solution, so 60−x liters is 60% solution.\nSalt equation: 10x + 60(60−x) = 45·60.\n(10-60)x = (45-60)·60 ⇒ x = 18.\nFinal answer: $18 L of 10% and 42 L of 60%$", "final_answer": "18 L of 10% and 42 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1f51e56cc2cc9202", "problem_id": "c4f09390e5e0cdc2", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡0 (mod 3), x≡3 (mod 13). Give the solution modulo 39.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 39.\n3) Compute the CRT solution to get x ≡ 3 (mod 39).\nFinal answer: $x\\equiv 3\\pmod{39}$", "final_answer": "x\\equiv 3\\pmod{39}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1a4ef91d01c32fc0", "problem_id": "c4f09390e5e0cdc2", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Solve the congruences: x≡0 (mod 3), x≡3 (mod 13). Give the solution modulo 39.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 39.\nCompute the CRT solution to get x ≡ 3 (mod 39).\nFinal answer: $x\\equiv 3\\pmod{39}$", "final_answer": "x\\equiv 3\\pmod{39}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7131c522edf7240a", "problem_id": "23d55f982b3a6262", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡0 (mod 5), x≡8 (mod 9). Give the solution modulo 45.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 45.\n3) Compute the CRT solution to get x ≡ 35 (mod 45).\nFinal answer: $x\\equiv 35\\pmod{45}$", "final_answer": "x\\equiv 35\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "65e6c8af161e55e3", "problem_id": "23d55f982b3a6262", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡0 (mod 5), x≡8 (mod 9). Give the solution modulo 45.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 45.\nCompute the CRT solution to get x ≡ 35 (mod 45).\nCheck: 35 mod 5=0 and mod 9=8.\nFinal answer: $x\\equiv 35\\pmod{45}$", "final_answer": "x\\equiv 35\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f6cc56883e39ae65", "problem_id": "a06f80626b41742d", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the multiplicative inverse of 26 modulo 29.", "output": "1) We want x such that 26x ≡ 1 (mod 29).\n2) Use extended Euclid to find s,t with 26s + 29t = 1.\n3) One Bezout identity is: 26·(-10) + 29·(9) = 1.\n4) Reduce s modulo 29: x ≡ 19 (mod 29).\nFinal answer: $x\\equiv 19\\pmod{29}$", "final_answer": "x\\equiv 19\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2a65d2b582603919", "problem_id": "a06f80626b41742d", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the multiplicative inverse of 26 modulo 29.", "output": "We want x such that 26x ≡ 1 (mod 29).\nUse extended Euclid to find s,t with 26s + 29t = 1.\nOne Bezout identity is: 26·(-10) + 29·(9) = 1.\nReduce s modulo 29: x ≡ 19 (mod 29).\nCheck: 26·19 mod 29 = 1.\nFinal answer: $x\\equiv 19\\pmod{29}$", "final_answer": "x\\equiv 19\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4c11ab560b5e8846", "problem_id": "7ed86c03b65fb5a8", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 36 x + 81$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} - 36 x + 81$ = $3 \\left(x - 9\\right) \\left(x - 3\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $9$.\n5) From the second factor: x = $3$.\nFinal answer: $x \\in \\{9, 3\\}$", "final_answer": "x \\in \\{9, 3\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a335fbf38694d459", "problem_id": "8dafeffed0e67c36", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} 2 x - 9$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to x - 9.\n3) Then the limit is -13.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: -17.\nFinal answer: $-17$", "final_answer": "-17", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "af722928595a064a", "problem_id": "9040480666dc6c2d", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the multiplicative inverse of 3 modulo 43.", "output": "1) We want x such that 3x ≡ 1 (mod 43).\n2) Use extended Euclid to find s,t with 3s + 43t = 1.\n3) One Bezout identity is: 3·(-14) + 43·(1) = 1.\n4) Reduce s modulo 43: x ≡ 29 (mod 43).\nFinal answer: $x\\equiv 29\\pmod{43}$", "final_answer": "x\\equiv 29\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "076320b8c2cf3eb2", "problem_id": "9040480666dc6c2d", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 3 modulo 43.", "output": "We want x such that 3x ≡ 1 (mod 43).\nUse extended Euclid to find s,t with 3s + 43t = 1.\nOne Bezout identity is: 3·(-14) + 43·(1) = 1.\nReduce s modulo 43: x ≡ 29 (mod 43).\nCheck: 3·29 mod 43 = 1.\nFinal answer: $x\\equiv 29\\pmod{43}$", "final_answer": "x\\equiv 29\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ac908549503c670d", "problem_id": "943b78b91edcbb7d", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 12 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=2, p=$\\frac{1}{6}$ and simplify.\n4) Here C(12,2) = 66.\nFinal answer: $\\frac{107421875}{362797056}$", "final_answer": "\\frac{107421875}{362797056}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "16024ebd6edbfae8", "problem_id": "6f207e9759e1c11d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{3} - \\frac{3 x}{2} + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}) = \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3}$.\n3) So $f'(x) = \\frac{x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} + \\frac{2 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} - \\frac{2 x^{4}}{2 x^{4} - 12 x^{2} + 18} + \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{4} - 6 x^{2} + 9} - \\frac{6 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} - \\frac{12 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} + \\frac{21 x^{2}}{2 x^{4} - 12 x^{2} + 18} - \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 6} - \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3} - \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3} + \\frac{9 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} + \\frac{18 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{4} - 12 x^{2} + 18} + \\frac{9}{2 x^{4} - 12 x^{2} + 18}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}) = - \\frac{2 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x^{2} - 3\\right)^{2}} + \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(x^{2} - 3\\right)} + \\frac{x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x^{2} - 3} + \\frac{2 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 3}$.\nTherefore the correct derivative is $f'(x) = \\frac{- 2 x \\left(- 2 x^{3} - 3 x + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 3\\right)}{2 \\left(x^{2} - 3\\right)^{2}}$.\nFinal answer: $\\frac{- 2 x \\left(- 2 x^{3} - 3 x + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 3\\right)}{2 \\left(x^{2} - 3\\right)^{2}}$", "final_answer": "\\frac{- 2 x \\left(- 2 x^{3} - 3 x + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}\\right) + \\left(x^{2} - 3\\right) \\left(- 6 x^{2} + 4 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x^{2} - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 3\\right)}{2 \\left(x^{2} - 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f263fa2864f228a2", "problem_id": "2a867728c94f8d85", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡10 (mod 11), x≡5 (mod 9). Give the solution modulo 99.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 99.\n3) Compute the CRT solution to get x ≡ 32 (mod 99).\nFinal answer: $x\\equiv 32\\pmod{99}$", "final_answer": "x\\equiv 32\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ebe137764976f89f", "problem_id": "2a867728c94f8d85", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the congruences: x≡10 (mod 11), x≡5 (mod 9). Give the solution modulo 99.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 99.\nCompute the CRT solution to get x ≡ 32 (mod 99).\nFinal answer: $x\\equiv 32\\pmod{99}$", "final_answer": "x\\equiv 32\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e99524848ed0e49c", "problem_id": "f0843cc4e8eb43c3", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "An urn has 24 red balls and 20 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(44,8).\n3) Favorable samples: choose k reds and n−k blues: C(24,2)·C(20,6).\n4) Compute totals: C(44,8) = 177232627, and favorable = 10697760.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{563040}{9328033}$", "final_answer": "\\frac{563040}{9328033}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ef8dc20973057579", "problem_id": "f0843cc4e8eb43c3", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "An urn has 24 red balls and 20 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(44,8).\nFavorable samples: choose k reds and n−k blues: C(24,2)·C(20,6).\nFinal answer: $\\frac{563040}{9328033}$", "final_answer": "\\frac{563040}{9328033}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dfa42f343bef3d06", "problem_id": "1c45b1a20f6a39de", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $3 x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "3\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} + 6\\cdot x\\cdot \\sin{\\left(x \\right)} + x\\cdot \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72911bd822a00fe0", "problem_id": "3ecb21c9d55ee786", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 2 x^{3} + 3 x^{2} + 2 x + \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{3 x - 5}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 5}) = \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 5}$.\n3) So $f'(x) = - \\frac{24 x^{3}}{18 x^{2} - 60 x + 50} + \\frac{9 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} - 60 x + 50} + \\frac{18 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} - 60 x + 50} + \\frac{78 x^{2}}{18 x^{2} - 60 x + 50} - \\frac{30 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} - 60 x + 50} - \\frac{60 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} - 60 x + 50} - \\frac{60 x}{18 x^{2} - 60 x + 50} + \\frac{9 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{2} - 30 x + 25} - \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x - 10} + \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 5} - \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 5} + \\frac{25 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} - 60 x + 50} + \\frac{50 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} - 60 x + 50} - \\frac{2}{18 x^{2} - 60 x + 50} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 5}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 5}) = \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(3 x - 5\\right)} + \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 5} - \\frac{9 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(3 x - 5\\right)^{2}} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 5}$.\nTherefore the correct derivative is $f'(x) = \\frac{12 x^{3} - 18 x^{2} - 12 x + \\left(3 x - 5\\right) \\left(- 12 x^{2} + 12 x + \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 6 \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 18}{2 \\left(3 x - 5\\right)^{2}}$.\nFinal answer: $\\frac{12 x^{3} - 18 x^{2} - 12 x + \\left(3 x - 5\\right) \\left(- 12 x^{2} + 12 x + \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 6 \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 18}{2 \\left(3 x - 5\\right)^{2}}$", "final_answer": "\\frac{12 x^{3} - 18 x^{2} - 12 x + \\left(3 x - 5\\right) \\left(- 12 x^{2} + 12 x + \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 6 \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 18}{2 \\left(3 x - 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "db221c01273666fa", "problem_id": "48dd31d83389d55d", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{7}$. If it is flipped 15 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=15, k=2, p=$\\frac{1}{7}$ and simplify.\n4) Here C(15,2) = 105.\nFinal answer: $\\frac{195910410240}{678223072849}$", "final_answer": "\\frac{195910410240}{678223072849}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d73e44cdc32f29e5", "problem_id": "48dd31d83389d55d", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{7}$. If it is flipped 15 times, what is the probability of exactly 2 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=15, k=2, p=$\\frac{1}{7}$ and simplify.\nHere C(15,2) = 105.\nCheck: expression simplifies to $\\frac{195910410240}{678223072849}$.\nFinal answer: $\\frac{195910410240}{678223072849}$", "final_answer": "\\frac{195910410240}{678223072849}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7c81dc9e2ba3691c", "problem_id": "c4850d10ec87edff", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_4=28$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=28 and k=4.\n4) Compute C(31,3) = 4495.\nFinal answer: $4495$", "final_answer": "4495", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a4477e02dc5f727d", "problem_id": "c4850d10ec87edff", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_4=28$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=28 and k=4.\nCompute C(31,3) = 4495.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $4495$", "final_answer": "4495", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ac84ea00c95656dc", "problem_id": "eb78ebf057f2582a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_9=32$?", "output": "Let y_i=x_i-3, then y_i≥0 and y_1+…+y_9=5.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=5 and k=9.\nCompute C(13,8) = 1287.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $1287$", "final_answer": "1287", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b07fecd5f6c581d9", "problem_id": "eb78ebf057f2582a", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_9=32$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_9=5.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=5 and k=9.\n4) Compute C(13,8) = 1287.\nFinal answer: $1287$", "final_answer": "1287", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a79cdeaa44e0e8e1", "problem_id": "52e9fe9c89ba2fa3", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the linear system:\n$- x$ = $-4$\n$- 3 x - 2 y$ = $-4$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-3$: $3 x$ = $12$.\n3) Multiply the second equation by $-1$: $3 x + 2 y$ = $4$.\n4) Subtract to eliminate x: $- 2 y$ = $8$.\n5) Solve for y: y = $-4$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, -4\\right)$", "final_answer": "(x,y)=\\left(4, -4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3d7b83e0e036587d", "problem_id": "694ac8f6f8da0488", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -4} - x^{2} + 7 x - 26$.\n\nStudent solution:\n1) Substitute x=-4 into the expression.\n2) (Mistake) Simplify the expression to - x^{2} + 6 x - 26.\n3) Then the limit is -66.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-4 (polynomial is continuous).\nCorrect limit value: -70.\nFinal answer: $-70$", "final_answer": "-70", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2da7e9dfe236db7a", "problem_id": "15dd4ba5214534fc", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "An urn has 11 red balls and 25 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(36,12).\n3) Favorable samples: choose k reds and n−k blues: C(11,3)·C(25,9).\n4) Compute totals: C(36,12) = 1251677700, and favorable = 337090875.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{264385}{981708}$", "final_answer": "\\frac{264385}{981708}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "00e41f34f29a65be", "problem_id": "15dd4ba5214534fc", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 11 red balls and 25 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(36,12).\nFavorable samples: choose k reds and n−k blues: C(11,3)·C(25,9).\nCompute totals: C(36,12) = 1251677700, and favorable = 337090875.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{264385}{981708}$.\nFinal answer: $\\frac{264385}{981708}$", "final_answer": "\\frac{264385}{981708}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ec53fb5b387bed2", "problem_id": "da7214d499fb8348", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{2 \\pi} - 2 x^{3} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{3} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 3$.\n2) Evaluate at the bounds: F(2 \\pi) and F(0).\n3) Subtract: F(2 \\pi) − F(0) = $2 \\pi \\left(- 4 \\pi^{3} + 3 + 3 \\pi\\right)$.\n4) Simplify the result.\nFinal answer: $2\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{3} + 3 + 3\\cdot \\pi\\right)$", "final_answer": "2\\cdot \\pi\\cdot \\left(- 4\\cdot \\pi^{3} + 3 + 3\\cdot \\pi\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d850f1ac35a3f78", "problem_id": "ef00248462a5f9c6", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} x^{2} - x + 2$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to x^{2} - 2 x + 2.\n3) Then the limit is 17.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: 14.\nFinal answer: $14$", "final_answer": "14", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6cf7327af03e8b15", "problem_id": "30df0b969d30f10f", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A store sells a backpack for $149. It is discounted by 20%. You buy 8 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "1) Discounted price = 149 × (1 − 20/100) = 149 × 80/100 = 596/5.\n2) Subtotal = 8 × 596/5 = 4768/5.\n3) Total with tax = 4768/5 × 105/100 = 25032/25.\nFinal answer: $\\frac{25032}{25}$", "final_answer": "\\frac{25032}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "486ce09ca5bcc4e6", "problem_id": "30df0b969d30f10f", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $149. It is discounted by 20%. You buy 8 backpacks. After the discount, a sales tax of 5% is applied to the subtotal. What is the total cost?", "output": "Discounted price = 149 × (1 − 20/100) = 149 × 80/100 = 596/5.\nSubtotal = 8 × 596/5 = 4768/5.\nTotal with tax = 4768/5 × 105/100 = 25032/25.\nCheck: recomputing the formula yields 25032/25.\nFinal answer: $\\frac{25032}{25}$", "final_answer": "\\frac{25032}{25}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d35f5961661d5440", "problem_id": "e01fc9f8c0f4812f", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_5=24$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_5=9.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=9 and k=5.\n4) Compute C(13,4) = 715.\nFinal answer: $715$", "final_answer": "715", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1694aa61a1716fd1", "problem_id": "e01fc9f8c0f4812f", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_5=24$?", "output": "Let y_i=x_i-3, then y_i≥0 and y_1+…+y_5=9.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=9 and k=5.\nCompute C(13,4) = 715.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $715$", "final_answer": "715", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4e92dc694c15d6fe", "problem_id": "5d1e6b69cb3a51eb", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 22 x + 60$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 22 x + 60$ = $2 \\left(x - 6\\right) \\left(x - 5\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $6$.\n5) From the second factor: x = $5$.\nFinal answer: $x \\in \\{6, 5\\}$", "final_answer": "x \\in \\{6, 5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4329ecaee9197f4e", "problem_id": "5d1e6b69cb3a51eb", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 22 x + 60$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} - 22 x + 60$ = $2 \\left(x - 6\\right) \\left(x - 5\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $6$.\nFrom the second factor: x = $5$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{6, 5\\}$", "final_answer": "x \\in \\{6, 5\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "918fd2d222682779", "problem_id": "7e8af12b3c014b5e", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=18$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=18 and k=5.\n4) Compute C(22,4) = 7315.\nFinal answer: $7315$", "final_answer": "7315", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a34c2f0af351dc95", "problem_id": "7e8af12b3c014b5e", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=18$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=18 and k=5.\nCompute C(22,4) = 7315.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $7315$", "final_answer": "7315", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7bcaf31d5e32f512", "problem_id": "7be2adec7e98faa9", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "An urn has 21 red balls and 22 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(43,8).\nFavorable samples: choose k reds and n−k blues: C(21,3)·C(22,5).\nFinal answer: $\\frac{204820}{848003}$", "final_answer": "\\frac{204820}{848003}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8f1f54dedbeea252", "problem_id": "7be2adec7e98faa9", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 21 red balls and 22 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(43,8).\n3) Favorable samples: choose k reds and n−k blues: C(21,3)·C(22,5).\n4) Compute totals: C(43,8) = 145008513, and favorable = 35024220.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{204820}{848003}$", "final_answer": "\\frac{204820}{848003}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8182d9911e91bc45", "problem_id": "d5242a107d8b5408", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "An urn has 11 red balls and 25 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(36,10).\n3) Favorable samples: choose k reds and n−k blues: C(11,5)·C(25,5).\n4) Compute totals: C(36,10) = 254186856, and favorable = 24546060.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{8855}{91698}$", "final_answer": "\\frac{8855}{91698}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a0db3ae63d73583", "problem_id": "d5242a107d8b5408", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 11 red balls and 25 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(36,10).\nFavorable samples: choose k reds and n−k blues: C(11,5)·C(25,5).\nCompute totals: C(36,10) = 254186856, and favorable = 24546060.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{8855}{91698}$.\nFinal answer: $\\frac{8855}{91698}$", "final_answer": "\\frac{8855}{91698}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "68146246de611a57", "problem_id": "89eaaafc648a2e3e", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 16 red balls and 6 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(22,4).\n3) Favorable samples: choose k reds and n−k blues: C(16,2)·C(6,2).\n4) Compute totals: C(22,4) = 7315, and favorable = 1800.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{360}{1463}$", "final_answer": "\\frac{360}{1463}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4dfa2830e0ca9a31", "problem_id": "89eaaafc648a2e3e", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 16 red balls and 6 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(22,4).\nFavorable samples: choose k reds and n−k blues: C(16,2)·C(6,2).\nCompute totals: C(22,4) = 7315, and favorable = 1800.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{360}{1463}$.\nFinal answer: $\\frac{360}{1463}$", "final_answer": "\\frac{360}{1463}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "86c7285c578b82fb", "problem_id": "4f29c974238077a4", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡0 (mod 7), x≡1 (mod 9). Give the solution modulo 63.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 63.\n3) Compute the CRT solution to get x ≡ 28 (mod 63).\nFinal answer: $x\\equiv 28\\pmod{63}$", "final_answer": "x\\equiv 28\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c41518cb44646876", "problem_id": "4f29c974238077a4", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡0 (mod 7), x≡1 (mod 9). Give the solution modulo 63.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 63.\nCompute the CRT solution to get x ≡ 28 (mod 63).\nCheck: 28 mod 7=0 and mod 9=1.\nFinal answer: $x\\equiv 28\\pmod{63}$", "final_answer": "x\\equiv 28\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fb2c1ae425414a03", "problem_id": "c8d1ed921fc78e77", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - x^{4} - 2 x^{3} - x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- x^{4} - 2 x^{3} - x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{205 \\pi^{5}}{32} - \\frac{255 \\pi^{4}}{32} - \\frac{65 \\pi^{3}}{24} - \\frac{15 \\pi}{2} - \\frac{15 \\pi^{2}}{8} - 1$.\n4) Simplify the result.\nFinal answer: $- \\frac{205\\cdot \\pi^{5}}{32} - \\frac{255\\cdot \\pi^{4}}{32} - \\frac{65\\cdot \\pi^{3}}{24} - \\frac{15\\cdot \\pi}{2} - \\frac{15\\cdot \\pi^{2}}{8} - 1$", "final_answer": "- \\frac{205\\cdot \\pi^{5}}{32} - \\frac{255\\cdot \\pi^{4}}{32} - \\frac{65\\cdot \\pi^{3}}{24} - \\frac{15\\cdot \\pi}{2} - \\frac{15\\cdot \\pi^{2}}{8} - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ac66d019351d186e", "problem_id": "c8d1ed921fc78e77", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} - x^{4} - 2 x^{3} - x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- x^{4} - 2 x^{3} - x^{2} - x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{205 \\pi^{5}}{32} - \\frac{255 \\pi^{4}}{32} - \\frac{65 \\pi^{3}}{24} - \\frac{15 \\pi}{2} - \\frac{15 \\pi^{2}}{8} - 1$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{205\\cdot \\pi^{5}}{32} - \\frac{255\\cdot \\pi^{4}}{32} - \\frac{65\\cdot \\pi^{3}}{24} - \\frac{15\\cdot \\pi}{2} - \\frac{15\\cdot \\pi^{2}}{8} - 1$", "final_answer": "- \\frac{205\\cdot \\pi^{5}}{32} - \\frac{255\\cdot \\pi^{4}}{32} - \\frac{65\\cdot \\pi^{3}}{24} - \\frac{15\\cdot \\pi}{2} - \\frac{15\\cdot \\pi^{2}}{8} - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d934b5d659a9a35", "problem_id": "d4a5b04780138155", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the linear system:\n$- x - 8 y$ = $39$\n$5 x - y$ = $10$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $5$: $- 5 x - 40 y$ = $195$.\n3) Multiply the second equation by $-1$: $- 5 x + y$ = $-10$.\n4) Subtract to eliminate x: $- 41 y$ = $205$.\n5) Solve for y: y = $-5$.\n6) Substitute back to find x: x = $1$.\nFinal answer: $(x,y)=\\left(1, -5\\right)$", "final_answer": "(x,y)=\\left(1, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b3df2f2b6d9020a9", "problem_id": "d4a5b04780138155", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$- x - 8 y$ = $39$\n$5 x - y$ = $10$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $5$: $- 5 x - 40 y$ = $195$.\nMultiply the second equation by $-1$: $- 5 x + y$ = $-10$.\nSubtract to eliminate x: $- 41 y$ = $205$.\nSolve for y: y = $-5$.\nSubstitute back to find x: x = $1$.\nCheck: plug (x,y)=(1,-5) into both equations.\nFinal answer: $(x,y)=\\left(1, -5\\right)$", "final_answer": "(x,y)=\\left(1, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "559d8e49ae1e34f9", "problem_id": "d6334c18f690852d", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} \\cos{\\left(x^{2} + 3 x + 1 \\right)} + 2 \\left(x + 1\\right) e^{x}$.", "output": "Let f(x) = $3 x^{2} \\cos{\\left(x^{2} + 3 x + 1 \\right)} + 2 \\left(x + 1\\right) e^{x}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $- 3 x^{2} \\left(2 x + 3\\right) \\sin{\\left(x^{2} + 3 x + 1 \\right)} + 6 x \\cos{\\left(x^{2} + 3 x + 1 \\right)} + 2 \\left(x + 1\\right) e^{x} + 2 e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $- 3\\cdot x^{2}\\cdot \\left(2\\cdot x + 3\\right)\\cdot \\sin{\\left(x^{2} + 3\\cdot x + 1 \\right)} + 6\\cdot x\\cdot \\cos{\\left(x^{2} + 3\\cdot x + 1 \\right)} + \\left(2\\cdot x + 2\\right)\\cdot e^{x} + 2\\cdot e^{x}$", "final_answer": "- 3\\cdot x^{2}\\cdot \\left(2\\cdot x + 3\\right)\\cdot \\sin{\\left(x^{2} + 3\\cdot x + 1 \\right)} + 6\\cdot x\\cdot \\cos{\\left(x^{2} + 3\\cdot x + 1 \\right)} + \\left(2\\cdot x + 2\\right)\\cdot e^{x} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "91bb97b606aa3ddf", "problem_id": "d6334c18f690852d", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} \\cos{\\left(x^{2} + 3 x + 1 \\right)} + 2 \\left(x + 1\\right) e^{x}$.", "output": "1) Let f(x) = $3 x^{2} \\cos{\\left(x^{2} + 3 x + 1 \\right)} + 2 \\left(x + 1\\right) e^{x}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $- 3 x^{2} \\left(2 x + 3\\right) \\sin{\\left(x^{2} + 3 x + 1 \\right)} + 6 x \\cos{\\left(x^{2} + 3 x + 1 \\right)} + 2 \\left(x + 1\\right) e^{x} + 2 e^{x}$.\nFinal answer: $- 3\\cdot x^{2}\\cdot \\left(2\\cdot x + 3\\right)\\cdot \\sin{\\left(x^{2} + 3\\cdot x + 1 \\right)} + 6\\cdot x\\cdot \\cos{\\left(x^{2} + 3\\cdot x + 1 \\right)} + \\left(2\\cdot x + 2\\right)\\cdot e^{x} + 2\\cdot e^{x}$", "final_answer": "- 3\\cdot x^{2}\\cdot \\left(2\\cdot x + 3\\right)\\cdot \\sin{\\left(x^{2} + 3\\cdot x + 1 \\right)} + 6\\cdot x\\cdot \\cos{\\left(x^{2} + 3\\cdot x + 1 \\right)} + \\left(2\\cdot x + 2\\right)\\cdot e^{x} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "27a21bff3284cc85", "problem_id": "d95e2d80ec48808a", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} - 2 x^{3} - 2 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 2 x^{3} - 2 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\pi).\n3) Subtract: F(2 \\pi) − F(- \\pi) = $- \\frac{15 \\pi^{4}}{2} - 6 \\pi^{3} - 3 \\pi$.\n4) Simplify the result.\nFinal answer: $- \\frac{15\\cdot \\pi^{4}}{2} - 6\\cdot \\pi^{3} - 3\\cdot \\pi$", "final_answer": "- \\frac{15\\cdot \\pi^{4}}{2} - 6\\cdot \\pi^{3} - 3\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6cb3eb966cb9e2ea", "problem_id": "d95e2d80ec48808a", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{2 \\pi} - 2 x^{3} - 2 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 2 x^{3} - 2 x^{2} + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 1$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\pi).\nSubtract: F(2 \\pi) − F(- \\pi) = $- \\frac{15 \\pi^{4}}{2} - 6 \\pi^{3} - 3 \\pi$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{15\\cdot \\pi^{4}}{2} - 6\\cdot \\pi^{3} - 3\\cdot \\pi$", "final_answer": "- \\frac{15\\cdot \\pi^{4}}{2} - 6\\cdot \\pi^{3} - 3\\cdot \\pi", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c9626da35abe69bc", "problem_id": "2f33041a0747a6b2", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_7=29$?", "output": "Let y_i=x_i-3, then y_i≥0 and y_1+…+y_7=8.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=8 and k=7.\nCompute C(14,6) = 3003.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $3003$", "final_answer": "3003", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "11c5f989f7882358", "problem_id": "2f33041a0747a6b2", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_7=29$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_7=8.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=8 and k=7.\n4) Compute C(14,6) = 3003.\nFinal answer: $3003$", "final_answer": "3003", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "54cdc77e229d3e32", "problem_id": "a6d73985fdff9722", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x + 1\\right) e^{x} + \\left(2 x^{3} - x^{2} - 3 x - 1\\right) \\cos{\\left(2 x - 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x^{2} \\cos{\\left(2 x - 2 \\right)}) = x^{2} \\sin{\\left(2 x - 2 \\right)}$.\n3) So $f'(x) = - 4 x^{3} \\sin{\\left(2 x - 2 \\right)} + x^{2} \\sin{\\left(2 x - 2 \\right)} + 6 x^{2} \\cos{\\left(2 x - 2 \\right)} + x e^{x} + 6 x \\sin{\\left(2 x - 2 \\right)} + 2 e^{x} + 2 \\sin{\\left(2 x - 2 \\right)} - 3 \\cos{\\left(2 x - 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x^{2} \\cos{\\left(2 x - 2 \\right)}) = 2 x^{2} \\sin{\\left(2 x - 2 \\right)} - 2 x \\cos{\\left(2 x - 2 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x + 1\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) - 3\\right) - 1\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(6 x^{2} - 2 x - 3\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}$.\nFinal answer: $\\left(x + 1\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) - 3\\right) - 1\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(6 x^{2} - 2 x - 3\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}$", "final_answer": "\\left(x + 1\\right) e^{x} - 2 \\left(x \\left(x \\left(2 x - 1\\right) - 3\\right) - 1\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(6 x^{2} - 2 x - 3\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "72bb6ce41082c05b", "problem_id": "46a40bc3e7cbfc1d", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(2 x^{2} - 2 x - 3\\right) \\cos{\\left(3 x^{2} + 3 x - 1 \\right)}$.", "output": "Let f(x) = $2 \\left(x + 2\\right) e^{x} + \\left(2 x^{2} - 2 x - 3\\right) \\cos{\\left(3 x^{2} + 3 x - 1 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $2 \\left(x + 2\\right) e^{x} + 2 \\left(2 x - 1\\right) \\cos{\\left(3 x^{2} + 3 x - 1 \\right)} - 3 \\left(2 x + 1\\right) \\left(2 x^{2} - 2 x - 3\\right) \\sin{\\left(3 x^{2} + 3 x - 1 \\right)} + 2 e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(4\\cdot x - 2\\right)\\cdot \\cos{\\left(3\\cdot x^{2} + 3\\cdot x - 1 \\right)} - \\left(6\\cdot x + 3\\right)\\cdot \\left(2\\cdot x^{2} - 2\\cdot x - 3\\right)\\cdot \\sin{\\left(3\\cdot x^{2} + 3\\cdot x - 1 \\right)} + 2\\cdot e^{x}$", "final_answer": "\\left(2\\cdot x + 4\\right)\\cdot e^{x} + \\left(4\\cdot x - 2\\right)\\cdot \\cos{\\left(3\\cdot x^{2} + 3\\cdot x - 1 \\right)} - \\left(6\\cdot x + 3\\right)\\cdot \\left(2\\cdot x^{2} - 2\\cdot x - 3\\right)\\cdot \\sin{\\left(3\\cdot x^{2} + 3\\cdot x - 1 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5008e7860e560f52", "problem_id": "6203bebe9ed132b3", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Find the multiplicative inverse of 12 modulo 47.", "output": "We want x such that 12x ≡ 1 (mod 47).\nUse extended Euclid to find s,t with 12s + 47t = 1.\nOne Bezout identity is: 12·(4) + 47·(-1) = 1.\nReduce s modulo 47: x ≡ 4 (mod 47).\nCheck: 12·4 mod 47 = 1.\nFinal answer: $x\\equiv 4\\pmod{47}$", "final_answer": "x\\equiv 4\\pmod{47}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c7a50f40a984216e", "problem_id": "67193628b2d1589d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 3 x + 7$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 2 x + 7.\n3) Then the limit is 11.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 13.\nFinal answer: $13$", "final_answer": "13", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "218d451bfdd1636d", "problem_id": "49b473059d3f7c1c", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "An urn has 26 red balls and 22 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(48,12).\nFavorable samples: choose k reds and n−k blues: C(26,3)·C(22,9).\nFinal answer: $\\frac{1309000}{70514711}$", "final_answer": "\\frac{1309000}{70514711}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3186b86db3f19875", "problem_id": "49b473059d3f7c1c", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 26 red balls and 22 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(48,12).\n3) Favorable samples: choose k reds and n−k blues: C(26,3)·C(22,9).\n4) Compute totals: C(48,12) = 69668534468, and favorable = 1293292000.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1309000}{70514711}$", "final_answer": "\\frac{1309000}{70514711}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "be5e34ee16c49b66", "problem_id": "1236b42ead0a0b09", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_4=15$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=15 and k=4.\n4) Compute C(18,3) = 816.\nFinal answer: $816$", "final_answer": "816", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff25b5e7517cb989", "problem_id": "1236b42ead0a0b09", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_4=15$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=15 and k=4.\nFinal answer: $816$", "final_answer": "816", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "988acd66beb896fe", "problem_id": "e6d50ef556d31101", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 20 red balls and 19 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(39,12).\nFavorable samples: choose k reds and n−k blues: C(20,3)·C(19,9).\nCompute totals: C(39,12) = 3910797436, and favorable = 105310920.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{6270}{232841}$.\nFinal answer: $\\frac{6270}{232841}$", "final_answer": "\\frac{6270}{232841}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8edaac3db17390f7", "problem_id": "e6d50ef556d31101", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "An urn has 20 red balls and 19 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(39,12).\n3) Favorable samples: choose k reds and n−k blues: C(20,3)·C(19,9).\n4) Compute totals: C(39,12) = 3910797436, and favorable = 105310920.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{6270}{232841}$", "final_answer": "\\frac{6270}{232841}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "31040acfa435a777", "problem_id": "6d46e9c1f1a4de7e", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 21 times, what is the probability of exactly 6 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=21, k=6, p=$\\frac{1}{5}$ and simplify.\nHere C(21,6) = 54264.\nCheck: expression simplifies to $\\frac{58265526337536}{476837158203125}$.\nFinal answer: $\\frac{58265526337536}{476837158203125}$", "final_answer": "\\frac{58265526337536}{476837158203125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "73e0671ee83b8fae", "problem_id": "6d46e9c1f1a4de7e", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 21 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=21, k=6, p=$\\frac{1}{5}$ and simplify.\n4) Here C(21,6) = 54264.\nFinal answer: $\\frac{58265526337536}{476837158203125}$", "final_answer": "\\frac{58265526337536}{476837158203125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dd061a88051d84c5", "problem_id": "ca7dbcaa083c4fbe", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A store sells a backpack for $26. It is discounted by 35%. You buy 6 backpacks. What is the total cost?", "output": "1) Discounted price = 26 × (1 − 35/100) = 26 × 65/100 = 169/10.\n2) Subtotal = 6 × 169/10 = 507/5.\nFinal answer: $\\frac{507}{5}$", "final_answer": "\\frac{507}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "24cf900af35fa2e2", "problem_id": "ca7dbcaa083c4fbe", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A store sells a backpack for $26. It is discounted by 35%. You buy 6 backpacks. What is the total cost?", "output": "Discounted price = 26 × (1 − 35/100) = 26 × 65/100 = 169/10.\nSubtotal = 6 × 169/10 = 507/5.\nCheck: recomputing the formula yields 507/5.\nFinal answer: $\\frac{507}{5}$", "final_answer": "\\frac{507}{5}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ce9b870dcf8da0b", "problem_id": "5d46e263bdd9dd9f", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 10 red balls and 16 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(26,4).\n3) Favorable samples: choose k reds and n−k blues: C(10,2)·C(16,2).\n4) Compute totals: C(26,4) = 14950, and favorable = 5400.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{108}{299}$", "final_answer": "\\frac{108}{299}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c93140683622d9ea", "problem_id": "5d46e263bdd9dd9f", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 10 red balls and 16 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(26,4).\nFavorable samples: choose k reds and n−k blues: C(10,2)·C(16,2).\nCompute totals: C(26,4) = 14950, and favorable = 5400.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{108}{299}$.\nFinal answer: $\\frac{108}{299}$", "final_answer": "\\frac{108}{299}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7ca73ff38f243b5f", "problem_id": "e2ff23ce63d6114b", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=26$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=26 and k=7.\n4) Compute C(32,6) = 906192.\nFinal answer: $906192$", "final_answer": "906192", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a41ca13171bb18bc", "problem_id": "e2ff23ce63d6114b", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=26$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=26 and k=7.\nCompute C(32,6) = 906192.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $906192$", "final_answer": "906192", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "08ee979d8a7d7c1a", "problem_id": "71de88d71798ed0c", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡3 (mod 7), x≡4 (mod 9). Give the solution modulo 63.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 63.\n3) Compute the CRT solution to get x ≡ 31 (mod 63).\nFinal answer: $x\\equiv 31\\pmod{63}$", "final_answer": "x\\equiv 31\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0c50060bb0235c58", "problem_id": "71de88d71798ed0c", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡3 (mod 7), x≡4 (mod 9). Give the solution modulo 63.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 63.\nCompute the CRT solution to get x ≡ 31 (mod 63).\nCheck: 31 mod 7=3 and mod 9=4.\nFinal answer: $x\\equiv 31\\pmod{63}$", "final_answer": "x\\equiv 31\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a6ffc5496dedf91d", "problem_id": "0a9fdca573340386", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the linear system:\n$5 x + 5 y$ = $-45$\n$- 4 x - 5 y$ = $44$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $- 20 x - 20 y$ = $180$.\n3) Multiply the second equation by $5$: $- 20 x - 25 y$ = $220$.\n4) Subtract to eliminate x: $5 y$ = $-40$.\n5) Solve for y: y = $-8$.\n6) Substitute back to find x: x = $-1$.\nFinal answer: $(x,y)=\\left(-1, -8\\right)$", "final_answer": "(x,y)=\\left(-1, -8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "407a7c52f3cb0f63", "problem_id": "0a9fdca573340386", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$5 x + 5 y$ = $-45$\n$- 4 x - 5 y$ = $44$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $- 20 x - 20 y$ = $180$.\nMultiply the second equation by $5$: $- 20 x - 25 y$ = $220$.\nSubtract to eliminate x: $5 y$ = $-40$.\nSolve for y: y = $-8$.\nSubstitute back to find x: x = $-1$.\nCheck: plug (x,y)=(-1,-8) into both equations.\nFinal answer: $(x,y)=\\left(-1, -8\\right)$", "final_answer": "(x,y)=\\left(-1, -8\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cc287e493161372f", "problem_id": "22021fc6e7c53782", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Find the equation of the tangent line to $y=x^{3} - 3 x^{2} + x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=-2$.", "output": "1) Differentiate: f'(x) = $3 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 1$.\n2) Slope at x=-2: m = f'(-2) = $2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e} + 25$.\n3) Point on curve: (-2, f(-2)) = (-2, $-21 + \\cos{\\left(4 \\right)} + e^{-1}$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -21 + \\cos{\\left(4 \\right)} + e^{-1} = 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e} + 25(x − -2).\n6) Optionally expand: y = $2 x \\sin{\\left(4 \\right)} + \\frac{x}{2 e} + 25 x + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 29$.\nFinal answer: $y = 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 25\\cdot x + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 29$", "final_answer": "y = 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 25\\cdot x + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 29", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9fa100caf875e867", "problem_id": "22021fc6e7c53782", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the equation of the tangent line to $y=x^{3} - 3 x^{2} + x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)} + 1$ at $x=-2$.", "output": "Differentiate: f'(x) = $3 x^{2} - 6 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} + 1$.\nSlope at x=-2: m = f'(-2) = $2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e} + 25$.\nPoint on curve: (-2, f(-2)) = (-2, $-21 + \\cos{\\left(4 \\right)} + e^{-1}$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − -21 + \\cos{\\left(4 \\right)} + e^{-1} = 2 \\sin{\\left(4 \\right)} + \\frac{1}{2 e} + 25(x − -2).\nOptionally expand: y = $2 x \\sin{\\left(4 \\right)} + \\frac{x}{2 e} + 25 x + 4 \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 29$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 25\\cdot x + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 29$", "final_answer": "y = 2\\cdot x\\cdot \\sin{\\left(4 \\right)} + \\frac{x}{2\\cdot e} + 25\\cdot x + 4\\cdot \\sin{\\left(4 \\right)} + \\cos{\\left(4 \\right)} + \\frac{2}{e} + 29", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0f3ed21c1024dca6", "problem_id": "b64424e706c75bcf", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the congruences: x≡8 (mod 9), x≡3 (mod 8). Give the solution modulo 72.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 72.\nCompute the CRT solution to get x ≡ 35 (mod 72).\nFinal answer: $x\\equiv 35\\pmod{72}$", "final_answer": "x\\equiv 35\\pmod{72}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b543097744dc83f6", "problem_id": "b64424e706c75bcf", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡8 (mod 9), x≡3 (mod 8). Give the solution modulo 72.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 72.\n3) Compute the CRT solution to get x ≡ 35 (mod 72).\nFinal answer: $x\\equiv 35\\pmod{72}$", "final_answer": "x\\equiv 35\\pmod{72}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c7b6d996347affef", "problem_id": "c12a721133905493", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Two friends split $1484 in the ratio 5:9. How much does each receive?", "output": "1) Total ratio parts = 5+9 = 14.\n2) Each part = 1484 ÷ 14 = 106.\n3) First share = 5 × 106 = 530.\n4) Second share = 9 × 106 = 954.\nFinal answer: $954$", "final_answer": "954", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e930704f85661bfe", "problem_id": "c12a721133905493", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Two friends split $1484 in the ratio 5:9. How much does each receive?", "output": "Total ratio parts = 5+9 = 14.\nEach part = 1484 ÷ 14 = 106.\nFirst share = 5 × 106 = 530.\nSecond share = 9 × 106 = 954.\nCheck: 530+954=1484.\nFinal answer: $954$", "final_answer": "954", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "48f4020b93164e89", "problem_id": "b3eaa0d106c0ab6e", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} - 3 x^{3} + 2 x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $- 3 x^{3} + 2 x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\n2) Evaluate at the bounds: F(\\pi) and F(0).\n3) Subtract: F(\\pi) − F(0) = $\\frac{\\pi \\left(- 9 \\pi^{3} - 24 \\pi - 36 + 8 \\pi^{2}\\right)}{12}$.\n4) Simplify the result.\nFinal answer: $\\frac{\\pi\\cdot \\left(- 9\\cdot \\pi^{3} - 24\\cdot \\pi - 36 + 8\\cdot \\pi^{2}\\right)}{12}$", "final_answer": "\\frac{\\pi\\cdot \\left(- 9\\cdot \\pi^{3} - 24\\cdot \\pi - 36 + 8\\cdot \\pi^{2}\\right)}{12}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec25eb8ac7c59c0a", "problem_id": "b3eaa0d106c0ab6e", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\pi} - 3 x^{3} + 2 x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $- 3 x^{3} + 2 x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\nEvaluate at the bounds: F(\\pi) and F(0).\nSubtract: F(\\pi) − F(0) = $\\frac{\\pi \\left(- 9 \\pi^{3} - 24 \\pi - 36 + 8 \\pi^{2}\\right)}{12}$.\nFinal answer: $\\frac{\\pi\\cdot \\left(- 9\\cdot \\pi^{3} - 24\\cdot \\pi - 36 + 8\\cdot \\pi^{2}\\right)}{12}$", "final_answer": "\\frac{\\pi\\cdot \\left(- 9\\cdot \\pi^{3} - 24\\cdot \\pi - 36 + 8\\cdot \\pi^{2}\\right)}{12}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1092962a317fa933", "problem_id": "30dc50f402258b40", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{11} x + \\frac{29 x}{5} - \\frac{6}{5} + \\frac{\\sqrt{11}}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$\\frac{29}{5} - \\sqrt{11}$, c=$- \\frac{6}{5} + \\frac{\\sqrt{11}}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{1236}{25} - \\frac{62 \\sqrt{11}}{5}$, so √Δ = $\\sqrt{\\frac{1236}{25} - \\frac{62 \\sqrt{11}}{5}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $- \\frac{29}{10} + \\frac{\\sqrt{1236 - 310 \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}$.\n7) x2 = $- \\frac{29}{10} - \\frac{\\sqrt{1236 - 310 \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{- \\frac{29}{10} + \\frac{\\sqrt{1236 - 310\\cdot \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}, - \\frac{29}{10} - \\frac{\\sqrt{1236 - 310\\cdot \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{29}{10} + \\frac{\\sqrt{1236 - 310\\cdot \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}, - \\frac{29}{10} - \\frac{\\sqrt{1236 - 310\\cdot \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "698cc0bb11328cbc", "problem_id": "30dc50f402258b40", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\sqrt{11} x + \\frac{29 x}{5} - \\frac{6}{5} + \\frac{\\sqrt{11}}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$\\frac{29}{5} - \\sqrt{11}$, c=$- \\frac{6}{5} + \\frac{\\sqrt{11}}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{1236}{25} - \\frac{62 \\sqrt{11}}{5}$, so √Δ = $\\sqrt{\\frac{1236}{25} - \\frac{62 \\sqrt{11}}{5}}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $- \\frac{29}{10} + \\frac{\\sqrt{1236 - 310 \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}$.\nx2 = $- \\frac{29}{10} - \\frac{\\sqrt{1236 - 310 \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{- \\frac{29}{10} + \\frac{\\sqrt{1236 - 310\\cdot \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}, - \\frac{29}{10} - \\frac{\\sqrt{1236 - 310\\cdot \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}\\}$", "final_answer": "x \\in \\{- \\frac{29}{10} + \\frac{\\sqrt{1236 - 310\\cdot \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}, - \\frac{29}{10} - \\frac{\\sqrt{1236 - 310\\cdot \\sqrt{11}}}{10} + \\frac{\\sqrt{11}}{2}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d4ce10d2edc4358d", "problem_id": "5e4349ae5ee20369", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "You want 60 liters of a 25% salt solution. You have a 20% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 60−x liters is 70% solution.\nSalt equation: 20x + 70(60−x) = 25·60.\n(20-70)x = (25-70)·60 ⇒ x = 54.\nThen 60−x = 6.\nCheck: 20·54 + 70·6 = 25·60.\nFinal answer: $54 L of 20% and 6 L of 70%$", "final_answer": "54 L of 20% and 6 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a39dd7f9fe4cbae2", "problem_id": "eb8cba1f44cc36a0", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- x^{2} - 2 x + \\left(3 x^{2} + x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{3 x^{2} + x - 4}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{3 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(3 x^{2} + x\\right) - 4}) = \\frac{3 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x^{2} + x - 4}$.\n3) So $f'(x) = \\frac{9 x^{4} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} + 12 x^{3} - 46 x^{2} - 16 x + 32} + \\frac{18 x^{4} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} + 12 x^{3} - 46 x^{2} - 16 x + 32} + \\frac{6 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} + 12 x^{3} - 46 x^{2} - 16 x + 32} + \\frac{12 x^{3} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} + 12 x^{3} - 46 x^{2} - 16 x + 32} + \\frac{18 x^{3} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{4} + 6 x^{3} - 23 x^{2} - 8 x + 16} - \\frac{23 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} + 12 x^{3} - 46 x^{2} - 16 x + 32} - \\frac{46 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} + 12 x^{3} - 46 x^{2} - 16 x + 32} + \\frac{10 x^{2}}{18 x^{4} + 12 x^{3} - 46 x^{2} - 16 x + 32} + \\frac{3 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{4} + 6 x^{3} - 23 x^{2} - 8 x + 16} - \\frac{3 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x^{2} + 2 x - 8} + \\frac{3 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x^{2} + x - 4} - \\frac{3 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(3 x^{2} + x\\right) - 4} - \\frac{8 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} + 12 x^{3} - 46 x^{2} - 16 x + 32} - \\frac{16 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} + 12 x^{3} - 46 x^{2} - 16 x + 32} + \\frac{40 x}{18 x^{4} + 12 x^{3} - 46 x^{2} - 16 x + 32} - \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(3 x^{2} + x\\right) - 4} + \\frac{16 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{4} + 12 x^{3} - 46 x^{2} - 16 x + 32} + \\frac{32 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{4} + 12 x^{3} - 46 x^{2} - 16 x + 32} + \\frac{20}{18 x^{4} + 12 x^{3} - 46 x^{2} - 16 x + 32}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{3 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(3 x^{2} + x\\right) - 4}) = \\frac{3 x^{2} \\left(- 6 x - 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(\\left(3 x^{2} + x\\right) - 4\\right)^{2}} + \\frac{3 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(\\left(3 x^{2} + x\\right) - 4\\right)} + \\frac{3 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{\\left(3 x^{2} + x\\right) - 4} + \\frac{6 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(3 x^{2} + x\\right) - 4}$.\nTherefore the correct derivative is $f'(x) = \\frac{2 \\left(- 6 x - 1\\right) \\left(- x^{2} - 2 x + \\left(x \\left(3 x + 1\\right) - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(x \\left(3 x + 1\\right) - 4\\right) \\left(- 4 x + 2 \\left(6 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(3 x + 1\\right) - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(3 x + 1\\right) - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 4\\right)}{2 \\left(x \\left(3 x + 1\\right) - 4\\right)^{2}}$.\nFinal answer: $\\frac{2 \\left(- 6 x - 1\\right) \\left(- x^{2} - 2 x + \\left(x \\left(3 x + 1\\right) - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(x \\left(3 x + 1\\right) - 4\\right) \\left(- 4 x + 2 \\left(6 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(3 x + 1\\right) - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(3 x + 1\\right) - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 4\\right)}{2 \\left(x \\left(3 x + 1\\right) - 4\\right)^{2}}$", "final_answer": "\\frac{2 \\left(- 6 x - 1\\right) \\left(- x^{2} - 2 x + \\left(x \\left(3 x + 1\\right) - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2\\right) + \\left(x \\left(3 x + 1\\right) - 4\\right) \\left(- 4 x + 2 \\left(6 x + 1\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + \\left(x \\left(3 x + 1\\right) - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x \\left(3 x + 1\\right) - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} - 4\\right)}{2 \\left(x \\left(3 x + 1\\right) - 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "6fd18cf7a63cdd7a", "problem_id": "b6ffdb3b77c97b2d", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the linear system:\n$- 4 x - 7 y$ = $26$\n$6 x + 5 y$ = $-6$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $6$: $- 24 x - 42 y$ = $156$.\nMultiply the second equation by $-4$: $- 24 x - 20 y$ = $24$.\nSubtract to eliminate x: $- 22 y$ = $132$.\nSolve for y: y = $-6$.\nSubstitute back to find x: x = $4$.\nCheck: plug (x,y)=(4,-6) into both equations.\nFinal answer: $(x,y)=\\left(4, -6\\right)$", "final_answer": "(x,y)=\\left(4, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a7539f5f9da16ae8", "problem_id": "b6ffdb3b77c97b2d", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the linear system:\n$- 4 x - 7 y$ = $26$\n$6 x + 5 y$ = $-6$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $6$: $- 24 x - 42 y$ = $156$.\n3) Multiply the second equation by $-4$: $- 24 x - 20 y$ = $24$.\n4) Subtract to eliminate x: $- 22 y$ = $132$.\n5) Solve for y: y = $-6$.\n6) Substitute back to find x: x = $4$.\nFinal answer: $(x,y)=\\left(4, -6\\right)$", "final_answer": "(x,y)=\\left(4, -6\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "14146e314c15ffb2", "problem_id": "b6919d05747282d0", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 3 x^{2} + 5 x + 11$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{2} + 4 x + 11.\n3) Then the limit is 31.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 33.\nFinal answer: $33$", "final_answer": "33", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5aca9c1fc763e8c3", "problem_id": "1a0e20276e40c31f", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "An urn has 18 red balls and 10 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(28,7).\n3) Favorable samples: choose k reds and n−k blues: C(18,1)·C(10,6).\n4) Compute totals: C(28,7) = 1184040, and favorable = 3780.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{21}{6578}$", "final_answer": "\\frac{21}{6578}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0cfd5bac0c80696d", "problem_id": "105923339bb65bff", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$2 y$ = $-6$\n$- 4 x - 10 y$ = $30$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-4$: $- 8 y$ = $24$.\n3) Multiply the second equation by $0$: $0$ = $0$.\n4) Subtract to eliminate x: $- 8 y$ = $24$.\n5) Solve for y: y = $-3$.\n6) Substitute back to find x: x = $0$.\nFinal answer: $(x,y)=\\left(0, -3\\right)$", "final_answer": "(x,y)=\\left(0, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "911ab67bdcf80fa6", "problem_id": "105923339bb65bff", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$2 y$ = $-6$\n$- 4 x - 10 y$ = $30$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-4$: $- 8 y$ = $24$.\nMultiply the second equation by $0$: $0$ = $0$.\nSubtract to eliminate x: $- 8 y$ = $24$.\nSolve for y: y = $-3$.\nSubstitute back to find x: x = $0$.\nCheck: plug (x,y)=(0,-3) into both equations.\nFinal answer: $(x,y)=\\left(0, -3\\right)$", "final_answer": "(x,y)=\\left(0, -3\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "011bd4b89122e7a8", "problem_id": "6ef66b574aa6386b", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_7=26$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_7=12.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=12 and k=7.\n4) Compute C(18,6) = 18564.\nFinal answer: $18564$", "final_answer": "18564", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6ba628909fdbb932", "problem_id": "6ef66b574aa6386b", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_7=26$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_7=12.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=12 and k=7.\nCompute C(18,6) = 18564.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $18564$", "final_answer": "18564", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "197905bbca890fb6", "problem_id": "62e8dfbac604fd61", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡0 (mod 4), x≡8 (mod 11). Give the solution modulo 44.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 44.\n3) Compute the CRT solution to get x ≡ 8 (mod 44).\nFinal answer: $x\\equiv 8\\pmod{44}$", "final_answer": "x\\equiv 8\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8dd8f5ae272420fd", "problem_id": "62e8dfbac604fd61", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡0 (mod 4), x≡8 (mod 11). Give the solution modulo 44.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 44.\nCompute the CRT solution to get x ≡ 8 (mod 44).\nCheck: 8 mod 4=0 and mod 11=8.\nFinal answer: $x\\equiv 8\\pmod{44}$", "final_answer": "x\\equiv 8\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4b7b4b1c317da52a", "problem_id": "1456cb9eaa26cb09", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "An urn has 8 red balls and 16 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(24,6).\n3) Favorable samples: choose k reds and n−k blues: C(8,3)·C(16,3).\n4) Compute totals: C(24,6) = 134596, and favorable = 31360.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1120}{4807}$", "final_answer": "\\frac{1120}{4807}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4f8592d22c1ce67a", "problem_id": "1456cb9eaa26cb09", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "An urn has 8 red balls and 16 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(24,6).\nFavorable samples: choose k reds and n−k blues: C(8,3)·C(16,3).\nFinal answer: $\\frac{1120}{4807}$", "final_answer": "\\frac{1120}{4807}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5f8d082dbfbc57fb", "problem_id": "d9e1345adbd94eec", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} 2 x^{3} + 3 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $2 x^{3} + 3 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\n3) Subtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{15 \\pi^{4}}{32} - \\frac{9 \\pi}{2} - \\frac{9 \\pi^{2}}{8} + 5 + \\frac{9 \\pi^{3}}{8}$.\n4) Simplify the result.\nFinal answer: $- \\frac{15\\cdot \\pi^{4}}{32} - \\frac{9\\cdot \\pi}{2} - \\frac{9\\cdot \\pi^{2}}{8} + 5 + \\frac{9\\cdot \\pi^{3}}{8}$", "final_answer": "- \\frac{15\\cdot \\pi^{4}}{32} - \\frac{9\\cdot \\pi}{2} - \\frac{9\\cdot \\pi^{2}}{8} + 5 + \\frac{9\\cdot \\pi^{3}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1291bde46059554c", "problem_id": "d9e1345adbd94eec", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} 2 x^{3} + 3 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $2 x^{3} + 3 x^{2} + 3 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} - 3$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{15 \\pi^{4}}{32} - \\frac{9 \\pi}{2} - \\frac{9 \\pi^{2}}{8} + 5 + \\frac{9 \\pi^{3}}{8}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{15\\cdot \\pi^{4}}{32} - \\frac{9\\cdot \\pi}{2} - \\frac{9\\cdot \\pi^{2}}{8} + 5 + \\frac{9\\cdot \\pi^{3}}{8}$", "final_answer": "- \\frac{15\\cdot \\pi^{4}}{32} - \\frac{9\\cdot \\pi}{2} - \\frac{9\\cdot \\pi^{2}}{8} + 5 + \\frac{9\\cdot \\pi^{3}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5ef18faf4d7871d8", "problem_id": "2aff119ccb94f44f", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nFinal answer: $x^{3}\\cdot \\cos{\\left(x \\right)} + 3\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot x\\cdot \\cos{\\left(x \\right)} - 2\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}$", "final_answer": "x^{3}\\cdot \\cos{\\left(x \\right)} + 3\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot x\\cdot \\cos{\\left(x \\right)} - 2\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e2d72172e8fd309d", "problem_id": "2aff119ccb94f44f", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 3 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $x^{3} \\cos{\\left(x \\right)} + 3 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} + 3 \\cos{\\left(x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $x^{3}\\cdot \\cos{\\left(x \\right)} + 3\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot x\\cdot \\cos{\\left(x \\right)} - 2\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}$", "final_answer": "x^{3}\\cdot \\cos{\\left(x \\right)} + 3\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - 2\\cdot x^{2}\\cdot \\cos{\\left(x \\right)} - 4\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot x\\cdot \\cos{\\left(x \\right)} - 2\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)} + 3\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e6edb4b05bb14fd9", "problem_id": "4166a418f2c3a4dd", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $x^{2}\\cdot \\cos{\\left(x \\right)} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} - x\\cdot \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "x^{2}\\cdot \\cos{\\left(x \\right)} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} - x\\cdot \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "75ec86a178ab238f", "problem_id": "4166a418f2c3a4dd", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.", "output": "Let f(x) = $x^{2} \\sin{\\left(x \\right)} - x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $x^{2}\\cdot \\cos{\\left(x \\right)} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} - x\\cdot \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "x^{2}\\cdot \\cos{\\left(x \\right)} + 2\\cdot x\\cdot \\sin{\\left(x \\right)} - x\\cdot \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)} - 6\\cdot \\sin{\\left(2\\cdot x \\right)} + 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ac41334d376ca0c4", "problem_id": "f63ddf19f7fb58bf", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the linear system:\n$- 6 x + 2 y$ = $-34$\n$- 5 x - 6 y$ = $-59$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $-5$: $30 x - 10 y$ = $170$.\n3) Multiply the second equation by $-6$: $30 x + 36 y$ = $354$.\n4) Subtract to eliminate x: $- 46 y$ = $-184$.\n5) Solve for y: y = $4$.\n6) Substitute back to find x: x = $7$.\nFinal answer: $(x,y)=\\left(7, 4\\right)$", "final_answer": "(x,y)=\\left(7, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0d4da43710e95379", "problem_id": "f63ddf19f7fb58bf", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$- 6 x + 2 y$ = $-34$\n$- 5 x - 6 y$ = $-59$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-5$: $30 x - 10 y$ = $170$.\nMultiply the second equation by $-6$: $30 x + 36 y$ = $354$.\nSubtract to eliminate x: $- 46 y$ = $-184$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $7$.\nCheck: plug (x,y)=(7,4) into both equations.\nFinal answer: $(x,y)=\\left(7, 4\\right)$", "final_answer": "(x,y)=\\left(7, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "73cddc0f5eb2f872", "problem_id": "36bb1f60e0847365", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A cyclist rides at 58 miles per hour for 3/2 hours, then at 23 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 58 × 3/2 = 87 miles.\nSecond leg distance = 23 × 2 = 46 miles.\nTotal distance = 87 + 46 = 133 miles.\nCheck: adding both legs gives 133.\nFinal answer: $133$", "final_answer": "133", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d1017325fef190f4", "problem_id": "36bb1f60e0847365", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A cyclist rides at 58 miles per hour for 3/2 hours, then at 23 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 58 × 3/2 = 87 miles.\n2) Second leg distance = 23 × 2 = 46 miles.\n3) Total distance = 87 + 46 = 133 miles.\nFinal answer: $133$", "final_answer": "133", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "75ca5377fba3a5f6", "problem_id": "e4c4822235dd393d", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Two friends split $114 in the ratio 1:2. How much does each receive?", "output": "1) Total ratio parts = 1+2 = 3.\n2) Each part = 114 ÷ 3 = 38.\n3) First share = 1 × 38 = 38.\n4) Second share = 2 × 38 = 76.\nFinal answer: $76$", "final_answer": "76", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ce4cc7f3b4b1b65", "problem_id": "e4c4822235dd393d", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Two friends split $114 in the ratio 1:2. How much does each receive?", "output": "Total ratio parts = 1+2 = 3.\nEach part = 114 ÷ 3 = 38.\nFirst share = 1 × 38 = 38.\nSecond share = 2 × 38 = 76.\nCheck: 38+76=114.\nFinal answer: $76$", "final_answer": "76", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "023811cad09e4fba", "problem_id": "20840d55a224a699", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 13 times, what is the probability of exactly 6 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=13, k=6, p=$\\frac{1}{5}$ and simplify.\nHere C(13,6) = 1716.\nCheck: expression simplifies to $\\frac{28114944}{1220703125}$.\nFinal answer: $\\frac{28114944}{1220703125}$", "final_answer": "\\frac{28114944}{1220703125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "87da5d5dd0693928", "problem_id": "20840d55a224a699", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 13 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=13, k=6, p=$\\frac{1}{5}$ and simplify.\n4) Here C(13,6) = 1716.\nFinal answer: $\\frac{28114944}{1220703125}$", "final_answer": "\\frac{28114944}{1220703125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2c63347164f8f1dc", "problem_id": "2ff973d4c0995a08", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Worker A finishes a job in 9 hours and Worker B finishes it in 22 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/9 job/hour. B's rate = 1/22 job/hour.\n2) Combined rate = 1/9 + 1/22 = 31/198 job/hour.\n3) Time = 1 ÷ (combined rate) = 198/31 hours.\nFinal answer: $\\frac{198}{31}$", "final_answer": "\\frac{198}{31}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "79c8720f084f319e", "problem_id": "02f5ba068679179f", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(- x^{3} + 2 x^{2} - 2 x - 2\\right) \\cos{\\left(2 x - 2 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 2 \\cos{\\left(2 x - 2 \\right)}) = 2 \\sin{\\left(2 x - 2 \\right)}$.\n3) So $f'(x) = 2 x^{3} \\sin{\\left(2 x - 2 \\right)} - 4 x^{2} \\sin{\\left(2 x - 2 \\right)} - 3 x^{2} \\cos{\\left(2 x - 2 \\right)} + x e^{x} + 4 x \\sin{\\left(2 x - 2 \\right)} + 4 x \\cos{\\left(2 x - 2 \\right)} - 2 e^{x} + 2 \\sin{\\left(2 x - 2 \\right)} - 2 \\cos{\\left(2 x - 2 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 2 \\cos{\\left(2 x - 2 \\right)}) = 4 \\sin{\\left(2 x - 2 \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(x - 3\\right) e^{x} - 2 \\left(x \\left(x \\left(2 - x\\right) - 2\\right) - 2\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(- 3 x^{2} + 4 x - 2\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}$.\nFinal answer: $\\left(x - 3\\right) e^{x} - 2 \\left(x \\left(x \\left(2 - x\\right) - 2\\right) - 2\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(- 3 x^{2} + 4 x - 2\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}$", "final_answer": "\\left(x - 3\\right) e^{x} - 2 \\left(x \\left(x \\left(2 - x\\right) - 2\\right) - 2\\right) \\sin{\\left(2 x - 2 \\right)} + \\left(- 3 x^{2} + 4 x - 2\\right) \\cos{\\left(2 x - 2 \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7da8fd1993e08e74", "problem_id": "09a1b48be6d0b0c0", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the equation of the tangent line to $y=3 x^{3} + 2 x^{2} - 3 x + e^{\\frac{x}{2}} + \\cos{\\left(2 x \\right)}$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $9 x^{2} + 4 x + \\frac{e^{\\frac{x}{2}}}{2} - 2 \\sin{\\left(2 x \\right)} - 3$.\n2) Slope at x=-1: m = f'(-1) = $\\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)} + 2$.\n3) Point on curve: (-1, f(-1)) = (-1, $\\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 2$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − \\cos{\\left(2 \\right)} + e^{- \\frac{1}{2}} + 2 = \\frac{1}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)} + 2(x − -1).\n6) Optionally expand: y = $\\frac{x}{2 e^{\\frac{1}{2}}} + 2 x \\sin{\\left(2 \\right)} + 2 x + \\cos{\\left(2 \\right)} + \\frac{3}{2 e^{\\frac{1}{2}}} + 2 \\sin{\\left(2 \\right)} + 4$.\nFinal answer: $y = \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + 2\\cdot x + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)} + 4$", "final_answer": "y = \\frac{x}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot x\\cdot \\sin{\\left(2 \\right)} + 2\\cdot x + \\cos{\\left(2 \\right)} + \\frac{3}{2\\cdot e^{\\frac{1}{2}}} + 2\\cdot \\sin{\\left(2 \\right)} + 4", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72911bd822a00fe0", "problem_id": "3ecb21c9d55ee786", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 2 x^{3} + 3 x^{2} + 2 x + \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 3}{3 x - 5}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 5}) = \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 5}$.\n3) So $f'(x) = - \\frac{24 x^{3}}{18 x^{2} - 60 x + 50} + \\frac{9 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} - 60 x + 50} + \\frac{18 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} - 60 x + 50} + \\frac{78 x^{2}}{18 x^{2} - 60 x + 50} - \\frac{30 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} - 60 x + 50} - \\frac{60 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} - 60 x + 50} - \\frac{60 x}{18 x^{2} - 60 x + 50} + \\frac{9 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{2} - 30 x + 25} - \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{6 x - 10} + \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 5} - \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 5} + \\frac{25 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} - 60 x + 50} + \\frac{50 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} - 60 x + 50} - \\frac{2}{18 x^{2} - 60 x + 50} - \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 5}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 5}) = \\frac{3 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 \\left(3 x - 5\\right)} + \\frac{3 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x - 5} - \\frac{9 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(3 x - 5\\right)^{2}} + \\frac{3 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x - 5}$.\nTherefore the correct derivative is $f'(x) = \\frac{12 x^{3} - 18 x^{2} - 12 x + \\left(3 x - 5\\right) \\left(- 12 x^{2} + 12 x + \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 6 \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 18}{2 \\left(3 x - 5\\right)^{2}}$.\nFinal answer: $\\frac{12 x^{3} - 18 x^{2} - 12 x + \\left(3 x - 5\\right) \\left(- 12 x^{2} + 12 x + \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 6 \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 18}{2 \\left(3 x - 5\\right)^{2}}$", "final_answer": "\\frac{12 x^{3} - 18 x^{2} - 12 x + \\left(3 x - 5\\right) \\left(- 12 x^{2} + 12 x + \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4\\right) - 6 \\left(3 x - 5\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 18}{2 \\left(3 x - 5\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "425402c7e07c7eab", "problem_id": "e9ea5739ee57406b", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_5=39$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_5=24.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=24 and k=5.\n4) Compute C(28,4) = 20475.\nFinal answer: $20475$", "final_answer": "20475", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9342384f1c3c8d61", "problem_id": "e9ea5739ee57406b", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_5=39$?", "output": "Let y_i=x_i-3, then y_i≥0 and y_1+…+y_5=24.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=24 and k=5.\nCompute C(28,4) = 20475.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $20475$", "final_answer": "20475", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a67e07c5b6a157e3", "problem_id": "497f04bad5ab4efa", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the linear system:\n$6 x + 7 y$ = $-71$\n$- 8 x - 5 y$ = $73$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-8$: $- 48 x - 56 y$ = $568$.\nMultiply the second equation by $6$: $- 48 x - 30 y$ = $438$.\nSubtract to eliminate x: $- 26 y$ = $130$.\nSolve for y: y = $-5$.\nSubstitute back to find x: x = $-6$.\nCheck: plug (x,y)=(-6,-5) into both equations.\nFinal answer: $(x,y)=\\left(-6, -5\\right)$", "final_answer": "(x,y)=\\left(-6, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c63766824903b6f4", "problem_id": "497f04bad5ab4efa", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the linear system:\n$6 x + 7 y$ = $-71$\n$- 8 x - 5 y$ = $73$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-8$: $- 48 x - 56 y$ = $568$.\nMultiply the second equation by $6$: $- 48 x - 30 y$ = $438$.\nFinal answer: $(x,y)=\\left(-6, -5\\right)$", "final_answer": "(x,y)=\\left(-6, -5\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f962a9fddc8ace74", "problem_id": "46f865a9a5db1e65", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "An urn has 18 red balls and 23 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(41,9).\n3) Favorable samples: choose k reds and n−k blues: C(18,4)·C(23,5).\n4) Compute totals: C(41,9) = 350343565, and favorable = 102965940.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{5796}{19721}$", "final_answer": "\\frac{5796}{19721}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f9333b4ba5b172f4", "problem_id": "bda48a2c50db7cae", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=x^{2} \\sin{\\left(x \\right)} - 3 x \\sin{\\left(x \\right)} - \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\sin{\\left(x \\right)}) = - \\cos{\\left(x \\right)}$.\n3) So $f'(x) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\sin{\\left(x \\right)}) = - \\cos{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$.\nFinal answer: $x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}$", "final_answer": "x^{2} \\cos{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} - \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "025fa9753d5138e6", "problem_id": "bd9839cba0c84ab9", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "An urn has 15 red balls and 15 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(30,8).\n3) Favorable samples: choose k reds and n−k blues: C(15,4)·C(15,4).\n4) Compute totals: C(30,8) = 5852925, and favorable = 1863225.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{637}{2001}$", "final_answer": "\\frac{637}{2001}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db9c858ead4a03c8", "problem_id": "bd9839cba0c84ab9", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "An urn has 15 red balls and 15 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(30,8).\nFavorable samples: choose k reds and n−k blues: C(15,4)·C(15,4).\nCompute totals: C(30,8) = 5852925, and favorable = 1863225.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{637}{2001}$.\nFinal answer: $\\frac{637}{2001}$", "final_answer": "\\frac{637}{2001}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ba42377e34c79ac7", "problem_id": "59623104ecfcad50", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "You want 18 liters of a 25% salt solution. You have a 20% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 18−x liters is 60% solution.\nSalt equation: 20x + 60(18−x) = 25·18.\n(20-60)x = (25-60)·18 ⇒ x = 63/4.\nFinal answer: $63/4 L of 20% and 9/4 L of 60%$", "final_answer": "63/4 L of 20% and 9/4 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ab94299882370e5e", "problem_id": "59623104ecfcad50", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "You want 18 liters of a 25% salt solution. You have a 20% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 18−x liters is 60% solution.\n2) Salt equation: 20x + 60(18−x) = 25·18.\n3) (20-60)x = (25-60)·18 ⇒ x = 63/4.\n4) Then 18−x = 9/4.\nFinal answer: $63/4 L of 20% and 9/4 L of 60%$", "final_answer": "63/4 L of 20% and 9/4 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d7eef107663e908c", "problem_id": "0ca74145d998a6ac", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 7 red balls and 13 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(20,4).\nFavorable samples: choose k reds and n−k blues: C(7,2)·C(13,2).\nCompute totals: C(20,4) = 4845, and favorable = 1638.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{546}{1615}$.\nFinal answer: $\\frac{546}{1615}$", "final_answer": "\\frac{546}{1615}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "984ac6e5476d876a", "problem_id": "0ca74145d998a6ac", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "An urn has 7 red balls and 13 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(20,4).\n3) Favorable samples: choose k reds and n−k blues: C(7,2)·C(13,2).\n4) Compute totals: C(20,4) = 4845, and favorable = 1638.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{546}{1615}$", "final_answer": "\\frac{546}{1615}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "94fb0fee0765d895", "problem_id": "cabd7697d16e382f", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=16$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=16 and k=7.\nCompute C(22,6) = 74613.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $74613$", "final_answer": "74613", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1f37b1ebdd18e379", "problem_id": "cabd7697d16e382f", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=16$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=16 and k=7.\n4) Compute C(22,6) = 74613.\nFinal answer: $74613$", "final_answer": "74613", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3eed66ea57121f43", "problem_id": "bbd90636f13742cb", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} x \\left(2 - 3 x\\right)$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $- 3 x^{3} - x^{2} + 2 x - 2$ and a = -1.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $- 9 x^{2} - 2 x + 2$.\nEvaluate at a: p'(-1) = $-5$.\nCheck: the identity gives $-5$.\nFinal answer: $-5$", "final_answer": "-5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec67044181a3a476", "problem_id": "bbd90636f13742cb", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} x \\left(2 - 3 x\\right)$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $- 3 x^{3} - x^{2} + 2 x - 2$ and a = -1.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $- 9 x^{2} - 2 x + 2$.\n5) Evaluate at a: p'(-1) = $-5$.\nFinal answer: $-5$", "final_answer": "-5", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c69a440d0dd53ab3", "problem_id": "6d6b24d23ff2edb8", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "A cyclist rides at 47 miles per hour for 1/2 hours, then at 40 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 47 × 1/2 = 47/2 miles.\n2) Second leg distance = 40 × 5/2 = 100 miles.\n3) Total distance = 47/2 + 100 = 247/2 miles.\nFinal answer: $\\frac{247}{2}$", "final_answer": "\\frac{247}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "01002533f12952d5", "problem_id": "6d6b24d23ff2edb8", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A cyclist rides at 47 miles per hour for 1/2 hours, then at 40 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 47 × 1/2 = 47/2 miles.\nSecond leg distance = 40 × 5/2 = 100 miles.\nTotal distance = 47/2 + 100 = 247/2 miles.\nCheck: adding both legs gives 247/2.\nFinal answer: $\\frac{247}{2}$", "final_answer": "\\frac{247}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3a957e79b8b5bb1f", "problem_id": "eb6d8be89961404f", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Find the multiplicative inverse of 32 modulo 53.", "output": "1) We want x such that 32x ≡ 1 (mod 53).\n2) Use extended Euclid to find s,t with 32s + 53t = 1.\n3) One Bezout identity is: 32·(5) + 53·(-3) = 1.\n4) Reduce s modulo 53: x ≡ 5 (mod 53).\nFinal answer: $x\\equiv 5\\pmod{53}$", "final_answer": "x\\equiv 5\\pmod{53}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "721046ac0846be79", "problem_id": "eb6d8be89961404f", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the multiplicative inverse of 32 modulo 53.", "output": "We want x such that 32x ≡ 1 (mod 53).\nUse extended Euclid to find s,t with 32s + 53t = 1.\nOne Bezout identity is: 32·(5) + 53·(-3) = 1.\nReduce s modulo 53: x ≡ 5 (mod 53).\nCheck: 32·5 mod 53 = 1.\nFinal answer: $x\\equiv 5\\pmod{53}$", "final_answer": "x\\equiv 5\\pmod{53}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "408d5e7bb700f3de", "problem_id": "914e361e287c2b6a", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Find the multiplicative inverse of 5 modulo 59.", "output": "1) We want x such that 5x ≡ 1 (mod 59).\n2) Use extended Euclid to find s,t with 5s + 59t = 1.\n3) One Bezout identity is: 5·(12) + 59·(-1) = 1.\n4) Reduce s modulo 59: x ≡ 12 (mod 59).\nFinal answer: $x\\equiv 12\\pmod{59}$", "final_answer": "x\\equiv 12\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "805d13e0543f5f63", "problem_id": "914e361e287c2b6a", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the multiplicative inverse of 5 modulo 59.", "output": "We want x such that 5x ≡ 1 (mod 59).\nUse extended Euclid to find s,t with 5s + 59t = 1.\nOne Bezout identity is: 5·(12) + 59·(-1) = 1.\nReduce s modulo 59: x ≡ 12 (mod 59).\nCheck: 5·12 mod 59 = 1.\nFinal answer: $x\\equiv 12\\pmod{59}$", "final_answer": "x\\equiv 12\\pmod{59}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c821c962a2ce892e", "problem_id": "bfc4f3cae26a47cb", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 11 red balls and 12 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 6 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(23,12).\n3) Favorable samples: choose k reds and n−k blues: C(11,6)·C(12,6).\n4) Compute totals: C(23,12) = 1352078, and favorable = 426888.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{30492}{96577}$", "final_answer": "\\frac{30492}{96577}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "221f94ace2ff09fa", "problem_id": "bfc4f3cae26a47cb", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 11 red balls and 12 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 6 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(23,12).\nFavorable samples: choose k reds and n−k blues: C(11,6)·C(12,6).\nCompute totals: C(23,12) = 1352078, and favorable = 426888.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{30492}{96577}$.\nFinal answer: $\\frac{30492}{96577}$", "final_answer": "\\frac{30492}{96577}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5496903a0170c5d0", "problem_id": "820f4fc5773abbde", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 29 red balls and 12 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(41,9).\n3) Favorable samples: choose k reds and n−k blues: C(29,4)·C(12,5).\n4) Compute totals: C(41,9) = 350343565, and favorable = 18810792.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{131544}{2449955}$", "final_answer": "\\frac{131544}{2449955}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "06d5d529c4b9394c", "problem_id": "820f4fc5773abbde", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 29 red balls and 12 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(41,9).\nFavorable samples: choose k reds and n−k blues: C(29,4)·C(12,5).\nCompute totals: C(41,9) = 350343565, and favorable = 18810792.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{131544}{2449955}$.\nFinal answer: $\\frac{131544}{2449955}$", "final_answer": "\\frac{131544}{2449955}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5ca9217726d5122b", "problem_id": "faa2b89452cf7f8b", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $460 in the ratio 6:4. How much does each receive?", "output": "Total ratio parts = 6+4 = 10.\nEach part = 460 ÷ 10 = 46.\nFirst share = 6 × 46 = 276.\nSecond share = 4 × 46 = 184.\nCheck: 276+184=460.\nFinal answer: $184$", "final_answer": "184", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "20fab5a5a2f2a765", "problem_id": "faa2b89452cf7f8b", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Two friends split $460 in the ratio 6:4. How much does each receive?", "output": "$184$", "final_answer": "184", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "39e85093e7325fd6", "problem_id": "5bc9b0efe00e7e36", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Worker A finishes a job in 10 hours and Worker B finishes it in 9 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "A's rate = 1/10 job/hour. B's rate = 1/9 job/hour.\nCombined rate = 1/10 + 1/9 = 19/90 job/hour.\nTime = 1 ÷ (combined rate) = 90/19 hours.\nFinal answer: $\\frac{90}{19}$", "final_answer": "\\frac{90}{19}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "56c49aae24d3cd5d", "problem_id": "5bc9b0efe00e7e36", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Worker A finishes a job in 10 hours and Worker B finishes it in 9 hours. If they work together at constant rates, how long does it take to finish the job?", "output": "1) A's rate = 1/10 job/hour. B's rate = 1/9 job/hour.\n2) Combined rate = 1/10 + 1/9 = 19/90 job/hour.\n3) Time = 1 ÷ (combined rate) = 90/19 hours.\nFinal answer: $\\frac{90}{19}$", "final_answer": "\\frac{90}{19}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "work_rates", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fd0d90c0d44c4168", "problem_id": "34c618045a490225", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int x^{3} + 2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "1) Identify an integration method (term-by-term, substitution, or a known pattern).\n2) Integrand: $x^{3} + 2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 2$.\n3) Compute an antiderivative: F(x) = $\\frac{x^{4}}{4} + \\frac{2 x^{3}}{3} + x^{2} - 2 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\n4) Include the constant of integration C.\nFinal answer: $C + \\frac{x^{4}}{4} + \\frac{2\\cdot x^{3}}{3} + x^{2} - 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{4}}{4} + \\frac{2\\cdot x^{3}}{3} + x^{2} - 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2b2a2bdcd8762490", "problem_id": "34c618045a490225", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Compute the indefinite integral:\n$\\displaystyle \\int x^{3} + 2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 2\\,dx$.", "output": "Identify an integration method (term-by-term, substitution, or a known pattern).\nIntegrand: $x^{3} + 2 x^{2} + 2 x + 2 \\sin{\\left(x \\right)} - 3 \\cos{\\left(2 x \\right)} - 2$.\nCompute an antiderivative: F(x) = $\\frac{x^{4}}{4} + \\frac{2 x^{3}}{3} + x^{2} - 2 x - \\frac{3 \\sin{\\left(2 x \\right)}}{2} - 2 \\cos{\\left(x \\right)}$.\nInclude the constant of integration C.\nCheck: differentiating F(x) gives the integrand.\nFinal answer: $C + \\frac{x^{4}}{4} + \\frac{2\\cdot x^{3}}{3} + x^{2} - 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}$", "final_answer": "C + \\frac{x^{4}}{4} + \\frac{2\\cdot x^{3}}{3} + x^{2} - 2\\cdot x - \\frac{3\\cdot \\sin{\\left(2\\cdot x \\right)}}{2} - 2\\cdot \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "indefinite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e22b9c9d7d7ca822", "problem_id": "d4455acbe7198374", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 16 times, what is the probability of exactly 2 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=16, k=2, p=$\\frac{2}{5}$ and simplify.\nHere C(16,2) = 120.\nCheck: expression simplifies to $\\frac{459165024}{30517578125}$.\nFinal answer: $\\frac{459165024}{30517578125}$", "final_answer": "\\frac{459165024}{30517578125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d6b2bac4af559b4", "problem_id": "d4455acbe7198374", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 16 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=16, k=2, p=$\\frac{2}{5}$ and simplify.\n4) Here C(16,2) = 120.\nFinal answer: $\\frac{459165024}{30517578125}$", "final_answer": "\\frac{459165024}{30517578125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8d60dbfd4347a86f", "problem_id": "3cd5f4f015c4ac78", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} 4 x^{4} - 4 x^{3} + 4 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $4 x^{4} - 4 x^{3} + 4 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2$.\nEvaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\nSubtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{255 \\pi^{4}}{16} - \\frac{15 \\pi^{2}}{4} - 1 + 5 \\pi + \\frac{65 \\pi^{3}}{6} + \\frac{205 \\pi^{5}}{8}$.\nFinal answer: $- \\frac{255\\cdot \\pi^{4}}{16} - \\frac{15\\cdot \\pi^{2}}{4} - 1 + 5\\cdot \\pi + \\frac{65\\cdot \\pi^{3}}{6} + \\frac{205\\cdot \\pi^{5}}{8}$", "final_answer": "- \\frac{255\\cdot \\pi^{4}}{16} - \\frac{15\\cdot \\pi^{2}}{4} - 1 + 5\\cdot \\pi + \\frac{65\\cdot \\pi^{3}}{6} + \\frac{205\\cdot \\pi^{5}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ff47df88524d7434", "problem_id": "3cd5f4f015c4ac78", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\frac{\\pi}{2}}^{2 \\pi} 4 x^{4} - 4 x^{3} + 4 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $4 x^{4} - 4 x^{3} + 4 x^{2} - 2 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)} + 2$.\n2) Evaluate at the bounds: F(2 \\pi) and F(- \\frac{\\pi}{2}).\n3) Subtract: F(2 \\pi) − F(- \\frac{\\pi}{2}) = $- \\frac{255 \\pi^{4}}{16} - \\frac{15 \\pi^{2}}{4} - 1 + 5 \\pi + \\frac{65 \\pi^{3}}{6} + \\frac{205 \\pi^{5}}{8}$.\n4) Simplify the result.\nFinal answer: $- \\frac{255\\cdot \\pi^{4}}{16} - \\frac{15\\cdot \\pi^{2}}{4} - 1 + 5\\cdot \\pi + \\frac{65\\cdot \\pi^{3}}{6} + \\frac{205\\cdot \\pi^{5}}{8}$", "final_answer": "- \\frac{255\\cdot \\pi^{4}}{16} - \\frac{15\\cdot \\pi^{2}}{4} - 1 + 5\\cdot \\pi + \\frac{65\\cdot \\pi^{3}}{6} + \\frac{205\\cdot \\pi^{5}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a53e7f07e3841a93", "problem_id": "c35700347231d7a3", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 7 red balls and 9 blue balls. If 4 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(16,4).\n3) Favorable samples: choose k reds and n−k blues: C(7,2)·C(9,2).\n4) Compute totals: C(16,4) = 1820, and favorable = 756.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{27}{65}$", "final_answer": "\\frac{27}{65}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ec9ff43afc123a0b", "problem_id": "3ef96dfabadc66d8", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 2 x^{3} + 5 x^{2} - 6 x + 8$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{3} + 5 x^{2} - 7 x + 8.\n3) Then the limit is 22.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: 21.\nFinal answer: $21$", "final_answer": "21", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e4be7ffe4268de8e", "problem_id": "855a8eedf256ec62", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x^{2} + 5 x + 7$.", "output": "1) Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\n2) Here p(x) = $x^{3} + 3 x^{2} - 3 x - 3$ and a = 2.\n3) Use the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\n4) Differentiate: p'(x) = $3 x^{2} + 6 x - 3$.\n5) Evaluate at a: p'(2) = $21$.\nFinal answer: $21$", "final_answer": "21", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6c1a53746ee60bd7", "problem_id": "855a8eedf256ec62", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x^{2} + 5 x + 7$.", "output": "Recognize the removable-discontinuity form (p(x)−p(a))/(x−a).\nHere p(x) = $x^{3} + 3 x^{2} - 3 x - 3$ and a = 2.\nUse the identity: lim_{x→a} (p(x)−p(a))/(x−a) = p'(a).\nDifferentiate: p'(x) = $3 x^{2} + 6 x - 3$.\nEvaluate at a: p'(2) = $21$.\nCheck: the identity gives $21$.\nFinal answer: $21$", "final_answer": "21", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e3222730826e33c7", "problem_id": "999358344841fc82", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 2 x - 144$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $2 x^{2} - 2 x - 144$ = $2 \\left(x - 9\\right) \\left(x + 8\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $9$.\n5) From the second factor: x = $-8$.\nFinal answer: $x \\in \\{9, -8\\}$", "final_answer": "x \\in \\{9, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "be85e9298b761ec5", "problem_id": "999358344841fc82", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - 2 x - 144$ = 0.", "output": "Because the roots are simple, factoring is the fastest method.\nFactor the polynomial: $2 x^{2} - 2 x - 144$ = $2 \\left(x - 9\\right) \\left(x + 8\\right)$.\nSet each factor equal to 0 and solve each linear equation.\nFrom the first factor: x = $9$.\nFrom the second factor: x = $-8$.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{9, -8\\}$", "final_answer": "x \\in \\{9, -8\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "de739f11c4bde767", "problem_id": "3c7f7b0b4815aa84", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 13 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=13, k=3, p=$\\frac{2}{5}$ and simplify.\n4) Here C(13,3) = 286.\nFinal answer: $\\frac{135104112}{1220703125}$", "final_answer": "\\frac{135104112}{1220703125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1d9e759ed47acf72", "problem_id": "3c7f7b0b4815aa84", "style": "answer_only", "instruction": "Final answer only.", "input": "A coin lands heads with probability p=$\\frac{2}{5}$. If it is flipped 13 times, what is the probability of exactly 3 heads?", "output": "$\\frac{135104112}{1220703125}$", "final_answer": "\\frac{135104112}{1220703125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9765f502c9d6e47c", "problem_id": "70c52b15d75c61c7", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "An urn has 6 red balls and 11 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(17,7).\n3) Favorable samples: choose k reds and n−k blues: C(6,1)·C(11,6).\n4) Compute totals: C(17,7) = 19448, and favorable = 2772.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{63}{442}$", "final_answer": "\\frac{63}{442}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f4e4c2962f2e64ba", "problem_id": "70c52b15d75c61c7", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 6 red balls and 11 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(17,7).\nFavorable samples: choose k reds and n−k blues: C(6,1)·C(11,6).\nCompute totals: C(17,7) = 19448, and favorable = 2772.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{63}{442}$.\nFinal answer: $\\frac{63}{442}$", "final_answer": "\\frac{63}{442}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "841397939ec3fb2e", "problem_id": "be0c6034c8b6bdd3", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{3} - 4 x^{2} + 2 x$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 3 x^{3}) = - 9 x^{3}$.\n3) So $f'(x) = - 9 x^{3} - 8 x + 2$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 3 x^{3}) = - 9 x^{2}$.\nTherefore the correct derivative is $f'(x) = - 9 x^{2} - 8 x + 2$.\nFinal answer: $- 9 x^{2} - 8 x + 2$", "final_answer": "- 9 x^{2} - 8 x + 2", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "907deffe56b55804", "problem_id": "4405a390abbb894d", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 18 times, what is the probability of exactly 9 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=18, k=9, p=$\\frac{1}{5}$ and simplify.\n4) Here C(18,9) = 48620.\nFinal answer: $\\frac{2549088256}{762939453125}$", "final_answer": "\\frac{2549088256}{762939453125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "60868097e6e7826b", "problem_id": "bd35c989d7e036b0", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "You want 27 liters of a 30% salt solution. You have a 25% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 25% solution, so 27−x liters is 60% solution.\n2) Salt equation: 25x + 60(27−x) = 30·27.\n3) (25-60)x = (30-60)·27 ⇒ x = 162/7.\n4) Then 27−x = 27/7.\nFinal answer: $162/7 L of 25% and 27/7 L of 60%$", "final_answer": "162/7 L of 25% and 27/7 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e980f1bea24e1499", "problem_id": "bd35c989d7e036b0", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "You want 27 liters of a 30% salt solution. You have a 25% solution and a 60% solution. How many liters of each should you mix?", "output": "162/7 L of 25% and 27/7 L of 60%", "final_answer": "162/7 L of 25% and 27/7 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "87c5310524e60af5", "problem_id": "4c8ceab1aa15af65", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=34$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=34 and k=5.\n4) Compute C(38,4) = 73815.\nFinal answer: $73815$", "final_answer": "73815", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b1bb665828688df8", "problem_id": "4c8ceab1aa15af65", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=34$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=34 and k=5.\nCompute C(38,4) = 73815.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $73815$", "final_answer": "73815", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a3f8d7e9f5fc7804", "problem_id": "996faca75abe07b4", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} - x^{2} - 2 x - 4$ at $x=-1$.", "output": "Differentiate: f'(x) = $- 6 x^{2} - 2 x - 2$.\nSlope at x=-1: m = f'(-1) = $-6$.\nPoint on curve: (-1, f(-1)) = (-1, $-1$).\nFinal answer: $y = - 6\\cdot x - 7$", "final_answer": "y = - 6\\cdot x - 7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e58b4e69049e9d36", "problem_id": "996faca75abe07b4", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Find the equation of the tangent line to $y=- 2 x^{3} - x^{2} - 2 x - 4$ at $x=-1$.", "output": "1) Differentiate: f'(x) = $- 6 x^{2} - 2 x - 2$.\n2) Slope at x=-1: m = f'(-1) = $-6$.\n3) Point on curve: (-1, f(-1)) = (-1, $-1$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -1 = -6(x − -1).\n6) Optionally expand: y = $- 6 x - 7$.\nFinal answer: $y = - 6\\cdot x - 7$", "final_answer": "y = - 6\\cdot x - 7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f5415802c30b46f8", "problem_id": "d4822a861691cbf1", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡4 (mod 5), x≡7 (mod 13). Give the solution modulo 65.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 65.\nCompute the CRT solution to get x ≡ 59 (mod 65).\nCheck: 59 mod 5=4 and mod 13=7.\nFinal answer: $x\\equiv 59\\pmod{65}$", "final_answer": "x\\equiv 59\\pmod{65}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9560c1db3b982ce5", "problem_id": "d4822a861691cbf1", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡4 (mod 5), x≡7 (mod 13). Give the solution modulo 65.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 65.\n3) Compute the CRT solution to get x ≡ 59 (mod 65).\nFinal answer: $x\\equiv 59\\pmod{65}$", "final_answer": "x\\equiv 59\\pmod{65}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3d6751477ac1adab", "problem_id": "3454bbd59efca687", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "An urn has 29 red balls and 16 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(45,9).\n3) Favorable samples: choose k reds and n−k blues: C(29,2)·C(16,7).\n4) Compute totals: C(45,9) = 886163135, and favorable = 4644640.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{6496}{1239389}$", "final_answer": "\\frac{6496}{1239389}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d2a37e7e230a0db2", "problem_id": "3454bbd59efca687", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "An urn has 29 red balls and 16 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(45,9).\nFavorable samples: choose k reds and n−k blues: C(29,2)·C(16,7).\nFinal answer: $\\frac{6496}{1239389}$", "final_answer": "\\frac{6496}{1239389}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f3f2eedf091c20a4", "problem_id": "44b84832e0d5c26a", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡3 (mod 7), x≡2 (mod 5). Give the solution modulo 35.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 35.\n3) Compute the CRT solution to get x ≡ 17 (mod 35).\nFinal answer: $x\\equiv 17\\pmod{35}$", "final_answer": "x\\equiv 17\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2603c65878e7eafa", "problem_id": "44b84832e0d5c26a", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡3 (mod 7), x≡2 (mod 5). Give the solution modulo 35.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 35.\nCompute the CRT solution to get x ≡ 17 (mod 35).\nCheck: 17 mod 7=3 and mod 5=2.\nFinal answer: $x\\equiv 17\\pmod{35}$", "final_answer": "x\\equiv 17\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "370cbeca87a2cae0", "problem_id": "81ffa9b878642f08", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "An urn has 25 red balls and 25 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(50,11).\n3) Favorable samples: choose k reds and n−k blues: C(25,5)·C(25,6).\n4) Compute totals: C(50,11) = 37353738800, and favorable = 9409323000.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{41745}{165722}$", "final_answer": "\\frac{41745}{165722}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "424d7d0203ea3632", "problem_id": "c1d34d67a269c9d7", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{2} - x + 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 3 x^{2}) = - 6 x^{2}$.\n3) So $f'(x) = - 6 x^{2} - 1$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 3 x^{2}) = - 6 x$.\nTherefore the correct derivative is $f'(x) = - 6 x - 1$.\nFinal answer: $- 6 x - 1$", "final_answer": "- 6 x - 1", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b07ac6f02515eb2e", "problem_id": "3eba6580e620083a", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 20 times, what is the probability of exactly 18 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=20, k=18, p=$\\frac{1}{2}$ and simplify.\nHere C(20,18) = 190.\nCheck: expression simplifies to $\\frac{95}{524288}$.\nFinal answer: $\\frac{95}{524288}$", "final_answer": "\\frac{95}{524288}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "90e7fc7fb6a5314b", "problem_id": "3eba6580e620083a", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{2}$. If it is flipped 20 times, what is the probability of exactly 18 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=20, k=18, p=$\\frac{1}{2}$ and simplify.\n4) Here C(20,18) = 190.\nFinal answer: $\\frac{95}{524288}$", "final_answer": "\\frac{95}{524288}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "805d671d812220ff", "problem_id": "6d8deff5dcfc9ff1", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 10 red balls and 14 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(24,12).\n3) Favorable samples: choose k reds and n−k blues: C(10,2)·C(14,10).\n4) Compute totals: C(24,12) = 2704156, and favorable = 45045.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{495}{29716}$", "final_answer": "\\frac{495}{29716}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1d3469da61899862", "problem_id": "6d8deff5dcfc9ff1", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 10 red balls and 14 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(24,12).\nFavorable samples: choose k reds and n−k blues: C(10,2)·C(14,10).\nCompute totals: C(24,12) = 2704156, and favorable = 45045.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{495}{29716}$.\nFinal answer: $\\frac{495}{29716}$", "final_answer": "\\frac{495}{29716}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5acb7e3a83e7fa1e", "problem_id": "2f10ccbd5e5e9222", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_9=42$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_9=15.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=15 and k=9.\n4) Compute C(23,8) = 490314.\nFinal answer: $490314$", "final_answer": "490314", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a59ac8d6dcbf8414", "problem_id": "2d7e3350e7114741", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{2} + 3 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{x - 4}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}) = - \\frac{4 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4}$.\n3) So $f'(x) = \\frac{x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{2 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{6 x^{2}}{2 x^{2} - 16 x + 32} - \\frac{8 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{16 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{48 x}{2 x^{2} - 16 x + 32} + \\frac{16 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{2 x^{2} - 16 x + 32} + \\frac{32 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{2 x^{2} - 16 x + 32} - \\frac{20}{2 x^{2} - 16 x + 32} - \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x^{2} - 8 x + 16} - \\frac{4 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} + \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4} + \\frac{4 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4}) = - \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{x - 4} - \\frac{4 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{x - 4} + \\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(x - 4\\right)^{2}}$.\nTherefore the correct derivative is $f'(x) = \\frac{6 x^{2} - 6 x + \\left(x - 4\\right) \\left(- 12 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4}{2 \\left(x - 4\\right)^{2}}$.\nFinal answer: $\\frac{6 x^{2} - 6 x + \\left(x - 4\\right) \\left(- 12 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4}{2 \\left(x - 4\\right)^{2}}$", "final_answer": "\\frac{6 x^{2} - 6 x + \\left(x - 4\\right) \\left(- 12 x + \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 2 \\left(x - 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 4}{2 \\left(x - 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "2567ac23cca801f2", "problem_id": "b799b69f96cd6ce0", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the multiplicative inverse of 20 modulo 41.", "output": "1) We want x such that 20x ≡ 1 (mod 41).\n2) Use extended Euclid to find s,t with 20s + 41t = 1.\n3) One Bezout identity is: 20·(-2) + 41·(1) = 1.\n4) Reduce s modulo 41: x ≡ 39 (mod 41).\nFinal answer: $x\\equiv 39\\pmod{41}$", "final_answer": "x\\equiv 39\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a3e18da13fdc1aa6", "problem_id": "b799b69f96cd6ce0", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the multiplicative inverse of 20 modulo 41.", "output": "We want x such that 20x ≡ 1 (mod 41).\nUse extended Euclid to find s,t with 20s + 41t = 1.\nOne Bezout identity is: 20·(-2) + 41·(1) = 1.\nReduce s modulo 41: x ≡ 39 (mod 41).\nCheck: 20·39 mod 41 = 1.\nFinal answer: $x\\equiv 39\\pmod{41}$", "final_answer": "x\\equiv 39\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b69127084a440df5", "problem_id": "cc488b3c7d666820", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{5} x}{2} - \\frac{5 x}{2} + \\frac{3}{2} + \\frac{3 \\sqrt{5}}{2}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$1$, b=$- \\frac{3 \\sqrt{5}}{2} - \\frac{5}{2}$, c=$\\frac{3}{2} + \\frac{3 \\sqrt{5}}{2}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{3 \\sqrt{5}}{2} + \\frac{23}{2}$, so √Δ = $\\sqrt{\\frac{3 \\sqrt{5}}{2} + \\frac{23}{2}}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $\\frac{5}{4} + \\frac{3 \\sqrt{5}}{4} + \\frac{\\sqrt{6 \\sqrt{5} + 46}}{4}$.\n7) x2 = $- \\frac{\\sqrt{6 \\sqrt{5} + 46}}{4} + \\frac{5}{4} + \\frac{3 \\sqrt{5}}{4}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{\\frac{5}{4} + \\frac{3\\cdot \\sqrt{5}}{4} + \\frac{\\sqrt{6\\cdot \\sqrt{5} + 46}}{4}, - \\frac{\\sqrt{6\\cdot \\sqrt{5} + 46}}{4} + \\frac{5}{4} + \\frac{3\\cdot \\sqrt{5}}{4}\\}$", "final_answer": "x \\in \\{\\frac{5}{4} + \\frac{3\\cdot \\sqrt{5}}{4} + \\frac{\\sqrt{6\\cdot \\sqrt{5} + 46}}{4}, - \\frac{\\sqrt{6\\cdot \\sqrt{5} + 46}}{4} + \\frac{5}{4} + \\frac{3\\cdot \\sqrt{5}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "eec7d2f2bfcb6988", "problem_id": "cc488b3c7d666820", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$x^{2} - \\frac{3 \\sqrt{5} x}{2} - \\frac{5 x}{2} + \\frac{3}{2} + \\frac{3 \\sqrt{5}}{2}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$1$, b=$- \\frac{3 \\sqrt{5}}{2} - \\frac{5}{2}$, c=$\\frac{3}{2} + \\frac{3 \\sqrt{5}}{2}$.\nCompute the discriminant Δ=b^2−4ac.\nFinal answer: $x \\in \\{\\frac{5}{4} + \\frac{3\\cdot \\sqrt{5}}{4} + \\frac{\\sqrt{6\\cdot \\sqrt{5} + 46}}{4}, - \\frac{\\sqrt{6\\cdot \\sqrt{5} + 46}}{4} + \\frac{5}{4} + \\frac{3\\cdot \\sqrt{5}}{4}\\}$", "final_answer": "x \\in \\{\\frac{5}{4} + \\frac{3\\cdot \\sqrt{5}}{4} + \\frac{\\sqrt{6\\cdot \\sqrt{5} + 46}}{4}, - \\frac{\\sqrt{6\\cdot \\sqrt{5} + 46}}{4} + \\frac{5}{4} + \\frac{3\\cdot \\sqrt{5}}{4}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "80e93f5330b4403a", "problem_id": "4dbbbc11d7faec95", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{5}$. If it is flipped 21 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=21, k=5, p=$\\frac{3}{5}$ and simplify.\n4) Here C(21,5) = 20349.\nFinal answer: $\\frac{324062871552}{476837158203125}$", "final_answer": "\\frac{324062871552}{476837158203125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a1279d4f45a00cc1", "problem_id": "4dbbbc11d7faec95", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{5}$. If it is flipped 21 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=21, k=5, p=$\\frac{3}{5}$ and simplify.\nFinal answer: $\\frac{324062871552}{476837158203125}$", "final_answer": "\\frac{324062871552}{476837158203125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "67af2aa0543b1ecd", "problem_id": "ffc7bdc580bee689", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=21$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=21 and k=6.\n4) Compute C(26,5) = 65780.\nFinal answer: $65780$", "final_answer": "65780", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "615ef6d86aaa83d4", "problem_id": "ffc7bdc580bee689", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=21$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=21 and k=6.\nFinal answer: $65780$", "final_answer": "65780", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c21d2f244bdd6d65", "problem_id": "5c9623ca99b5d6bd", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "An urn has 25 red balls and 15 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(40,12).\n3) Favorable samples: choose k reds and n−k blues: C(25,4)·C(15,8).\n4) Compute totals: C(40,12) = 5586853480, and favorable = 81402750.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{626175}{42975796}$", "final_answer": "\\frac{626175}{42975796}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fa7b8df87d100f5b", "problem_id": "5c9623ca99b5d6bd", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 25 red balls and 15 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(40,12).\nFavorable samples: choose k reds and n−k blues: C(25,4)·C(15,8).\nCompute totals: C(40,12) = 5586853480, and favorable = 81402750.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{626175}{42975796}$.\nFinal answer: $\\frac{626175}{42975796}$", "final_answer": "\\frac{626175}{42975796}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "73a902985a5823dc", "problem_id": "4ac309b40bb60a72", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 10 times, what is the probability of exactly 2 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=10, k=2, p=$\\frac{3}{7}$ and simplify.\nFinal answer: $\\frac{26542080}{282475249}$", "final_answer": "\\frac{26542080}{282475249}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "47ed03bcd1cff578", "problem_id": "4ac309b40bb60a72", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{7}$. If it is flipped 10 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=10, k=2, p=$\\frac{3}{7}$ and simplify.\n4) Here C(10,2) = 45.\nFinal answer: $\\frac{26542080}{282475249}$", "final_answer": "\\frac{26542080}{282475249}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0b63bd73d0e0f5ff", "problem_id": "920194bd46f3c588", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 23 red balls and 18 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(41,9).\n3) Favorable samples: choose k reds and n−k blues: C(23,3)·C(18,6).\n4) Compute totals: C(41,9) = 350343565, and favorable = 32876844.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{13524}{144115}$", "final_answer": "\\frac{13524}{144115}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8a3d2e99dbf89130", "problem_id": "920194bd46f3c588", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 23 red balls and 18 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(41,9).\nFavorable samples: choose k reds and n−k blues: C(23,3)·C(18,6).\nCompute totals: C(41,9) = 350343565, and favorable = 32876844.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{13524}{144115}$.\nFinal answer: $\\frac{13524}{144115}$", "final_answer": "\\frac{13524}{144115}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7758166e85c9c784", "problem_id": "fba225ea2c6ae479", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Find the multiplicative inverse of 37 modulo 43.", "output": "We want x such that 37x ≡ 1 (mod 43).\nUse extended Euclid to find s,t with 37s + 43t = 1.\nOne Bezout identity is: 37·(7) + 43·(-6) = 1.\nReduce s modulo 43: x ≡ 7 (mod 43).\nCheck: 37·7 mod 43 = 1.\nFinal answer: $x\\equiv 7\\pmod{43}$", "final_answer": "x\\equiv 7\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b1610c2a256d6f2b", "problem_id": "fba225ea2c6ae479", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the multiplicative inverse of 37 modulo 43.", "output": "1) We want x such that 37x ≡ 1 (mod 43).\n2) Use extended Euclid to find s,t with 37s + 43t = 1.\n3) One Bezout identity is: 37·(7) + 43·(-6) = 1.\n4) Reduce s modulo 43: x ≡ 7 (mod 43).\nFinal answer: $x\\equiv 7\\pmod{43}$", "final_answer": "x\\equiv 7\\pmod{43}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d52eba3ab92aad50", "problem_id": "ef74617df9773018", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the multiplicative inverse of 42 modulo 53.", "output": "We want x such that 42x ≡ 1 (mod 53).\nUse extended Euclid to find s,t with 42s + 53t = 1.\nOne Bezout identity is: 42·(24) + 53·(-19) = 1.\nReduce s modulo 53: x ≡ 24 (mod 53).\nCheck: 42·24 mod 53 = 1.\nFinal answer: $x\\equiv 24\\pmod{53}$", "final_answer": "x\\equiv 24\\pmod{53}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d142e6b385b2b737", "problem_id": "ef74617df9773018", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the multiplicative inverse of 42 modulo 53.", "output": "1) We want x such that 42x ≡ 1 (mod 53).\n2) Use extended Euclid to find s,t with 42s + 53t = 1.\n3) One Bezout identity is: 42·(24) + 53·(-19) = 1.\n4) Reduce s modulo 53: x ≡ 24 (mod 53).\nFinal answer: $x\\equiv 24\\pmod{53}$", "final_answer": "x\\equiv 24\\pmod{53}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "35f8e3b2ebcff417", "problem_id": "ebf113ea76e367ff", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡1 (mod 8), x≡0 (mod 11). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 33 (mod 88).\nFinal answer: $x\\equiv 33\\pmod{88}$", "final_answer": "x\\equiv 33\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "87e73e3a5425f2b1", "problem_id": "b6247d562fb05d69", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{2} - 4 x + 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 4 x) = - 4 x$.\n3) So $f'(x) = 0$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 4 x) = -4$.\nTherefore the correct derivative is $f'(x) = 4 \\left(x - 1\\right)$.\nFinal answer: $4 \\left(x - 1\\right)$", "final_answer": "4 \\left(x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c93e7226ca0086c9", "problem_id": "504b9911f491f9da", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "You want 34 liters of a 35% salt solution. You have a 15% solution and a 70% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 34−x liters is 70% solution.\nSalt equation: 15x + 70(34−x) = 35·34.\n(15-70)x = (35-70)·34 ⇒ x = 238/11.\nFinal answer: $238/11 L of 15% and 136/11 L of 70%$", "final_answer": "238/11 L of 15% and 136/11 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "13204da5c925254d", "problem_id": "504b9911f491f9da", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "You want 34 liters of a 35% salt solution. You have a 15% solution and a 70% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 34−x liters is 70% solution.\n2) Salt equation: 15x + 70(34−x) = 35·34.\n3) (15-70)x = (35-70)·34 ⇒ x = 238/11.\n4) Then 34−x = 136/11.\nFinal answer: $238/11 L of 15% and 136/11 L of 70%$", "final_answer": "238/11 L of 15% and 136/11 L of 70%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f4bddc2e5d31fc2a", "problem_id": "ca37e305e6636670", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{3 x^{2} - 3 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{2 x - 3}$.", "output": "Let f(x) = $\\frac{3 x^{2} - 3 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{2 x - 3}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $\\frac{- 12 x^{2} + 12 x + \\left(2 x - 3\\right) \\left(12 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6\\right) - 4 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8}{2 \\left(2 x - 3\\right)^{2}}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $\\frac{- 12\\cdot x^{2} + 12\\cdot x + \\left(2\\cdot x - 3\\right)\\cdot \\left(12\\cdot x + \\left(2\\cdot x - 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 6\\right) - \\left(8\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 8}{2\\cdot \\left(2\\cdot x - 3\\right)^{2}}$", "final_answer": "\\frac{- 12\\cdot x^{2} + 12\\cdot x + \\left(2\\cdot x - 3\\right)\\cdot \\left(12\\cdot x + \\left(2\\cdot x - 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 6\\right) - \\left(8\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 8}{2\\cdot \\left(2\\cdot x - 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5e2efa09b4a52f5b", "problem_id": "ca37e305e6636670", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{3 x^{2} - 3 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{2 x - 3}$.", "output": "1) Let f(x) = $\\frac{3 x^{2} - 3 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 2}{2 x - 3}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $\\frac{- 12 x^{2} + 12 x + \\left(2 x - 3\\right) \\left(12 x + \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6\\right) - 4 \\left(2 x - 3\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 8}{2 \\left(2 x - 3\\right)^{2}}$.\nFinal answer: $\\frac{- 12\\cdot x^{2} + 12\\cdot x + \\left(2\\cdot x - 3\\right)\\cdot \\left(12\\cdot x + \\left(2\\cdot x - 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 6\\right) - \\left(8\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 8}{2\\cdot \\left(2\\cdot x - 3\\right)^{2}}$", "final_answer": "\\frac{- 12\\cdot x^{2} + 12\\cdot x + \\left(2\\cdot x - 3\\right)\\cdot \\left(12\\cdot x + \\left(2\\cdot x - 3\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + \\left(4\\cdot x - 6\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\cos{\\left(x \\right)} + 4\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} - 6\\right) - \\left(8\\cdot x - 12\\right)\\cdot e^{\\frac{x}{2}}\\cdot \\sin{\\left(x \\right)} + 8}{2\\cdot \\left(2\\cdot x - 3\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9cdb82ba6162157d", "problem_id": "11c57d2eba1ef4dd", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the equation of the tangent line to $y=- 4 x^{3} - 2 x^{2} + 2 x + 1$ at $x=2$.", "output": "1) Differentiate: f'(x) = $- 12 x^{2} - 4 x + 2$.\n2) Slope at x=2: m = f'(2) = $-54$.\n3) Point on curve: (2, f(2)) = (2, $-35$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − -35 = -54(x − 2).\n6) Optionally expand: y = $73 - 54 x$.\nFinal answer: $y = 73 - 54\\cdot x$", "final_answer": "y = 73 - 54\\cdot x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b435d5a3b56ebcaa", "problem_id": "39a036e8e4b3525b", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -3} 3 x - 12$.\n\nStudent solution:\n1) Substitute x=-3 into the expression.\n2) (Mistake) Simplify the expression to 2 x - 12.\n3) Then the limit is -18.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-3 (polynomial is continuous).\nCorrect limit value: -21.\nFinal answer: $-21$", "final_answer": "-21", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "5574ea2763a20f33", "problem_id": "e979932fef773bfc", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A cyclist rides at 47 miles per hour for 1/2 hours, then at 60 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 47 × 1/2 = 47/2 miles.\n2) Second leg distance = 60 × 5/2 = 150 miles.\n3) Total distance = 47/2 + 150 = 347/2 miles.\nFinal answer: $\\frac{347}{2}$", "final_answer": "\\frac{347}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9e961584fb07a599", "problem_id": "e979932fef773bfc", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A cyclist rides at 47 miles per hour for 1/2 hours, then at 60 miles per hour for 5/2 hours. How many miles does the cyclist travel in total?", "output": "First leg distance = 47 × 1/2 = 47/2 miles.\nSecond leg distance = 60 × 5/2 = 150 miles.\nTotal distance = 47/2 + 150 = 347/2 miles.\nFinal answer: $\\frac{347}{2}$", "final_answer": "\\frac{347}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "77be2cd885f3be62", "problem_id": "a927cbb2deccfed9", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=16$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=16 and k=5.\nFinal answer: $4845$", "final_answer": "4845", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d78efc826d4faecf", "problem_id": "a927cbb2deccfed9", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=16$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=16 and k=5.\n4) Compute C(20,4) = 4845.\nFinal answer: $4845$", "final_answer": "4845", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d3bb1444e742cac6", "problem_id": "4b3ff6396ab45d89", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_6=32$?", "output": "1) Let y_i=x_i-3, then y_i≥0 and y_1+…+y_6=14.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=14 and k=6.\n4) Compute C(19,5) = 11628.\nFinal answer: $11628$", "final_answer": "11628", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f63ef18d3cec612e", "problem_id": "4b3ff6396ab45d89", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "How many solutions in integers with $x_i\\ge 3$ satisfy $x_1+\\cdots+x_6=32$?", "output": "Let y_i=x_i-3, then y_i≥0 and y_1+…+y_6=14.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=14 and k=6.\nFinal answer: $11628$", "final_answer": "11628", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ba63374f95e03fb2", "problem_id": "279ad9c2dd326cd1", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A cyclist rides at 62 miles per hour for 1 hours, then at 70 miles per hour for 3/2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 62 × 1 = 62 miles.\n2) Second leg distance = 70 × 3/2 = 105 miles.\n3) Total distance = 62 + 105 = 167 miles.\nFinal answer: $167$", "final_answer": "167", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0c217c8a19159712", "problem_id": "0a1be025f503c38a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{2} + x + 3$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{2} + 3.\n3) Then the limit is 11.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 13.\nFinal answer: $13$", "final_answer": "13", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ad637bea11b06085", "problem_id": "27ad70de5c9eb5fc", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡10 (mod 11), x≡0 (mod 7). Give the solution modulo 77.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 77.\nCompute the CRT solution to get x ≡ 21 (mod 77).\nCheck: 21 mod 11=10 and mod 7=0.\nFinal answer: $x\\equiv 21\\pmod{77}$", "final_answer": "x\\equiv 21\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "74e225b1e49f8bfa", "problem_id": "bdfc7ed1e6d5f604", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\left(x - 3\\right) e^{x} + \\left(x^{3} + x^{2} + 3\\right) \\cos{\\left(x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(3 \\cos{\\left(x \\right)}) = - 3 \\sin{\\left(x \\right)}$.\n3) So $f'(x) = - x^{3} \\sin{\\left(x \\right)} - x^{2} \\sin{\\left(x \\right)} + 3 x^{2} \\cos{\\left(x \\right)} + x e^{x} + 2 x \\cos{\\left(x \\right)} - 2 e^{x} - 3 \\sin{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(3 \\cos{\\left(x \\right)}) = - 3 \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = x \\left(3 x + 2\\right) \\cos{\\left(x \\right)} + \\left(x - 3\\right) e^{x} - \\left(x^{2} \\left(x + 1\\right) + 3\\right) \\sin{\\left(x \\right)} + e^{x}$.\nFinal answer: $x \\left(3 x + 2\\right) \\cos{\\left(x \\right)} + \\left(x - 3\\right) e^{x} - \\left(x^{2} \\left(x + 1\\right) + 3\\right) \\sin{\\left(x \\right)} + e^{x}$", "final_answer": "x \\left(3 x + 2\\right) \\cos{\\left(x \\right)} + \\left(x - 3\\right) e^{x} - \\left(x^{2} \\left(x + 1\\right) + 3\\right) \\sin{\\left(x \\right)} + e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "a6a655acdc04afe6", "problem_id": "8158c0a081c9f7f4", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 8 times, what is the probability of exactly 5 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=8, k=5, p=$\\frac{2}{7}$ and simplify.\n4) Here C(8,5) = 56.\nFinal answer: $\\frac{32000}{823543}$", "final_answer": "\\frac{32000}{823543}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "364e818c2238cec9", "problem_id": "8158c0a081c9f7f4", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 8 times, what is the probability of exactly 5 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=8, k=5, p=$\\frac{2}{7}$ and simplify.\nFinal answer: $\\frac{32000}{823543}$", "final_answer": "\\frac{32000}{823543}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "646eb9970f52ead5", "problem_id": "1ac9a7fbc4ac8111", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A store sells a backpack for $505. It is discounted by 15%. You buy 10 backpacks. What is the total cost?", "output": "Discounted price = 505 × (1 − 15/100) = 505 × 85/100 = 1717/4.\nSubtotal = 10 × 1717/4 = 8585/2.\nCheck: recomputing the formula yields 8585/2.\nFinal answer: $\\frac{8585}{2}$", "final_answer": "\\frac{8585}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9fdd4286c22ca07c", "problem_id": "1ac9a7fbc4ac8111", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A store sells a backpack for $505. It is discounted by 15%. You buy 10 backpacks. What is the total cost?", "output": "1) Discounted price = 505 × (1 − 15/100) = 505 × 85/100 = 1717/4.\n2) Subtotal = 10 × 1717/4 = 8585/2.\nFinal answer: $\\frac{8585}{2}$", "final_answer": "\\frac{8585}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1438909827110dbc", "problem_id": "2465acd71850f425", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Two friends split $1057 in the ratio 3:4. How much does each receive?", "output": "Total ratio parts = 3+4 = 7.\nEach part = 1057 ÷ 7 = 151.\nFirst share = 3 × 151 = 453.\nFinal answer: $604$", "final_answer": "604", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "da179b7699cd9a0d", "problem_id": "2465acd71850f425", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Two friends split $1057 in the ratio 3:4. How much does each receive?", "output": "1) Total ratio parts = 3+4 = 7.\n2) Each part = 1057 ÷ 7 = 151.\n3) First share = 3 × 151 = 453.\n4) Second share = 4 × 151 = 604.\nFinal answer: $604$", "final_answer": "604", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "de164f06ea4a09ab", "problem_id": "58f5ca97da3c4ecd", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "An urn has 23 red balls and 16 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 6 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(39,12).\nFavorable samples: choose k reds and n−k blues: C(23,6)·C(16,6).\nFinal answer: $\\frac{116886}{565471}$", "final_answer": "\\frac{116886}{565471}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3577a419d20c2ccf", "problem_id": "58f5ca97da3c4ecd", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "An urn has 23 red balls and 16 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 6 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(39,12).\n3) Favorable samples: choose k reds and n−k blues: C(23,6)·C(16,6).\n4) Compute totals: C(39,12) = 3910797436, and favorable = 808383576.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{116886}{565471}$", "final_answer": "\\frac{116886}{565471}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c8ec5f579752b40d", "problem_id": "0b96da7994872d16", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "An urn has 15 red balls and 20 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(35,8).\n3) Favorable samples: choose k reds and n−k blues: C(15,2)·C(20,6).\n4) Compute totals: C(35,8) = 23535820, and favorable = 4069800.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1710}{9889}$", "final_answer": "\\frac{1710}{9889}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6a0d9e383dd21d53", "problem_id": "0b96da7994872d16", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 15 red balls and 20 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(35,8).\nFavorable samples: choose k reds and n−k blues: C(15,2)·C(20,6).\nCompute totals: C(35,8) = 23535820, and favorable = 4069800.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1710}{9889}$.\nFinal answer: $\\frac{1710}{9889}$", "final_answer": "\\frac{1710}{9889}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5af6bef5e372855f", "problem_id": "973ae2ce03ff8da6", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "An urn has 15 red balls and 29 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(44,8).\n3) Favorable samples: choose k reds and n−k blues: C(15,1)·C(29,7).\n4) Compute totals: C(44,8) = 177232627, and favorable = 23411700.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1800900}{13633279}$", "final_answer": "\\frac{1800900}{13633279}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "531d94b6f65735fc", "problem_id": "46a40bc3e7cbfc1d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 2\\right) e^{x} + \\left(2 x^{2} - 2 x - 3\\right) \\cos{\\left(3 x^{2} + 3 x - 1 \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x^{2} \\cos{\\left(\\left(3 x^{2} + 3 x\\right) - 1 \\right)}) = - 2 x^{2} \\sin{\\left(3 x^{2} + 3 x - 1 \\right)}$.\n3) So $f'(x) = 10 x^{2} \\sin{\\left(3 x^{2} + 3 x - 1 \\right)} + 2 x e^{x} + 24 x \\sin{\\left(3 x^{2} + 3 x - 1 \\right)} + 6 e^{x} + 9 \\sin{\\left(3 x^{2} + 3 x - 1 \\right)} - 2 \\cos{\\left(\\left(3 x^{2} + 3 x\\right) - 1 \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x^{2} \\cos{\\left(\\left(3 x^{2} + 3 x\\right) - 1 \\right)}) = - 2 x^{2} \\left(6 x + 3\\right) \\sin{\\left(3 x^{2} + 3 x - 1 \\right)} + 4 x \\cos{\\left(\\left(3 x^{2} + 3 x\\right) - 1 \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(x + 2\\right) e^{x} + 2 \\left(2 x - 1\\right) \\cos{\\left(\\left(3 x^{2} + 3 x\\right) - 1 \\right)} - 3 \\left(2 x + 1\\right) \\left(2 x \\left(x - 1\\right) - 3\\right) \\sin{\\left(3 x^{2} + 3 x - 1 \\right)} + 2 e^{x}$.\nFinal answer: $2 \\left(x + 2\\right) e^{x} + 2 \\left(2 x - 1\\right) \\cos{\\left(\\left(3 x^{2} + 3 x\\right) - 1 \\right)} - 3 \\left(2 x + 1\\right) \\left(2 x \\left(x - 1\\right) - 3\\right) \\sin{\\left(3 x^{2} + 3 x - 1 \\right)} + 2 e^{x}$", "final_answer": "2 \\left(x + 2\\right) e^{x} + 2 \\left(2 x - 1\\right) \\cos{\\left(\\left(3 x^{2} + 3 x\\right) - 1 \\right)} - 3 \\left(2 x + 1\\right) \\left(2 x \\left(x - 1\\right) - 3\\right) \\sin{\\left(3 x^{2} + 3 x - 1 \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "1163ab5a52b63729", "problem_id": "b46a395b5e0740c9", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡0 (mod 5), x≡11 (mod 13). Give the solution modulo 65.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 65.\nCompute the CRT solution to get x ≡ 50 (mod 65).\nCheck: 50 mod 5=0 and mod 13=11.\nFinal answer: $x\\equiv 50\\pmod{65}$", "final_answer": "x\\equiv 50\\pmod{65}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "484dcabc5a68a8ef", "problem_id": "b46a395b5e0740c9", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡0 (mod 5), x≡11 (mod 13). Give the solution modulo 65.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 65.\n3) Compute the CRT solution to get x ≡ 50 (mod 65).\nFinal answer: $x\\equiv 50\\pmod{65}$", "final_answer": "x\\equiv 50\\pmod{65}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2fcfd6efca2f36b3", "problem_id": "749e08d09de5f6f6", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the congruences: x≡3 (mod 7), x≡7 (mod 11). Give the solution modulo 77.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 77.\n3) Compute the CRT solution to get x ≡ 73 (mod 77).\nFinal answer: $x\\equiv 73\\pmod{77}$", "final_answer": "x\\equiv 73\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d5e57ae66493a4ba", "problem_id": "749e08d09de5f6f6", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the congruences: x≡3 (mod 7), x≡7 (mod 11). Give the solution modulo 77.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 77.\nCompute the CRT solution to get x ≡ 73 (mod 77).\nFinal answer: $x\\equiv 73\\pmod{77}$", "final_answer": "x\\equiv 73\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c87e13a0936ad87c", "problem_id": "93ea02d030252b55", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡3 (mod 5), x≡2 (mod 7). Give the solution modulo 35.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 35.\n3) Compute the CRT solution to get x ≡ 23 (mod 35).\nFinal answer: $x\\equiv 23\\pmod{35}$", "final_answer": "x\\equiv 23\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7e66c25b233ab1c6", "problem_id": "d2266df0b4a0f3d1", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(2 x^{3} - 3 x^{2} - 3 x\\right) \\cos{\\left(3 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 3 x \\cos{\\left(3 x \\right)}) = 3 x \\sin{\\left(3 x \\right)}$.\n3) So $f'(x) = - 6 x^{3} \\sin{\\left(3 x \\right)} + 9 x^{2} \\sin{\\left(3 x \\right)} + 6 x^{2} \\cos{\\left(3 x \\right)} + 2 x e^{x} + 3 x \\sin{\\left(3 x \\right)} - 6 x \\cos{\\left(3 x \\right)} - 4 e^{x}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 3 x \\cos{\\left(3 x \\right)}) = 9 x \\sin{\\left(3 x \\right)} - 3 \\cos{\\left(3 x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 3 x \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x \\right)} + 2 \\left(x - 3\\right) e^{x} + 3 \\left(2 x^{2} - 2 x - 1\\right) \\cos{\\left(3 x \\right)} + 2 e^{x}$.\nFinal answer: $- 3 x \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x \\right)} + 2 \\left(x - 3\\right) e^{x} + 3 \\left(2 x^{2} - 2 x - 1\\right) \\cos{\\left(3 x \\right)} + 2 e^{x}$", "final_answer": "- 3 x \\left(x \\left(2 x - 3\\right) - 3\\right) \\sin{\\left(3 x \\right)} + 2 \\left(x - 3\\right) e^{x} + 3 \\left(2 x^{2} - 2 x - 1\\right) \\cos{\\left(3 x \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "c4b7c9cb2c8f2577", "problem_id": "c86a22f2a66a086e", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many binary strings of length 18 contain exactly 4 ones with no two ones adjacent?", "output": "1) Reserve 3 mandatory zeros between the 4 ones.\n2) This leaves n−k+1 = 15 valid slots for the ones.\n3) Choose the slots: C(15,4) = 1365.\nFinal answer: $1365$", "final_answer": "1365", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8e5a926568b7a5ed", "problem_id": "c86a22f2a66a086e", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "How many binary strings of length 18 contain exactly 4 ones with no two ones adjacent?", "output": "Reserve 3 mandatory zeros between the 4 ones.\nThis leaves n−k+1 = 15 valid slots for the ones.\nChoose the slots: C(15,4) = 1365.\nFinal answer: $1365$", "final_answer": "1365", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c9eae30c64a50b5d", "problem_id": "f7144afbf9f31eb5", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} 3 x^{4} + 4 x^{3} - 3 x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{4} + 4 x^{3} - 3 x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{15 \\pi^{4}}{16} - \\frac{9 \\pi^{3}}{8} + 5 + \\frac{3 \\pi^{2}}{2} + \\frac{99 \\pi^{5}}{160}$.\nFinal answer: $- \\frac{15\\cdot \\pi^{4}}{16} - \\frac{9\\cdot \\pi^{3}}{8} + 5 + \\frac{3\\cdot \\pi^{2}}{2} + \\frac{99\\cdot \\pi^{5}}{160}$", "final_answer": "- \\frac{15\\cdot \\pi^{4}}{16} - \\frac{9\\cdot \\pi^{3}}{8} + 5 + \\frac{3\\cdot \\pi^{2}}{2} + \\frac{99\\cdot \\pi^{5}}{160}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2b1276a3cd1ce801", "problem_id": "f7144afbf9f31eb5", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{- \\pi}^{\\frac{\\pi}{2}} 3 x^{4} + 4 x^{3} - 3 x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{4} + 4 x^{3} - 3 x^{2} - 4 x + 3 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(- \\pi).\nSubtract: F(\\frac{\\pi}{2}) − F(- \\pi) = $- \\frac{15 \\pi^{4}}{16} - \\frac{9 \\pi^{3}}{8} + 5 + \\frac{3 \\pi^{2}}{2} + \\frac{99 \\pi^{5}}{160}$.\nSimplify the result.\nCheck: SymPy differentiation confirms the antiderivative.\nFinal answer: $- \\frac{15\\cdot \\pi^{4}}{16} - \\frac{9\\cdot \\pi^{3}}{8} + 5 + \\frac{3\\cdot \\pi^{2}}{2} + \\frac{99\\cdot \\pi^{5}}{160}$", "final_answer": "- \\frac{15\\cdot \\pi^{4}}{16} - \\frac{9\\cdot \\pi^{3}}{8} + 5 + \\frac{3\\cdot \\pi^{2}}{2} + \\frac{99\\cdot \\pi^{5}}{160}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dfecd195a85385aa", "problem_id": "40e31712fa479dfe", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_10=34$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_10=24.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=24 and k=10.\n4) Compute C(33,9) = 38567100.\nFinal answer: $38567100$", "final_answer": "38567100", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db24fae274407b43", "problem_id": "40e31712fa479dfe", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_10=34$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_10=24.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=24 and k=10.\nCompute C(33,9) = 38567100.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $38567100$", "final_answer": "38567100", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cc2583ae44c8af07", "problem_id": "a72375a5dac2642a", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 18 times, what is the probability of exactly 8 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=18, k=8, p=$\\frac{1}{6}$ and simplify.\n4) Here C(18,8) = 43758.\nFinal answer: $\\frac{23740234375}{5642219814912}$", "final_answer": "\\frac{23740234375}{5642219814912}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "947135967db46bfd", "problem_id": "a72375a5dac2642a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{6}$. If it is flipped 18 times, what is the probability of exactly 8 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=18, k=8, p=$\\frac{1}{6}$ and simplify.\nHere C(18,8) = 43758.\nCheck: expression simplifies to $\\frac{23740234375}{5642219814912}$.\nFinal answer: $\\frac{23740234375}{5642219814912}$", "final_answer": "\\frac{23740234375}{5642219814912}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ee939339b5b52d06", "problem_id": "febc99a6b7483739", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Two friends split $1683 in the ratio 9:2. How much does each receive?", "output": "1) Total ratio parts = 9+2 = 11.\n2) Each part = 1683 ÷ 11 = 153.\n3) First share = 9 × 153 = 1377.\n4) Second share = 2 × 153 = 306.\nFinal answer: $306$", "final_answer": "306", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "18e498a4ee79d409", "problem_id": "febc99a6b7483739", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Two friends split $1683 in the ratio 9:2. How much does each receive?", "output": "Total ratio parts = 9+2 = 11.\nEach part = 1683 ÷ 11 = 153.\nFirst share = 9 × 153 = 1377.\nSecond share = 2 × 153 = 306.\nCheck: 1377+306=1683.\nFinal answer: $306$", "final_answer": "306", "metadata": {"format": "word", "domain": "arithmetic", "topic": "ratio_split", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cbd34c4bcbfd841c", "problem_id": "a36290a850a1f779", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Find the equation of the tangent line to $y=4 x^{3} + 2 x^{2} + x$ at $x=1$.", "output": "1) Differentiate: f'(x) = $12 x^{2} + 4 x + 1$.\n2) Slope at x=1: m = f'(1) = $17$.\n3) Point on curve: (1, f(1)) = (1, $7$).\n4) Use point-slope form y − y0 = m(x − x0).\n5) Plug in values: y − 7 = 17(x − 1).\n6) Optionally expand: y = $17 x - 10$.\nFinal answer: $y = 17\\cdot x - 10$", "final_answer": "y = 17\\cdot x - 10", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5ea9f69fc0e2413e", "problem_id": "a36290a850a1f779", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Find the equation of the tangent line to $y=4 x^{3} + 2 x^{2} + x$ at $x=1$.", "output": "Differentiate: f'(x) = $12 x^{2} + 4 x + 1$.\nSlope at x=1: m = f'(1) = $17$.\nPoint on curve: (1, f(1)) = (1, $7$).\nUse point-slope form y − y0 = m(x − x0).\nPlug in values: y − 7 = 17(x − 1).\nOptionally expand: y = $17 x - 10$.\nCheck: the line passes through the point and has matching slope.\nFinal answer: $y = 17\\cdot x - 10$", "final_answer": "y = 17\\cdot x - 10", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "tangent_line", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ac278b3e7e94f742", "problem_id": "af5b3e56404cf222", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Find the multiplicative inverse of 16 modulo 41.", "output": "1) We want x such that 16x ≡ 1 (mod 41).\n2) Use extended Euclid to find s,t with 16s + 41t = 1.\n3) One Bezout identity is: 16·(18) + 41·(-7) = 1.\n4) Reduce s modulo 41: x ≡ 18 (mod 41).\nFinal answer: $x\\equiv 18\\pmod{41}$", "final_answer": "x\\equiv 18\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dbb087ec548752bb", "problem_id": "af5b3e56404cf222", "style": "answer_only", "instruction": "Final answer only.", "input": "Find the multiplicative inverse of 16 modulo 41.", "output": "$x\\equiv 18\\pmod{41}$", "final_answer": "x\\equiv 18\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fee32210790806d6", "problem_id": "c54b1448176f53e5", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_3=12$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=12 and k=3.\nCompute C(14,2) = 91.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $91$", "final_answer": "91", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "628bfccf532eaaff", "problem_id": "c54b1448176f53e5", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_3=12$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=12 and k=3.\n4) Compute C(14,2) = 91.\nFinal answer: $91$", "final_answer": "91", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a9ca00233809d641", "problem_id": "21991906fb6bc90c", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 26 red balls and 18 blue balls. If 9 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(44,9).\n3) Favorable samples: choose k reds and n−k blues: C(26,2)·C(18,7).\n4) Compute totals: C(44,9) = 708930508, and favorable = 10342800.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{198900}{13633279}$", "final_answer": "\\frac{198900}{13633279}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ff3e2d14958746b", "problem_id": "8eb129c8ffc74457", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -2} 3 x^{2} - 5 x + 8$.\n\nStudent solution:\n1) Substitute x=-2 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{2} - 6 x + 8.\n3) Then the limit is 32.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-2 (polynomial is continuous).\nCorrect limit value: 30.\nFinal answer: $30$", "final_answer": "30", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ac4f1fdfeefe68a2", "problem_id": "ea41a0ac86d7b0b1", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the congruences: x≡8 (mod 9), x≡6 (mod 8). Give the solution modulo 72.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 72.\n3) Compute the CRT solution to get x ≡ 62 (mod 72).\nFinal answer: $x\\equiv 62\\pmod{72}$", "final_answer": "x\\equiv 62\\pmod{72}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d8b0bc319e0c965f", "problem_id": "ea41a0ac86d7b0b1", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡8 (mod 9), x≡6 (mod 8). Give the solution modulo 72.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 72.\nCompute the CRT solution to get x ≡ 62 (mod 72).\nCheck: 62 mod 9=8 and mod 8=6.\nFinal answer: $x\\equiv 62\\pmod{72}$", "final_answer": "x\\equiv 62\\pmod{72}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "82d6befb9ae6a8ef", "problem_id": "eceecb64db313126", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 15 red balls and 14 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(29,7).\n3) Favorable samples: choose k reds and n−k blues: C(15,1)·C(14,6).\n4) Compute totals: C(29,7) = 1560780, and favorable = 45045.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{77}{2668}$", "final_answer": "\\frac{77}{2668}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "32cd067620624704", "problem_id": "acdd413b5fdca8a9", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x - 2$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 3 x - 2.\n3) Then the limit is -8.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -6.\nFinal answer: $-6$", "final_answer": "-6", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "3d1375a9b7a4adbd", "problem_id": "15916205b37074a6", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the multiplicative inverse of 14 modulo 47.", "output": "1) We want x such that 14x ≡ 1 (mod 47).\n2) Use extended Euclid to find s,t with 14s + 47t = 1.\n3) One Bezout identity is: 14·(-10) + 47·(3) = 1.\n4) Reduce s modulo 47: x ≡ 37 (mod 47).\nFinal answer: $x\\equiv 37\\pmod{47}$", "final_answer": "x\\equiv 37\\pmod{47}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5b1237f5d1bf887f", "problem_id": "15916205b37074a6", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the multiplicative inverse of 14 modulo 47.", "output": "We want x such that 14x ≡ 1 (mod 47).\nUse extended Euclid to find s,t with 14s + 47t = 1.\nOne Bezout identity is: 14·(-10) + 47·(3) = 1.\nReduce s modulo 47: x ≡ 37 (mod 47).\nCheck: 14·37 mod 47 = 1.\nFinal answer: $x\\equiv 37\\pmod{47}$", "final_answer": "x\\equiv 37\\pmod{47}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e769c8eebeac8635", "problem_id": "d4bf820fbc4853a1", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡6 (mod 7), x≡4 (mod 11). Give the solution modulo 77.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 77.\n3) Compute the CRT solution to get x ≡ 48 (mod 77).\nFinal answer: $x\\equiv 48\\pmod{77}$", "final_answer": "x\\equiv 48\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f91f5cb54280a7e0", "problem_id": "98e765fee2d42e08", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 2 x^{2} - 4 x + 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x^{2}) = - 4 x^{2}$.\n3) So $f'(x) = - 4 x^{2} - 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x^{2}) = - 4 x$.\nTherefore the correct derivative is $f'(x) = 4 \\left(- x - 1\\right)$.\nFinal answer: $4 \\left(- x - 1\\right)$", "final_answer": "4 \\left(- x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "74584fef159b8fd5", "problem_id": "5aff7bd4b2533b6b", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 8 red balls and 8 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(16,8).\n3) Favorable samples: choose k reds and n−k blues: C(8,3)·C(8,5).\n4) Compute totals: C(16,8) = 12870, and favorable = 3136.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1568}{6435}$", "final_answer": "\\frac{1568}{6435}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f63f36a832239345", "problem_id": "5aff7bd4b2533b6b", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "An urn has 8 red balls and 8 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "$\\frac{1568}{6435}$", "final_answer": "\\frac{1568}{6435}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "61cd2280c722d510", "problem_id": "b6d4871a3a430780", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "An urn has 10 red balls and 25 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(35,12).\n3) Favorable samples: choose k reds and n−k blues: C(10,5)·C(25,7).\n4) Compute totals: C(35,12) = 834451800, and favorable = 121136400.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{28842}{198679}$", "final_answer": "\\frac{28842}{198679}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "757be645afa325db", "problem_id": "5657bd67d9d2a4a2", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{2} \\sin{\\left(x \\right)} - 3 x \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 3 x \\sin{\\left(x \\right)}) = - 3 x \\cos{\\left(x \\right)}$.\n3) So $f'(x) = 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 3 x \\sin{\\left(x \\right)}) = - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$", "final_answer": "3 x^{2} \\cos{\\left(x \\right)} + 6 x \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} - 3 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "aad48a53c27b80c6", "problem_id": "07d25db6d5d26df6", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{4}{7}$. If it is flipped 16 times, what is the probability of exactly 12 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=16, k=12, p=$\\frac{4}{7}$ and simplify.\n4) Here C(16,12) = 1820.\nFinal answer: $\\frac{353328168960}{4747561509943}$", "final_answer": "\\frac{353328168960}{4747561509943}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b1936538adf02075", "problem_id": "07d25db6d5d26df6", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{4}{7}$. If it is flipped 16 times, what is the probability of exactly 12 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=16, k=12, p=$\\frac{4}{7}$ and simplify.\nHere C(16,12) = 1820.\nCheck: expression simplifies to $\\frac{353328168960}{4747561509943}$.\nFinal answer: $\\frac{353328168960}{4747561509943}$", "final_answer": "\\frac{353328168960}{4747561509943}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fa326f848618e8cd", "problem_id": "071fad801159a5a8", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} x^{3} - x^{2} - 2 x - 5$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to x^{3} - x^{2} - 3 x - 5.\n3) Then the limit is -8.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: -7.\nFinal answer: $-7$", "final_answer": "-7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "886222cbc8db0801", "problem_id": "83bb28a90d06ff0f", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 30 red balls and 28 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(58,8).\n3) Favorable samples: choose k reds and n−k blues: C(30,1)·C(28,7).\n4) Compute totals: C(58,8) = 1916797311, and favorable = 35521200.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{9200}{496451}$", "final_answer": "\\frac{9200}{496451}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c6088b2d4c686576", "problem_id": "3ae33964133bfe4d", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the congruences: x≡0 (mod 8), x≡2 (mod 13). Give the solution modulo 104.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 104.\n3) Compute the CRT solution to get x ≡ 80 (mod 104).\nFinal answer: $x\\equiv 80\\pmod{104}$", "final_answer": "x\\equiv 80\\pmod{104}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c708f2f9ef414dbf", "problem_id": "3ae33964133bfe4d", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡0 (mod 8), x≡2 (mod 13). Give the solution modulo 104.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 104.\nCompute the CRT solution to get x ≡ 80 (mod 104).\nCheck: 80 mod 8=0 and mod 13=2.\nFinal answer: $x\\equiv 80\\pmod{104}$", "final_answer": "x\\equiv 80\\pmod{104}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "741f4a5341593ae1", "problem_id": "3d3f3bd548e3d22c", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡7 (mod 8), x≡5 (mod 13). Give the solution modulo 104.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 104.\nCompute the CRT solution to get x ≡ 31 (mod 104).\nCheck: 31 mod 8=7 and mod 13=5.\nFinal answer: $x\\equiv 31\\pmod{104}$", "final_answer": "x\\equiv 31\\pmod{104}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "afa99d3e24357608", "problem_id": "3d3f3bd548e3d22c", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡7 (mod 8), x≡5 (mod 13). Give the solution modulo 104.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 104.\n3) Compute the CRT solution to get x ≡ 31 (mod 104).\nFinal answer: $x\\equiv 31\\pmod{104}$", "final_answer": "x\\equiv 31\\pmod{104}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "09224f0126f8ead1", "problem_id": "846db50033fe6cc2", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "An urn has 15 red balls and 27 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(42,10).\n3) Favorable samples: choose k reds and n−k blues: C(15,2)·C(27,8).\n4) Compute totals: C(42,10) = 1471442973, and favorable = 233107875.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{77625}{489991}$", "final_answer": "\\frac{77625}{489991}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "07fb54b42be8c3fc", "problem_id": "846db50033fe6cc2", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "An urn has 15 red balls and 27 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(42,10).\nFavorable samples: choose k reds and n−k blues: C(15,2)·C(27,8).\nFinal answer: $\\frac{77625}{489991}$", "final_answer": "\\frac{77625}{489991}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9b1a4535f4357933", "problem_id": "09f4640484a08da4", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "An urn has 11 red balls and 13 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(24,5).\n3) Favorable samples: choose k reds and n−k blues: C(11,1)·C(13,4).\n4) Compute totals: C(24,5) = 42504, and favorable = 7865.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{715}{3864}$", "final_answer": "\\frac{715}{3864}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d94f524f3bc7472", "problem_id": "09f4640484a08da4", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 11 red balls and 13 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(24,5).\nFavorable samples: choose k reds and n−k blues: C(11,1)·C(13,4).\nCompute totals: C(24,5) = 42504, and favorable = 7865.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{715}{3864}$.\nFinal answer: $\\frac{715}{3864}$", "final_answer": "\\frac{715}{3864}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "be721171eea7fa91", "problem_id": "2d33ec124a227c72", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "You want 55 liters of a 35% salt solution. You have a 15% solution and a 60% solution. How many liters of each should you mix?", "output": "Let x = liters of 15% solution, so 55−x liters is 60% solution.\nSalt equation: 15x + 60(55−x) = 35·55.\n(15-60)x = (35-60)·55 ⇒ x = 275/9.\nThen 55−x = 220/9.\nCheck: 15·275/9 + 60·220/9 = 35·55.\nFinal answer: $275/9 L of 15% and 220/9 L of 60%$", "final_answer": "275/9 L of 15% and 220/9 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b02ad44e07f6a98d", "problem_id": "2d33ec124a227c72", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "You want 55 liters of a 35% salt solution. You have a 15% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 15% solution, so 55−x liters is 60% solution.\n2) Salt equation: 15x + 60(55−x) = 35·55.\n3) (15-60)x = (35-60)·55 ⇒ x = 275/9.\n4) Then 55−x = 220/9.\nFinal answer: $275/9 L of 15% and 220/9 L of 60%$", "final_answer": "275/9 L of 15% and 220/9 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2c2b23c2f961d4ff", "problem_id": "046f2807e22ccc9d", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Differentiate with respect to x:\n$f(x)=3 x^{3} \\sin{\\left(x \\right)} - x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.", "output": "1) Let f(x) = $3 x^{3} \\sin{\\left(x \\right)} - x^{2} \\sin{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $3 x^{3} \\cos{\\left(x \\right)} + 9 x^{2} \\sin{\\left(x \\right)} - x^{2} \\cos{\\left(x \\right)} - 2 x \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)}$.\nFinal answer: $3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} + 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - x^{2}\\cdot \\cos{\\left(x \\right)} - 2\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot x\\cdot \\cos{\\left(x \\right)} - 2\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)}$", "final_answer": "3\\cdot x^{3}\\cdot \\cos{\\left(x \\right)} + 9\\cdot x^{2}\\cdot \\sin{\\left(x \\right)} - x^{2}\\cdot \\cos{\\left(x \\right)} - 2\\cdot x\\cdot \\sin{\\left(x \\right)} - 2\\cdot x\\cdot \\cos{\\left(x \\right)} - 2\\cdot \\sin{\\left(x \\right)} - 4\\cdot \\sin{\\left(2\\cdot x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ae06c8edc650afb1", "problem_id": "622f9230972b3858", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to -1} - 3 x^{2} + 5 x - 4$.\n\nStudent solution:\n1) Substitute x=-1 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{2} + 4 x - 4.\n3) Then the limit is -11.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=-1 (polynomial is continuous).\nCorrect limit value: -12.\nFinal answer: $-12$", "final_answer": "-12", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "e7a6135960852e1c", "problem_id": "adfe202ac964bfdc", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{4} + 3 x^{3} + 3 x^{2} - 3 x + 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(3 x^{3}) = 9 x^{3}$.\n3) So $f'(x) = 25 x^{3} + 6 x - 3$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(3 x^{3}) = 9 x^{2}$.\nTherefore the correct derivative is $f'(x) = 16 x^{3} + 9 x^{2} + 6 x - 3$.\nFinal answer: $16 x^{3} + 9 x^{2} + 6 x - 3$", "final_answer": "16 x^{3} + 9 x^{2} + 6 x - 3", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "95ebf757a404e430", "problem_id": "47b155cf20e901f4", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the linear system:\n$4 x - 3 y$ = $16$\n$- 5 x - 5 y$ = $-55$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $-5$: $- 20 x + 15 y$ = $-80$.\nMultiply the second equation by $4$: $- 20 x - 20 y$ = $-220$.\nSubtract to eliminate x: $35 y$ = $140$.\nSolve for y: y = $4$.\nSubstitute back to find x: x = $7$.\nCheck: plug (x,y)=(7,4) into both equations.\nFinal answer: $(x,y)=\\left(7, 4\\right)$", "final_answer": "(x,y)=\\left(7, 4\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "466e0fef3d84cafc", "problem_id": "83a3b0f783dfd312", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 4 x^{3} + 3 x^{2} - 4 x - 1$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 4 x^{3}) = - 12 x^{3}$.\n3) So $f'(x) = - 12 x^{3} + 6 x - 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 4 x^{3}) = - 12 x^{2}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- 6 x^{2} + 3 x - 2\\right)$.\nFinal answer: $2 \\left(- 6 x^{2} + 3 x - 2\\right)$", "final_answer": "2 \\left(- 6 x^{2} + 3 x - 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "7715f4b71ef82fcb", "problem_id": "574ab399bd67014c", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 11 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=11, k=3, p=$\\frac{3}{8}$ and simplify.\n4) Here C(11,3) = 165.\nFinal answer: $\\frac{1740234375}{8589934592}$", "final_answer": "\\frac{1740234375}{8589934592}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fcc0331d9feb2eb4", "problem_id": "2ef3c3984923aac6", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡5 (mod 9), x≡7 (mod 13). Give the solution modulo 117.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 117.\nCompute the CRT solution to get x ≡ 59 (mod 117).\nCheck: 59 mod 9=5 and mod 13=7.\nFinal answer: $x\\equiv 59\\pmod{117}$", "final_answer": "x\\equiv 59\\pmod{117}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "663d3e17b00ea03f", "problem_id": "2ef3c3984923aac6", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the congruences: x≡5 (mod 9), x≡7 (mod 13). Give the solution modulo 117.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 117.\n3) Compute the CRT solution to get x ≡ 59 (mod 117).\nFinal answer: $x\\equiv 59\\pmod{117}$", "final_answer": "x\\equiv 59\\pmod{117}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "04ee0958d728475b", "problem_id": "90b5c9913946f531", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x + 1\\right) e^{x} + \\left(- x^{2} - x - 2\\right) \\cos{\\left(2 x^{2} + 2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- x \\cos{\\left(2 x^{2} + 2 x \\right)}) = x \\sin{\\left(2 x \\left(x + 1\\right) \\right)}$.\n3) So $f'(x) = 4 x^{3} \\sin{\\left(2 x^{2} + 2 x \\right)} + 2 x^{2} \\sin{\\left(2 x^{2} + 2 x \\right)} + 2 x e^{x} + x \\sin{\\left(2 x^{2} + 2 x \\right)} + 8 x \\sin{\\left(2 x^{2} + 2 x \\right)} - 2 x \\cos{\\left(2 x^{2} + 2 x \\right)} + 4 e^{x} + 4 \\sin{\\left(2 x^{2} + 2 x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- x \\cos{\\left(2 x^{2} + 2 x \\right)}) = x \\left(4 x + 2\\right) \\sin{\\left(2 x^{2} + 2 x \\right)} - \\cos{\\left(2 x^{2} + 2 x \\right)}$.\nTherefore the correct derivative is $f'(x) = \\left(- 2 x - 1\\right) \\cos{\\left(2 x^{2} + 2 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 2 \\left(2 x + 1\\right) \\left(x \\left(- x - 1\\right) - 2\\right) \\sin{\\left(2 x^{2} + 2 x \\right)} + 2 e^{x}$.\nFinal answer: $\\left(- 2 x - 1\\right) \\cos{\\left(2 x^{2} + 2 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 2 \\left(2 x + 1\\right) \\left(x \\left(- x - 1\\right) - 2\\right) \\sin{\\left(2 x^{2} + 2 x \\right)} + 2 e^{x}$", "final_answer": "\\left(- 2 x - 1\\right) \\cos{\\left(2 x^{2} + 2 x \\right)} + 2 \\left(x + 1\\right) e^{x} - 2 \\left(2 x + 1\\right) \\left(x \\left(- x - 1\\right) - 2\\right) \\sin{\\left(2 x^{2} + 2 x \\right)} + 2 e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b8b3cf6fb7a40596", "problem_id": "b87aaedf7379ba98", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "An urn has 12 red balls and 7 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(19,8).\n3) Favorable samples: choose k reds and n−k blues: C(12,4)·C(7,4).\n4) Compute totals: C(19,8) = 75582, and favorable = 17325.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1925}{8398}$", "final_answer": "\\frac{1925}{8398}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "65b7cd9affe21120", "problem_id": "fb73f2f1eebf0ec6", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_6=30$?", "output": "$33649$", "final_answer": "33649", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "66a84fbd064afaec", "problem_id": "fb73f2f1eebf0ec6", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_6=30$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_6=18.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=18 and k=6.\n4) Compute C(23,5) = 33649.\nFinal answer: $33649$", "final_answer": "33649", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e835ace73b8d9df9", "problem_id": "54bdc676434eaa21", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 21 red balls and 19 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(40,7).\nFavorable samples: choose k reds and n−k blues: C(21,1)·C(19,6).\nCompute totals: C(40,7) = 18643560, and favorable = 569772.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{147}{4810}$.\nFinal answer: $\\frac{147}{4810}$", "final_answer": "\\frac{147}{4810}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "52d24ada739eedfd", "problem_id": "54bdc676434eaa21", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "An urn has 21 red balls and 19 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(40,7).\n3) Favorable samples: choose k reds and n−k blues: C(21,1)·C(19,6).\n4) Compute totals: C(40,7) = 18643560, and favorable = 569772.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{147}{4810}$", "final_answer": "\\frac{147}{4810}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "47f86611861b2413", "problem_id": "aebb8861a0a1237e", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "An urn has 13 red balls and 10 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(23,6).\n3) Favorable samples: choose k reds and n−k blues: C(13,3)·C(10,3).\n4) Compute totals: C(23,6) = 100947, and favorable = 34320.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1040}{3059}$", "final_answer": "\\frac{1040}{3059}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "acc36c08621983ea", "problem_id": "aebb8861a0a1237e", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 13 red balls and 10 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(23,6).\nFavorable samples: choose k reds and n−k blues: C(13,3)·C(10,3).\nCompute totals: C(23,6) = 100947, and favorable = 34320.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{1040}{3059}$.\nFinal answer: $\\frac{1040}{3059}$", "final_answer": "\\frac{1040}{3059}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0af35260a157e618", "problem_id": "5f0b4a0a5bed4ae8", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡0 (mod 8), x≡4 (mod 13). Give the solution modulo 104.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 104.\n3) Compute the CRT solution to get x ≡ 56 (mod 104).\nFinal answer: $x\\equiv 56\\pmod{104}$", "final_answer": "x\\equiv 56\\pmod{104}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "622469a1461292ea", "problem_id": "5f0b4a0a5bed4ae8", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡0 (mod 8), x≡4 (mod 13). Give the solution modulo 104.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 104.\nCompute the CRT solution to get x ≡ 56 (mod 104).\nCheck: 56 mod 8=0 and mod 13=4.\nFinal answer: $x\\equiv 56\\pmod{104}$", "final_answer": "x\\equiv 56\\pmod{104}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "22c9d60d95668ab1", "problem_id": "4362b873551e3941", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "An urn has 12 red balls and 11 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(23,5).\nFavorable samples: choose k reds and n−k blues: C(12,2)·C(11,3).\nFinal answer: $\\frac{990}{3059}$", "final_answer": "\\frac{990}{3059}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ffb3dd18e0fe4417", "problem_id": "4362b873551e3941", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "An urn has 12 red balls and 11 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(23,5).\n3) Favorable samples: choose k reds and n−k blues: C(12,2)·C(11,3).\n4) Compute totals: C(23,5) = 33649, and favorable = 10890.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{990}{3059}$", "final_answer": "\\frac{990}{3059}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5760b876c4592368", "problem_id": "6e87b57c0d751690", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 3), x≡2 (mod 7). Give the solution modulo 21.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 21.\nCompute the CRT solution to get x ≡ 2 (mod 21).\nCheck: 2 mod 3=2 and mod 7=2.\nFinal answer: $x\\equiv 2\\pmod{21}$", "final_answer": "x\\equiv 2\\pmod{21}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6946ba056e90e622", "problem_id": "6e87b57c0d751690", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the congruences: x≡2 (mod 3), x≡2 (mod 7). Give the solution modulo 21.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 21.\n3) Compute the CRT solution to get x ≡ 2 (mod 21).\nFinal answer: $x\\equiv 2\\pmod{21}$", "final_answer": "x\\equiv 2\\pmod{21}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "20f9be2580aa08cf", "problem_id": "3e98b70675deb8ca", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A cyclist rides at 31 miles per hour for 1 hours, then at 37 miles per hour for 2 hours. How many miles does the cyclist travel in total?", "output": "1) First leg distance = 31 × 1 = 31 miles.\n2) Second leg distance = 37 × 2 = 74 miles.\n3) Total distance = 31 + 74 = 105 miles.\nFinal answer: $105$", "final_answer": "105", "metadata": {"format": "word", "domain": "arithmetic", "topic": "rate_time_distance", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "db655db7eabf6968", "problem_id": "f7d4f6af5c069b4a", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{2} + 3 x - 3$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- x^{2}) = - 2 x^{2}$.\n3) So $f'(x) = 3 - 2 x^{2}$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- x^{2}) = - 2 x$.\nTherefore the correct derivative is $f'(x) = 3 - 2 x$.\nFinal answer: $3 - 2 x$", "final_answer": "3 - 2 x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "265b0feffa73d921", "problem_id": "b55dbfebb935bdd6", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "An urn has 15 red balls and 12 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(27,5).\n3) Favorable samples: choose k reds and n−k blues: C(15,2)·C(12,3).\n4) Compute totals: C(27,5) = 80730, and favorable = 23100.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{770}{2691}$", "final_answer": "\\frac{770}{2691}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5a6d6fbea6d73843", "problem_id": "b55dbfebb935bdd6", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "An urn has 15 red balls and 12 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(27,5).\nFavorable samples: choose k reds and n−k blues: C(15,2)·C(12,3).\nCompute totals: C(27,5) = 80730, and favorable = 23100.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{770}{2691}$.\nFinal answer: $\\frac{770}{2691}$", "final_answer": "\\frac{770}{2691}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d1ffd1885af186c9", "problem_id": "e189efdf110f02ff", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A store sells a backpack for $101. It is discounted by 25%. You buy 6 backpacks. What is the total cost?", "output": "1) Discounted price = 101 × (1 − 25/100) = 101 × 75/100 = 303/4.\n2) Subtotal = 6 × 303/4 = 909/2.\nFinal answer: $\\frac{909}{2}$", "final_answer": "\\frac{909}{2}", "metadata": {"format": "word", "domain": "arithmetic", "topic": "percent_discount", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f86810d3be7d6815", "problem_id": "d8cc166d19e77d96", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 2 x^{3} - 5 x^{2} - 8 x - 14$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 2 x^{3} - 5 x^{2} - 9 x - 14.\n3) Then the limit is -68.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -66.\nFinal answer: $-66$", "final_answer": "-66", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f1d7385a7f04ba65", "problem_id": "0b4e0293ede371eb", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 14 red balls and 14 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(28,5).\n3) Favorable samples: choose k reds and n−k blues: C(14,2)·C(14,3).\n4) Compute totals: C(28,5) = 98280, and favorable = 33124.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{91}{270}$", "final_answer": "\\frac{91}{270}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "410ce834e3207f3f", "problem_id": "0b4e0293ede371eb", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 14 red balls and 14 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(28,5).\nFavorable samples: choose k reds and n−k blues: C(14,2)·C(14,3).\nCompute totals: C(28,5) = 98280, and favorable = 33124.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{91}{270}$.\nFinal answer: $\\frac{91}{270}$", "final_answer": "\\frac{91}{270}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "85bfcc75975560af", "problem_id": "b704b78d11cc6dee", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡3 (mod 5), x≡6 (mod 7). Give the solution modulo 35.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 35.\n3) Compute the CRT solution to get x ≡ 13 (mod 35).\nFinal answer: $x\\equiv 13\\pmod{35}$", "final_answer": "x\\equiv 13\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c7267865bbf4cbd5", "problem_id": "b704b78d11cc6dee", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡3 (mod 5), x≡6 (mod 7). Give the solution modulo 35.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 35.\nCompute the CRT solution to get x ≡ 13 (mod 35).\nCheck: 13 mod 5=3 and mod 7=6.\nFinal answer: $x\\equiv 13\\pmod{35}$", "final_answer": "x\\equiv 13\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "07c81dd471c44508", "problem_id": "b05ba14b152670c3", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡6 (mod 8), x≡7 (mod 11). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 62 (mod 88).\nFinal answer: $x\\equiv 62\\pmod{88}$", "final_answer": "x\\equiv 62\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a287b9c3100892b6", "problem_id": "b05ba14b152670c3", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the congruences: x≡6 (mod 8), x≡7 (mod 11). Give the solution modulo 88.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 88.\nCompute the CRT solution to get x ≡ 62 (mod 88).\nFinal answer: $x\\equiv 62\\pmod{88}$", "final_answer": "x\\equiv 62\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f2b9152d8361288e", "problem_id": "ff9c25a0bacdee7f", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_6=37$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_6=31.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=31 and k=6.\nCompute C(36,5) = 376992.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $376992$", "final_answer": "376992", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1c6cc9107a2656f2", "problem_id": "ff9c25a0bacdee7f", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_6=37$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_6=31.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=31 and k=6.\n4) Compute C(36,5) = 376992.\nFinal answer: $376992$", "final_answer": "376992", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8fa23b38fbe052ce", "problem_id": "8118fdfb606c5b07", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 9 red balls and 11 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(20,5).\nFavorable samples: choose k reds and n−k blues: C(9,1)·C(11,4).\nCompute totals: C(20,5) = 15504, and favorable = 2970.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{495}{2584}$.\nFinal answer: $\\frac{495}{2584}$", "final_answer": "\\frac{495}{2584}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5f03cad8242fe9ec", "problem_id": "8118fdfb606c5b07", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "An urn has 9 red balls and 11 blue balls. If 5 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(20,5).\n3) Favorable samples: choose k reds and n−k blues: C(9,1)·C(11,4).\n4) Compute totals: C(20,5) = 15504, and favorable = 2970.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{495}{2584}$", "final_answer": "\\frac{495}{2584}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "576276b400c0efb3", "problem_id": "32c78add8d7bd0c6", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$3 x^{2} - 3 x - 270$ = 0.", "output": "1) Because the roots are simple, factoring is the fastest method.\n2) Factor the polynomial: $3 x^{2} - 3 x - 270$ = $3 \\left(x - 10\\right) \\left(x + 9\\right)$.\n3) Set each factor equal to 0 and solve each linear equation.\n4) From the first factor: x = $10$.\n5) From the second factor: x = $-9$.\nFinal answer: $x \\in \\{10, -9\\}$", "final_answer": "x \\in \\{10, -9\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d01272b354e4c7b2", "problem_id": "bfa05cd85d8d5b07", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} - 3 x^{3} - 8 x^{2} - 16 x - 30$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to - 3 x^{3} - 8 x^{2} - 17 x - 30.\n3) Then the limit is -120.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: -118.\nFinal answer: $-118$", "final_answer": "-118", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "44edaf6e94b09212", "problem_id": "2549865e501952b5", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the linear system:\n$- 3 x$ = $15$\n$12 x + 11 y$ = $-82$", "output": "Eliminate x by taking a linear combination of the equations.\nMultiply the first equation by $12$: $- 36 x$ = $180$.\nMultiply the second equation by $-3$: $- 36 x - 33 y$ = $246$.\nSubtract to eliminate x: $33 y$ = $-66$.\nSolve for y: y = $-2$.\nSubstitute back to find x: x = $-5$.\nCheck: plug (x,y)=(-5,-2) into both equations.\nFinal answer: $(x,y)=\\left(-5, -2\\right)$", "final_answer": "(x,y)=\\left(-5, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "054247b473b74b4a", "problem_id": "2549865e501952b5", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the linear system:\n$- 3 x$ = $15$\n$12 x + 11 y$ = $-82$", "output": "1) Eliminate x by taking a linear combination of the equations.\n2) Multiply the first equation by $12$: $- 36 x$ = $180$.\n3) Multiply the second equation by $-3$: $- 36 x - 33 y$ = $246$.\n4) Subtract to eliminate x: $33 y$ = $-66$.\n5) Solve for y: y = $-2$.\n6) Substitute back to find x: x = $-5$.\nFinal answer: $(x,y)=\\left(-5, -2\\right)$", "final_answer": "(x,y)=\\left(-5, -2\\right)", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "linear_system_2x2", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "36c8e00304cc1a74", "problem_id": "e79afbd900514b6c", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "You want 64 liters of a 45% salt solution. You have a 20% solution and a 50% solution. How many liters of each should you mix?", "output": "Let x = liters of 20% solution, so 64−x liters is 50% solution.\nSalt equation: 20x + 50(64−x) = 45·64.\n(20-50)x = (45-50)·64 ⇒ x = 32/3.\nThen 64−x = 160/3.\nCheck: 20·32/3 + 50·160/3 = 45·64.\nFinal answer: $32/3 L of 20% and 160/3 L of 50%$", "final_answer": "32/3 L of 20% and 160/3 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0a572fe22ae473ae", "problem_id": "e79afbd900514b6c", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "You want 64 liters of a 45% salt solution. You have a 20% solution and a 50% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 20% solution, so 64−x liters is 50% solution.\n2) Salt equation: 20x + 50(64−x) = 45·64.\n3) (20-50)x = (45-50)·64 ⇒ x = 32/3.\n4) Then 64−x = 160/3.\nFinal answer: $32/3 L of 20% and 160/3 L of 50%$", "final_answer": "32/3 L of 20% and 160/3 L of 50%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c570f1d292a998f1", "problem_id": "f77f9699392d2eca", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡1 (mod 9), x≡2 (mod 10). Give the solution modulo 90.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 90.\nCompute the CRT solution to get x ≡ 82 (mod 90).\nCheck: 82 mod 9=1 and mod 10=2.\nFinal answer: $x\\equiv 82\\pmod{90}$", "final_answer": "x\\equiv 82\\pmod{90}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4a3a4dec99a9ef63", "problem_id": "f77f9699392d2eca", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡1 (mod 9), x≡2 (mod 10). Give the solution modulo 90.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 90.\n3) Compute the CRT solution to get x ≡ 82 (mod 90).\nFinal answer: $x\\equiv 82\\pmod{90}$", "final_answer": "x\\equiv 82\\pmod{90}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "02359514f7e0aba1", "problem_id": "01aa935e080e7121", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡3 (mod 5). Give the solution modulo 15.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 15.\n3) Compute the CRT solution to get x ≡ 13 (mod 15).\nFinal answer: $x\\equiv 13\\pmod{15}$", "final_answer": "x\\equiv 13\\pmod{15}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fc9acb90e7ccc89a", "problem_id": "01aa935e080e7121", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡3 (mod 5). Give the solution modulo 15.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 15.\nCompute the CRT solution to get x ≡ 13 (mod 15).\nCheck: 13 mod 3=1 and mod 5=3.\nFinal answer: $x\\equiv 13\\pmod{15}$", "final_answer": "x\\equiv 13\\pmod{15}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "11d4101c07c6ea82", "problem_id": "a9b833a2631ac6b3", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "An urn has 28 red balls and 12 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(40,10).\n3) Favorable samples: choose k reds and n−k blues: C(28,5)·C(12,5).\n4) Compute totals: C(40,10) = 847660528, and favorable = 77837760.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{34020}{370481}$", "final_answer": "\\frac{34020}{370481}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7336d1e3f7673cfc", "problem_id": "a9b833a2631ac6b3", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "An urn has 28 red balls and 12 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(40,10).\nFavorable samples: choose k reds and n−k blues: C(28,5)·C(12,5).\nCompute totals: C(40,10) = 847660528, and favorable = 77837760.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{34020}{370481}$.\nFinal answer: $\\frac{34020}{370481}$", "final_answer": "\\frac{34020}{370481}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e5c0abcda7f0d2fe", "problem_id": "ccd04c37bdde9670", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 12 times, what is the probability of exactly 9 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=12, k=9, p=$\\frac{3}{8}$ and simplify.\nHere C(12,9) = 220.\nCheck: expression simplifies to $\\frac{135320625}{17179869184}$.\nFinal answer: $\\frac{135320625}{17179869184}$", "final_answer": "\\frac{135320625}{17179869184}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "35dda8ed9eaa740d", "problem_id": "ccd04c37bdde9670", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 12 times, what is the probability of exactly 9 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=9, p=$\\frac{3}{8}$ and simplify.\n4) Here C(12,9) = 220.\nFinal answer: $\\frac{135320625}{17179869184}$", "final_answer": "\\frac{135320625}{17179869184}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "17d3f96facd0d6b1", "problem_id": "0a445f268081b360", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_6=19$?", "output": "Let y_i=x_i-2, then y_i≥0 and y_1+…+y_6=7.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=7 and k=6.\nCompute C(12,5) = 792.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $792$", "final_answer": "792", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "882e0196012b0fc4", "problem_id": "0a445f268081b360", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_6=19$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_6=7.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=7 and k=6.\n4) Compute C(12,5) = 792.\nFinal answer: $792$", "final_answer": "792", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "898e69161ba26b3e", "problem_id": "33a14802f4ebe4d6", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the congruences: x≡1 (mod 11), x≡2 (mod 9). Give the solution modulo 99.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 99.\n3) Compute the CRT solution to get x ≡ 56 (mod 99).\nFinal answer: $x\\equiv 56\\pmod{99}$", "final_answer": "x\\equiv 56\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "847cfbaeecdb5436", "problem_id": "33a14802f4ebe4d6", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡1 (mod 11), x≡2 (mod 9). Give the solution modulo 99.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 99.\nCompute the CRT solution to get x ≡ 56 (mod 99).\nCheck: 56 mod 11=1 and mod 9=2.\nFinal answer: $x\\equiv 56\\pmod{99}$", "final_answer": "x\\equiv 56\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1613d383385105b4", "problem_id": "c24a7a139b8ea49a", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 21 red balls and 22 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(43,7).\nFavorable samples: choose k reds and n−k blues: C(21,1)·C(22,6).\nCompute totals: C(43,7) = 32224114, and favorable = 1566873.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{82467}{1696006}$.\nFinal answer: $\\frac{82467}{1696006}$", "final_answer": "\\frac{82467}{1696006}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1a0a9464dd9b88a3", "problem_id": "c24a7a139b8ea49a", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "An urn has 21 red balls and 22 blue balls. If 7 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(43,7).\n3) Favorable samples: choose k reds and n−k blues: C(21,1)·C(22,6).\n4) Compute totals: C(43,7) = 32224114, and favorable = 1566873.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{82467}{1696006}$", "final_answer": "\\frac{82467}{1696006}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bd3be24545a66fb2", "problem_id": "18e1f403f3e1e80a", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=35$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=35 and k=6.\n4) Compute C(40,5) = 658008.\nFinal answer: $658008$", "final_answer": "658008", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4396afb43661f1b4", "problem_id": "18e1f403f3e1e80a", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=35$?", "output": "$658008$", "final_answer": "658008", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fdf182f5a06d3250", "problem_id": "ca7251de7dd7e0e9", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡3 (mod 11), x≡12 (mod 13). Give the solution modulo 143.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 143.\n3) Compute the CRT solution to get x ≡ 25 (mod 143).\nFinal answer: $x\\equiv 25\\pmod{143}$", "final_answer": "x\\equiv 25\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7e9cbe5adfd74a2d", "problem_id": "ca7251de7dd7e0e9", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡3 (mod 11), x≡12 (mod 13). Give the solution modulo 143.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 143.\nCompute the CRT solution to get x ≡ 25 (mod 143).\nCheck: 25 mod 11=3 and mod 13=12.\nFinal answer: $x\\equiv 25\\pmod{143}$", "final_answer": "x\\equiv 25\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9b509c746d09de7f", "problem_id": "f750d2c4734cd320", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "How many binary strings of length 21 contain exactly 9 ones with no two ones adjacent?", "output": "1) Reserve 8 mandatory zeros between the 9 ones.\n2) This leaves n−k+1 = 13 valid slots for the ones.\n3) Choose the slots: C(13,9) = 715.\nFinal answer: $715$", "final_answer": "715", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "counting_no_adjacent", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d6b278945b14fa6a", "problem_id": "ec3295382b14f1e2", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=\\frac{- 3 x^{3} + 2 x^{2} + 3 x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 1}{3 x + 4}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(\\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 4}) = \\frac{4 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 4}$.\n3) So $f'(x) = - \\frac{36 x^{3}}{18 x^{2} + 48 x + 32} + \\frac{9 x^{2} e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{18 x^{2} e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} + 48 x + 32} - \\frac{60 x^{2}}{18 x^{2} + 48 x + 32} + \\frac{24 x e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{48 x e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{32 x}{18 x^{2} + 48 x + 32} + \\frac{16 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{32 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{18 x^{2} + 48 x + 32} + \\frac{18}{18 x^{2} + 48 x + 32} + \\frac{12 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{9 x^{2} + 24 x + 16} + \\frac{4 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 4} - \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 4} - \\frac{4 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 4}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(\\frac{4 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 4}) = \\frac{2 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{3 x + 4} + \\frac{4 e^{\\frac{x}{2}} \\cos{\\left(x \\right)}}{3 x + 4} - \\frac{12 e^{\\frac{x}{2}} \\sin{\\left(x \\right)}}{\\left(3 x + 4\\right)^{2}}$.\nTherefore the correct derivative is $f'(x) = \\frac{18 x^{3} - 12 x^{2} - 18 x + \\left(3 x + 4\\right) \\left(- 18 x^{2} + 8 x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 6 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6}{2 \\left(3 x + 4\\right)^{2}}$.\nFinal answer: $\\frac{18 x^{3} - 12 x^{2} - 18 x + \\left(3 x + 4\\right) \\left(- 18 x^{2} + 8 x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 6 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6}{2 \\left(3 x + 4\\right)^{2}}$", "final_answer": "\\frac{18 x^{3} - 12 x^{2} - 18 x + \\left(3 x + 4\\right) \\left(- 18 x^{2} + 8 x + \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 2 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\cos{\\left(x \\right)} + 6 e^{\\frac{x}{2}} \\sin{\\left(x \\right)} + 6\\right) - 6 \\left(3 x + 4\\right) e^{\\frac{x}{2}} \\sin{\\left(x \\right)} - 6}{2 \\left(3 x + 4\\right)^{2}}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "ec965b8c6bec5572", "problem_id": "f60ba8b19ca231e7", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 19 times, what is the probability of exactly 11 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=19, k=11, p=$\\frac{1}{5}$ and simplify.\n4) Here C(19,11) = 75582.\nFinal answer: $\\frac{4953341952}{19073486328125}$", "final_answer": "\\frac{4953341952}{19073486328125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b11fc96413f8db18", "problem_id": "f60ba8b19ca231e7", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{5}$. If it is flipped 19 times, what is the probability of exactly 11 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=19, k=11, p=$\\frac{1}{5}$ and simplify.\nFinal answer: $\\frac{4953341952}{19073486328125}$", "final_answer": "\\frac{4953341952}{19073486328125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fc24db0e9b81e476", "problem_id": "b278388e8dd3b228", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 16 red balls and 6 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(22,6).\n3) Favorable samples: choose k reds and n−k blues: C(16,3)·C(6,3).\n4) Compute totals: C(22,6) = 74613, and favorable = 11200.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1600}{10659}$", "final_answer": "\\frac{1600}{10659}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6e7493e39dfccee9", "problem_id": "b278388e8dd3b228", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "An urn has 16 red balls and 6 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(22,6).\nFavorable samples: choose k reds and n−k blues: C(16,3)·C(6,3).\nFinal answer: $\\frac{1600}{10659}$", "final_answer": "\\frac{1600}{10659}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "547dcceb9e1425ac", "problem_id": "646700bcb2d6d300", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "1) Compute an antiderivative F(x) of the integrand $3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\n2) Evaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\n3) Subtract: F(\\frac{\\pi}{2}) − F(0) = $2 + \\pi + \\frac{3 \\pi^{2}}{8} + \\frac{\\pi^{3}}{8}$.\n4) Simplify the result.\nFinal answer: $2 + \\pi + \\frac{3\\cdot \\pi^{2}}{8} + \\frac{\\pi^{3}}{8}$", "final_answer": "2 + \\pi + \\frac{3\\cdot \\pi^{2}}{8} + \\frac{\\pi^{3}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "aa3ea3b8d31d41aa", "problem_id": "646700bcb2d6d300", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Compute the definite integral:\n$\\displaystyle \\int_{0}^{\\frac{\\pi}{2}} 3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2\\,dx$.", "output": "Compute an antiderivative F(x) of the integrand $3 x^{2} + 3 x + 2 \\sin{\\left(x \\right)} - \\cos{\\left(2 x \\right)} + 2$.\nEvaluate at the bounds: F(\\frac{\\pi}{2}) and F(0).\nSubtract: F(\\frac{\\pi}{2}) − F(0) = $2 + \\pi + \\frac{3 \\pi^{2}}{8} + \\frac{\\pi^{3}}{8}$.\nFinal answer: $2 + \\pi + \\frac{3\\cdot \\pi^{2}}{8} + \\frac{\\pi^{3}}{8}$", "final_answer": "2 + \\pi + \\frac{3\\cdot \\pi^{2}}{8} + \\frac{\\pi^{3}}{8}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "definite_integral", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "7962e4b7b1a54d02", "problem_id": "c655d5ecc9d19e4c", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 13 times, what is the probability of exactly 9 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=13, k=9, p=$\\frac{1}{4}$ and simplify.\nHere C(13,9) = 715.\nCheck: expression simplifies to $\\frac{57915}{67108864}$.\nFinal answer: $\\frac{57915}{67108864}$", "final_answer": "\\frac{57915}{67108864}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "72b461aa29c3c2a0", "problem_id": "c655d5ecc9d19e4c", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 13 times, what is the probability of exactly 9 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=13, k=9, p=$\\frac{1}{4}$ and simplify.\n4) Here C(13,9) = 715.\nFinal answer: $\\frac{57915}{67108864}$", "final_answer": "\\frac{57915}{67108864}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c1a874590b049a50", "problem_id": "4f8541eb81577040", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "An urn has 30 red balls and 17 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(47,10).\n3) Favorable samples: choose k reds and n−k blues: C(30,3)·C(17,7).\n4) Compute totals: C(47,10) = 5178066751, and favorable = 78958880.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{552160}{36210257}$", "final_answer": "\\frac{552160}{36210257}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "67cf3008dd1afce4", "problem_id": "4f8541eb81577040", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 30 red balls and 17 blue balls. If 10 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(47,10).\nFavorable samples: choose k reds and n−k blues: C(30,3)·C(17,7).\nCompute totals: C(47,10) = 5178066751, and favorable = 78958880.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{552160}{36210257}$.\nFinal answer: $\\frac{552160}{36210257}$", "final_answer": "\\frac{552160}{36210257}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "90667bd6962f0245", "problem_id": "3efe3f7a693c6b45", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Find the multiplicative inverse of 24 modulo 41.", "output": "1) We want x such that 24x ≡ 1 (mod 41).\n2) Use extended Euclid to find s,t with 24s + 41t = 1.\n3) One Bezout identity is: 24·(12) + 41·(-7) = 1.\n4) Reduce s modulo 41: x ≡ 12 (mod 41).\nFinal answer: $x\\equiv 12\\pmod{41}$", "final_answer": "x\\equiv 12\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "97aaba126ec2ca27", "problem_id": "3efe3f7a693c6b45", "style": "answer_only", "instruction": "Solve the problem and give only the final answer.", "input": "Find the multiplicative inverse of 24 modulo 41.", "output": "$x\\equiv 12\\pmod{41}$", "final_answer": "x\\equiv 12\\pmod{41}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1555a9d2d19f3564", "problem_id": "61b9ceee65d4afb7", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡10 (mod 11), x≡1 (mod 8). Give the solution modulo 88.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 88.\nCompute the CRT solution to get x ≡ 65 (mod 88).\nCheck: 65 mod 11=10 and mod 8=1.\nFinal answer: $x\\equiv 65\\pmod{88}$", "final_answer": "x\\equiv 65\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c18b2a7b72171487", "problem_id": "61b9ceee65d4afb7", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the congruences: x≡10 (mod 11), x≡1 (mod 8). Give the solution modulo 88.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 88.\nCompute the CRT solution to get x ≡ 65 (mod 88).\nFinal answer: $x\\equiv 65\\pmod{88}$", "final_answer": "x\\equiv 65\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "0023ecdbb1be3650", "problem_id": "295e84eba4dac351", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡4 (mod 5). Give the solution modulo 15.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 15.\n3) Compute the CRT solution to get x ≡ 4 (mod 15).\nFinal answer: $x\\equiv 4\\pmod{15}$", "final_answer": "x\\equiv 4\\pmod{15}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4c81a3c62734bc8d", "problem_id": "295e84eba4dac351", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the congruences: x≡1 (mod 3), x≡4 (mod 5). Give the solution modulo 15.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 15.\nCompute the CRT solution to get x ≡ 4 (mod 15).\nFinal answer: $x\\equiv 4\\pmod{15}$", "final_answer": "x\\equiv 4\\pmod{15}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9c54065dfe25d178", "problem_id": "df40fd5b799b119d", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡2 (mod 11), x≡9 (mod 10). Give the solution modulo 110.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 110.\n3) Compute the CRT solution to get x ≡ 79 (mod 110).\nFinal answer: $x\\equiv 79\\pmod{110}$", "final_answer": "x\\equiv 79\\pmod{110}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1e918ffb9c50cbb1", "problem_id": "aed1a0eaae1d9cef", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} 2 x^{3} + x^{2} + 4 x + 6$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to 2 x^{3} + x^{2} + 3 x + 6.\n3) Then the limit is 32.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 34.\nFinal answer: $34$", "final_answer": "34", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "07850f20e66a315e", "problem_id": "a31264c58a3e47f7", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡8 (mod 9), x≡7 (mod 11). Give the solution modulo 99.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 99.\n3) Compute the CRT solution to get x ≡ 62 (mod 99).\nFinal answer: $x\\equiv 62\\pmod{99}$", "final_answer": "x\\equiv 62\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "48af0da9586bb878", "problem_id": "a31264c58a3e47f7", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the congruences: x≡8 (mod 9), x≡7 (mod 11). Give the solution modulo 99.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 99.\nCompute the CRT solution to get x ≡ 62 (mod 99).\nFinal answer: $x\\equiv 62\\pmod{99}$", "final_answer": "x\\equiv 62\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "559470f5e016fff0", "problem_id": "0c169141be83c022", "style": "concise", "instruction": "Provide a compact reasoning and the final answer.", "input": "Solve the congruences: x≡3 (mod 11), x≡0 (mod 10). Give the solution modulo 110.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 110.\nCompute the CRT solution to get x ≡ 80 (mod 110).\nFinal answer: $x\\equiv 80\\pmod{110}$", "final_answer": "x\\equiv 80\\pmod{110}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fc01fa4eebecc7f2", "problem_id": "0c169141be83c022", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡3 (mod 11), x≡0 (mod 10). Give the solution modulo 110.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 110.\n3) Compute the CRT solution to get x ≡ 80 (mod 110).\nFinal answer: $x\\equiv 80\\pmod{110}$", "final_answer": "x\\equiv 80\\pmod{110}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1e81dec3a55913c9", "problem_id": "303abfbf507cb79d", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{2} \\sin{\\left(x \\right)} - 2 \\sin{\\left(x \\right)} + 2 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(- 2 \\sin{\\left(x \\right)}) = - 2 \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(- 2 \\sin{\\left(x \\right)}) = - 2 \\cos{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$.\nFinal answer: $- 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}$", "final_answer": "- 3 x^{2} \\cos{\\left(x \\right)} - 6 x \\sin{\\left(x \\right)} - 4 \\sin{\\left(2 x \\right)} - 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "31de340658a8a9d8", "problem_id": "4a2dc867e87a85fb", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "You want 24 liters of a 25% salt solution. You have a 10% solution and a 60% solution. How many liters of each should you mix?", "output": "1) Let x = liters of 10% solution, so 24−x liters is 60% solution.\n2) Salt equation: 10x + 60(24−x) = 25·24.\n3) (10-60)x = (25-60)·24 ⇒ x = 84/5.\n4) Then 24−x = 36/5.\nFinal answer: $84/5 L of 10% and 36/5 L of 60%$", "final_answer": "84/5 L of 10% and 36/5 L of 60%", "metadata": {"format": "word", "domain": "arithmetic", "topic": "mixture", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c1f228bca0854f67", "problem_id": "cd52d17bb2c3ffc9", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡1 (mod 7), x≡0 (mod 5). Give the solution modulo 35.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 35.\nCompute the CRT solution to get x ≡ 15 (mod 35).\nCheck: 15 mod 7=1 and mod 5=0.\nFinal answer: $x\\equiv 15\\pmod{35}$", "final_answer": "x\\equiv 15\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8eddc5002114d8f0", "problem_id": "cd52d17bb2c3ffc9", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡1 (mod 7), x≡0 (mod 5). Give the solution modulo 35.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 35.\n3) Compute the CRT solution to get x ≡ 15 (mod 35).\nFinal answer: $x\\equiv 15\\pmod{35}$", "final_answer": "x\\equiv 15\\pmod{35}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4bed1684cd08a13c", "problem_id": "1856ee23d6a69feb", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 19 times, what is the probability of exactly 9 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=19, k=9, p=$\\frac{1}{8}$ and simplify.\n4) Here C(19,9) = 92378.\nFinal answer: $\\frac{13047249276061}{72057594037927936}$", "final_answer": "\\frac{13047249276061}{72057594037927936}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e74a83df219a459a", "problem_id": "1856ee23d6a69feb", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{8}$. If it is flipped 19 times, what is the probability of exactly 9 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=19, k=9, p=$\\frac{1}{8}$ and simplify.\nHere C(19,9) = 92378.\nCheck: expression simplifies to $\\frac{13047249276061}{72057594037927936}$.\nFinal answer: $\\frac{13047249276061}{72057594037927936}$", "final_answer": "\\frac{13047249276061}{72057594037927936}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d84b9506915e6c33", "problem_id": "b22d136a7d250dd5", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 23 red balls and 24 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(47,8).\nFavorable samples: choose k reds and n−k blues: C(23,1)·C(24,7).\nCompute totals: C(47,8) = 314457495, and favorable = 7960392.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{10488}{414305}$.\nFinal answer: $\\frac{10488}{414305}$", "final_answer": "\\frac{10488}{414305}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4fe40de84961ab60", "problem_id": "b22d136a7d250dd5", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 23 red balls and 24 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(47,8).\n3) Favorable samples: choose k reds and n−k blues: C(23,1)·C(24,7).\n4) Compute totals: C(47,8) = 314457495, and favorable = 7960392.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{10488}{414305}$", "final_answer": "\\frac{10488}{414305}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ded4373a5a07ef97", "problem_id": "c72038a4faf08663", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡2 (mod 3), x≡0 (mod 8). Give the solution modulo 24.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 24.\n3) Compute the CRT solution to get x ≡ 8 (mod 24).\nFinal answer: $x\\equiv 8\\pmod{24}$", "final_answer": "x\\equiv 8\\pmod{24}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "088f4e3e99bdb66e", "problem_id": "c72038a4faf08663", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡2 (mod 3), x≡0 (mod 8). Give the solution modulo 24.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 24.\nCompute the CRT solution to get x ≡ 8 (mod 24).\nCheck: 8 mod 3=2 and mod 8=0.\nFinal answer: $x\\equiv 8\\pmod{24}$", "final_answer": "x\\equiv 8\\pmod{24}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2fef411ce1d89ea6", "problem_id": "3bcced58a55327b0", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡0 (mod 9), x≡3 (mod 5). Give the solution modulo 45.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 45.\n3) Compute the CRT solution to get x ≡ 18 (mod 45).\nFinal answer: $x\\equiv 18\\pmod{45}$", "final_answer": "x\\equiv 18\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d3fde46d823d11c8", "problem_id": "3bcced58a55327b0", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡0 (mod 9), x≡3 (mod 5). Give the solution modulo 45.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 45.\nCompute the CRT solution to get x ≡ 18 (mod 45).\nCheck: 18 mod 9=0 and mod 5=3.\nFinal answer: $x\\equiv 18\\pmod{45}$", "final_answer": "x\\equiv 18\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a19185d4edd592c0", "problem_id": "afb40cc69eb2ea7e", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 20 times, what is the probability of exactly 10 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=20, k=10, p=$\\frac{2}{7}$ and simplify.\n4) Here C(20,10) = 184756.\nFinal answer: $\\frac{1847560000000000}{79792266297612001}$", "final_answer": "\\frac{1847560000000000}{79792266297612001}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a7151b6b4f88ce4c", "problem_id": "afb40cc69eb2ea7e", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{7}$. If it is flipped 20 times, what is the probability of exactly 10 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=20, k=10, p=$\\frac{2}{7}$ and simplify.\nFinal answer: $\\frac{1847560000000000}{79792266297612001}$", "final_answer": "\\frac{1847560000000000}{79792266297612001}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "46f32d24d1075651", "problem_id": "56ad0eba96ec14b0", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Find the multiplicative inverse of 35 modulo 37.", "output": "1) We want x such that 35x ≡ 1 (mod 37).\n2) Use extended Euclid to find s,t with 35s + 37t = 1.\n3) One Bezout identity is: 35·(18) + 37·(-17) = 1.\n4) Reduce s modulo 37: x ≡ 18 (mod 37).\nFinal answer: $x\\equiv 18\\pmod{37}$", "final_answer": "x\\equiv 18\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fb2630a45dbf7107", "problem_id": "56ad0eba96ec14b0", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Find the multiplicative inverse of 35 modulo 37.", "output": "We want x such that 35x ≡ 1 (mod 37).\nUse extended Euclid to find s,t with 35s + 37t = 1.\nOne Bezout identity is: 35·(18) + 37·(-17) = 1.\nReduce s modulo 37: x ≡ 18 (mod 37).\nCheck: 35·18 mod 37 = 1.\nFinal answer: $x\\equiv 18\\pmod{37}$", "final_answer": "x\\equiv 18\\pmod{37}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "76fb797fa9b89c56", "problem_id": "9ef4bedea90b0247", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(2 x^{2} - 3\\right) \\cos{\\left(x^{2} + x + 3 \\right)}$.", "output": "1) Let f(x) = $2 \\left(x - 3\\right) e^{x} + \\left(2 x^{2} - 3\\right) \\cos{\\left(x^{2} + x + 3 \\right)}$.\n2) Differentiate using product/chain/quotient rules as needed.\n3) Simplify the derivative to get f'(x) = $4 x \\cos{\\left(x^{2} + x + 3 \\right)} + 2 \\left(x - 3\\right) e^{x} - \\left(2 x + 1\\right) \\left(2 x^{2} - 3\\right) \\sin{\\left(x^{2} + x + 3 \\right)} + 2 e^{x}$.\nFinal answer: $4\\cdot x\\cdot \\cos{\\left(x^{2} + x + 3 \\right)} + \\left(2\\cdot x - 6\\right)\\cdot e^{x} - \\left(2\\cdot x + 1\\right)\\cdot \\left(2\\cdot x^{2} - 3\\right)\\cdot \\sin{\\left(x^{2} + x + 3 \\right)} + 2\\cdot e^{x}$", "final_answer": "4\\cdot x\\cdot \\cos{\\left(x^{2} + x + 3 \\right)} + \\left(2\\cdot x - 6\\right)\\cdot e^{x} - \\left(2\\cdot x + 1\\right)\\cdot \\left(2\\cdot x^{2} - 3\\right)\\cdot \\sin{\\left(x^{2} + x + 3 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "59a0cff39b53cb2b", "problem_id": "9ef4bedea90b0247", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 \\left(x - 3\\right) e^{x} + \\left(2 x^{2} - 3\\right) \\cos{\\left(x^{2} + x + 3 \\right)}$.", "output": "Let f(x) = $2 \\left(x - 3\\right) e^{x} + \\left(2 x^{2} - 3\\right) \\cos{\\left(x^{2} + x + 3 \\right)}$.\nDifferentiate using product/chain/quotient rules as needed.\nSimplify the derivative to get f'(x) = $4 x \\cos{\\left(x^{2} + x + 3 \\right)} + 2 \\left(x - 3\\right) e^{x} - \\left(2 x + 1\\right) \\left(2 x^{2} - 3\\right) \\sin{\\left(x^{2} + x + 3 \\right)} + 2 e^{x}$.\nCheck: symbolic differentiation in SymPy matches the derived expression.\nFinal answer: $4\\cdot x\\cdot \\cos{\\left(x^{2} + x + 3 \\right)} + \\left(2\\cdot x - 6\\right)\\cdot e^{x} - \\left(2\\cdot x + 1\\right)\\cdot \\left(2\\cdot x^{2} - 3\\right)\\cdot \\sin{\\left(x^{2} + x + 3 \\right)} + 2\\cdot e^{x}$", "final_answer": "4\\cdot x\\cdot \\cos{\\left(x^{2} + x + 3 \\right)} + \\left(2\\cdot x - 6\\right)\\cdot e^{x} - \\left(2\\cdot x + 1\\right)\\cdot \\left(2\\cdot x^{2} - 3\\right)\\cdot \\sin{\\left(x^{2} + x + 3 \\right)} + 2\\cdot e^{x}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "305ca45c7a6522ab", "problem_id": "6b9ad19b3a6f1b96", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 29 red balls and 14 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 1 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(43,11).\n3) Favorable samples: choose k reds and n−k blues: C(29,1)·C(14,10).\n4) Compute totals: C(43,11) = 5752004349, and favorable = 29029.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{319}{63208839}$", "final_answer": "\\frac{319}{63208839}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "65a227fe7ba44ee0", "problem_id": "4154d331602d7bf5", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} 3 x^{2} + 4 x + 3$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to 3 x^{2} + 3 x + 3.\n3) Then the limit is 9.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: 10.\nFinal answer: $10$", "final_answer": "10", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "16be566916b4d6d6", "problem_id": "b3e6b9de088c963c", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=18$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=18 and k=6.\nCompute C(23,5) = 33649.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $33649$", "final_answer": "33649", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d6bc382c7c09024b", "problem_id": "b3e6b9de088c963c", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=18$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=18 and k=6.\n4) Compute C(23,5) = 33649.\nFinal answer: $33649$", "final_answer": "33649", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cc9b12c24c23a517", "problem_id": "0718941980cd7780", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 15 times, what is the probability of exactly 6 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=15, k=6, p=$\\frac{1}{10}$ and simplify.\nFinal answer: $\\frac{387807909489}{200000000000000}$", "final_answer": "\\frac{387807909489}{200000000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "508945256fde5605", "problem_id": "0718941980cd7780", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{10}$. If it is flipped 15 times, what is the probability of exactly 6 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=15, k=6, p=$\\frac{1}{10}$ and simplify.\n4) Here C(15,6) = 5005.\nFinal answer: $\\frac{387807909489}{200000000000000}$", "final_answer": "\\frac{387807909489}{200000000000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "69413ad05e15b95f", "problem_id": "7cc5fe3a440356b3", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 12 times, what is the probability of exactly 9 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=12, k=9, p=$\\frac{2}{3}$ and simplify.\nFinal answer: $\\frac{112640}{531441}$", "final_answer": "\\frac{112640}{531441}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b301456294df3df3", "problem_id": "7cc5fe3a440356b3", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "A coin lands heads with probability p=$\\frac{2}{3}$. If it is flipped 12 times, what is the probability of exactly 9 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=12, k=9, p=$\\frac{2}{3}$ and simplify.\n4) Here C(12,9) = 220.\nFinal answer: $\\frac{112640}{531441}$", "final_answer": "\\frac{112640}{531441}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2ed4075b6fec43ca", "problem_id": "0192067c9aa035bb", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "An urn has 25 red balls and 18 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(43,12).\n3) Favorable samples: choose k reds and n−k blues: C(25,5)·C(18,7).\n4) Compute totals: C(43,12) = 15338678264, and favorable = 1690809120.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{136620}{1239389}$", "final_answer": "\\frac{136620}{1239389}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9115a7d89f023d39", "problem_id": "0192067c9aa035bb", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "An urn has 25 red balls and 18 blue balls. If 12 balls are drawn without replacement, what is the probability that exactly 5 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(43,12).\nFavorable samples: choose k reds and n−k blues: C(25,5)·C(18,7).\nCompute totals: C(43,12) = 15338678264, and favorable = 1690809120.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{136620}{1239389}$.\nFinal answer: $\\frac{136620}{1239389}$", "final_answer": "\\frac{136620}{1239389}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ea219218d9d80bb9", "problem_id": "ae986cdff3fe460c", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 10 red balls and 10 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(20,6).\n3) Favorable samples: choose k reds and n−k blues: C(10,2)·C(10,4).\n4) Compute totals: C(20,6) = 38760, and favorable = 9450.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{315}{1292}$", "final_answer": "\\frac{315}{1292}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5ed422ee64b333e8", "problem_id": "ae986cdff3fe460c", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "An urn has 10 red balls and 10 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(20,6).\nFavorable samples: choose k reds and n−k blues: C(10,2)·C(10,4).\nFinal answer: $\\frac{315}{1292}$", "final_answer": "\\frac{315}{1292}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "01cae1005f0cbcd5", "problem_id": "f29af530ecffa939", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{5}$. If it is flipped 6 times, what is the probability of exactly 3 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=6, k=3, p=$\\frac{3}{5}$ and simplify.\n4) Here C(6,3) = 20.\nFinal answer: $\\frac{864}{3125}$", "final_answer": "\\frac{864}{3125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "806549846ad44fc3", "problem_id": "f29af530ecffa939", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{5}$. If it is flipped 6 times, what is the probability of exactly 3 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=6, k=3, p=$\\frac{3}{5}$ and simplify.\nFinal answer: $\\frac{864}{3125}$", "final_answer": "\\frac{864}{3125}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "efbca7816940b6fc", "problem_id": "391bf82ada6159a6", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the congruences: x≡8 (mod 11), x≡1 (mod 7). Give the solution modulo 77.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 77.\nCompute the CRT solution to get x ≡ 8 (mod 77).\nFinal answer: $x\\equiv 8\\pmod{77}$", "final_answer": "x\\equiv 8\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9d0771af38984635", "problem_id": "391bf82ada6159a6", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡8 (mod 11), x≡1 (mod 7). Give the solution modulo 77.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 77.\n3) Compute the CRT solution to get x ≡ 8 (mod 77).\nFinal answer: $x\\equiv 8\\pmod{77}$", "final_answer": "x\\equiv 8\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "928516f591f8b77e", "problem_id": "63d42fd31aa3bc5b", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 21 times, what is the probability of exactly 7 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=21, k=7, p=$\\frac{3}{8}$ and simplify.\n4) Here C(21,7) = 116280.\nFinal answer: $\\frac{194018829345703125}{1152921504606846976}$", "final_answer": "\\frac{194018829345703125}{1152921504606846976}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8f04a568558cf00b", "problem_id": "63d42fd31aa3bc5b", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{8}$. If it is flipped 21 times, what is the probability of exactly 7 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=21, k=7, p=$\\frac{3}{8}$ and simplify.\nHere C(21,7) = 116280.\nCheck: expression simplifies to $\\frac{194018829345703125}{1152921504606846976}$.\nFinal answer: $\\frac{194018829345703125}{1152921504606846976}$", "final_answer": "\\frac{194018829345703125}{1152921504606846976}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fd213cabdbabbd7f", "problem_id": "5dc4cd35c84c7b79", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "An urn has 30 red balls and 15 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(45,8).\n3) Favorable samples: choose k reds and n−k blues: C(30,2)·C(15,6).\n4) Compute totals: C(45,8) = 215553195, and favorable = 2177175.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{1015}{100491}$", "final_answer": "\\frac{1015}{100491}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ab6757378dbb566a", "problem_id": "5dc4cd35c84c7b79", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "An urn has 30 red balls and 15 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(45,8).\nFavorable samples: choose k reds and n−k blues: C(30,2)·C(15,6).\nFinal answer: $\\frac{1015}{100491}$", "final_answer": "\\frac{1015}{100491}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "648238482905ecb7", "problem_id": "7e785b8444c11f2a", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡3 (mod 4), x≡4 (mod 11). Give the solution modulo 44.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 44.\n3) Compute the CRT solution to get x ≡ 15 (mod 44).\nFinal answer: $x\\equiv 15\\pmod{44}$", "final_answer": "x\\equiv 15\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ac5b1a2090ad9854", "problem_id": "7e785b8444c11f2a", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the congruences: x≡3 (mod 4), x≡4 (mod 11). Give the solution modulo 44.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 44.\nCompute the CRT solution to get x ≡ 15 (mod 44).\nFinal answer: $x\\equiv 15\\pmod{44}$", "final_answer": "x\\equiv 15\\pmod{44}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "913cf6c204438f14", "problem_id": "297acbbd73f89cdd", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "Solve the congruences: x≡2 (mod 5), x≡3 (mod 11). Give the solution modulo 55.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 55.\n3) Compute the CRT solution to get x ≡ 47 (mod 55).\nFinal answer: $x\\equiv 47\\pmod{55}$", "final_answer": "x\\equiv 47\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d746d43d4e6a7ff3", "problem_id": "297acbbd73f89cdd", "style": "concise", "instruction": "Compute the result with a short explanation and give the final answer.", "input": "Solve the congruences: x≡2 (mod 5), x≡3 (mod 11). Give the solution modulo 55.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 55.\nCompute the CRT solution to get x ≡ 47 (mod 55).\nFinal answer: $x\\equiv 47\\pmod{55}$", "final_answer": "x\\equiv 47\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f02886d897917ba7", "problem_id": "a1ee05b52a7bc7b1", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the congruences: x≡4 (mod 11), x≡6 (mod 10). Give the solution modulo 110.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 110.\nCompute the CRT solution to get x ≡ 26 (mod 110).\nFinal answer: $x\\equiv 26\\pmod{110}$", "final_answer": "x\\equiv 26\\pmod{110}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a74f0404f32699a2", "problem_id": "a1ee05b52a7bc7b1", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡4 (mod 11), x≡6 (mod 10). Give the solution modulo 110.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 110.\n3) Compute the CRT solution to get x ≡ 26 (mod 110).\nFinal answer: $x\\equiv 26\\pmod{110}$", "final_answer": "x\\equiv 26\\pmod{110}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "08bffa1716633bb6", "problem_id": "62d8adacb7505d93", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_5=37$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=37 and k=5.\n4) Compute C(41,4) = 101270.\nFinal answer: $101270$", "final_answer": "101270", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9ce5153d324c62c5", "problem_id": "660f83bc358400a6", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=4 x^{3} - 3 x^{2} - 2$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(4 x^{3}) = 12 x^{3}$.\n3) So $f'(x) = 12 x^{3} - 6 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(4 x^{3}) = 12 x^{2}$.\nTherefore the correct derivative is $f'(x) = 6 x \\left(2 x - 1\\right)$.\nFinal answer: $6 x \\left(2 x - 1\\right)$", "final_answer": "6 x \\left(2 x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "316ac139ddd765e6", "problem_id": "5b38505b62b60b20", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 11 red balls and 10 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(21,6).\n3) Favorable samples: choose k reds and n−k blues: C(11,2)·C(10,4).\n4) Compute totals: C(21,6) = 54264, and favorable = 11550.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{275}{1292}$", "final_answer": "\\frac{275}{1292}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "51ef633db56f7fc0", "problem_id": "5b38505b62b60b20", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "An urn has 11 red balls and 10 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(21,6).\nFavorable samples: choose k reds and n−k blues: C(11,2)·C(10,4).\nFinal answer: $\\frac{275}{1292}$", "final_answer": "\\frac{275}{1292}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "556fdc78244e131a", "problem_id": "2a1ae2dc844d3766", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "How many solutions in integers with $x_i\\ge 2$ satisfy $x_1+\\cdots+x_9=43$?", "output": "1) Let y_i=x_i-2, then y_i≥0 and y_1+…+y_9=25.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=25 and k=9.\n4) Compute C(33,8) = 13884156.\nFinal answer: $13884156$", "final_answer": "13884156", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "60588b03b3be929f", "problem_id": "5baffeeac5a4af65", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{2} - 4$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- x^{2}) = - 2 x^{2}$.\n3) So $f'(x) = - 2 x^{2}$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- x^{2}) = - 2 x$.\nTherefore the correct derivative is $f'(x) = - 2 x$.\nFinal answer: $- 2 x$", "final_answer": "- 2 x", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "f20069e12120615d", "problem_id": "53c996d8859059ee", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 25 red balls and 23 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 4 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(48,8).\nFavorable samples: choose k reds and n−k blues: C(25,4)·C(23,4).\nCompute totals: C(48,8) = 377348994, and favorable = 112015750.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{221375}{745749}$.\nFinal answer: $\\frac{221375}{745749}$", "final_answer": "\\frac{221375}{745749}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "583429959354af03", "problem_id": "5271d56e18843f66", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 4 x^{2} - 4 x - 2$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 4 x) = - 4 x$.\n3) So $f'(x) = - 12 x$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 4 x) = -4$.\nTherefore the correct derivative is $f'(x) = 4 \\left(- 2 x - 1\\right)$.\nFinal answer: $4 \\left(- 2 x - 1\\right)$", "final_answer": "4 \\left(- 2 x - 1\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "df551036224de0e1", "problem_id": "55df449ba7f9a447", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{16 x}{5} - \\frac{42}{5}$ = 0.", "output": "1) Identify coefficients a, b, c in ax^2+bx+c=0.\n2) Here a=$2$, b=$- \\frac{16}{5}$, c=$- \\frac{42}{5}$.\n3) Compute the discriminant Δ=b^2−4ac.\n4) Δ = $\\frac{1936}{25}$, so √Δ = $\\frac{44}{5}$.\n5) Apply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\n6) x1 = $3$.\n7) x2 = $- \\frac{7}{5}$.\n8) Collect the solutions into a set.\nFinal answer: $x \\in \\{3, - \\frac{7}{5}\\}$", "final_answer": "x \\in \\{3, - \\frac{7}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "af4aa422324aaf70", "problem_id": "55df449ba7f9a447", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the quadratic equation over the real numbers:\n$2 x^{2} - \\frac{16 x}{5} - \\frac{42}{5}$ = 0.", "output": "Identify coefficients a, b, c in ax^2+bx+c=0.\nHere a=$2$, b=$- \\frac{16}{5}$, c=$- \\frac{42}{5}$.\nCompute the discriminant Δ=b^2−4ac.\nΔ = $\\frac{1936}{25}$, so √Δ = $\\frac{44}{5}$.\nApply the quadratic formula $x=\\frac{-b\\pm\\sqrt{\\Delta}}{2a}$.\nx1 = $3$.\nx2 = $- \\frac{7}{5}$.\nCollect the solutions into a set.\nCheck: substituting each solution into the polynomial gives 0.\nFinal answer: $x \\in \\{3, - \\frac{7}{5}\\}$", "final_answer": "x \\in \\{3, - \\frac{7}{5}\\}", "metadata": {"format": "symbolic", "domain": "algebra", "topic": "quadratic_equation", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "45be89b4e3f5c032", "problem_id": "db881649275da537", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the congruences: x≡6 (mod 7), x≡8 (mod 10). Give the solution modulo 70.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 70.\nCompute the CRT solution to get x ≡ 48 (mod 70).\nFinal answer: $x\\equiv 48\\pmod{70}$", "final_answer": "x\\equiv 48\\pmod{70}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fa3a319778421b2f", "problem_id": "db881649275da537", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡6 (mod 7), x≡8 (mod 10). Give the solution modulo 70.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 70.\n3) Compute the CRT solution to get x ≡ 48 (mod 70).\nFinal answer: $x\\equiv 48\\pmod{70}$", "final_answer": "x\\equiv 48\\pmod{70}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a589f6bae048c6ec", "problem_id": "80b56951684bb166", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "An urn has 24 red balls and 23 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(47,8).\n3) Favorable samples: choose k reds and n−k blues: C(24,2)·C(23,6).\n4) Compute totals: C(47,8) = 314457495, and favorable = 27861372.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{36708}{414305}$", "final_answer": "\\frac{36708}{414305}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2fcf1ecfa3b2b12f", "problem_id": "80b56951684bb166", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "An urn has 24 red balls and 23 blue balls. If 8 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(47,8).\nFavorable samples: choose k reds and n−k blues: C(24,2)·C(23,6).\nCompute totals: C(47,8) = 314457495, and favorable = 27861372.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{36708}{414305}$.\nFinal answer: $\\frac{36708}{414305}$", "final_answer": "\\frac{36708}{414305}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "01b437df633d47b4", "problem_id": "354c9edb8bba55e3", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 7 times, what is the probability of exactly 2 heads?", "output": "1) Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\n2) Use $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\n3) Substitute n=7, k=2, p=$\\frac{1}{4}$ and simplify.\n4) Here C(7,2) = 21.\nFinal answer: $\\frac{5103}{16384}$", "final_answer": "\\frac{5103}{16384}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "47070b67c331a32f", "problem_id": "354c9edb8bba55e3", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{1}{4}$. If it is flipped 7 times, what is the probability of exactly 2 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=7, k=2, p=$\\frac{1}{4}$ and simplify.\nHere C(7,2) = 21.\nCheck: expression simplifies to $\\frac{5103}{16384}$.\nFinal answer: $\\frac{5103}{16384}$", "final_answer": "\\frac{5103}{16384}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5330a76ab9aa279d", "problem_id": "f3fbe5f75fc6bb02", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "An urn has 25 red balls and 14 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(39,11).\n3) Favorable samples: choose k reds and n−k blues: C(25,2)·C(14,9).\n4) Compute totals: C(39,11) = 1676056044, and favorable = 600600.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{3850}{10743949}$", "final_answer": "\\frac{3850}{10743949}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8902239c0e20a21d", "problem_id": "f3fbe5f75fc6bb02", "style": "answer_only", "instruction": "Provide only the final answer (no steps).", "input": "An urn has 25 red balls and 14 blue balls. If 11 balls are drawn without replacement, what is the probability that exactly 2 are red?", "output": "$\\frac{3850}{10743949}$", "final_answer": "\\frac{3850}{10743949}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "19dfba3cf6b38ba3", "problem_id": "fbd5cd6ebb3e228b", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "An urn has 13 red balls and 9 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "1) Use a hypergeometric counting argument.\n2) Total equally likely samples: C(22,6).\n3) Favorable samples: choose k reds and n−k blues: C(13,3)·C(9,3).\n4) Compute totals: C(22,6) = 74613, and favorable = 24024.\n5) Divide favorable by total and simplify.\nFinal answer: $\\frac{104}{323}$", "final_answer": "\\frac{104}{323}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ef784187eeeef03f", "problem_id": "fbd5cd6ebb3e228b", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "An urn has 13 red balls and 9 blue balls. If 6 balls are drawn without replacement, what is the probability that exactly 3 are red?", "output": "Use a hypergeometric counting argument.\nTotal equally likely samples: C(22,6).\nFavorable samples: choose k reds and n−k blues: C(13,3)·C(9,3).\nCompute totals: C(22,6) = 74613, and favorable = 24024.\nDivide favorable by total and simplify.\nCheck: computed probability simplifies to $\\frac{104}{323}$.\nFinal answer: $\\frac{104}{323}$", "final_answer": "\\frac{104}{323}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "hypergeometric", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fb700dfef2344ace", "problem_id": "ef8b18d9cfe420e7", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Find the multiplicative inverse of 12 modulo 31.", "output": "1) We want x such that 12x ≡ 1 (mod 31).\n2) Use extended Euclid to find s,t with 12s + 31t = 1.\n3) One Bezout identity is: 12·(13) + 31·(-5) = 1.\n4) Reduce s modulo 31: x ≡ 13 (mod 31).\nFinal answer: $x\\equiv 13\\pmod{31}$", "final_answer": "x\\equiv 13\\pmod{31}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d68a959fd3c9d57f", "problem_id": "d3b067b3b14a6674", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- 3 x^{3} \\sin{\\left(x \\right)} + 2 x \\sin{\\left(x \\right)} + \\sin{\\left(x \\right)} + 3 \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x \\sin{\\left(x \\right)}) = 2 x \\cos{\\left(x \\right)}$.\n3) So $f'(x) = - 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x \\sin{\\left(x \\right)}) = 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = - 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$.\nFinal answer: $- 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}$", "final_answer": "- 3 x^{3} \\cos{\\left(x \\right)} - 9 x^{2} \\sin{\\left(x \\right)} + 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} - 6 \\sin{\\left(2 x \\right)} + \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d3a338f673515d08", "problem_id": "b39c1a91a816552e", "style": "concise", "instruction": "Solve and give a compact solution, then the final answer.", "input": "Solve the congruences: x≡3 (mod 11), x≡7 (mod 13). Give the solution modulo 143.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 143.\nCompute the CRT solution to get x ≡ 124 (mod 143).\nFinal answer: $x\\equiv 124\\pmod{143}$", "final_answer": "x\\equiv 124\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "9e24bfdbe89985c1", "problem_id": "b39c1a91a816552e", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡3 (mod 11), x≡7 (mod 13). Give the solution modulo 143.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 143.\nCompute the CRT solution to get x ≡ 124 (mod 143).\nCheck: 124 mod 11=3 and mod 13=7.\nFinal answer: $x\\equiv 124\\pmod{143}$", "final_answer": "x\\equiv 124\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a0545474f6e88a58", "problem_id": "989df28cc09af7b6", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_4=16$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=16 and k=4.\n4) Compute C(19,3) = 969.\nFinal answer: $969$", "final_answer": "969", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "55cc091e4cf6907e", "problem_id": "989df28cc09af7b6", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_4=16$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=16 and k=4.\nCompute C(19,3) = 969.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $969$", "final_answer": "969", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bbb586c445d455f1", "problem_id": "34f54e5ed7a5834a", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡0 (mod 3), x≡5 (mod 7). Give the solution modulo 21.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 21.\n3) Compute the CRT solution to get x ≡ 12 (mod 21).\nFinal answer: $x\\equiv 12\\pmod{21}$", "final_answer": "x\\equiv 12\\pmod{21}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "a651593380a05353", "problem_id": "34f54e5ed7a5834a", "style": "verifier", "instruction": "Solve the problem, then verify the result with a quick check. Provide the final answer.", "input": "Solve the congruences: x≡0 (mod 3), x≡5 (mod 7). Give the solution modulo 21.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 21.\nCompute the CRT solution to get x ≡ 12 (mod 21).\nCheck: 12 mod 3=0 and mod 7=5.\nFinal answer: $x\\equiv 12\\pmod{21}$", "final_answer": "x\\equiv 12\\pmod{21}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e1fd938c8004754d", "problem_id": "8bb6c802a0bd2cdf", "style": "tutor", "instruction": "Show your work in clear steps (with intermediate values) and finish with a final answer.", "input": "Solve the congruences: x≡1 (mod 5), x≡0 (mod 11). Give the solution modulo 55.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 55.\n3) Compute the CRT solution to get x ≡ 11 (mod 55).\nFinal answer: $x\\equiv 11\\pmod{55}$", "final_answer": "x\\equiv 11\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5d7bebd0eb5e0f77", "problem_id": "5377178775cdd578", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=- x^{2} - 4 x$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- x^{2}) = - 2 x^{2}$.\n3) So $f'(x) = - 2 x^{2} - 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- x^{2}) = - 2 x$.\nTherefore the correct derivative is $f'(x) = 2 \\left(- x - 2\\right)$.\nFinal answer: $2 \\left(- x - 2\\right)$", "final_answer": "2 \\left(- x - 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "cf9b324192e833ce", "problem_id": "90c50fcb74a6ea49", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_7=43$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=43 and k=7.\n4) Compute C(49,6) = 13983816.\nFinal answer: $13983816$", "final_answer": "13983816", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "475cc78c267d0d72", "problem_id": "9d4cd9c05c99030c", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 2} x^{3} + 4 x^{2} + 7 x + 13$.\n\nStudent solution:\n1) Substitute x=2 into the expression.\n2) (Mistake) Simplify the expression to x^{3} + 4 x^{2} + 6 x + 13.\n3) Then the limit is 49.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=2 (polynomial is continuous).\nCorrect limit value: 51.\nFinal answer: $51$", "final_answer": "51", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "b08f1e64150350e3", "problem_id": "0e76c1b819c8c9ff", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡7 (mod 11), x≡0 (mod 5). Give the solution modulo 55.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 55.\n3) Compute the CRT solution to get x ≡ 40 (mod 55).\nFinal answer: $x\\equiv 40\\pmod{55}$", "final_answer": "x\\equiv 40\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d5a88b52e2adf857", "problem_id": "0e76c1b819c8c9ff", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡7 (mod 11), x≡0 (mod 5). Give the solution modulo 55.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 55.\nCompute the CRT solution to get x ≡ 40 (mod 55).\nCheck: 40 mod 11=7 and mod 5=0.\nFinal answer: $x\\equiv 40\\pmod{55}$", "final_answer": "x\\equiv 40\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b0259f5aa1c158dd", "problem_id": "7e8ae5a1ffa3976d", "style": "concise", "instruction": "Provide a brief derivation and the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{10}$. If it is flipped 9 times, what is the probability of exactly 7 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=9, k=7, p=$\\frac{3}{10}$ and simplify.\nFinal answer: $\\frac{964467}{250000000}$", "final_answer": "\\frac{964467}{250000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3b4e17f9be4b4a0d", "problem_id": "7e8ae5a1ffa3976d", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "A coin lands heads with probability p=$\\frac{3}{10}$. If it is flipped 9 times, what is the probability of exactly 7 heads?", "output": "Let X be the number of heads in n independent flips. Then X ~ Bin(n,p).\nUse $P(X=k)=\\binom{n}{k}p^k(1-p)^{n-k}$.\nSubstitute n=9, k=7, p=$\\frac{3}{10}$ and simplify.\nHere C(9,7) = 36.\nCheck: expression simplifies to $\\frac{964467}{250000000}$.\nFinal answer: $\\frac{964467}{250000000}$", "final_answer": "\\frac{964467}{250000000}", "metadata": {"format": "symbolic", "domain": "probability", "topic": "binomial_probability", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "874a852da4c09ac2", "problem_id": "230da4c41ab57047", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡5 (mod 11), x≡0 (mod 13). Give the solution modulo 143.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 143.\n3) Compute the CRT solution to get x ≡ 104 (mod 143).\nFinal answer: $x\\equiv 104\\pmod{143}$", "final_answer": "x\\equiv 104\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4e025f3483e75dd5", "problem_id": "230da4c41ab57047", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡5 (mod 11), x≡0 (mod 13). Give the solution modulo 143.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 143.\nCompute the CRT solution to get x ≡ 104 (mod 143).\nCheck: 104 mod 11=5 and mod 13=0.\nFinal answer: $x\\equiv 104\\pmod{143}$", "final_answer": "x\\equiv 104\\pmod{143}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f74f600273dab56e", "problem_id": "46c842cba097b97b", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡3 (mod 4), x≡1 (mod 9). Give the solution modulo 36.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 36.\n3) Compute the CRT solution to get x ≡ 19 (mod 36).\nFinal answer: $x\\equiv 19\\pmod{36}$", "final_answer": "x\\equiv 19\\pmod{36}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ffa6cfec71bcd36e", "problem_id": "46c842cba097b97b", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡3 (mod 4), x≡1 (mod 9). Give the solution modulo 36.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 36.\nCompute the CRT solution to get x ≡ 19 (mod 36).\nCheck: 19 mod 4=3 and mod 9=1.\nFinal answer: $x\\equiv 19\\pmod{36}$", "final_answer": "x\\equiv 19\\pmod{36}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6cf606d7395ff53a", "problem_id": "a03cecb587f179af", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=13$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=13 and k=6.\n4) Compute C(18,5) = 8568.\nFinal answer: $8568$", "final_answer": "8568", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f3b6697b20b77372", "problem_id": "a03cecb587f179af", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=13$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=13 and k=6.\nCompute C(18,5) = 8568.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $8568$", "final_answer": "8568", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d279fcab7547e212", "problem_id": "0d4699c29e15c8be", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡3 (mod 4), x≡0 (mod 7). Give the solution modulo 28.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 28.\n3) Compute the CRT solution to get x ≡ 7 (mod 28).\nFinal answer: $x\\equiv 7\\pmod{28}$", "final_answer": "x\\equiv 7\\pmod{28}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e53347a5420acddd", "problem_id": "e1636336dac3aceb", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{4} - 2 x^{3} + 2 x^{2} + 4 x + 2$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim: one term $\\frac{d}{dx}(- 2 x^{3}) = - 6 x^{3}$.\n3) So $f'(x) = 2 x^{3} + 4 x + 4$.", "output": "First incorrect step: 2\nThe error is differentiating a power: the exponent must decrease by 1.\nCorrect: $\\frac{d}{dx}(- 2 x^{3}) = - 6 x^{2}$.\nTherefore the correct derivative is $f'(x) = 2 \\left(4 x^{3} - 3 x^{2} + 2 x + 2\\right)$.\nFinal answer: $2 \\left(4 x^{3} - 3 x^{2} + 2 x + 2\\right)$", "final_answer": "2 \\left(4 x^{3} - 3 x^{2} + 2 x + 2\\right)", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "d4e9f019f9bc5cc6", "problem_id": "5948e360f6cfcc01", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "Solve the congruences: x≡3 (mod 5), x≡4 (mod 9). Give the solution modulo 45.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 45.\n3) Compute the CRT solution to get x ≡ 13 (mod 45).\nFinal answer: $x\\equiv 13\\pmod{45}$", "final_answer": "x\\equiv 13\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4963720c1fed8ead", "problem_id": "5948e360f6cfcc01", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡3 (mod 5), x≡4 (mod 9). Give the solution modulo 45.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 45.\nCompute the CRT solution to get x ≡ 13 (mod 45).\nCheck: 13 mod 5=3 and mod 9=4.\nFinal answer: $x\\equiv 13\\pmod{45}$", "final_answer": "x\\equiv 13\\pmod{45}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "71ad8a9b3cd99352", "problem_id": "07a26bdf3d31706b", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "Find the multiplicative inverse of 6 modulo 29.", "output": "1) We want x such that 6x ≡ 1 (mod 29).\n2) Use extended Euclid to find s,t with 6s + 29t = 1.\n3) One Bezout identity is: 6·(5) + 29·(-1) = 1.\n4) Reduce s modulo 29: x ≡ 5 (mod 29).\nFinal answer: $x\\equiv 5\\pmod{29}$", "final_answer": "x\\equiv 5\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "3fd23abc27c872db", "problem_id": "07a26bdf3d31706b", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Find the multiplicative inverse of 6 modulo 29.", "output": "We want x such that 6x ≡ 1 (mod 29).\nUse extended Euclid to find s,t with 6s + 29t = 1.\nOne Bezout identity is: 6·(5) + 29·(-1) = 1.\nReduce s modulo 29: x ≡ 5 (mod 29).\nCheck: 6·5 mod 29 = 1.\nFinal answer: $x\\equiv 5\\pmod{29}$", "final_answer": "x\\equiv 5\\pmod{29}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "471220ec90b1c5c9", "problem_id": "dc4a53249735b962", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_8=41$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=41 and k=8.\nCompute C(48,7) = 73629072.\nCheck: matches the standard stars-and-bars formula.\nFinal answer: $73629072$", "final_answer": "73629072", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "2053f4f46064d47c", "problem_id": "dc4a53249735b962", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_8=41$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=41 and k=8.\n4) Compute C(48,7) = 73629072.\nFinal answer: $73629072$", "final_answer": "73629072", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "dfabd15101e0dbc8", "problem_id": "796dda9e4de67fe2", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Find the multiplicative inverse of 24 modulo 53.", "output": "1) We want x such that 24x ≡ 1 (mod 53).\n2) Use extended Euclid to find s,t with 24s + 53t = 1.\n3) One Bezout identity is: 24·(-11) + 53·(5) = 1.\n4) Reduce s modulo 53: x ≡ 42 (mod 53).\nFinal answer: $x\\equiv 42\\pmod{53}$", "final_answer": "x\\equiv 42\\pmod{53}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b9919886fc13f190", "problem_id": "796dda9e4de67fe2", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Find the multiplicative inverse of 24 modulo 53.", "output": "We want x such that 24x ≡ 1 (mod 53).\nUse extended Euclid to find s,t with 24s + 53t = 1.\nOne Bezout identity is: 24·(-11) + 53·(5) = 1.\nReduce s modulo 53: x ≡ 42 (mod 53).\nCheck: 24·42 mod 53 = 1.\nFinal answer: $x\\equiv 42\\pmod{53}$", "final_answer": "x\\equiv 42\\pmod{53}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "modular_inverse", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c4a468e7fa244e12", "problem_id": "0f668de8c40d7715", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡3 (mod 5), x≡2 (mod 11). Give the solution modulo 55.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 55.\n3) Compute the CRT solution to get x ≡ 13 (mod 55).\nFinal answer: $x\\equiv 13\\pmod{55}$", "final_answer": "x\\equiv 13\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "f067218191dda17a", "problem_id": "0f668de8c40d7715", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡3 (mod 5), x≡2 (mod 11). Give the solution modulo 55.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 55.\nCompute the CRT solution to get x ≡ 13 (mod 55).\nCheck: 13 mod 5=3 and mod 11=2.\nFinal answer: $x\\equiv 13\\pmod{55}$", "final_answer": "x\\equiv 13\\pmod{55}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1663cbbe08cede96", "problem_id": "fea9e13c4d62a1b4", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the congruences: x≡0 (mod 11), x≡3 (mod 9). Give the solution modulo 99.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 99.\n3) Compute the CRT solution to get x ≡ 66 (mod 99).\nFinal answer: $x\\equiv 66\\pmod{99}$", "final_answer": "x\\equiv 66\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d908e64ee75bc185", "problem_id": "fea9e13c4d62a1b4", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡0 (mod 11), x≡3 (mod 9). Give the solution modulo 99.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 99.\nCompute the CRT solution to get x ≡ 66 (mod 99).\nCheck: 66 mod 11=0 and mod 9=3.\nFinal answer: $x\\equiv 66\\pmod{99}$", "final_answer": "x\\equiv 66\\pmod{99}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "cc815ec26695f575", "problem_id": "67f008ad6d8fa665", "style": "tutor", "instruction": "Show a full solution with explicit computations and a final answer line.", "input": "Solve the congruences: x≡0 (mod 8), x≡4 (mod 5). Give the solution modulo 40.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 40.\n3) Compute the CRT solution to get x ≡ 24 (mod 40).\nFinal answer: $x\\equiv 24\\pmod{40}$", "final_answer": "x\\equiv 24\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8fafd4f614ab175c", "problem_id": "67f008ad6d8fa665", "style": "concise", "instruction": "Solve the problem and keep the explanation concise but complete. Provide the final answer.", "input": "Solve the congruences: x≡0 (mod 8), x≡4 (mod 5). Give the solution modulo 40.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 40.\nCompute the CRT solution to get x ≡ 24 (mod 40).\nFinal answer: $x\\equiv 24\\pmod{40}$", "final_answer": "x\\equiv 24\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "74a84809fd9bd220", "problem_id": "19f5bee37889f6af", "style": "tutor", "instruction": "Give a detailed derivation, showing intermediate expressions, and then the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_6=39$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=39 and k=6.\n4) Compute C(44,5) = 1086008.\nFinal answer: $1086008$", "final_answer": "1086008", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "fa326f848618e8cd", "problem_id": "071fad801159a5a8", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Evaluate the limit:\n$\\displaystyle \\lim_{x\\to 1} x^{3} - x^{2} - 2 x - 5$.\n\nStudent solution:\n1) Substitute x=1 into the expression.\n2) (Mistake) Simplify the expression to x^{3} - x^{2} - 3 x - 5.\n3) Then the limit is -8.", "output": "First incorrect step: 2\nThe error is dropping/altering terms before taking the limit.\nUse the original expression and substitute x=1 (polynomial is continuous).\nCorrect limit value: -7.\nFinal answer: $-7$", "final_answer": "-7", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "limit", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "80352f8c51947f3b", "problem_id": "983be6f721f254ba", "style": "verifier", "instruction": "Compute the answer and add a brief verification calculation. Provide the final answer.", "input": "Solve the congruences: x≡3 (mod 5), x≡4 (mod 13). Give the solution modulo 65.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 65.\nCompute the CRT solution to get x ≡ 43 (mod 65).\nCheck: 43 mod 5=3 and mod 13=4.\nFinal answer: $x\\equiv 43\\pmod{65}$", "final_answer": "x\\equiv 43\\pmod{65}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "bd88f0b84533b42e", "problem_id": "983be6f721f254ba", "style": "tutor", "instruction": "Write a complete step-by-step solution and conclude with the final answer.", "input": "Solve the congruences: x≡3 (mod 5), x≡4 (mod 13). Give the solution modulo 65.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 65.\n3) Compute the CRT solution to get x ≡ 43 (mod 65).\nFinal answer: $x\\equiv 43\\pmod{65}$", "final_answer": "x\\equiv 43\\pmod{65}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "e1e7e9a2e68cfae9", "problem_id": "7e99da9d5364a799", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡5 (mod 7), x≡1 (mod 9). Give the solution modulo 63.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 63.\n3) Compute the CRT solution to get x ≡ 19 (mod 63).\nFinal answer: $x\\equiv 19\\pmod{63}$", "final_answer": "x\\equiv 19\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "239329f1f4413589", "problem_id": "7e99da9d5364a799", "style": "verifier", "instruction": "Compute the answer and include a short verification step. Give the final answer.", "input": "Solve the congruences: x≡5 (mod 7), x≡1 (mod 9). Give the solution modulo 63.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 63.\nCompute the CRT solution to get x ≡ 19 (mod 63).\nCheck: 19 mod 7=5 and mod 9=1.\nFinal answer: $x\\equiv 19\\pmod{63}$", "final_answer": "x\\equiv 19\\pmod{63}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 1, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "8cf6c140d20904c3", "problem_id": "3b104a320d2c4320", "style": "tutor", "instruction": "Provide a detailed solution with numbered steps and a final answer line.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_10=35$?", "output": "1) Let y_i=x_i-1, then y_i≥0 and y_1+…+y_10=25.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=25 and k=10.\n4) Compute C(34,9) = 52451256.\nFinal answer: $52451256$", "final_answer": "52451256", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1fee33264e95a68c", "problem_id": "3b104a320d2c4320", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "How many solutions in integers with $x_i\\ge 1$ satisfy $x_1+\\cdots+x_10=35$?", "output": "Let y_i=x_i-1, then y_i≥0 and y_1+…+y_10=25.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=25 and k=10.\nFinal answer: $52451256$", "final_answer": "52451256", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 4, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "b664b6bfbe21b17c", "problem_id": "23093d2102e6fe2a", "style": "tutor", "instruction": "Explain your reasoning clearly and show all important intermediate computations. End with the final answer.", "input": "Solve the congruences: x≡3 (mod 8), x≡10 (mod 11). Give the solution modulo 88.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 88.\n3) Compute the CRT solution to get x ≡ 43 (mod 88).\nFinal answer: $x\\equiv 43\\pmod{88}$", "final_answer": "x\\equiv 43\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d153d60f8636360b", "problem_id": "23093d2102e6fe2a", "style": "verifier", "instruction": "Show the solution, then validate it with a quick check. End with the final answer.", "input": "Solve the congruences: x≡3 (mod 8), x≡10 (mod 11). Give the solution modulo 88.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 88.\nCompute the CRT solution to get x ≡ 43 (mod 88).\nCheck: 43 mod 8=3 and mod 11=10.\nFinal answer: $x\\equiv 43\\pmod{88}$", "final_answer": "x\\equiv 43\\pmod{88}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "21ac11dbd29abf1b", "problem_id": "bb58e7dd0582f569", "style": "tutor", "instruction": "Solve like a tutor: derive step-by-step and finish with the final answer.", "input": "Solve the congruences: x≡2 (mod 7), x≡7 (mod 11). Give the solution modulo 77.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 77.\n3) Compute the CRT solution to get x ≡ 51 (mod 77).\nFinal answer: $x\\equiv 51\\pmod{77}$", "final_answer": "x\\equiv 51\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "65e277eb88f33994", "problem_id": "bb58e7dd0582f569", "style": "concise", "instruction": "Solve with minimal but sufficient steps, then provide the final answer.", "input": "Solve the congruences: x≡2 (mod 7), x≡7 (mod 11). Give the solution modulo 77.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 77.\nCompute the CRT solution to get x ≡ 51 (mod 77).\nFinal answer: $x\\equiv 51\\pmod{77}$", "final_answer": "x\\equiv 51\\pmod{77}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ace8515e46f06547", "problem_id": "bdea911768e94544", "style": "tutor", "instruction": "Solve carefully and do not skip algebra/arithmetic steps. End with the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_3=15$?", "output": "1) We are counting nonnegative integer solutions to a sum constraint.\n2) Stars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\n3) Here N=15 and k=3.\n4) Compute C(17,2) = 136.\nFinal answer: $136$", "final_answer": "136", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "6864c05ceb10d0b4", "problem_id": "bdea911768e94544", "style": "concise", "instruction": "Give a short solution with the key steps and the final answer.", "input": "How many solutions in nonnegative integers satisfy $x_1+\\cdots+x_3=15$?", "output": "We are counting nonnegative integer solutions to a sum constraint.\nStars and bars: the number of solutions to y1+…+yk=N with yi≥0 is C(N+k−1,k−1).\nHere N=15 and k=3.\nFinal answer: $136$", "final_answer": "136", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "stars_and_bars", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "922b41ea900def15", "problem_id": "5928c43e7508f11f", "style": "concise", "instruction": "Compute and present the key steps only, then the final answer.", "input": "Solve the congruences: x≡0 (mod 8), x≡1 (mod 5). Give the solution modulo 40.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 40.\nCompute the CRT solution to get x ≡ 16 (mod 40).\nFinal answer: $x\\equiv 16\\pmod{40}$", "final_answer": "x\\equiv 16\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1c2ce2de894dee2e", "problem_id": "5928c43e7508f11f", "style": "tutor", "instruction": "Work through the problem carefully with explicit steps and intermediate values. Finish with the final answer.", "input": "Solve the congruences: x≡0 (mod 8), x≡1 (mod 5). Give the solution modulo 40.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 40.\n3) Compute the CRT solution to get x ≡ 16 (mod 40).\nFinal answer: $x\\equiv 16\\pmod{40}$", "final_answer": "x\\equiv 16\\pmod{40}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "af049e0caeb483b5", "problem_id": "f4b7ec4f3e55ef27", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the congruences: x≡2 (mod 8), x≡11 (mod 13). Give the solution modulo 104.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 104.\n3) Compute the CRT solution to get x ≡ 50 (mod 104).\nFinal answer: $x\\equiv 50\\pmod{104}$", "final_answer": "x\\equiv 50\\pmod{104}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "1f0d518ee42c73f7", "problem_id": "f4b7ec4f3e55ef27", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 8), x≡11 (mod 13). Give the solution modulo 104.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 104.\nCompute the CRT solution to get x ≡ 50 (mod 104).\nCheck: 50 mod 8=2 and mod 13=11.\nFinal answer: $x\\equiv 50\\pmod{104}$", "final_answer": "x\\equiv 50\\pmod{104}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "5336bcd494a9e58a", "problem_id": "12717b20a9259407", "style": "verifier_negative", "instruction": "A student attempted the solution below. Identify the FIRST incorrect step, explain the mistake precisely, then give the corrected solution and final answer.", "input": "Differentiate with respect to x:\n$f(x)=2 x^{3} \\sin{\\left(x \\right)} - 2 x^{2} \\sin{\\left(x \\right)} + x \\sin{\\left(x \\right)} + 2 \\sin{\\left(x \\right)} + \\cos{\\left(2 x \\right)}$.\n\nStudent solution:\n1) Differentiate term-by-term.\n2) Claim (missing chain rule): $\\frac{d}{dx}(2 x^{3} \\sin{\\left(x \\right)}) = 2 x^{3} \\cos{\\left(x \\right)}$.\n3) So $f'(x) = 2 x^{3} \\cos{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.", "output": "First incorrect step: 2\nThe error is omitting the chain-rule factor (derivative of the inside function).\nCorrect derivative of that term is $\\frac{d}{dx}(2 x^{3} \\sin{\\left(x \\right)}) = 2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)}$.\nTherefore the correct derivative is $f'(x) = 2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$.\nFinal answer: $2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}$", "final_answer": "2 x^{3} \\cos{\\left(x \\right)} + 6 x^{2} \\sin{\\left(x \\right)} - 2 x^{2} \\cos{\\left(x \\right)} - 4 x \\sin{\\left(x \\right)} + x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} - 2 \\sin{\\left(2 x \\right)} + 2 \\cos{\\left(x \\right)}", "metadata": {"format": "symbolic", "domain": "calculus", "topic": "derivative", "difficulty": 3, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0", "task": "error_localization", "is_negative": true}}
{"id": "dcdc4fb9569dd8e1", "problem_id": "3d472c36dd2badf6", "style": "tutor", "instruction": "Explain each step clearly and end with the final answer.", "input": "Solve the congruences: x≡8 (mod 9), x≡4 (mod 10). Give the solution modulo 90.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 90.\n3) Compute the CRT solution to get x ≡ 44 (mod 90).\nFinal answer: $x\\equiv 44\\pmod{90}$", "final_answer": "x\\equiv 44\\pmod{90}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "c8c81f9a69aef14c", "problem_id": "7255aafecc2c075c", "style": "tutor", "instruction": "Solve step-by-step like a math tutor. Show intermediate calculations and end with a final answer.", "input": "Solve the congruences: x≡2 (mod 7), x≡12 (mod 13). Give the solution modulo 91.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 91.\n3) Compute the CRT solution to get x ≡ 51 (mod 91).\nFinal answer: $x\\equiv 51\\pmod{91}$", "final_answer": "x\\equiv 51\\pmod{91}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "ed1daf3b4ec67134", "problem_id": "7255aafecc2c075c", "style": "verifier", "instruction": "Solve step-by-step and include a verification (e.g., plug-in/check totals). Provide the final answer.", "input": "Solve the congruences: x≡2 (mod 7), x≡12 (mod 13). Give the solution modulo 91.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 91.\nCompute the CRT solution to get x ≡ 51 (mod 91).\nCheck: 51 mod 7=2 and mod 13=12.\nFinal answer: $x\\equiv 51\\pmod{91}$", "final_answer": "x\\equiv 51\\pmod{91}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 5, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "4893980eeacf8455", "problem_id": "09f5e2c4f465c239", "style": "tutor", "instruction": "Provide a thorough solution as if teaching, including intermediate computations, then give the final answer.", "input": "Solve the congruences: x≡4 (mod 9), x≡7 (mod 8). Give the solution modulo 72.", "output": "1) Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\n2) The solution is unique modulo m1·m2 = 72.\n3) Compute the CRT solution to get x ≡ 31 (mod 72).\nFinal answer: $x\\equiv 31\\pmod{72}$", "final_answer": "x\\equiv 31\\pmod{72}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}
{"id": "d42b88e0ac1b6c8e", "problem_id": "09f5e2c4f465c239", "style": "verifier", "instruction": "Solve and add a check (substitution/count/sanity check). Provide the final answer.", "input": "Solve the congruences: x≡4 (mod 9), x≡7 (mod 8). Give the solution modulo 72.", "output": "Since gcd(m1,m2)=1, the Chinese Remainder Theorem applies.\nThe solution is unique modulo m1·m2 = 72.\nCompute the CRT solution to get x ≡ 31 (mod 72).\nCheck: 31 mod 9=4 and mod 8=7.\nFinal answer: $x\\equiv 31\\pmod{72}$", "final_answer": "x\\equiv 31\\pmod{72}", "metadata": {"format": "symbolic", "domain": "discrete", "topic": "chinese_remainder_theorem", "difficulty": 2, "verified": true, "source": "synthetic_v3_open", "sympy_version": "1.14.0"}}